Firstly, an equivalent characterization of ϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between the ϵ-Henig saddle point of the Lagrangian set-valued map and the ϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.We study  Xia and Qiu , λk ∈ K.  K is called absorbent if and only if 0 ∈ cor(K). Let K be a nonempty subset in Y. The vector closure of K is the setLet K is vectorially closed (v-closed) if vcl(K) = K. Following Adán and Novo , K is veB  be a nonempty convex subset in Y. B is a base of C if and only if C = cone(B) and there exists a balanced, absorbent, and convex set V such that 0 ∉ B + V in Y. Write CV(B): = cone(V + B).Let  C) × cor(D) ≠ ∅ and B is a base of C. We recall a notion of  ϵ-Henig properly efficient point introduced by Zhou et al. [From now on, we suppose that cor) if and only if there exists a balanced, absorbent, and convex set V with 0 ∉ B + V such that ϵ-Henig properly maximal efficient point with respect to  B  : = ⋃x∈AF(x), 〈F(x), y*〉: = {〈y, y*〉∣y ∈ F(x)}, and 〈F(A), y*〉: = ⋃x∈A〈F(x), y*〉. The meanings of G(A), 〈G(x), z*〉, and 〈G(A), z*〉 are similar to those of F(A), 〈F(x), y*〉, and 〈F(A), y*〉, respectively. Let F : A⇉Y is called generalized C-subconvexlike on A if and only if cone(F(A)) + cor(C)  is a convex set in Y. A set-valued map Z be a linear space, and let M, N⊆Z be two nonempty sets such that M − N is a convex set in Z. If  cor(M − N) ≠ ∅ and 0 ∉ vcl(M − N), then there exists z* ∈ Z*∖{0} such that supz2∈N〈z2, z*〉<infz1∈M〈z1, z*〉.Let ϵ-Henig proper efficiency.In this section, we will establish approximate saddle point theorems of set-valued optimization problems in the sense of F : A⇉Y and G : A⇉Z be two set-valued maps on A. We consider the following vector optimization problem with set-valued maps:S : = {x ∈ A | G(x)∩(−D) ≠ ∅}.  Let ϵ ∈ C. ϵ-Henig properly efficient solution of (VP) if and only if there exists Hmin(F(S), B). The pair ϵ-Henig properly efficient element of (VP).Let L the set of all linear operators from Z to Y. A subset L+ of L is defined as L+: = {T ∈ L | T(D)⊆C}. The Lagrangian set-valued map of (VP) is defined byWe denote by Consider the following unconstrained vector optimization problem with set-valued maps:ϵ ∈ C,  ϵ-Henig properly efficient element of ϵ-Henig properly efficient element of (VP).Let ϵ-Henig proper saddle point of the Lagrangian set-valued map L in linear spaces. Now, we will introduce a new notion called ϵ-Henig proper saddle point of the Lagrangian set-valued map L if and only ifϵ-Henig proper saddle point of the Lagrangian set-valued map L. The following proposition is an important equivalent characterization for an  D be v-closed and ϵ ∈ C. Then, ϵ-Henig proper saddle point of the Lagrangian set-valued map V with 0 ∉ B + V such that ;;;Let Necessity. Let ϵ-Henig proper saddle point of L. Then, there exist V with 0 ∉ B + V such thatT = 0 in  = C, it follows from  and B + V, b1 ≠ 0. Therefore,z1 ∉ −D. Similar to the above proof, there exists z2* ∈ D+∖{0} such that 〈z1, z2*〉>0. Taking b2 ∈ B, we define a vector-valued map T2 : Z → Y as follows:uch thaty¯+T¯(z¯),T¯),B),y¯+T¯(z¯)uch thatcone∉ϵ we have−T¯(z¯)∈Cuch thatcone,T2(z1)−T¯Sufficiency. Since r1 > 0,   r2 > 0,  T3 ∈ L+,   v ∈ V,  and  b3 ∈ B such that z′ ∈ −D. Since T3 ∈ L+,  −T3(z′) ∈ C. Therefore, there exist r3 ≥ 0 and b4 ∈ B such thatr2/(r2 + r1r3) ≤ 1, it follows from the balance of  V  thatB thatV + B and r2 + r1r3 > 0, it follows from =rws from (r1(y′−y¯−f B thatr2r2+r1r3e assertcone(⋃T∈ove thaty¯+T¯(z¯)ϵ-strictly efficient point is equivalent to the notion of ϵ-Henig properly efficient point in locally convex spaces. Moreover, the generalized subconvexlikeness of the set-valued map F is equivalent to ic-cone convexlikeness of the set-valued map F introduced by Sach [C) ≠ ∅. Therefore,  According to Theorem 1 in , the not by Sach  when theD be v-closed and ϵ ∈ C. If ϵ-Henig proper saddle point of the Lagrangian set-valued map L, then there exist Let ϵ-Henig proper saddle point of the Lagrangian set-valued map L, it follows from V with  0 ∉ B + V such that conditions (i)–(iv) hold. By condition (ii), U with r1 > 0,  r2 > 0,  x1 ∈ S,  y1 ∈ F(S),  b1 ∈ B, and  u1 ∈ U such thatc ∈ C such thatB is a base of C, there exist r3 ≥ 0 and b2 ∈ B such that c = r3b2. By ; C × D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z.  Then, there exists ϵ-Henig properly efficient element of Let By D be v-closed, ϵ ∈ C,  ϵ-Henig properly efficient element of (VP); C × D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z;  Then, there exists ϵ-Henig proper saddle point of  L.Let ϵ-Henig proper efficiency of set-valued optimization problems in linear spaces. In this section, we will give several duality theorems characterized by ϵ ∈ C and let B be a base of C. The set-valued map Φ : L+⇉Y, defined by Φ(T) = ϵ-Hmin, B), is called an ϵ-Henig properly dual map of (VP).Let Now, we construct the following duality problem of the primal problem (VP):ϵ ∈ C.ϵ-efficient point of (VD) if and only ifLet ϵ ∈ C,  Let V with 0 ∉ B + V such that C∖{0}⊆CV(B)∖{0}. Therefore, Since ϵ ∈ C andϵ-Henig properly efficient element of (VP) and ϵ-efficient point of (VD). Let ϵ-Henig properly efficient element of ϵ-Henig properly efficient element of (VP). Because ϵ-efficient point of (VD).  Since ϵ-Henig properly efficient element of (VP); C × D-subconvexlike on A, where vcl(cone(G(A) + D)) = Z.  Then,ϵ-efficient point of (VD). Let ϵ-Henig properly efficient element of ϵ-efficient point of (VD).  According to ϵ-Henig saddle point of the set-valued map in linear spaces. The relationships between the ϵ-Henig saddle point of the set-valued map and the ϵ-Henig properly efficient element of the set-valued optimization problem are established. Some duality theorems are obtained in the sense of ϵ-Henig proper efficiency. When ϵ-Henig proper efficiency is replaced by ϵ-super efficiency in linear spaces, whether the conclusions of this paper hold is an interesting topic.Based on , we intr


G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)| ≤ 2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL or PSL or SL or SL.Let  G having exactly four conjugacy classes of noncyclic proper subgroups is nonsolvable if and only if G≅PSL or SL. As a generalization of the above result, we showed that any group having at most three conjugacy classes of nonnormal noncyclic proper subgroups is solvable, and a group G having exactly four conjugacy classes of nonnormal noncyclic proper subgroups is nonsolvable if and only if G≅PSL or SL  the set of the numbers of conjugates of noncyclic proper subgroups of G. It is clear that a group G with 𝒩𝒞(G) = ∅ is either a cyclic group or a minimal noncyclic group, and a group G with 𝒩𝒞(G) = {1} is a group in which every noncyclic proper subgroup is normal. In [G in which every noncyclic proper subgroup is nonnormal; all such groups G satisfy 1 ∉ 𝒩𝒞(G).Let rmal. In , we also𝒩𝒞(G)| we denote the order of 𝒩𝒞(G). Note that we cannot ensure that 1 ∈ 𝒩𝒞(G) for any solvable group G with |𝒩𝒞(G)| = n ≥ 1. For example, let G≅Dpn2 be a dihedral group of order 2pn, where n ≥ 1 and p is an odd prime. Then 𝒩𝒞(Dpn2) = {p, p2,…, pn}, so 1 ∉ 𝒩𝒞(Dpn2). For the nonsolvable group of the smallest order PSL, it is easy to see that 𝒩𝒞) = {5, 6, 10}, and so |𝒩𝒞)| = 3.By |𝒩𝒞(G)| on the solvability of groups, we have the following result, the proof of which is given in For the influence of |G be a group.𝒩𝒞(G)|≤2, then G is solvable.If |G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL or PSL or SL or SL.  Let The following two corollaries are direct consequences of G be a group with |𝒩𝒞(G)|≤3. Then G is nonsolvable if and only if 𝒩𝒞(G) = {5,6, 10} or {14,78,91}.  Let G be a group and 𝒩𝒯(G) the set of the numbers of conjugates of nontrivial subgroups of G.𝒩𝒯(G)|≤2, then G is solvable.If |G is a nonsolvable group with |𝒩𝒯(G)| = 3 if and only if G≅PSL.  Let G be a group and 𝒩𝒞*(G) the set of the numbers of conjugates of nonnormal noncyclic proper subgroups of G. Obviously 𝒩𝒞*(G)⊆𝒩𝒞(G).Let Arguing as in the proof of G be a group. If |𝒩𝒞*(G)|≤2, then G is solvable.  Let G is a nonsolvable group with |𝒩𝒞*(G)| = 3, we cannot get that Φ(G) = Z(G). For example, let G≅PSL × ℤp, where p ≥ 7 is a prime. It is easy to see that |𝒩𝒞*(G)| = 3. But Φ(G) = 1 and Z(G) = ℤp.If we assume that G be a group and 𝒩𝒜(G) the set of the numbers of conjugates of nonabelian proper subgroups of G. Obviously 𝒩𝒜(G)⊆𝒩𝒞(G). Arguing as in the proof of Let G be a group. If |𝒩𝒜(G)|≤2, then G is solvable.  Let In this section, we collect some essential lemmas needed in the sequel.G be a group. If all nonnormal maximal subgroups of G have the same order, then G is solvable.  Let G be a nonsolvable group having exactly two classes of nonnormal maximal subgroups of the same order; then G/S(G)≅PSL, where S(G) is the largest solvable normal subgroup of G.  Let G be a group having exactly n classes of maximal subgroups of the same order, where 1 ≤ n ≤ 3; then one of the following statements holds:G is a group with n = 1, and then G is a p-group for some prime p;suppose that G is a nonsolvable group with n = 2, and then G/Φ(G)≅) × ℤ7j, where i, j = 0, 1,…, and ℤ2i3⋊PSL is a semidirect product of the normal subgroup ℤ2i3 and the subgroup PSL;suppose that G is a nonsolvable group with n = 3, and then G/S(G)≅A6; PSL, q = 11, 13, 23, 59, 61; PSL; U3(3); PSL; PSL, and f is a prime; PSL × PSL × ⋯×PSL.suppose that  Let The proof of G be a group. If |𝒩𝒞(G)|≤2, then G is solvable.  Let G is nonsolvable. Then by [G are noncyclic. Let ℳ𝒮(G) be the set of the numbers of conjugates of maximal subgroups of G. It follows that ℳ𝒮(G)⊆𝒩𝒞(G). Then |ℳ𝒮(G)|≤2.ℳ𝒮(G). Since G is nonsolvable, G must have nonnormal maximal subgroups. Let M be any nonnormal maximal subgroup of G; one has |G : NG(M)| = |G : M|. Since |ℳ𝒮(G)|≤2, we know that G has at most one class of nonnormal maximal subgroups of the same order. It follows that G is solvable by Suppose that 1 ∈ ℳ𝒮(G). It follows that all maximal subgroups of G are nonnormal. By the hypothesis, G has at most two classes of maximal subgroups of the same order. Since G is nonsolvable and G has no normal maximal subgroups, one has G/Φ(G)≅ℤ2i3⋊PSL by i = 0, 1,…. It is easy to see that 𝒩𝒞(G/Φ(G))⊆𝒩𝒞(G) and |𝒩𝒞)|>2. It follows that |𝒩𝒞(G)|>2, a contradiction.Suppose that 1 ∉  Assume that G is solvable.Thus, our assumption is not true, so G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL or PSL or SL or SL.  