In this paper, we use the properties of error estimation and the analytic method to study the reciprocal sums of higher power of higher-order sequences. Then we establish several new and interesting identities relating to the infinite and finite sums.Let { Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways; see [The so-called Fibonacci zeta function and Lucas zeta function, defined byζF(s)=∑n=ays; see . Ohtsukaays; see  studied Further, Wu and Zhang , 5 generk=n∞(1/vk))−1. Specifically, suppose that ||x|| = ⌊x + (1/2)⌋ (the nearest integer function) and {vn}n≥0 is an integer sequence satisfying the recurrence formulap ≥ q and n > k. Then we can conclude that there exists a positive integer n0 such thatn > n0.Recently, some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other well-known sequences and obtained several meaningful results; see –16. In pIn , Wu and n > m, the mth-order linear recursive sequence {un} is defined as follows:ui ∈ ℕ for 0 ≤ i < m and at least one of them not being zero. For any positive real number β > 2 and any positive integer a1 ≥ a2 ≥ ⋯≥am ≥ 1, there exists a positive integer n1 such thatβ → +∞, there exists a positive integer n2 such thatFor any positive integer k=n∞(1/uks))−1 for all integers s ≥ 2, because it is quite unclear a priori what the shape of the result might be. In [It seems difficult to deal with  of reciprocal sums of some recursive sequences. The main purpose of this paper is using the properties of error estimation and the analytic method to study the higher power of the reciprocal sums of {un} and obtain several new and interesting identities. The results are as follows.To find and prove this result is a substantial achievement since such a complex formula would not be clear beforehand that a result would even be possible. However, there is no research considering the higher power f(x) has exactly one positive real zero α with a1 < α < a1 + 1;polynomial (II)m − 1 zeros of f(x) lie within the unit circle in the complex plane.other Let See Lemma 1 of .m ≥ 2 and {un}n≥0 be an integer sequence satisfying the recurrence formula .Let  formula . Then foun is given byc > 0, d > 1, a1 < α < a1 + 1, and α is the positive real zero of f(x). Now we prove s = 2. Suppose that for all integers 2 ≤ s ≤ k we haves = k + 1 we haveFrom Lemma 2 of , the clogiven byun=cαn+O, (1/αndn)}.In this section, we shall complete the proof of Taking the reciprocal of this expression yieldsh = (a1s⌊βn⌋−sn/αs⌊βn⌋−sn), then for any real number β > 2 and positive integer s we haveIf h = (1/αndn), for any positive integer a1 ≥ 2, 1 < (α/a1) < αd holds. Then for any positive integer s withIf β > 2 and positive integer 1 ≤ s < ⌊logα/a1)k=nβn⌋⌊((−a1)sk/uks))−1 − (−1)sn(uns/a1sn + un−1s/a1sn−s)|| = 0, (n ≥ n8). ||(∑(b)k=nβn⌋⌊(a1spk+sq/upk+qs))−1 − (upn+qs/a1spn+sq − upn−p+qs/a1spn+sq−sp)|| = 0, (n ≥ n9). ||(∑(c)k=nβn⌋⌊((−a1)spk+sq/upk+qs))−1 − (−1)spn+sq(upn+qs/a1spn+sq + upn−p+qs/a1spn+sq−sp)|| = 0, (n ≥ n10). ||k=n∞((−a1)sk/uks))−1 − (−1)sn(uns/a1sn + un−1s/a1sn−s)|| = 0, (n ≥ n11). ||(∑(e)k=n∞(a1spk+sq/upk+qs))−1 − (upn+qs/a1spn+sq − upn−p+qs/a1spn+sq−sp)|| = 0, (n ≥ n12). ||(∑(f)k=n∞((−a1)spk+sq/upk+qs))−1 − (−1)spn+sq(upn+qs/a1spn+sq + upn−p+qs/a1spn+sq−sp)|| = 0, (n ≥ n13). ||(∑Let {fined by  with theh = max {(a1s⌊βn⌋−sn/αs⌊βn⌋−sn), (1/αndn)}.We shall prove only (c) in Taking the reciprocal of this expression yieldsCase  1. If h = (a1s⌊βn⌋−sn)/(αs⌊βn⌋−sn), then for any real number β > 2 and positive integer s we haveCase  2. If h = (1/αndn), for any positive integer a1 ≥ 2, 1 < (α/a1) < αd holds. Then for any positive integer s withβ > 2 and positive integer 1 ≤ s < ⌊logα/a1)(αpdp⌋, there exists n ≥ n10 sufficiently large so that the modulus of the last error term of identity (In both cases, it follows that for any real number identity  becomes


Triticum aestivum L.) during domestication, and this enables them to grow productively in a wide range of environments. Several major genes controlling flowering time have been identified in wheat with mutant alleles having sequence changes such as insertions, deletions or point mutations. We investigated genetic variants in commercial varieties of wheat that regulate flowering by altering photoperiod response  or vernalization requirement  and for which no candidate mutation was found within the gene sequence. Genetic and genomic approaches showed that in both cases alleles conferring altered flowering time had an increased copy number of the gene and altered gene expression. Alleles with an increased copy number of Ppd-B1 confer an early flowering day neutral phenotype and have arisen independently at least twice. Plants with an increased copy number of Vrn-A1 have an increased requirement for vernalization so that longer periods of cold are required to potentiate flowering. The results suggest that copy number variation (CNV) plays a significant role in wheat adaptation.The timing of flowering during the year is an important adaptive character affecting reproductive success in plants and is critical to crop yield. Flowering time has been extensively manipulated in crops such as wheat ( Triticum aestivum) is an allohexaploid combining A, B and D genomes from three diploid ancestors Cultivated bread wheat (Photoperiod-1 (Ppd-1) loci on chromosomes 2A, 2B and 2D. The Ppd-1 genes are members of the pseudo-response regulator (PRR) family orthologous to the Ppd-H1 gene of barley Wheat's ancestors and many modern varieties are described as photoperiod sensitive. That is, they flower rapidly in long days but are late flowering in short days. Day neutral (photoperiod insensitive) varieties flower rapidly in short or long days. This allows production in environments where appropriate temperature and rainfall coincide with short day conditions or where early flowering avoids high summer temperatures, as in Southern Europe a suffix  while wild type alleles have a b suffix Ppd-A1a alleles and the single Ppd-D1a allele are deletions that remove a shared region of the promoter likely to be important for regulation Four day neutral alleles have been characterized in wheat previously. They are given an Ppd-1b (wild type) alleles have a marked diurnal fluctuation in expression, with very low transcript levels at dawn, a peak 3–6 h after dawn and a subsequent drop to very low levels in the dark. Ppd-1a alleles that have been studied show elevated expression throughout the 24 h period but particularly at dawn and during the dark period. This is associated with expression of TaFT1 (the wheat orthologue of the flowering inducer FLOWERING LOCUS T) in short days Ppd-1 and the induction of TaFT1 without the normal long day cue.Ppd-B1a alleles identified by genetic mapping in five diverse sources: ‘Chinese Spring’ , Vrn-2 or TaFT1  genes Vrn-1 encodes a MADS-box transcription factor whose expression increases quantitatively during vernalization. This potentiates flowering by inhibiting expression of the flowering repressor Vrn-2Vrn-A1, Vrn-B1 or Vrn-D1 genes  are the predominant cause of spring types in wheat Vernalization is another important flowering time control. The ancestors of wheat and many modern varieties are winter types that require vernalization, but during domestication there has been selection for spring types that do not. Spring types result from mutation of the Vrn-A1 was the cause; specifically a single base difference in exon 4 Vrn-A1 as the cause but in contrast to In addition to the winter/spring difference, winter types vary in the duration of cold necessary to complete vernalization. We refer to this as variation in vernalization requirement. This aspect of vernalization is much less studied, but recently it was proposed that Ppd-B1a and Vrn-A1 alleles we investigated, the changes in flowering time were associated with increased gene copy number and not with specific sequence polymorphisms. Copy number variation (CNV) is a type of mutation that has been widely studied in human genetics Arabidopsis thaliana accessions For the Ppd-B1. The photoperiod sensitive hexaploid spring wheat variety ‘Paragon’ was the recurrent parent for introgression lines containing Ppd-1a alleles from ‘Chinese Spring’, ‘Sonora64’ or ‘Récital’. Two independent lines of each allele were developed and from each of these ten plants at BC3F4  or BC4F4  were grown in short days  in a photoperiod glasshouse. Days to ear emergence on the main stem were recorded for each plant , while ‘Chinese Spring’ had four copies. Furthermore, the other genotypes with a day neutral Ppd-B1a allele also had elevated copy numbers; two in ‘Récital’ and three in ‘Sonora64’, ‘Timstein’ and ‘C591’ (Ppd-B1a) replaced the ‘Mercia’ 2B had the same increased copy number as ‘Chinese Spring’ itself. Conversely, a line in which the 2B chromosome from the photoperiod sensitive variety ‘Marquis’ (Ppd-1Bb) replaced the ‘Chinese Spring’ chromosome had a low copy number, showing that all the additional gene copies were on chromosome 2B. These data suggested that increased copy number could be the basis of the Ppd-B1a alleles.d ‘C591’ . A ‘MercA BAC library Ppd-B1 gene flanked by retrotransposons. 97J10 and 170M01 did not assemble readily because of sequence differences at the ends of approximately 35 kb regions containing either an intact or truncated gene. Digestion with NotI allowed BAC clone inserts to be sized. 170M01, which did not contain the truncated copy, would contain three repeat units of approximately 35 kb. PFGE of 97J10 using AarI, which does not cut within the truncated copy, placed the truncated copy at the left end of the repeating structure (copy_1), showing that the Ppd-B1a locus of ‘Chinese Spring’ comprised one truncated copy and three intact copies of the Ppd-B1 gene, all in the same orientation, within a 185 kb region  from the other varieties in Tsp509I CAPS assay of amplicons from ‘Chinese Spring’ and individual BAC clones showed that all copies had the A form. Although exon and intron sequences were identical apart from the truncation in copy_1, different haplotypes were found upstream as shown by six nucleotides below each gene copy  which was correlated with copy number, showing the copies to be genetically linked. This suggests that the ‘Récital’ allele is derived from a third independent amplification event or from the ‘Sonora64’/‘Timstein’ type by further mutation.‘Récital’ had a haploid copy number of two  and the Ppd-B1b alleles , ‘Cheyenne’ and ‘Renan’ which have wild type  alleles , S2. Thi alleles . This suPpd-B1 specific quantitative RT-PCR assay. First, ‘Mercia’ (Ppd-B1b) was compared with a ‘Mercia(Chinese Spring 2B)’ (Ppd-B1a) single chromosome substitution line. Plants were grown in short days  for 21 days and RNA samples were taken from three biological replicates, each of four plants, at 3 h intervals over a 24 h period. Expression of the ‘Mercia’ allele was similar to that of Ppd-1b alleles in previous studies Ppd-B1a allele had significantly higher levels at dawn, a higher expression peak, and elevated expression throughout the dark period ’, ‘Paragon(Sonora64 Ppd-B1a)’ and ‘Paragon’ introgression lines were grown in short days and sampled at the end of the dark period and at three time points in the lit period as described above  or 0 to 6 weeks  after which they were transferred to a heated lit glasshouse providing long days (18 h light). Day length was sufficient to satisfy photoperiod response, and genotyping showed that no Days to ear emergence on the main stem were recorded for five plants of each genotype per time point. Unvernalized ‘Claire’ plants flowered about 40 days before ‘Malacca’ but the acceleration of flowering per week of vernalization was greater in ‘Malacca’ so that flowering times were similar after 4 or more weeks of cold. Unvernalized plants of ‘Hereward’ and ‘Malacca’ flowered at a similar time but the 1 to 4 week treatments in ‘Hereward’ had less effect on flowering, after which the response was similar to ‘Malacca’ .Vrn-A1 on chromosome 5A To investigate the genetic basis of long vernalization requirement in ‘Hereward’ we selected a 4 week vernalization treatment  to study a population of 77 doubled haploid (DH) lines from a ‘Malacca’×‘Hereward’ cross. Three plants of each DH line and four plants of each parent were vernalized and then grown in a heated glasshouse with supplementary lighting providing long days (18 h light). Days to ear emergence on the main stem were recorded and averaged for each genotype. QTL mapping showed one significant peak  and ‘Hereward’ (JF965397) by direct sequencing of overlapping PCR amplicons generated from gene specific primers based on the ‘Triple Dirk C’ winter allele  only fitted a 1∶1 ratio while for ‘Hereward’ (51∶87) they only fitted a 1∶2 ratio, suggesting two and three copies of Vrn-A1, respectively. Full length cDNA clones from RNA of vernalized ‘Malacca’ and ‘Hereward’ plants also had the C and T forms showing that two intact variants of the gene were present and expressed. The C/T difference in exon 4 is likely to be that previously reported as distinguishing ‘Jagger’ from ‘2174’ Approximately 12 kb of AY747601 . No diffAY747601 . CloningVrn-A1  confirmed a haploid copy number of two for ‘Malacca’ and three for ‘Hereward’ . The latter has a spring allele of 2 population to be used for an additional test of the genetic behaviour of the ‘Hereward’ Vrn-A1 copies  increased most rapidly in ‘Claire’ and was slowest in ‘Hereward’, while ‘Malacca’ was intermediate ’ lines were developed as follows. ‘Récital’ carries Ppd-B1a and Ppd-D1a alleles 1F2 plants were assayed for Ppd-D1a as described in Ppd-D1a carriers discarded. The remainder were grown in short days . Early flowering plants would have Ppd-B1a and these were backcrossed to ‘Paragon’. BC3F2 families in short days showed 3∶1 segregation for flowering time and early flowering plants had higher Ppd-B1 copy number, showing that Ppd-B1a was the major determinant of flowering time. BC3F3 families that were all early flowering in short days  were selected. BC3F4 plants were compared to ‘Paragon’ and previously described ‘Paragon’ BC4F4 introgression lines homozygous for the ‘GS-100’ Ppd-A1a, ‘Chinese Spring’ Ppd-B1a, ‘Sonora64’ Ppd-B1a or ‘Sonora64’ Ppd-D1a alleles ‘Mercia’ and ‘Mercia(Chinese Spring 2B)’ lines were from the John Innes Centre Germplasm Resources Unit. ‘Paragon, annealing (55°C for 20 sec) and polymerisation (72°C for 1 min per kb of amplicon).50 ng genomic DNA in 20 µl reactions comprising 1× PCR Buffer and 0.4 units Ppd-1 genes from a ‘Chinese Spring’ BAC library Bsu36I or BsiW1 followed by re-ligation and selection with chloramphenicol allowed recovery of the rightmost and leftmost Ppd-B1 copies, respectively, in resulting subclones. The haplotypes of individual copies was determined by standard PCR (above) and Sanger sequencing (see sequencing of Vrn-A1).Filter hybridization was used to select clones containing catttctcccgcttgtaagg and caacgttacagcttcggtca, respectively) was conducted using the Roche Expand Long template kit . This produced an approximately 9 kb fragment in ‘Sonora64’ and ‘Timstein’, which was analyzed by Sanger sequencing (below) using internal primers in a stepwise manner.Long range PCR using primers facing outwards from the ends of the gene sequence  and 97J10R2 (ccggaacctgaggatcatc) gave a 425 bp product; (b) the junction between intact Ppd-B1 copies in the ‘Chinese Spring’ allele. Primers PpdB1_F25 (aaaacattatgcatatagcttgtgtc) and PpdB1_R70 (cagacatggactcggaacac) gave a 994 bp product; (c) the junction between intact Ppd-B1 copies in the ‘Sonora64’/‘Timstein’ allele. Primers PpdB1_F31 (ccaggcgagtgatttacaca) and PpdB1_R36 (gggcacgttaacacaccttt) gave a 223 bp product. Junction sequences and primer positions for assays are given in PCR with “Hot Start” polymerase and buffer (Qiagen) and an initial denaturation step of 95°C for 5 minutes was used to detect; (a) the truncated vrn-A1 sequence (AY747601) was used to design a series overlapping PCR amplicons spanning the gene. Amplicons were obtained from genomic DNA using the standard PCR protocol and were directly sequenced using ABI Big Dye Mix v3.1 (Applied Biosystems Inc) under the manufacturer's conditions, with products resolved on an ABI 3730 capillary electrophoresis instrument. Segments of Vrn-A1 from ‘IL369’ were amplified using primers IL369_F1 (tgaattcatgatcggagcag) and IL369_R1 (tgctgaacttctctgcaagtg) or IL396_F2 (tgctgaacttctctgcaagtg) and IL369_R1 (‘Triple Dirk D’ IL369_R1  and direPpd-B1 were gcgtaagttactatctctcatggtgtatc and tttgttttagtacccagtaccataccag (0.2 µM) and the probe sequence was FAM-ctgctgcttcagttcctagtttcacttgtgtcc-TAMRA (0.1 µM). Forward and reverse primers for Vrn-A1 were gcagcccacttttggtctcta and tctgccctctcgcctgtt (0.2 µM) and the probe sequence was FAM-tgtgttcgctttggttgtgcagca-TAMRA (0.1 µM). Forward and reverse primers for the TaCO2 control were tgctaaccgtgtggcatcac and ggtacatagtgctgctgcatctg (0.1 µM) and the probe sequence was VIC-catgagcgtgtgcgtgtctgcg-TAMRA (0.1 µM). Ppd-B1 plus control or Vrn-A1 plus control were analyzed together in multiplexed reactions.20 µl reactions comprised water (3 µl), AbGeneQPCR mix (10 µl), Probe plus primers (2 µl) and DNA (5 µl). Samples were denatured at 95°C for 15 min followed by 40 cycles of . Forward and reverse primers for www.kbioscience.co.uk) in reactions containing water (2 µl), KASPar mix (4 µl), primers (0.1 µl), 50 mM MgCl2 (0.064 µl) and DNA (2 µl). An activation time  was followed by 20 cycles of  followed by 24 cycles of . Fluorescence was read as an end point reading at 25°C. Primer combinations were;Polymorphisms were scored using KBioscience KASP reagents . Generic primer ccagttgctgcaactccttgagatt (0.4 µM).Exon4_C/T SNP specific primers: gaaggtgaccaagttcatgctgagtttgatcttgctgcgccG, gaaggtcggagtcaacggattctgagtttgatcttgctgcgccA (0.16 µM). Generic primer cttccccacagctcgtggagaa (0.4 µM).Exon7_C/T SNP specific primers: Ppd-B1a)’, ‘Paragon(Sonora64 Ppd-B1a)’ and ‘Paragon’ introgression lines in the same way.RNA of ‘Mercia’ and ‘Mercia (Chinese Spring 2B)’ was extracted from seedlings grown for 20 days after germination in a controlled environment room with short days . Three replicate samples from each genotype were harvested into liquid nitrogen at each three-hourly time point over 24 h. RNA was sampled from 20 day old seedlings of ‘Paragon’ and ‘Paragon(Chinese Spring http://www.invitrogen.com). DNA was removed by DNAseI digestion. cDNA was synthesized with Superscript II (Invitrogen) using the manufacturer's protocols with 5 µg of total RNA as template and a mixture of OligodT (12–18) (250 ng) and random hexamers (150 ng) as primers. 1/40 by volume of the final cDNA aliquot was used for real-time PCR as described below. In the case of 18S rRNA analysis, cDNA samples were diluted 1∶100 and 1/40 of this dilution was used as template.RNA was extracted using Trizol reagent . The forward primer was specific for Ppd-B1, and genome-specificity was confirmed using ‘Chinese Spring’ nullisomic-tetrasomic lines. The primers generate a 549 bp amplicon from cDNA. Similarly, primers taactgctcctcacaagtgc and ccggaacctgaggatcatc were used to detect expression of the truncated copy in ‘Chinese Spring’ . Primer pairs were:Samples of leaf tissue from plants of each genotype/vernalisation time combination  were harvested in the vernalisation chamber at the end of each treatment 3–4 h after dawn and frozen in liquid nitrogen. RNA and cDNA were prepared as above, and VrnA1-all: cttgaacggtatgagcgctat and gcatgaaggaagaagatgaag, product size 396 bp.VrnA1-exon 4 C form: gcatctcatgggagaggatC and gaatagtacgcctgtatgggctggat, product size 436 bp.VrnA1-exon 4 T form: gcatctcatgggagaggatT and gaatagtacgcctgtatgggctggat, product size 436 bp. This assay produced a low level signal in ‘Claire’, which does not contain the T form  The ‘Chinese Spring’ Ppd-B1a allele. (B) Hypothetical structure of the ‘Sonora64’/‘Timstein’ Ppd-B1a allele. (C) Hypothetical structure of the ‘Récital’ Ppd-B1a allele. Ppd-B1 copies are shown as large dark green rectangles (exons plus introns), the solid black line shows upstream and downstream regions and small coloured rectangles show transposable elements as in (TIF)Click here for additional data file.Figure S2PCR assays detecting junction sequences in Ppd-B1a  alleles and sequences of the junction regions. (A) to (C) Reverse colour images of PCR products from genotypes with known Ppd-B1 alleles. Genotypes with a known Ppd-B1a allele are boxed. ‘Chinese Spring(N2BT2D)’ is nullisomic for chromosome 2B (no Ppd-B1 gene is present) and tetrasomic for chromosome 2D. ‘Chinese Spring(N2AT2B)’ is nullisomic for chromosome 2A and tetrasomic for chromosome 2B. (A) Assay for the gene/transposon junction  in the truncated Ppd-B1 copy of ‘Chinese Spring’. Primer positions for this assay are underlined in the sequence below. Ppd-B1 gene sequence is to the left of the arrow  and the TREP 3161 WIS element is to the right of the arrow. A 425 bp band is produced when the junction is present. …TAACTGCTCCTCACAAGTGCCGGAAGGGAAAGACGCCGACCGTGAGAACGCCATGCCATATCTTGAGCTGAGCCTAAAGAGGTCGAGATCGACCACGGAGGGTGCGGATGCGATCCAGGAGGAACAGAGGAACGTCGTGAGACGATCAGACCTCTCGGCATTCACGAGgtgcaaagcataatatcagtgtcctttgtgaatccttaaatcatccatatgttgcatactaaccgttttcattctttgcaagGTACAATACGTGCTCGTTCTCCAATCAAGGCGGGGCAGGGTTCGTCGGGAGCTGTTCGCCCA↑cgtgactgccaagcgttcataacgtcttggttctatgggatgggtgcttcacctagcggtccttctaggacatatgctttcttggcagctatgaggatgatcctcaggttccgg. The same primers were used for RT-PCR where a 343 bp product was produced from cDNA. The product was cloned and sequenced, confirming that intron 6 was correctly spliced. The transcript has a stop codon (double underline) close to the break point which gives a predicted protein lacking a CCT domain. (B) Assay for the junction between intact Ppd-B1 gene copies in the ‘Chinese Spring’ allele. Primer positions for this assay are underlined in the sequence below. The left primer spans the junction between Ppd-B1 gene sequence and the TREP 3457 Danae element. The junction  marks the start of the TREP 3161 WIS element sequence which contains the right primer. A 994 bp band is generated when the junction is present. …aaacattatgcatatagcttgtgtcggtgtacaaaagtaggagctctgcttttgacccctttacttgtgcacgagcagtcagagccacccgccacggccacgcataacagggcagggaagggaagccggagagaagccgaaggcacaagacaaacaaagcaacgacaaggaccaaggctacaaagtgcagatgggcgaagcaggtttcccctgcaagacccttgccgggggcagcctccgcagccccggcaagcccctttgccggggcaactcgcccacaccaatggagagagccacccttgaacccacggcctccaatgtcaaccaccacgttggaccagggctcgagaggcacctccatggtggcatgcagatctttgtgaagacatagaatgctcaagaacatgtgatgattggagggtaacaatcctcaccaagatcctcaccgaataaacccacaagacccccccggcaagatccttgccgaggacggcaagcgccacggcaagacccttgccgggccacccaacgagaccctcgccagaggcaaccacgagcccactgccaggcccgcaccaaccaactccccgccgccgttcgcatgcagctgccaacccaaccagctgagcaggcacctgcgtggcaacatgcagcttctaggcccactcagcacacacctgtgtggcggcatgcagatcttcgtggaggctccaccaccgcaccacctcagctgcttgcctgcctacatggcaccgcatgcatcgctggccagggcgcgtgtcgaagcaaggaggagcggcgacggatgggacgggcgtcgctcccgtccccgataaagtgagggacacctaagccatgcat↑gttcatcatgtgagacggactagtcatcatcggtgaacatctccatgttgatcgtatcttccatacgactcatgttcgaactttcggtcccttgtgttccgagtccatgtctgtacatgctaggctcgtcaagttaacataagtgttttgcatgtgtaaatctgtcttacacccattgtatgtggacgttggaatctatcacacccgatcatcacgtggtgcttcgaaacaatgaacttcacacaagcattatacctggaaccgtagcataattttcctcaacattttgagctacgtgtatggagtttgcattgttatcttcaacatttctcccgcttgtaaggcctgca… (C) Assay for the junction between intact Ppd-B1 gene copies in the ‘Sonora64’/‘Timstein’/‘C591’ allele. Primer positions for this assay are underlined in the sequence below. The junction is shown by a vertical arrow. The TREP 3457 Danae element is to the left of the arrow, the TREP 3173 Derami element is to the right. A 223 bp band is generated when the junction is present. …ccaggcgagtgatttacacatcacccctcagaaggaatgagaaaacttgcgcaagtacatccaaaggttcagccgggtgcagtacaacatccccgatgttcatcccgccgtcgtgatcagtgtgttccattagaatgtgcgcaaccgcaagatgcgtgaagagctagcgatgaacaa↑aagttaaattttcatgaagtgaaataaaaggtgtgttaacgtgcccatgccatctattttcaaaaataaataaataatctttaagaaaagttaaatgcataaaaaagtccaatagtttacatatacattggctcaatatagtggtgaggcatatttatttatattactcatggtcttttcatggttgatgcacttgccggtggattccatctcttcttgggcggatctgtcccatgagtttgttggcgcctttaccggaggtcaccaagctcatgg… Descriptions of repeat sequences and TREP numbers in (A) to (C) follow (TIF)Click here for additional data file.Figure S3QTL Cartographer plot of days to ear emergence in ‘Malacca’×‘Hereward’ DH lines. Plants were vernalized for 4 weeks  and grown in lit glasshouse (18 h light). For each DH line the score was the mean of three plants. One region exceeded the significance threshold (black line at LOD 2.5) and this was on a 20 cM linkage group containing marker barc151 previously shown to be linked to Vrn-A1Vrn-A1 was scored qualitatively based on copy number scores from the TaqMan® assay (Vrn-A1 position.(TIF)Click here for additional data file.Figure S4Example trace files of directly sequenced PCR amplicons from the region of Vrn-A1 exon 4 containing the C/T variants. ‘Hereward’ and ‘Malacca’ had a C/T double peak (arrowed) while ‘Claire’ and ‘Chinese Spring’ had only the C form.(TIF)Click here for additional data file.Figure S5C and T variants in exon 4 of Vrn-A1. (A) Predicted amino acid sequence of part of the K box region of VRN-1 and related MADS-box genes. Protein sequences are from related MADS-box genes Vrn-A1 from ‘IL369’. Intr1/C/F, Intr1/AB/R, Intr1/A/F2 and Intr1/A/R3 are previously published primers for PCR assays (TIF)Click here for additional data file.File S1Quantification of Ppd-B1 expression.(DOC)Click here for additional data file.