A group The sufficiency part is evident, and we only need to prove the necessity part.ℳ𝒮(G)|≤3. We claim thatBy the hypothesis, |ℳ𝒮(G). Then G has at most two classes of nonnormal maximal subgroups of the same order. Since G is nonsolvable, one has G/S(G)≅PSL by Lemmas 𝒩𝒞(G/S(G))⊆𝒩𝒞(G) and |𝒩𝒞)|>3. It follows that |𝒩𝒞(G)|>3, a contradiction. Thus, 1 ∉ ℳ𝒮(G).Otherwise, assume that 1 ∈ (G)≅PSL2, by Lemmaℳ𝒮(G)|≤3, we have that G has at most three classes of maximal subgroups of the same order.Since |G cannot have exactly one class of maximal subgroups of the same order.By G has exactly two classes of maximal subgroups of the same order, according toG/Φ(G)≅ℤ2i3⋊PSL since G has no normal maximal subgroups, where i = 0, 1,…. Since |𝒩𝒞)|>3, it follows that |𝒩𝒞(G)|>3, a contradiction.If G has exactly three classes of maximal subgroups of the same order. By G/S(G) might be isomorphic to A6 or PSL, q = 11, 13, 23, 59, 61 or PSL or U3(3) or PSL or PSL, and f is a prime or PSL × PSL×⋯×PSL. If G/S(G) is an isomorphism to A6 or PSL, q = 11, 23, 59, 61 or PSL or U3(3) or PSL or PSL, and f is an odd prime or PSL × PSL × ⋯×PSL. It is easy to see that |𝒩𝒞(G/S(G))|>3 by [𝒩𝒞(G)|>3, a contradiction. Thus, G/S(G)≅PSL≅PSL or PSL.Thus, ))|>3 by , 9, whicℳ𝒮(G) and |ℳ𝒮(G)| = |𝒩𝒞(G)| = 3. It follows that 1 ∉ 𝒩𝒞(G), so S(G) is cyclic. We claim thatNote that 1 ∉ G) < S(G). Let M be a maximal subgroup of G such that S(G)≰M. Then G = S(G)M. It is obvious that S(G)∩M⊴M. Moreover, S(G)∩M⊴S(G), since S(G) is cyclic. It follows that S(G)∩M⊴G. Therefore, G/(S(G)∩M) = S(G)/(S(G)∩M)⋊M/(S(G)∩M). Let M⊴G; this contradicts that all maximal subgroups of G are nonnormal. Thus, our assumption is not true, so Φ(G) = S(G).Otherwise, assume that Φ(G/Φ(G)≅PSL or PSL.It follows that G) = 1, then G≅PSL or PSL.If Φ(G) ≠ 1. Let p be any prime divisor of |Φ(G)|. We claim that p≯2. Otherwise, assume that p > 2. Let T be a subgroup of Φ(G) such that Φ(G)/T≅ℤp. That is, Φ(G/T)≅ℤp. Then (G/T)/ℤp≅(G/T)/Φ(G/T) = (G/T)/(Φ(G)/T)≅G/Φ(G)≅PSL or PSL. Since p > 2 and Schur multipliers of both PSL and PSL are ℤ2, we have that G/T≅PSL × ℤp or PSL × ℤp. Note that |𝒩𝒞 × ℤp)|>3 and |𝒩𝒞 × ℤp)|>3. It follows that |𝒩𝒞(G)|>3, a contradiction. Thus, p≯2, so Φ(G) is a cyclic 2-group. If |Φ(G)| = 2n > 2, let L be a subgroup of Φ(G) such that Φ(G)/L≅ℤ2. Then (G/L)/ℤ2≅(G/L)/Φ(G/L) = (G/L)/(Φ(G)/L)≅G/Φ(G)≅PSL or PSL. We have that G/L≅SL or SL. Let M be a subgroup of L such that L/M≅ℤ2. Then (G/M)/ℤ2≅(G/M)/(L/M)≅G/L≅SL or SL. Since Schur multipliers of both SL and SL are trivial, we have that G/M≅SL × ℤ2 or SL × ℤ2; this contradicts that all maximal subgroups of G are nonnormal. Thus, |Φ(G)| = 2. It follows that G≅SL or SL. Next, suppose that Φ( Lemmas


We present a fixed-point iterative method for solving systems of nonlinear equations. The convergence theorem of the proposed method is proved under suitable conditions. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. One of the basic problems in mathematics is how to solve nonlinear equationsf(x)=0.Iues; see  and the The zeros of a nonlinear equation cannot in general be expressed in closed form; thus we have to use approximate methods. Nowadays, we often use iterative methods to get the approximate solution of the system ; the besIn this paper, we will present a new fixed point iterative method for solving the system  and provThis paper is organized as follows. In α* is a simple root of (f(α*) = 0. For α, αk ∈ , using Taylor's formula, we haver = 1 in the above equality, we getf′(αk + t(α − αk)) in the interval  is replaced with its value in t = 0, that is, with f′(αk), then we havef′(αk + t(α − αk)) in the interval  by its value in t = 1, that is, by f′(α), then we haveR[f(α)] (still use “=”), we haveαk+1 be the solution of (We now consider the following nonlinear equation:f(x)=0, x we have∫01f′=f(αky using (f(α)≅f(αkf(x)=0, xod from =f(αk we havef(α)=f. Choose the initial value α0 ∈ N, where α* is certain real zero of nonlinear mapping f(α) and δ > 0 is a sufficiently small constant. Take the stopping criteria ϵ1, ϵ2 > 0. Set k : = 0.Step 1 (the predictor step). Compute the predictorStep 2 (the corrector step). Computing the correctorStep 3. If |f(αk+1)|≤ϵ1 or |αk+1 − αk | ≤ϵ2 then stop; otherwise, set αk : = αk+1, k : = k + 1; go to Step 1.In this section, we consider the convergence and convergent rate of α* ∈ I be a simple zero of sufficiently differentiable function f : ⊆R → R. If α0 is sufficiently close to α*, then the two-step iterative method defined by (ek = αk − α* and c1 = f′′′(α*)/6f′(α*).Let fined by -12) has hasα* ∈ f′(βk), we can getα = α* in (f(α*) = 0; we havef′(αk), we can getα = βk in (f′(βk) at the point αk, we haveBy  we get( = α* in  and f(α*4).From  we get( can getf(αk)f′(α3).From , 18), a, a(14)ek we havef′(βk)=f′);thus,f(αk)=f′(e obtainf(βk)=12f can getf′(βk)=f′ can getf′(βk)ek+ituting (f′(βk)ek+n-dimensional case of the method, and we also study these iterative methods' order of convergence. Consider the system of nonlinear equationsfi, i = 1,2,…, n, maps a vector x = T of the n-dimensional space Rn into the real line R. The system  = (f1(x), f2(x),…, fn(x))T. Thus, the system ; that is, F(x*) = 0. For any x, xk ∈ D, we may write Taylor's expansion for F as follows ) in the interval  by its value in t = 0, that is, by F′(xk), then we haveF′(xk + t(x − xk)) in the interval  by its value in t = 1, that is, by F′(x), then we haveR[F(x)] (still use “=”), we havexk+1 be the solution of :(32)F( we have∫01F′(xk+ we haveF(x)=F(xky using (F(x)≅F(xk system (F(x)=0.Lod from (xk+1=xk−F we have∫01 F′(x we haveF(x)=F(xk we haveF(x)=F40). Let uce from  and (40)ce from . Choose the initial value x0 ∈ N, α ∈  and β = 1 − α, where x* is certain real zero of nonlinear mapping F(x) and δ > 0 is a sufficiently small constant. Take the stopping criterions ϵ1, ϵ2 > 0. Set k : = 0.Step 1 (the predictor step). Compute the predictorStep 2 (the corrector step). Computing the correctorStep 3. If ||F(xk+1)|| ≤ ϵ1 or ||xk+1 − xk|| ≤ ϵ2 then stop; otherwise, set k : = k + 1; go to Step 1.β = 0, α = 1, our algorithms (n2 + 2n.If we take gorithms  and 43)β = 0, α In this section, we consider the convergence and convergent rate of F : D⊆Rn → Rn be a sufficiently differentiable function on a convex set D⊆Rn containing a root x* of the nonlinear system (Let r system . The iter system -43) has hasF : Dek = xk − x*, from (F′(yk), we can getx = x* in (F(x*) = 0; we haveF′(xk)−1 to the two sides of the above equation, we obtainx = yk in (yk − xk = −F′(xk)−1F(xk). It follows from the above equation thatF′(yk) at the point xk, we haveDefining x*, from  and 43)ek = xk = x* in  and F.From  we get ye obtainF′(xk)−1F|).From , 50), a, aek =  we haveF′(yk)=F′);thus,F(xk)=F′(e