The well-posedness of this problem is established in Banach spaces C0β,γ(Eα−β) of all Eα−β-valued continuous functions φ(t) on  satisfying a Hölder condition with a weight (t + τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.The nonlocal boundary value problem for the parabolic differential equation  It is known that ⊂E→E be a linear unbounded operator densely defined in E. The operator A is said to be positive in the Banach space E if its spectrum σA lies in the interior of the sector of angle ϕ, 0 < 2ϕ < 2π, symmetric with respect to the real axis, and if, on the edges of this sector, S1(ϕ) = {ρeiϕ : 0 ≤ ρ ≤ ∞} and S2(ϕ) = {ρeiϕ− : 0 ≤ ρ ≤ ∞}, and outside of the sector the resolvent (λ−A)−1 is subject to the boundϕ is called the spectral angle of the positive operator A and is denoted by φ. We call A strongly positive in the Banach space E if its spectral angle φ < π/2.Let A in the Banach space E, let us introduce the fractional spaces Eβ = Eβ (0 < β < 1) consisting of those ν ∈ E for which the normFor positive operator E with the strongly positive operator A was established. The importance of coercive inequalities (well-posedness) is well known . The derivative at the endpoints of the segment is understood as the appropriate unilateral derivatives.v(t) belongs to D = D(A(t)) for all t ∈ , and the function A(t)v(t) is continuous on .The element v(t) satisfies the equation and the initial condition (ondition .A function  problem  if the fC(E) = C. Here, C(E) stands for the Banach space of all continuous functions φ(t) defined on  with values in E equipped with the normf(t) ∈ C(E) and v0 ∈ D.A solution of problem  defined C(E) if the following conditions are satisfied.(1)f(t) ∈ C(E) and any v0 ∈ D. This means that an additive and homogeneous operator v(t) = v, v0) is defined which acts from C(E) × D to C(E) and gives the solution of problem (C(E).Problem  is uniqu problem  in C(E).(2)v, v0), regarded as an operator from C(E) × D to C(E), is continuous. Here, C(E) × D is understood as the normed space of the pairs (f(t), v0), f(t) ∈ C(E), and v0 ∈ D, equipped with the normBy Banach's theorem, in C(E) and these properties, one has coercive inequalityMC  (1 ≤ MC < +∞) does not depend on  v0  and f(t).We say that the problem  is well C(E) for (A(t) = A, then the coercivity inequality implies the analyticity of the semigroup exp{−sA}  (s ≥ 0), that is, the following estimatesM ∈ , the operator −A(t) generates an analytic semigroup exp{−sA(t)}  (s ≥ 0) with exponentially decreasing norm, when s → +∞, that is, the following estimatesM ∈ , and v0 ∈ D, it is easy to show that the formulaIf the function  problem .sA(t)}  (s ≥ 0) and the fundamental solution v of  of  and theot, s) of  which wis < s + τ < T,  0 ≤ t ≤ T, and 0 ≤ α ≤ 1, one has the inequalityM does not depend on α, t, s, and τ.For any 0 < s,   τ,  t ≤ T and 0 ≤ ε ≤ 1, the following estimates hold:M≥0 and δ > 0 do not depend on ε, t, s, and τ.For any 0 ≤ s<t ≤ T and u ∈ D, the following identities hold:For any 0 ≤ s<t ≤ t + r ≤ T, the following estimates hold:M≥0 does not depend on t and s.For any 0 ≤ v(t) is said to be a solution of problem (F(E) if it is a solution of this problem in C(E) and the functions  v′(t) and A(t)v(t) belong to F(E).A function  problem  in F(E) C(E), we say that the problem (F(E) if the following two conditions are satisfied.(1)f ∈ F(E) and v0 ∈ D, there exists the unique solution v(t) = v, v0) in F(E) of problem , v0) is defined which acts from F(E) × D to F(E) and gives the solution of (F(E).For any  problem . This meution of  in F(E).(2)v, v0), regarded as an operator from F(E) × D to F(E), is continuous. Here, F(E) × D is understood as the normed space of the pairs (f(t), v0),f(t) ∈ F(E), and v0 ∈ D, equipped with the normAs in the case of the space  problem  is well F(E) equal to C0β,γ(E),  space, obtained by completion of the set of all smooth E-valued functions φ(t) on  with respect to the normC0β,γ(Eα−β), 0 ≤ γ ≤ β ≤ α, 0 < α < 1 from  for any 0 ≤ t ≤ t + λ. Then, for any 0 ≤ s < t ≤ t + λ and u ∈ D, the following identity holdsAssume that I − v in E and the following estimate holds:Under the assumption of v(t) is called a solution of the problem (v(t) is continuously differentiable on the segment .v(t) belongs to D for all t ∈ , and the function A(t)v(t) is continuous on .The element v(t) satisfies the equation and the nonlocal boundary condition (ondition .A function  problem  if the fC(E) if the following conditions are satisfied.(1)f(t) ∈ C(E) and any φ ∈ D. This means that an additive and homogeneous operator v(t) = v, φ) is does not depend on defined which acts from C(E) × D to C(E) and gives the solution of problem (C(E).Problem  is uniqu problem  in C(E).(2)v, φ), regarded as an operator from C(E) × D to C(E), is continuous. Here, C(E) × D is understood as the normed space of the pairs (f(t), φ),  f(t) ∈ C(E), and φ ∈ D, equipped with the normBy Banach's theorem in C(E) and these properties, one has coercive inequalityMC  (1 ≤ MC < +∞) does not depend on φ  and f(t).We say that the problem  is well C(E) for (A(t) = A, then the coercivity inequality implies the analyticity of the semigroup exp{−sA}  (s ≥ 0). Thus, the analyticity of the semigroup exp{−sA}  (s ≥ 0) is necessary for the well-posedness of problem (C(E). Unfortunately, the analyticity of the semigroup exp{−sA}  (s ≥ 0) is not sufficient for the well-posedness of problem (C(E).Inequality  is calleC(E) for .  If A(t problem  in C(E). problem  in C(E).v(t) is said to be a solution of problem (F(E) if it is a solution of this problem in C(E) and the functions  v′(t) and A(t)v(t) belong to F(E).A function  problem  in F(E) C(E), we say that the problem (F(E) if the following two conditions are satisfied.(1)f ∈ F(E) and φ ∈ D, there exists the unique solution v(t) = v, φ) in F(E) of problem , φ) is defined which acts from F(E) × D to F(E) and gives the solution of (F(E).For any  problem . This meution of  in F(E).(2)v, φ), regarded as an operator from F(E) × D to F(E), is continuous. Here, F(E) × D is understood as the normed space of the pairs (f(t), φ),f(t) ∈ F(E), and φ ∈ D, equipped with the normIf v(t) is a solution in C0β,γ(E) of problem (C(E) of this problem. Hence, by , 0 ≤ γ ≤ β ≤ α, 0 < α < 1.As in the case of the space  problem  is well  problem , then itence, by , we get ence, by :34)v(t).Using  and formv(t).Usidness of  in the sf(t) ∈ C0β,γ(Eα−β)  . Suppose that the assumptions (α ≤ ε < 1. If A(0)φ + f(λ) − f(0) ∈ Eα−γ, then for the solution v(t) in C0β,γ(Eα−β) of the nonlocal boundary value problem (M(λ) does not depend on α, β, γ, φ, and f(t).Suppose umptions  and 10)f(t) ∈ C problem  the coerKm for any m = 1,2, 3,4 in Eα−γ, separately. We start with K1. Applying the inequality , a, a38)||stimates , we obtaerefore,||K11||Eαerefore,||K12||Eαquality , astimates , we obtaove thatz1−α+γ∫0λquality (||K2||Eα−timates (z1−(α−γ)|ove thatz1−α+γ∫0λequality , we getequality , we getstimates , (19), astimates , we obtatimates (z1−(α−γ)|quality ,  0 ≤ γ ≤ β ≤ α,  0 < α < 1.Furthermore, the method of proof of f(t) ∈ C0β,γ(Eα−β)  . Suppose that the assumptions (α ≤ ε < 1. If A(0)φ + f(λ) − f(0) ∈ Eα−ββ,γ, then for the solution v(t) in C0β,γ(Eα−β) of the nonlocal boundary value problem (M(λ) does not depend on α, β, γ, φ, and f(t).Suppose umptions  and 10)f(t) ∈ C problem  the coerC0β,γ(Eα−β)  , in which coercive solvability has been established, depend on the parameters α, β, and γ. However, the constants in the coercive inequalities depend only on α. Hence, we can choose the parameters β and γ freely, which increases the number of functional spaces in which problem    which is established in the paper  is the space of the all continuous functions φ(x) defined on  with the usual normAt,x acting in Cβ with the domain Cβ+2 and satisfying nonlocal boundary conditions v = v, vx = vx for any 0 ≤ t ≤ T.We introduce the Banach spaces E = C with a strongly positive operator At,x defined by formula  does not depend on γ, β,  f,  φ(x).For the solution of nonlocal boundary value problem , the folEα.The proof of Eα = Cα+μ2 for all 0 < 2α + μ < 1,  0 ≤ μ ≤ 1 and 0 ≤ x ≤ 1,  0 ≤ t ≤ T [ ≤ t ≤ T .t ≤ T, x ∈ ℝn}, we consider the nonlocal boundary value problem for the 2mth order multidimensional parabolic equationar = ar  and φ(x)(x ∈ ℝn),  ar, and f  , x ∈ ℝn) are given sufficiently smooth functions. Here, δ > 0 is a sufficiently large number.Second, on the range {0 ≤ Rn. Here, r ∈ Rn is a vector with nonnegative integer components,  |r| = r1 + ⋯+rn. If φ(y)   ∈ Rn) is an infinitely differentiable function that decays at infinity together with all its derivatives, then, by means of the Fourier transformation one establishes the equalityB(ξ) is called the symbol of the operator B and is given by  n, satisfies the inequalitiesξ ≠ 0 and for any 0 ≤ t ≤ T. Assume that the all compatibility conditions hold which guarantees that the problem . This allows us to reduce the nonlocal boundary value problem (E = Cμ(Rn) of all continuous bounded functions defined on ℝn satisfying a Hölder condition with the indicator μ ∈  with a strongly positive operator At,x = Bt,x + δI defined by  does not depend on  γ,  β,  f, φ(x).For the solution of the nonlocal boundary value problem , the folAt,x in Cμ(Rn), and on the following theorem on the structure of fractional spaces Eα(Cμ(Rn), At,x).The proof of Eα(Cμ(Rn), At,x) = Cmα+μ2(Rn) for all 0 < 2mα + μ < 1,  0 ≤ μ ≤ 1  and x ∈ ℝn,  0 ≤ t ≤ T [ ≤ t ≤ T .In the present study, the well-posedness of the nonlocal boundary value parabolic problem with dependent coefficients in Hölder spaces with a weight is established. In practice, new Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are obtained. Moreover, applying the result of the monograph , the hig