obtainF(yk)=12F can getF′(yk)=F′ we haveαI+βF′(yk).From (F(xk)+F(ye obtainF(yk)=12F can getF′(yk)ek+that is,ek+1=F′(yp is computed approximately by the following formula:In this section we present some examples to illustrate the efficiency and the performance of the newly developed method -43) (pr (pr43) :(65)FT. We test this problem by using initial value x0 = T as a starting point. The test results are listed in The test function is as follows see , 13, 14), 1413, 1x0 = T. The test results are listed in The test function is as follows (see ):(67)x1F(x) = (f1(x), f2(x), f3(x), f4(x))T, where x = T ∈ R4 and fi : R4 → R, i = 1,2, 3,4, such that T (the iterative sequence converges to x*) and x0 = T (the iterative sequence converges to x**) as starting point, respectively. The test results are listed in Tables The test function is hat see , 10, 13), 13F(x) x* = T and x** = T. We test this problem by using x0 = T (the iterative sequence converges to x*) and x0 = T (the iterative sequence converges to x**) as starting point, respectively. The test results are listed in Tables n = 101.The test function is as follows see , 13, 14), 1413, 10Fi(x)=xif : Rn → R is defined byF(x) = (F1(x), F2(x),…, Fn(x))T,x0 = T as starting point. The test results are listed in n = 512.Consider the unrestraint optimum problem (see )(71)minA is an n × n tridiagonal matrix defined byG(x) = (G1(x), G2(x),…, Gn(x))T, and Gi(x) = sinxi − 1, i = 1,2,…, n. We test this problem by using x0 = T as starting point. The test results are listed in n = 1024.Consider the discrete two-point boundary value problems (see ):(75)F(p that appears in Tables From the seven examples in


I-convergence of real sequences was introduced by Kostyrko et al., (2000/01) and also independently by Nuray and Ruckle (2000). In this paper, we introduce the concepts of -statistical convergence of order α and strong -Cesàro summability of order α of real sequences and investigated their relationship.The idea of  The idea of statistical convergence was given by Zygmund  in the fI-convergence of real sequences was introduced by Kostyrko et al. [I-convergence was studied by Das et al. [The idea of o et al.  and alsoo et al.  -statistical convergence of order α and strong -Cesàro summability of order α of real sequences and investigated their relationship. In p-Cesàro summability, I-convergence, and difference sequences. In Sα-convergent sequences and wα-summable sequences. In wα-summable sequences for different α's. In Sα-convergent sequences for different α's.In this paper, we introduce the concepts of  with the usual norm ||x|| = sup|xk|, where k ∈ ℕ = {1,2,…}, the set of positive integers. Also by bs, cs, ℓ1, and ℓp, we denote the spaces of all bounded, convergent, and absolutely and p-absolutely convergent series, respectively.Let p-Cesàro convergence of a sequence of real numbers were introduced in the literature independently of one another and followed different lines of development since their first appearance. It turns out, however, that the two definitions can be simply related to one another in general and are equivalent for bounded sequences. The idea of statistical convergence depends on the density of subsets of the set ℕ. The density of a subset E of ℕ is defined byThe definitions of statistical convergence and strong χE is the characteristic function of E. It is clear that any finite subset of ℕ has zero natural density and δ(Ec) = 