Several interesting consequences of our theorems are also given.We introduce the concept of triangular  They also gave some examples and applications of the obtained results to ordinary differential equations. In this paper, we will introduce the concept of triangular αc-admissible mappings (pair of mappings) with respect to ηc nonself-mappings and establish the existence of PPF dependent fixed point theorems for contraction mappings involving triangular αc-admissible mappings (pair of mappings) with respect to ηc nonself-mappings in Razumikhin class.The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities, and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions: modifying Banach's contractive condition, changing the space, or extending single-valued mapping to multivalued mapping  is a Banach space, I denotes a closed interval  in ℝ, and E0 =  denotes the sets of all continuous E-valued functions on I equipped with the supremum norm ||·||E0 defined byThroughout this paper, we assume that  for some c ∈ I.A mapping S : E0 → E0 and T : E0 → E. A point ϕ ∈ E0 is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if Tϕ = (Sϕ)(c) for some c ∈ I.Let T : E0 → E is called a Banach type contraction if there exists k ∈ . Define T : E0 → E by Tϕ = 2ϕ(1) for all ϕ ∈ E0 and α, η : E × E → [0, +∞) byη = x2 + y2 + |x||y| + 1/2. Then T is a triangular αc-admissible mapping with respect to ηc. Indeed, if α(ϕ(1), ξ(1)) ≥ η(ϕ(1), ξ(1)), then ϕ(1) ≥ ξ(1) and so 2ϕ(1) ≥ 2ξ(1). That is, Tϕ ≥ Tξ which implies α ≥ η. Also, ifϕ(c) ≥ π(c) and π(c) ≥ ξ(c) and so ϕ(c) ≥ ξ(c). That is, α(ϕ(c), ξ(c)) ≥ η(ϕ(c), ξ(c)).Let The following lemma is necessary later on.T be a triangular αc-admissible mapping with respect to ηc. Define the sequence {ϕn} by the following way:n ∈ ℕ, where ϕ0 ∈ ℛc is such that α(ϕ0(c), Tϕ0) ≥ η(ϕ0(c), Tϕ0). ThenLet T is a triangular αc-admissible mapping with respect to ηc,α(ϕm(c), ϕm+2(c)) ≥ η(ϕm(c), ϕm+2(c)). By continuing this process, we getSince then by  we get αℱ denote the class of all functions β : [0, +∞)→[0,1) satisfying the following condition:One of our main theorems is a result of Geraghty type  obtainedT : E0 → E, α, η : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular αc-admissible mapping with respect to ηc;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and α(ϕn(c), ϕn+1(c)) ≥ η(ϕn(c), ϕn+1(c)) for all n ∈ ℕ ∪ 0, then α(ϕn(c), ϕ(c)) ≥ η(ϕn(c), ϕ(c)) for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ η(ϕ0(c), Tϕ0).there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ η(ϕ0(c), Tϕ0). Since Tϕ0 ∈ E, there exists x1 ∈ E such that Tϕ0 = x1. Choose ϕ1 ∈ ℛc such thatϕn} in ℛc⊆E such that,ℛc is algebraically closed with respect to difference, it follows thatn ∈ ℕ. This implies that the sequence {||ϕn−ϕn+1||E0} is decreasing in ℝ+. Then, there exists r ≥ 0 such that limn→+∞||ϕn−ϕn+1||E0 = r. Assume r > 0. Now, by taking limit as n → +∞ in (n→+∞β(||ϕn−1−ϕn||E0). That is,β ∈ ℱ, limn→+∞||ϕn−1−ϕn||E0 = 0 which is a contradiction. Hence, r = 0. That is,Let , we get||ϕn−ϕn+1ϕn} is Cauchy in ℛc. Assume the contrary; then there exist ɛ > 0 and two sequences {mk} and {nk} with k ≤ mk < nk such thatk → +∞, we getNow, we prove that the sequence {On the other hand, by (iii) and , we havek → +∞ in the above inequality and applying (k→+∞β(||ϕmk−ϕnk||E0) = 1 and since β ∈ ℱ, we deduceϕn} is a Cauchy sequence in ℛc⊆E0. By the completeness of E0 we get that {ϕn} converges to a point ϕ* ∈ E0; that is, ϕn → ϕ* as n → +∞. Since ℛc is topologically closed, we deduce ϕ* ∈ ℛc. From (iv) we have α(ϕn(c), ϕ*(c)) ≥ η(ϕn(c), ϕ*(c)) for all n ∈ ℕ ∪ 0. Then, from (iii) we getn ∈ ℕ. Taking limit as n → +∞ in the above inequality, we getϕ* is a PPF dependent fixed point of T in ℛc.Therefore, we getapplying  and 31)(34)||ϕmkη(ϕ(c), ξ(c)) = 1 for all ϕ, ξ ∈ E0, then we deduce the following corollary.If in T : E0 → E and α : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular αc-admissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and α(ϕn(c), ϕn+1(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α(ϕn(c), ϕ(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let α(ϕ(c), ξ(c)) = 1 for all ϕ, ξ ∈ E0, then we deduce the following corollary.If in T : E0 → E and η : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular ηc-subadmissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and η(ϕn(c), ϕn+1(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η(ϕn(c), ϕ) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η(ϕ0(c), Tϕ0) ≤ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let c ∈ I and S : E0 → E0, T : E0 → E, α, η : E × E → [0, +∞). We say that  is a triangular αc-admissible pair with respect to ηc if, for ϕ, ξ, π ∈ E0,η(ϕ(c), ξ(c)) = 1, then, we say that  is a triangular αc-admissible pair. Also, if we take α(ϕ(c), ξ(c)) = 1, then we say that  is a triangular ηc-subadmissible pair.Let The following theorem gives a result of existence of PPF dependent coincidence points.S : E0 → E0, T : E0 → E, and α, η : E × E → [0, +∞) be four mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-admissible pair with respect to ηc;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc;there exists (iv)Sϕn} is a sequence in ℛc such that Sϕn → Sϕ as n → +∞ and α((Sϕn)(c), (Sϕn+1)(c)) ≥ η((Sϕn)(c), (Sϕn+1)(c)) for all n ∈ ℕ ∪ 0, then α((Sϕn)(c), (Sϕ)(c)) ≥ η((Sϕn)(c), (Sϕ)(c)) for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ η((Sϕ0)(c), Tϕ0);there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, there exists ϕ* ∈ ℛc such that Sϕ* ∈ ℛc is a PPF dependent fixed point of T and hence ϕ* is a PPF dependent coincidence point of S and T.Let ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ η((Sϕ0)(c), Tϕ0). By condition (vii), there exists ϕ1 ∈ ℛc such thatϕn} in ℛc such thatS(ℛc) is algebraically closed with respect to difference, it follows thatn ∈ ℕ. This implies that the sequence {||Sϕn−Sϕn+1||E0} is decreasing in ℝ+. Then, there exists r ≥ 0 such that limn→+∞||Sϕn−Sϕn+1||E0 = r. Assume r > 0. Now by taking limit as n → +∞ in (n→+∞β(||Sϕn−1−Sϕn||E0). That is,β ∈ ℱ, limn→+∞||Sϕn−1−Sϕn||E0 = 0 which is a contradiction. Hence, r = 0. That is,Let , we get||Sϕn−SϕnSϕn} is Cauchy in S(ℛc). Assume the contrary; then there exist ɛ > 0 and two sequences {mk} and {nk} with k ≤ mk < nk such thatk → +∞, we getNow, we prove that the sequence {On the other hand, by (iii) and , we havek → +∞ in the above inequality and applying (k→+∞β(||Sϕmk−Sϕnk||E0) = 1 and since β ∈ ℱ, we deduceSϕn} is a Cauchy sequence in S(ℛc) ⊂ ℛc. By the completeness of S(ℛc), there exists ϕ* ∈ ℛc such that Sϕn → Sϕ* as n → +∞. From (iv), we have α((Sϕn)(c), (Sϕ*)(c)) ≥ η((Sϕn)(c), (Sϕ*)(c)) for all n ∈ ℕ ∪ 0. Then from (iii) we getn ∈ ℕ. Taking limit as n → +∞ in the above inequality, we getSϕ* is a PPF dependent fixed point of T in S(ℛc) and hence ϕ* is a PPF dependent coincidence point of S and T.Therefore, we getapplying  and 57)(60)||Sϕmη(ϕ(c), ξ(c)) = 1 for all ϕ, ξ ∈ E0, then we deduce the following corollary.If in S : E0 → E0, T : E0 → E, and α : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-admissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc;there exists (iv)Sϕn} is a sequence in ℛc such that Sϕn → Sϕ as n → +∞ and α((Sϕn)(c), (Sϕn+1)(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α((Sϕn)(c), (Sϕ)(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let α(ϕ(c), ξ(c)) = 1 for all ϕ, ξ ∈ E0, then we deduce the following corollary.If in S : E0 → E0, T : E0 → E, and η : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-subadmissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and η((Sϕn)(c), (Sϕn+1)(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η((Sϕn)(c), (Sϕ)(c)) ≤ 1;if {(v)ϕ0 ∈ ℛc such that η((Sϕ0)(c), Tϕ0) ≤ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let T : E0 → E and α : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular αc-admissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and α(ϕn(c), ϕn+1(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α(ϕn(c), ϕ(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let α(ϕ(c), ξ(c)) ≥ 1; then by (iii) we haveT has a PPF dependent fixed point ϕ* ∈ ℛc.Let T : E0 → E and α : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular αc-admissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0, where ϵ ≥ 1;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and α(ϕn(c), ϕn+1(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α(ϕn(c), ϕ(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let α(ϕ(c), ξ(c)) ≥ 1; then by (iii) we haveTϕ−Tξ||E ≤ β(||ϕ−ξ||E0)||ϕ−ξ||E0. That is, all conditions of T has a PPF dependent fixed point ϕ* ∈ ℛc.Let T : E0 → E and α : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular αc-admissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0, where 1 < ϵ ≤ σ;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and α(ϕn(c), ϕn+1(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α(ϕn(c), ϕ(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α(ϕ0(c), Tϕ0) ≥ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let α(ϕ(c), ξ(c)) ≥ 1; then by (iii) we haveTϕ−Tξ||E ≤ β(||ϕ−ξ||E0)||ϕ−ξ||E0. That is, all conditions of T has a PPF dependent fixed point ϕ* ∈ ℛc.Let T : E0 → E and η : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular ηc-subadmissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → ∞ and η(ϕn(c), ϕn+1(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η(ϕn(c), ϕ(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η(ϕ0(c), Tϕ0) ≤ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let T : E0 → E and η : E × E → [0, ∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular ηc-subadmissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0, where ϵ ≥ 1 and ψ ∈ Ψ;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and η(ϕn(c), ϕn+1(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η(ϕn(c), ϕ(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η(ϕ0(c), Tϕ0) ≤ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let T : E0 → E and η : E × E → [0, +∞) be two mappings satisfying the following assertions:(i)c ∈ I such that ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)T is a triangular ηc-subadmissible mapping;(iii)β ∈ ℱ such thatϕ, ξ ∈ E0, where 1 < ϵ ≤ σ and ψ ∈ Ψ;there exists (iv)ϕn} is a sequence in E0 such that ϕn → ϕ as n → +∞ and η(ϕn(c), ϕn+1(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η(ϕn(c), ϕ(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η(ϕ0(c), Tϕ0) ≤ 1.there exists Then, T has a PPF dependent fixed point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and α : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-admissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc;there exists (iv)Sϕn} is a sequence in ℛc such that Sϕn → Sϕ as n → +∞ and α((Sϕn)(c), (Sϕn+1)(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α((Sϕn)(c), (Sϕ)(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and α : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-admissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc where ϵ ≥ 1;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and α((Sϕn)(c), (Sϕn+1)(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α((Sϕn)(c), (Sϕ)(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and α : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-admissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc, where 1 < ϵ ≤ σ;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and α((Sϕn)(c), (Sϕn+1)(c)) ≥ 1 for all n ∈ ℕ ∪ 0, then α((Sϕn)(c), (Sϕ)(c)) ≥ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that α((Sϕ0)(c), Tϕ0) ≥ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and η : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-subadmissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and η((Sϕn)(c), (Sϕn+1)(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η((Sϕn)(c), (Sϕ)(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η((Sϕ0)(c), Tϕ0) ≤ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and η : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-subadmissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc, where ϵ ≥ 1;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and η((Sϕn)(c), (Sϕn+1)(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η((Sϕn)(c), (Sϕ)(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η((Sϕ0)(c), Tϕ0) ≤ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let S : E0 → E0, T : E0 → E, and η : E × E → [0, +∞) be three mappings satisfying the following assertions:(i)c ∈ I such that S(ℛc) ⊂ ℛc is topologically closed and algebraically closed with respect to difference;there exists (ii)S, T) is a triangular αc-subadmissible pair;((iii)β ∈ ℱ such thatϕ, ξ ∈ ℛc, where 1 < ϵ ≤ σ;there exists (iv)Sϕn} is a sequence in E0 such that Sϕn → Sϕ as n → +∞ and η((Sϕn)(c), (Sϕn+1)(c)) ≤ 1 for all n ∈ ℕ ∪ 0, then η((Sϕn)(c), (Sϕ)(c)) ≤ 1 for all n ∈ ℕ ∪ 0;if {(v)ϕ0 ∈ ℛc such that η((Sϕ0)(c), Tϕ0) ≤ 1;there exists (vi)S(ℛc) is complete in ℛc;(vii)T(ℛc)⊂{(Sϕ)(c) : ϕ ∈ ℛc}.Then, S and T have a PPF dependent coincidence point ϕ* ∈ ℛc.Let


A predator-prey system was studied that has a discrete delay, stage-structure, and Beddington-DeAngelis functional response, where predator species has three stages, immature, mature, and old age stages. By using of Mawhin's continuous theorem of coincidence degree theory, a sufficient condition is obtained for the existence of a positive periodic solution. The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. The traditional predator-prey model has been studied extensively , x2(t), and x3(t) are the densities of immature, mature, and the old age predator at time t, respectively; y(t) represents the density of prey at time t. τ = maxt∈{τ1(t), τ2(t)}, τ1(t) ≥ 0 is delay due to prey densities and τ2(t) ≥ 0 is delay due to gestation of predator. The death rates of the immature, mature, and old age predator are proportional to the fact that existing immature, mature, and old age predator population with respective proportionality d1(t), d2(t), d3(t), r1(t) denote the rate of transformation from immature predator into mature predator; r2(t) is the rate of transformation from mature predator into old age predator; k(t) = (d(t)/c(t)) (0 < k(t) ≤ 1) is the transformation coefficient from prey into the immature predators. Let d1(t) + r1(t) = m(t) and d2(t) + r2(t) = n(t).In the present paper, motivated by the above work, we investigate a time delay predator-prey system with three-stage-structure for predator and Beddington-DeAngelis functional response of the form(H1)a(t), b(t), c(t), d(t), di(t) , α(t), β(t), γ(t), e2(t) and ri(t)  are all positive ω-periodic continuous function in R = .For ecological reasons, we consider the initial function of the system  only in f(t), t ∈  | ui(t)|; then X and Y are Banach spaces with the norm ||·||.Define Nu = (N1(t), N2(t), N3(t), N4(t))T, u ∈ X, Pu = (1/ω)∫0ωu(t)dt, u ∈ X, and Qy = (1/ω)∫0ωy(t)dt, y ∈ Y, whereLu = Nu,  u ∈ X.Let e system  can be wL = R4, ImL = {u(t) ∈ Z, ∫0ωu(t)dt = 0} is closed in Y and dimKerL = 4 = CodimImL and P, Q are continuous projectors such that ImP = KerL, KerQ = ImL = Im(I − Q). Therefore, L is a Fredholm mapping of index zero. Furthermore, the inverse KP : ImL → Ket P∩DomL exists and has the formQN and KP(I − Q)N are continuous by the Lebesgue theorem; it is not difficult to show that Ω ⊂ X by the Arzela-Ascoli theorem. Hence N is L-compact on Ω ⊂ X.Obviously, KerΩ ⊂ X. Corresponding to the operator equation, Lu = λNu, λ ∈ , we haveIn order to apply u = u(t) ∈ X is a solution to . Integrating (ω] leads toeu1(t) and integrating it over  lead toeu2(t) and integrating it over  give0ωeu2(t)dt)2 ≤ ω∫0ωeu2(t)2dt and from (eu3(t) and integrating it over  lead toω] leads tou(t) ∈ X, there exist ξi, ηi ∈  such thatt ∈ g(x) = x/(x + l) leads toA5 = e2MγM((r1LdLA4/mM) − nM(αM + βMA4)). From  is independent of λ. From condition of ζi ∈ , i = 1,2, such that the algebraic equations,v1*, v2*, v3*, v4*)T.Suppose that ution to  for a ceegrating  over theleads to∫0ωd(t)eudt=d−3ω,∫0ω[a(t)eation of  by eu1(ation of  by eu2( impliesmL∫0ωeu1(h yieldse2L∫0ωe2u impliesmL∫0ωeu1(].Hence∫0ωeu2(t)ation of  by eu3(].Hence∫0ωeu2(t)ation of  over .Hence∫0ωeu2(t)2.From (∫0ωeu1(t)er with (∫0ωeu3(t) implies∫0ωeu4(t)m−ω=:l1,∫0ω|u˙2(t].Hence∫0ωeu2(t)ws from (∫0ω|u˙1(tu2(t)dt,∫0ω|u˙3≤lndt=d−3ω,∫0ω → X, and a direct calculation shows thatΩ satisfies all the conditions of Lu = Nu has at least one solution in ω-periodic solution in Ω. Then, there exists at least one ω-periodic solution for the system  ≡ 1, β(t) ≡ m(constant), and γ(t) ≡ 0, the corresponding subsystems of the system (Chen et al.  considere system  are syste system ; that ise system . The cone system  remain te system .In a word, despite considering the three-stage-structure, people can more protect the beneficial insect from dying out and under the preconditions; the actualization of the periodic changes is more convenient and realistic. The results accord with the change in the natural environments, so the consideration of stage-structure is necessary and important.


Lx)(t) = x′(t) + ∑i=1mpi(t)x(t − τi(t)) = f(t),  t ∈ , x(tj) = βjx(tj − 0),  j = 1,…, k,  a = t0 < t1 < t2 < ⋯<tk < tk+1 = b,  x(ζ) = 0,  ζ ∉ , with nonlocal boundary condition lx = ∫abφ(s)x′(s)ds + θx(a) = c, φ ∈ L∞; θ,  c ∈ R. Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained. The impulsive delay differential equation is considered ( Mathematical models with impulsive differential equations attract the topic attention of many researchers see –7); many; many7])Various comparison theorems for solutions of the Cauchy and periodic problems for ordinary differential equations with impulses have been obtained in , 21–24. Equations with nonlocal boundary conditions are applied in modelling different processes . NonlocIn this paper we consider the following impulsive equation:We develop the ideas presented in  and we hD of piecewise continuous functions x :  → R, isomorphic to the topological product L × R, where L is the space of measurable essential bounded functions z :  → R, by the following equality:a = t0 < t1 < t2 < ⋯<tk < tk+1 = b.Define the space x(t) is absolutely continuous in , i = 1,…, k + 1, satisfying the equality x(ti) = βix(ti − 0). It is clear that x(t) has discontinuity of the first kind and is continuous from the right at points ti, i = 1,…, k.It is clear that By , the genD for every essential bounded f and c ∈ R, then its solution can be represented in the formG of this problem isC = 0  for t < s. If the boundary value problem –5) is u is u5) iFrom the boundary condition , we obtaSubstituting  into 9)9), we obC can be represented in the following form:Cij = C in the rectangle ti ≤ t < ti+1, si ≤ s < si+1.The Cauchy function Let us take derivative in :(15)x′ = 1, we obtainSince, according to the definition of the Cauchy function, Let us substitute  into 1313 as folC = 0, s > t, it follows that ∫atCds = ∫atCds + ∫tbCds = ∫abCds.Since Changing the order of integration in the third double integral, we getG of this problem is of the following form:C = 0, for t < s.Substituting now  into 1212, we obx(a)=cθ−1tituting , we obtaOne hasC of impulsive equations (quations –4)..C c = 0, φ(s) ≡ γ, θ = γ − 1 we obtain the Corollary.Substituting W is as follows:If generalized periodic problem –4), 7), (7)4), , (7)4), C for several simple equations.Let us construct the Cauchy function Consider now the following auxiliary equation:K. It is known that K = estp(ζ)dζ−∫. For every s this function is absolutely continuous function with respect to t, and K = 0 for s > t.Let us denote the Cauchy function of the nonimpulsive equation  by K for every fixed s as a function of t is a solution ofC = 1, we obtain the following theorem.Using the fact that the Cauchy function C of (Ha(t), Heaviside function, is defined by the following equality:The Cauchy function t, s) of  and 2)  C Now let us consider the following auxiliary equation:C0 of the problem , an, anC0 of the problem  and andC0 Green's function of problem  Green's function of generalized periodic problem , an, anG0..G0 = θ + ∫abφ(t))t′dt.Let us denote β = Col{β1,…, βk}. The computation of integrals in formula  = φ(s) + ∫sbφ(t))t′dt, we obtain for a ≤ s < t1Denoting I2 = φ(s) + ∫sbφ(t))t′dt, we obtain for t1 ≤ s < t2Denoting Ik+1 = φ(s) + ∫sbφ(t))t′dt, we obtain for tk ≤ s < bDenoting J = (γ − 1) + γ∫ab)t′dt.Let us denote The computation of integrals in formula  leads usJ1 = γ∫sb)t′dt + γ, we obtain for a ≤ s < t1Denoting J2 = γ∫sb)t′dt + γ, we obtain for t1 ≤ s < t2Denoting Jk+1(β) = γ∫sb)t′dt + γ, we obtain for tk ≤ s < bDenoting I≠0; then Green's function G0 of problem (k = 3) is described by Let us assume that  problem , 5) in  in I ≠ 0; then Green's function W0 of generalized periodic problem (k = 3) is described by Let us assume that  problem , 7) in  in J|)|(1 − βk+1)/(1 − β)|((2k + 1)/2)Δ where Δ = (maxi≤k+11≤|ti−ti−1|)2,Let us denote βi, 1 ≤ i ≤ k,  I ≠ 0 andf and real c.Let 0 <  problem  and thisBy ψ(t) = ∫abG0f(s)ds + C0(c/(θ + ∫abφ(s))s′ds)) = ∫abG0f(s)ds + /I).Let us denote M : D → D by the equality (Mx)(t) = −∫abG0∑i=1mpi(s)x(s − τi(s))ds.Define the operator We can write equation which is equivalent to problem –5):(43:(435):M|| < 1, then there exists the unique solution of (x(t) = (I − M)−1ψ(t).If ||ution of  which cax|| = maxa≤t≤b|x(t)|. Thus we have got the following condition for existence of unique solution of problem (It is clear that problem :(45)||MC0 can be written asDenoteIt is clear thatEstimation of the solution of the problem –5) can  can 5) cLet us denoteThen the problem –5) has  has 5) hIt is clear thatEstimating the integrals in formula , we get G0 = G1 + G2, where G1 is upper triangular and G2 lower triangular parts of G0 and estimate ||G0||. One hasLet us write From estimation of the integrals, we obtainBy (56)Q≤m(Substituting  into 5151 and 5 we obtaiΩ2 = |1 − βk+1|(2k + 1)Δ/2|1 − β||J|,Let us denote   βi, 1 ≤ i ≤ k,  J ≠ 0 andf.Let 0 <  problem , 7) and andβi, By ψ(t) = ∫abW0f(s)ds.Let us denote T : D → D by the equality (Tx)(t) = −∫abW0∑i=1mpi(s)x(s − τi(s))ds.Define the operator We can write equation which is equivalent to problem –4), 7), (7)4), , (7)4), T|| < 1, then there exists the unique solution of (x(t) = (I − M)−1ψ(t).If ||ution of , 7) whi whiT|| <x|| = maxa≤t≤b|x(t)|. Thus we have got the following condition for existence of unique solution of problem (It is clear that6||Tx||≤|| problem :(65)||TEstimation of the solution of the problem –4), 7), (7)4), , (7)4), Let us denoteThen the problem –4), 7), (7)4), , (7)4), It is clear thatEstimating the integrals in formula , we get W0 = W1 + W2, where W1 is upper triangular and W2, lower triangular parts, and estimate ||W0||:Let us write We obtainIt is clear thatWe obtain the following solution estimation:The proofs of the following two assertions follow from the construction of Green's functions.G0 of problem  is  is (75)0G0 of problem  is  is (76)0Estimation of Green's function  leads uspi(t) ≤ 0 for i = 1,…, m; then problem  is positive for t, s ∈ .Let conditions  and 75)75) be fuc = 0.Without losing generality we assume that Solution of problem –5) can  can 5) cM is positive.Using M|| < 1.The conditions of Now it is clear thatM that for every nonnegative f we get according to (ψ(t) ≥ 0 and x(t) ≥ ψ(t) ≥ 0.It follows from the positivity of the operator rding to  ψ(t) ≥ 0It is clear thatx(t) for every nonnegative f(t), we obtain that G ≥ G0 ≥ 0.From nonnegativity of pi(t) ≥ 0 for i = 1,…, m then problem  is negative for t, s ∈ .Let conditions  and 76)76) be fu problem  is uniquWe prove this assertion analogously to the proof of The proofs of the following two assertions follow from the construction of Green's functions.W0 is negative for t, s ∈ .Let problem , 7) W0 W0(81)0W0 is positive for t, s ∈ .Let problem , 7) W0 W0(82)0The proof of the following two theorems can be obtained analogously from the proof of pi(t) ≤ 0 for i = 1,…, m; then problem  is positive for t, s ∈ .Let conditions  and 82)82) be fu problem , 7) is  is pi(tpi(t) ≥ 0 for i = 1,…, m; then problem  is negative for t, s ∈ .Let conditions  and 81)81) be fu problem , 7) is  is pi(t


This paper attempts to present a multivariable extension of generalized Humbert polynomials. The results obtained here include various families of multilinear and multilateral generating functions, miscellaneous properties, and also some special cases for these multivariable polynomials. Thix = ,  y = , m =  and mi   is a positive integer. It follows from (λ)k : = λ(λ + 1) ⋯ (λ + k − 1), (λ)0 : = 1 is the Pochhammer symbol.In this paper, we consider the following multivariable extension of the generalized Humbert polynomials which are completely different from the polynomials introduced in . This cl of s complex variables z1,…, zs  (s ∈ ℕ) and of complex order μ, letak ≠ 0,   μ, ν ∈ ℂ); z = . Then, for p ∈ ℕ; x = ; y = ; m = , mi ∈ ℕ , one hasCorresponding to an identically nonvanishing function  one has∑n1,…,nr=S denote the first member of the assertion  of s complex variables z1,…, zs  (s ∈ ℕ) and for pi ∈ ℕ, λi, ηi ∈ ℂ, z = , w = , λ : = ,  η : =  letak1,…,kr ≠ 0; ni,  ∈ ℕ0; ℕ0 : = ℕ ∪ {0}. Then, one hasFor a nonvanishing function  one has∑l1=0n1⋯∑Ωμ+νk(z),  z = ,  k ∈ ℕ0, s ∈ ℕ, is expressed in terms of simpler functions of one and more variables, then we can give further applications of the above theorems. We first setμ+νkβ1,…,βr)((z) [As an application of the above theorems, when the multivariable function …,βr)(z)  are genecitly by .μ,ν: = ∑k=0∞ak g μ+νkβ1,…,βr)((z)wk, ak ≠ 0, μ, ν ∈ ℂ, z =  then, one hasIf Λ one has∑n1,…,nr=ak = 1,  μ = 0, ν = 1, we find thatUsing the generating relation  for the s = 2r  andλi, ηi   ∈ ℕ0,  u = , v =  in On the other hand, choosing 2r  and1φλ1+η1k1,ak1,…,kr ≠ 0; pi ∈ ℕ; ni, λi, ηi ∈ ℕ0, i = 1,2,…, r, then, we getIf, we get∑l1=0n1⋯∑ak  (k ∈ ℕ0), if the multivariable functions Ωμ+νk(y) and φλ1+η1k1,…,λr+ηrkr(y),  y = , (s ∈ ℕ), are expressed as an appropriate product of several simpler functions, the assertions of Theorems Furthermore, for every suitable choice of the coefficients In this section, we now discuss some further properties of the family of multivariable polynomials given by . We starn1,…,nr be a family of functions generated byni ≥ mi − 1,  i = 1,2,…, r; nj ≥ 0,  j ≠ i andni = 0,1,…, mi − 2, i = 1,2,…, r; nj ≥ 0, j ≠ i. Also, one finds thatni ≥ mi − 1, i = 1,2,…, r; nj ≥ 0, j ≠ i andni = 0,1,…, mi − 2,  i = 1,2,…, r; nj ≥ 0, j ≠ i.Let Ψi = 1,2,…, r. Then, by differentiating i = 1,2,…rentiate  with resrelation .n1,…,nr is a family of functions generated by 29), respSimilarly, as a consequence of n1,…,nrα) areni ≥ mi, i = 1,2,…, r; nj ≥ 0, j ≠ i andni = 1,2,…, mi − 1, i = 1,2,…, r; nj ≥ 0, j ≠ i.Other recurrence relations for the family of multivariable polynomials ΘThe generating relation  yields tni by ni − ki, i = 1,2,…, r,   the right-hand side of the last equality ist1n1 ⋯ trnr completes the proof.From , we haven1,…,nrα) given explicitly by  in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as followsExpansions of Θni − miki instead of ni in the last equality, we havet1n1 ⋯ trnr gives the desired relation.By , we getn1,…,nrα) in series of Gegenbauer, Hermite, and Laguerre polynomials.In a similar manner, in , using tnα) and give their several properties.In this section, we discuss some special cases of the family of multivariable polynomials Θr = 2. Equation  is a polynomial of degree ni with respect to the fixed variable xi  .   Thus, Cn1,…,nrα) is a polynomial of total degree (n1 + ⋯+nr) with respect to the variables x1,…, xr. Equation  is a polynomial of degree (n1 + ⋯+nr − 2) with respect to the variables x1,…, xr. In , one hasni, i = 1,2,…, r, is odd; thenFor the polynomials xi = 0, i = 1,2,…, r in .From the theorems and corollaries given in ni ≥ 1,  i = 1,2,…, r.By Corollaries Cn1,…,nrα) satisfy the following addition formula:From Cn1,…,nrα) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are as follows:As a result of Cn1,…,nrα) given by  have the following hypergeometric representation:FBr)(α)equality  can be wr = 2, For This case yields that in .Un1,…,nrα) satisfyni ≥ mi − 1, i = 1,2,…, r; nj ≥ 0, j ≠ i.From Corollaries Un1,…,nrα) have the following addition formula:As result of r-variable analogue of Lagrange polynomials  which ials (see (64) satisfyi = 1,2,…, r andFrom Corollaries Gn1,…,nrα) have the following addition formula:By Gn1,…,nrα) in series of Legendre, Gegenbauer, Hermite, and Laguerre polynomials are given byFrom Gn1,…,nrα).We can discuss some generating functions for the multivariable polynomials Gn1,…,nrα),   the following generating function holds true for k ∈ ℕ0:i = 1,2,…, r, wherenα,β) denotes the classical Lagrange polynomials defined by [For the polynomials fined by (72) satisfyni ≥ i − 1,  i = 1,2,…, r; nj ≥ 0, j ≠ i.By Corollaries Hn1,…,nrα) have the following addition formula:From mi = i, xi = 0, yi = −xi, i = 1,2,…, r in Hn1,…,nrα).Furthermore, setting ak1,…,kr ≠ 0; pi ∈ ℕ; ni, λi, ηi ∈ ℕ0, i = 1,2,…, r, and λ = , η = ,w = , then we haveIf


We present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle. Stability theory in the sense of Lyapunov is now well known. Its basic theory and applications can be found in the monographs of Lasalle and Lefschetz  and RoucThe practical stability is a very important problem in the field of application, which deals with the question of whether the system state evolves within certain subsets of the state-space. It is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Thus practical stability is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature. There are some relative results for practical stability of various dynamic systems. We can refer to the monograph of Lakshmikantham et al.  and the Recently, the study of set differential equations in a semilinear metric space has gained much attention due to its applicability to multivalued differential inclusions and fuzzy differential equations and its inclusion of ordinary differential systems as a special case , and som𝕋 be a time scale with t0 ≥ 0 as minimal element and no maximal element. Firstly, we give some relative definitions, which can be found in  denote the set of rd-continuous mappings from 𝕋 to ℝn.Let f : 𝕋 × ℝn → ℝn is said to be right-dense (rd) continuous and is denoted by f ∈ Crd ift, x) with right-dense or maximal t;it is continuous at each  = lims,y)→ and limy→xf exist at each  with left-dense t.the limits The mapping K(ℝn) denote the collection of all nonempty, compact, and convex subsets of ℝn. Define the Hausdorff metricd = inf[d : y ∈ A], A, B are bounded sets in ℝn. We note that K(ℝn) with this metric is a complete metric space.Let K(ℝn) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then K(ℝn) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.It is known that if the space A, B, C ∈ K(ℝn) and λ ∈ ℝ+.The Hausdorff metric  satisfieA, B ∈ K(ℝn). The set C ∈ K(ℝn) satisfying A = B + C is known as the Hukuhara difference of the sets A and B and is denoted by the symbol A − B.Let F: 𝕋 → K(ℝn), denoted by ΔF, is defined byF is defined asThe forward derivative of F, G : 𝕋 → K(ℝn) be differentiable at t; then Δ(F + G)(t) = ΔF(t) + ΔG(t);let αF)(t) = α(ΔF(t)), α ∈ ℝ+.Δ are singletons only, then there is only one selector possible, namely, F itself. In this case, the integral reduces to the generalized integral from time scales into (ℝn). The assumption n = 1 generalizes into the integral for time scales. When 𝕋 is everywhere right-dense, then we have Δt = dt, which results in the conventional formulation for integration of set valued functions. Moreover, for a, b, c ∈ 𝕋, we have acF(t)Δt = ∫abF(t)Δt + ∫bcF(t)Δt;∫abF(t)Δt is closed and convex but need not be necessarily compact.∫If the sets in F, G: 𝕋 → K(ℝn). Then the Hausdorff distance between F and G, D: 𝕋 → ℝ+, is Δ-integrable andLet F: 𝕋 → K(ℝn) is set valued function and t ∈ 𝕋k. Let ΔHF(t) be an element of K(ℝn) (provided it exists) with the property that given any ϵ > 0, there exists a neighborhood Ut of t ⋂𝕋 for some δ > 0) such thatt − h, t + h) ∈ Ut with 0 ≤ h < δ, where μ(t) = σ(t) − t. We call ΔHF(t) the ΔH-derivative of F at t. We say that F is ΔH-differentiable at t if its ΔH-derivative exists at t. Moreover, we say F is ΔH-differentiable on 𝕋k if its ΔH-derivative exists at each t ∈ 𝕋k. The multivalued function ΔHF(t): 𝕋k → K(ℝn) is then called the ΔH-derivative of F on 𝕋k.Assume Let  F: 𝕋 → K(ℝn), ΔHF(t) ∈ K(ℝn)  (provided it exists). Then some easy and useful relationships concerning the ΔH-derivative hold.(i)If the  ΔH-derivative of  F  at  t  exists, then it is unique. Hence, the  ΔH-derivative is well defined.(ii)Assume  F:  𝕋 → K(ℝn) is a multivalued function and let  t ∈ 𝕋k. Then one has the following.(1)If  F  is  ΔH-differentiable at  t, then  F  is continuous at  t.(2)If  F  is continuous at  t  and  t  is right-scattered, then  F  is  ΔH-differentiable at  t  with(3)If   t  is right-dense, then  F  is  ΔH-differentiable at  t  if the limits exist and satisfy the equations (4)If  F  is differentiable at  t, thenF: 𝕋 → K(ℝn), ΔHF(t) ∈ K(ℝn) (provided it exists). We consider the two cases 𝕋 = ℝ and 𝕋 = ℤ, where ℤ stands for the set consisting of all integers. (1)𝕋 = ℝ, then F: ℝ → K(ℝn) is ΔH-differentiable at t ∈ ℝ if and only ifIf exists, that is, if and only if F is differentiable in the Hukuhara sense at t. In this case, we have(2)𝕋 = ℤ, then F: ℤ → K(ℝn) is ΔH-differentiable at t ∈ ℤ withIf where Δ is the usual forward multivalued difference operator.Let K(ℝn)N:X, Y ∈ Kc(Rn)N, Xi, Yi ∈ Kc(Rn), i = 1,2,…, N.Consider the spaceF ∈ Crd, X ∈ K(ℝn)N, X =  such that for each i, i = 1,2,…, N, Xi ∈ K(ℝn).Consider the initial value problemFirst of all, we define the following classes of function:In order to discuss the stability of the solution of set valued differential systems , we statV ∈ Crd along with the solutions of DG: 𝕋 × ℝ+N → ℝN  (N ≥ 1) is said to be upper quasi-monotone nondecreasing in U, if U, W ∈ ℝ+N and ||U||M ≤ ||W||M imply ||G||M ≤ ||G||M, where ||·||M = maxi≤N1≤Ui.A function In the following, we will prove the comparison result in terms of vector Lyapunov functions relative to the set differential system on time scales.H1)(V ∈ Crd, V is locally Lipschitzian in X; that is, for X, Y ∈ K(ℝn)N, one has A is an N × N matrix of nonnegative elements, |V − V| =  − V1|, |V2 − V2|,…, |VN − VN|). Here, by |V| one means the vector |, |V2|,…, |VN|), where Vi are the components of V, i = 1,2,…, N;H2) = U is the solution ofThen for any solution X(t) of ||M ≤ ||U0||M.Assume that  X(t) of , one hasU(t) be the solution of  = V) as an application of the properties of Hausdorff metric; we obtain the estimationX(σ(t)) = X(t) + Y(t), where Y(t) is the Hukuhara difference of X(σ(t)) and X(t) for small h > 0 and is assumed to exist, hence,DΔ+m(t) is the right-derivative of m(t). Hence, we haveV||M ≤ ||U0||M, we can obtain thatLet ution of  for t ≥ Now, we list some definitions about stability which will be used in the following discussion.h0, h ∈ Γ, t ∈ 𝕋. Then one says that (i)h0 is finer than h if there exists a ρ > 0 and a function ϕ ∈ 𝒞𝒦 such that(ii)h0 is uniformly finer than h if in (i) ϕ is independent of t.Let λ, A, B be positive constants . The system ||M < A, t ≥ t0 for t0, t ∈ 𝕋, where X(t) = X is any solution of ||M < B, t ≥ t0 + T, t ∈ 𝕋;practically quasi-stable if for any strongly practically stable if (i) and (ii) hold simultaneously;ϵ > 0 there exists T0 > 0 such that t0 + T0 ∈ 𝕋 and ||X0|| < λ implies ||X(t)|| < ϵ, t ≥ t0 + T0.practically asymptotically stable if (i) holds and for any Let e system  is said λ, A, B be positive constants , h, h0 ∈ Γ. The system ((PS1)h0, h)-practically stable if for any 0 < λ < A, the condition h0 < λ implies h) < A, t ≥ t0, for some t0, t ∈ 𝕋, where X(t) = X is any solution of h0, h)-practically quasi-stable if for any λ, B, T > 0 and some t0 ∈ 𝕋 with t0 + T ∈ 𝕋, the condition h0 < λ implies h) < B, t ≥ t0 + T, t ∈ 𝕋;((PS3)h0, h)-strongly practically stable if (PS1) and (PS2) hold simultaneously;((PS4)h0, h)-practically asymptotically stable if (PS1) holds and for any ϵ > 0 there exists T0 > 0 such that t0 + T0 ∈ 𝕋 and h0 < λ implies h) < ϵ, t ≥ t0 + T0., Q0, Q ∈ Σ. Then we say that the system -practically stable if for any 0 < λ < A, the condition Q0(||U0||M) < λ implies Q(||U(t)||M) < A, t ≥ t0, t ∈ 𝕋, where U(t) = U is any solution of  ≤ ϕ), ϕ ∈ 𝒦, where h0 < λ;A3)(V ∈ Crd, such that V is locally Lipschitzian in X for each right-dense t ∈ 𝕋 and for a, b ∈ 𝒦. It holds thatthere exists A4) ∈ S,for  is the solution of (A5) (ϕ(λ) < A, a(λ) < b(A). Assume that ution of ; (A5)ϕ-practical stability properties of , b(A)), we haveA2) and (A5), it follows that h ≤ ϕ) < ϕ(λ) < A. We claim that h < A.Assume that  is practX(t) of  < λ and t1 > t0, t1 ∈ 𝕋, such that h) ≥ A, h) < A, t0 ≤ t < t1. As h0 < λ, A2), (A3) implies thath0 < λ implies h < A, t ≥ t0.Indeed, if this were not true, there would exist a solution X(t) of  with h0h0, h)-strongly practically stable. For given positive numbers λ, A, B, and T, suppose that (a(λ), b(A), b(B), and T; this means we only need to prove -practical quasi-stability of system (U0||M < a(λ) implies ||U(t)||M < b(B), t ≥ t0 + T with t0 + T ∈ 𝕋.Next we prove that system  is )||M < a(λ), if h0 < λ, we have b) ≤ ||V||M ≤ ||U(t)||M < b(B) for all t ≥ t0 + T if h0 < λ; thus we have h) < B, t ≥ t0 + T provided h0 < λ. Hence system -strongly practically stable.From the foregoing argument, since ||e system  is -practically asymptotically stable. Now, let us suppose that  < λ implies h) < B, t ≥ t0 + T0 for system ||M < a(λ) whenever h0 < λ, we obtaint ≥ t0 + T0, if h0 < λ. Thus we have h < B, t ≥ t0 + T0, provided h0 < λ. Hence system -strongly practically stable.Finally, we show that system  is  is replaced by A6)-practical stability properties of system -practical stability properties of the system -practically stable. Then we have (a(λ), b(A)), such thath0, h)-practical stability of (A2), (A6), we can show thatA6), we haveAssume that  is (Q0,ility of  does not


We consider the class of those distributions that satisfy Gauss's principle (the maximum likelihood estimator of the mean is the sample mean) and have a parameter orthogonal to the mean. It is shown that this so-called “mean orthogonal class” is closed under convolution. A previous characterization of the compound gamma characterization of random sums is revisited and clarified. A new characterization of the compound distribution with multiparameter Hermite count distribution and gamma severity distribution is obtained. Gauss's principle by Campbell } exists and denotes the variance by σ2 = σ2. The standard regularity conditions for maximum likelihood estimation are supposed to hold. The vector X =  denotes a random sample of size n, which realizes the random variable X, and gk = gk, k = 1,…, m, h = h, such that the following equivalent partial differential equations hold :(1):(1)X beμ is called orthogonal to the parameter vector θ, denoted by μ⊥θ, if one has E[(∂2ℓ/∂μ∂θk)] = E[(∂ℓ/∂μ)(∂ℓ/∂θk)] = 0, k = 1,…, m.The mean μ, θ). In this respect, one is interested in the subclass mean orthogonal property μ⊥θ. This so-called mean orthogonal class is characterized as follows.The original motivation for parameter orthogonality is improvement of maximum likelihood estimation by reparameterization. In the class tions in  are elemX be a random variable with cgf C satisfying the above assumptions. Then, one has Let This is shown in Hürlimann . Hudson  = ∑n=0∞p(n)sn of N, it is very useful to consider the associated so-called cumulant pgf defined byTo show this, some preliminaries are required. First, we review conditions under which g(k), k = 1,2,…, is the unique solution of the system of equations  ≥ 0, k = 1,2,…, the distribution of N is compound Poisson with parameter λN and severity distribution h(k) = g(k)/λN, k = 1,2,…. Otherwise, one speaks of the so-called pseudo-compound Poisson representation of the distribution.The sequence s  of the form  to , and set λN = −ln Pr(N = 0). Then μN = μN⊥θN is equivalent to the following conditions: Let the form . SupposeThe condition  is a resσN2·∂CN(tatements  and 11)(12)σN2·∂atements .σN2 · (∂λN/∂μN) = μN holds.If λN = G(1) = ∑k=1∞g(k), CN′(0) = ∑k=1∞kg(k) = μN. Now, using (σN2 · (∂g(k)/∂μN) = kg(k), k = 1,2,…. It follows that σN2 · (∂λN/∂μN) = ∑k=1∞σN2 · (∂g(k)/∂μN) = ∑k=1∞kg(k) = μN. The representation  implies w, using  one seesw, using  is equivMY(t) = exp{CY(t)} be the moment generating function of Y. Expressed in terms of the mean scaled severity Z = Y/μ one has MY(t) = MZ(μt). The relationship (CN(t) yields the series expansion:σ2 · (∂C/∂μ) − ((∂C/∂t) − μ) = 0 is satisfied. With the series representation for C(t) and the assumption σ2 · (∂λN/∂μ) = μ, this equation is equivalent to the following condition (use that MZ(t) does not depend on μ):σ2 · (∂λN/∂μ) = μ, implies thatMZ(t) = (1 − κ2t)α−. Since κ2μ = μY/α, one sees thatLet tionship  for the natural parameterization  of the compound gamma distribution. It is interesting to obtain explicit parameters orthogonal to the means of N, Y, and X. By the assumption μN = μN⊥θN, and since Y is gamma distributed, one has μY⊥α. It remains to construct a parameter vector orthogonal to the mean of X such thatθ = θ must be determined. This task can be solved in a unified way for a lot of counting distributions , θN > 0, p ∈ , be a negative binomial random variable. Its cumulant pgf (G(s) = −θN · ln(1 − p · s), λN = −θN · ln(1 − p). One has the following identity (see [s = 1 that p(∂λN/∂p) = μN. Together, this shows that (μN = θN · p/(1 − p)⊥θN. Now, by σ2 · (∂λN/∂μ) = μ is satisfied. Written in terms of the parameter α = (μNκ2)−1, κ = σ/μ, the latter equation is equivalent to the condition (1/μN)·(∂λN/∂μ) = α/μ. With ∂λN/∂μ = (∂λN/∂p)·(∂p/∂μ) = (μN/p)·(∂p/∂μ) one obtains the differential equation (1/p)·(∂p/∂μ) = α/μ, which has the solution p = θ · μα for some θ. In the coordinates , this constant is equal toμ = μN · μY⊥.Let lant pgf  reads G, B2(p),…, be independent and identically distributed Bernoulli random variables with probability of success p ∈ , i = 1,2,…. Then N(p) = ∑i=1NBi(p) (X(p) = 0  if  N = 0) is called an independent p-thinning of N.Let F be a family of count distributions. It is called closed under binomial subsampling if, for any random variable N with distribution in F, all its independent p-thinnings, for all p ∈  and factorial cumulant generating function (fcgf) CNf(s) = lnP(s + 1). For any integer n ≥ 1 the nth factorial cumulant of N is defined and denoted by κn)( = dnCNf(s)/dsn|s=0.Let There is only one count distribution family closed under convolution and binomial subsampling.F be a family of count distributions parameterized by its r first factorial cumulants θ = (κ(1),…, κr) and assume that its pgf P(s) is continuous in θ over its parameter space. Then F is closed under convolution and binomial subsampling if and only if the pgf is of the formLet See Puig and Valero , proof or = 1 corresponds to the Poisson distribution, r = 2 is the Hermite distribution , k = 1,2,…, r, solves the system in  > 0, g(r) ≥ 0, g(k) = 0, k = 2,…, r − 1 is the generalized Hermite by Gupta and Jain [g(k), k = 1,2,…, r, under which (g(k) < 0 is preceded by a positive value and followed by at least two positive values. In particular, if at least g(1), g(r − 1), g(r) are nonzero, then g(1) > 0, g(r − 1) > 0, g(r) > 0, are necessary conditions for (g(k) ≥ 0 for k = 1,2,…, r, then the multiparameter Hermite is compound Poisson with parameter λN = ∑k=1rg(k) and severity h(k) = g(k)/λN, thus infinitely divisible by Feller [Some comments are in order. The case n . For arWestcott . In termystem in , that isstem in (n·p(n)=∑kd Valero . The speand Jain . The muland Jain  family oestcott [P(s)=expestcott [P(s)=expck(θ), k = 1,2,…, r, be continuous real functions in the parameter vector θ over some parameter space, and set g(k) = ck(θ) · pk, k = 1,2,…, r, for a parameter p > 0. Assume that the cumulant pgf G(s) = ∑k=1rck(θ)·(ps)k defines a feasible multiparameter Hermite random variable N of order r over the parameter space. Then μN = μN = ∑k=1rk · ck(θ) · pk⊥θ.Let λN = G(1) = ∑k=1rck(θ) · pk. Then one hasSet ows that  is satisκ(1), κ(2). Since κ(1) = μN, κ(2) = σN2 − μN, it can equivalently be parameterized by its mean μN and variance σN2. Consider a parameterization p > 0, θN > 0 such that g(k) = θN · pk, k = 1,2. There exists a one-to-one mapping between  and . Since μN = g(1) + 2g(2),σN2 = g(1) + 4g(2), it is determined by the coordinate transformation:G(s) = θ · (ps + p2s2) defines a feasible two-parameter Hermite distribution such that the corresponding random variable belongs to μN = μN = θN · p · (1 + 2p)⊥θN. Since σN2 > μN one notes that the Hermite distribution is necessarily overdispersed. As noted by Puig and Valero [r ≥ 2. Therefore, it should be useful to analyze data with this property .Suppose the Hermite distribution is parameterized by its first two factorial cumulants d Valero  overdispWe are ready for the following new characterization result.N be a counting random variable parameterized by its r first factorial cumulants ξ = (κ(1),…, κr) and assume that its cgf CN is continuous in ξ over its parameter space and set μN = κ(1), λN = −ln Pr(N = 0). Suppose the cgf CY(t) of the severity Y exists, and let C(t) = CN(CY(t)) be the cgf of the random sum X = ∑i=1NYi. Assume the cgf of the mean scaled severity CZ(t) = CY(t/μ) is functionally independent of μ, and set α = (μNκ2)−1, κ = σ/μ. Assume N is closed under convolution and binomial subsampling, σ2 · (∂λN/∂μ) = μ. Then N is a multiparameter Hermite distribution of order r, and Y is gamma distributed with cgf CY(t) = α · ln{α/(α − μYt)}. Furthermore, there exists a parameterization  of N such that its cumulant pgf reads G(s) = ∑k=1rck(θN)·(ps)k. One has μN = μN = ∑k=1rk · ck(θN) · pk⊥θN, μY⊥α, μ = μN · μY⊥, and, in the coordinates , the constant θ is equal toLet μN, μY, μ follows along the same arguments as in The result follows by combining Theorems


Our result generalizes some useful results in the literature. We provide an example to support our result.We define the concept of  A large variety of the problems of analysis and applied mathematics relate to finding solutions of nonlinear functional equations which can be formulated in terms of finding the fixed points of nonlinear mappings. Heilpern  first inFuzzy sets” . There are basically two understandings of the meaning of L, one is when L is a complete lattice equipped with a multiplication ∗ operator satisfying certain conditions as shown in the initial paper αL, where αL ∈ L∖{0L}. The point z ∈ X is called a common L-fuzzy fixed point of S and T if z ∈ [Sz]αL∩[Tz]αL.Let . We say that T is β-admissible if for all x, y ∈ X we haveLet X be a nonempty set, T : X → 2X, where 2X is a collection of subset of X, β : X × X → αL(x), where αL(x) ∈ L∖{0L}, with β ≥ 1, we have β ≥ 1 for all z ∈ [Ty]αL(y) ≠ ϕ, where αL(y) ∈ L∖{0L};for each  (ii)x ∈ X and y ∈ [Tx]αL(x), where αL(x) ∈ L∖{0L}, with β ≥ 1, we have β ≥ 1 for all z ∈ [Sy]αL(y) ≠ ϕ, where αL(y) ∈ L∖{0L}.for each Let  is βℱL-admissible, then  is also βℱL-admissible.It is easy to see that if  be a complete metric space, β : X × X → αL(x), [Tx]αL(x) are nonempty closed bounded subsets of X and for x0 ∈ X, there exists x1 ∈ [Sx0]αL(x0) with β ≥ 1.For each (b) x, y ∈ X, we haveFor all  a1, a2, a3, a4, and a5 are nonnegative real numbers and ∑i=15ai < 1 and either a1 = a2 or a3 = a4.where (c) S, T) is βℱL-admissible pair.((d) xn} ∈ X, such that β ≥ 1 and xn → x, then β ≥ 1.If {Let (z ∈ X such that z ∈ [Sz]αL(z)∩[Tz]αL(z).Then there exists a1 + a3 + a5 = 0,a2 + a4 + a5 = 0,a1 + a3 + a5 ≠ 0 and a2 + a4 + a5 ≠ 0.We will prove the above result by considering the following three cases:Case 1. For x0 ∈ X in condition (a), there exist αL(x0) ∈ L∖{0L} and x1 ∈ [Sx0]αL(x0) such that β ≥ 1 and also there exists αL(x1) ∈ L∖{0L} such that [Sx0]αL(x0) and [Tx1]αL(x1) are nonempty closed bounded subsets of X. From Now, inequality  implies a1 + a3 + a5 = 0 together with the fact that d) = 0, we getUsing x1 ∈ [Tx1]αL(x1), which further implies thatIt follows that x0 ∈ X and x1 ∈ [Sx0]αL(x0) such that β ≥ 1, we have β ≥ 1 for all z ∈ [Tx1]αL(x1). Since x1 ∈ [Tx1]αL(x1), therefore β ≥ 1 and henceBy condition (c), for Again, inequality  implies a1 + a3 + a5 = 0 and d) = 0, we getx1 ∈ [Sx1]αL(x1) and henceSince Case 2. For x0 ∈ X in condition (a), there exist αL(x0) ∈ L∖{0L} and x1 ∈ [Sx0]αL(x0) such that β ≥ 1 and also there exists αL(x1) ∈ L∖{0L} such that [Sx0]αL(x0) and [Tx1]αL(x1) are nonempty closed bounded subsets of X. By condition (c), we have β ≥ 1 for all x2 ∈ [Tx1]αL(x1). From a2 + a4 + a5 = 0 together with the fact that d) = 0, we getUsing x2 ∈ [Sx2]αL(x2), which further implies thatIt follows that β ≥ 1, and henceBy condition (c), we have Again, inequality  implies a2 + a4 + a5 = 0 and d) = 0, we getx2 ∈ [Tx2]αL(x2) and henceSince Case 3. Let λ = ((a1 + a3 + a5)/(1 − a2 − a3)) and μ = ((a2 + a4 + a5)/(1 − a1 − a4)). Next, we show that if a1 = a2 or a3 = a4, then 0 < λμ < 1.a3 = a4, then λ, μ < 1 and so 0 < λμ < 1. Now if a1 = a2, thenx1 ∈ X, there exists αL(x1) ∈ L∖{0L} such that [Tx1]αL(x1) is a nonempty closed bounded subset of X. Since a1 + a3 + a5 > 0, by x2 ∈ [Tx1]αL(x1) such thatx2 ∈ X, there exists αL(x2) ∈ L∖{0L} such that [Sx2]αL(x2) is a nonempty closed bounded subset of X. Since a2 + a4 + a5 > 0, by x3 ∈ [Sx2]αL(x2) such thatx0 ∈ X and x1 ∈ [Sx0]αL(x0) such that β ≥ 1, we have β ≥ 1 for x2 ∈ [Tx1]αL(x1). So we havex3 ∈ X, there exists αL(x3) ∈ L∖{0L} such that [Tx3]αL(x3) is a nonempty closed bounded subset of X. From x4 ∈ [Tx3]αL(x3) such thatx1 ∈ X and x2 ∈ [Tx1]αL(x1) such that β ≥ 1, we have β ≥ 1 for x3 ∈ [Sx2]αL(x2). So we havexn} in X such thatk = 0,1, 2,…,m < n, we haveλμ < 1, so by Cauchy's root test, we get ∑(2i + 1)(λμ)i and ∑2i(λμ)i are convergent series. Therefore, {xn} is a Cauchy sequence in X. Now, from the completeness of X, there exists z ∈ X such that xn → z as n → ∞. By condition (d), we have β ≥ 1 for all n ∈ ℕ. Now, we haven → ∞, we have d) = 0. It implies that z ∈ [Sz]αL(z). Similarly, by usingz ∈ [Tz]αL(z). Therefore, z ∈ [Sz]αL(z)∩[Tz]αL(z). This completes the proof.If ies thatd = |x − y|, whenever x, y ∈ X; then  is a complete metric space. Let L = {δ, ω, τ, κ} with δ ≾L ω ≾L κ, δ ≾L τ ≾L κ, ω and τ are not comparable; then  is a complete distributive lattice. Define a pair of mappings S, T : X → ℱL(X) as follows:β : X × X → αL(0)∩[T0]αL(0).Let X, d) be a complete metric space, β : X × X → α(x) are nonempty closed bounded subsets of X and for x0 ∈ X, there exists x1 ∈ [Sx0]α(x0) with β ≥ 1.For each (b) x, y ∈ X, we haveFor all  a1, a2, a3, a4, and a5 are nonnegative real numbers and ∑i=15ai < 1 and either a1 = a2 or a3 = a4.where (c) S, T) is βℱ-admissible pair.((d) xn} ∈ X, such that β ≥ 1 and xn → x then β ≥ 1.If {Let (z ∈ X such that z ∈ [Sz]α(z)∩[Tz]α(z).Then there exists L-fuzzy mapping A : X → ℱL(X) defined byαL ∈ L∖{0L}, we haveConsider an β = 1 for all x, y ∈ X in If we set X, d) be a complete metric space and S, T fuzzy mappings from X into ℱ(X) satisfying the following conditions:(a) x ∈ X, there exists α(x) ∈ , [Tx]α(x) are nonempty closed bounded subsets of X;for each (b) x, y ∈ X, we havefor all  a1, a2, a3, a4, and a5 are nonnegative real numbers and ∑i=15ai < 1 and either a1 = a2 or a3 = a4.where Let (z ∈ X such that z ∈ [Sz]α(z)∩[Tz]α(z).Then there exists


Moreover, we derive the equation describing the flow on the center manifold; then we give the formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the theoretical results and chaotic behaviors are observed. Finally, using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of the periodic solutions.A modified Leslie-Gower predator-prey system with two delays is investigated. By choosing  The parameters r1 and r2 are the intrinsic growth rates of the prey and the predator. The value K is the carrying capacity of the prey, and γx takes on the role of a prey-dependent carrying capacity for the predator; the parameter γ is a measure of the quality of the prey as food for the predator. However, this model has attracted the attention of some authors  → R2 such thatφ = T ∈ C, defineU = (X(t), Y(t))T and Ut(θ) = U(t + θ), θ ∈ . For ψ ∈ C*), define A(0) = A and the adjoint operator A* of A asηT is the transpose of the matrix η.In this section, we show that the system undergoes the Hopf bifurcation for different combinations of t al. in , we studn system  can be w system (8U˙(t)= and ψ ∈ C*), in order to normalize the eigenvectors of operator A and adjoint operator A*, we define a bilinear inner productη(θ) = η.For iω0τ02′ are eigenvalues of A, they will also be the eigenvalues of A*. The eigenvectors of A and A* are calculated corresponding to the eigenvalues +iω0τ02′ and −iω0τ02′.Since ±q(θ) = Teiω0τ02′θ is the eigenvector of A corresponding to +iω0τ02′; q*(s) = (1/D)Teiω0τ02′s is the eigenvector of A* corresponding to −iω0τ02′ andR = (R(1), R(2)) ∈ R2 and S = (S(1), S(2)) ∈ R2 are constant vectors, computed asFollowing the algorithms explained in Hassard et al. , we can W20(θ) and W11(θ); then gij is determined by the parameters and delays τ02′ and τ1*. Thus, we can compute the following quantities:τ2 = τ02′ and when Re{λ′(τ02′)} > 0 which can be stated as follows: μ2 gives the direction of the Hopf bifurcation: if μ2 > 0  (μ2 < 0), the Hopf bifurcation is supercritical ;β2 determines the stability of bifurcating periodic solution: the periodic solutions are stable (unstable) if β2 < 0  (β2 > 0);T2 denotes the period of bifurcating period solutions: if T2 > 0  (T2 < 0), periodic solutions increase (decrease).As a result, we know r1 = 0.8, r2 = 1, a = 1.3, K = 0.7, γ = 1, and m = 0.5. It is easy to show that system . By calculation, when τ1 = 0, the critical delay for τ2 is obtained as τ02 = 1.3507 and τ01 = 5.8228 when τ2 = 0.To demonstrate the algorithm for determining the existence of the Hopf bifurcation in x˙(t)=0.8E∗ is asymptotically stable at τ1 = 0, τ2 = 1.1 < τ02 = 1.3507, while from E∗ loses stability and the Hopf bifurcation occurs at τ1 = 0, τ2 = 1.5 > τ02 = 1.3507. From E∗ is asymptotically stable when τ1 = 2.8 < τ01 = 5.8228, τ2 = 0, while from E∗ loses stability and the Hopf bifurcation occurs when τ1 = 6.5 > τ01 = 5.8228, τ2 = 0.We can see from τ1 = 1.28, when τ2 = 1.32 < τ02′ = 1.9507, E∗ is also stable . Further, the method we used here is based on the global Hopf bifurcating theorem for general functional differential equations introduced by Wu [τ = τ2 and write system (zt = (zt1(θ), zt2(θ))T = (z1(t + θ), z2(t + θ))T ∈ C. Following the work of Wu [  X = C,  Cl{(z(t), τ, p) ∈ X × R+ × R+, z(t) is a p-periodic solution of  in [m = 1,2, 3,…, where mth crossing number of Assume that 1–A4) in . Denote z-space and onto p-space are bounded, then the projection of τ-space is unbounded. Further, we show that the projection of τ-space is away from zero; then the projection of τ-space must include [τ, ∞). Following this idea, we can prove our results on the global continuation of the local Hopf bifurcation.It is well known that if (ii) of the theorem is not true, then B + E > 0 hold, nontrivial periodic solutions of (If (tions of  are unifx(t), y(t) be a nontrivial solution of system  at t = 0 with φ(0) > 0, ψ(0) > 0. Then it follows from , y(t)) is a nontrivial solution of (x(t) > 0, y(t) > 0, then we havex(t) < x(t − τ1)er1τ1 for t > τ1; then if t > τ1, we obtainε > 0, there exists a T > 0 such that when t > T, we havet > T + τ,y(t) < γ(1 − m)δer1τ for sufficiently large t. Thus, the nontrivial periodic solutions lying in the first quadrant of system  < K, 0 < y(t) < γK(1 − m), respectively. System ; we defineV is well defined and continuous for all x(t) > 0, y(t) > 0. The function V satisfiesx∗, y∗) is the only extremum of the function V in the first quadrant. It is easy to see that the point  is a minimum, sincex∗, y∗) is the global minimum; that is,x(t) > 0, y(t) > 0.Assume that system  has a nol systemx˙(t)=r1xatisfies∂V∂x=1x < x(t), we haveV satisfies Lyapunov's asymptotic stability theorem; we conclude thatCalculate the derivative of f system . Use Razt system  has periB + E > 0 hold, let ω0 and τj2′   be defined in τ > τj2′  (j ≥ 1), system , j = 1,2,…, are isolated centers.It is easy to know that the characteristic matrix of system  at the pτ − τj2′| ≤ δ and ∈∂Ωϵ, then the necessary and sufficient conditions for det(η + i(2π/p))) = 0 are η = 0, τ = τj2′, and p = 2π/ω0.Letlz*,τj2′,2π/ω0) in Σ is nonempty. Meanwhile, we havelz*,τj2′,2π/ω0). Noting that 2π/ω0 < τj2′ and applying p < τ* for  belonging to lz*,τj2′,2π/ω0) is bounded, which is a contradiction. This implies that the projection of lz*,τj2′,2π/ω0) and show the global existence of the periodic solutions.In this paper, we investigate the effect of the time delays f system  and deri