1 − δ(E).where α and strong p-Cesàro summability of order α were studied by Çolak [The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan in  and thenby Çolak .X = ℓ∞, c, or c0, where m ∈ ℕ, Δ0x = (xk), Δmx = (Δm−1xk − Δm−1xk+1), and so m(X) are Banach spaces normed byX = ℓ∞, c, or c0. Let X be any sequence spaces, if x ∈ Δm(X), then there exists one and only one y = (yk) ∈ X such thatk; for instance, k > 2m. We use this fact in the following examples.The notion of difference sequence spaces was introduced by Kızmaz  and it wRecently, the difference sequence spaces have been studied in , 36–39.X be nonempty set. Then a family of sets I ⊂ 2X (power sets of X) is said to be an ideal if I is additive, that is, A, B ∈ I implies A ∪ B ∈ I, and hereditary; that is, A ∈ I, B ⊂ A implies B ∈ I.Let F ⊂ 2X is said to be a filter of X if and only if (i) ϕ ∉ F, (ii) A, B ∈ F implies A∩B ∈ F, and (iii) A ∈ F, A ⊂ B implies B ∈ F.A nonempty family of sets I ⊂ 2X is called nontrivial if I ≠ 2X.An ideal I is said to be admissible if I⊃{{x} : x ∈ X}.A nontrivial ideal I is a nontrivial ideal in X(X ≠ ϕ), then the family of sets F(I) = {M ⊂ X : (∃A ∈ I)(M = X∖A)} is a filter of X, called the filter associated with I.If I will stand for a nontrivial admissible ideal of ℕ.Throughout the paper We now introduce our main definitions.x ∈ w is said to be I-convergent if there exists L ∈ ℂ such that, for all ε > 0, the set {n ∈ ℕ : |Δmxk − L| ≥ ε} ∈ I. In this case, one writes  − limxk = L. The set of all -convergent sequences will be denoted by c.A sequence α ∈ -statistical convergence of order α -convergence) if there is a real number L such thatLet Sα − limxk = L or Sα(I) − limΔmxk = L. The set of all -statistically convergent sequences of order α will be denoted by Sα. In the special case α = 1, we will write S instead of Sα.In this case, we write m, I)-statistical convergence of order α is well defined for 0 < α ≤ 1, but it is not well defined for α > 1 in general. For this x = (xk) is defined as follows:(Δε > 0 and α > 1 we haveδ > 0For every x = (xk) is -statistically convergent of order α, both to 1 and 0; that is, Sα − limxk = 1 and Sα − limxk = 0. But this is impossible.Therefore, the sequence m, I)-convergent sequence is -statistically convergent of order α(0 < α ≤ 1), but converse does not hold. For this, consider a sequence x = (xk) defined byIt is easy to see that every  for α ∈ .It is clear that α ∈ -Cesàro summable of order α -summable) if there is a real number L such thatLet wα − limxk = L. The set of all strong -Cesàro summable sequences of order α to L will be denoted by wα. In the special case α = 1, we will write w instead of wα.In this case, we write Sα-convergent sequences and wα-summable sequences. In wα-summable sequences for different α's. In Sα- convergent sequences for different α's.In this section, we give the main results of this paper. In α ∈  − limxk = L1, Sα  limyk = L2, and c ∈ ℝ; thenSα − limcxk = cL1,Sα − lim(xk + yk) = L1 + L2. Let Sα − limxk = L1 and c ∈ ℝ; thenSα − lim(cxk) = cL1.(i) Suppose that Sα − limxk = L1 and Sα − limyk = L2; then we haveA1∩A2 ≠ ϕ. Now for all n ∈ A1∩A2, we haveSα − lim(xk + yk) = L1 + L2.(ii) Now suppose that The proofs of the following two theorems are easy and thus omitted.α ∈ -convergent sequence is uniquely determined.Let x = (xk), y = (yk), and z = (zk) be real sequences such that Δmxk ≤ Δmyk ≤ Δmzk. If Sα − limxk = L = Sα − limzk, then Sα − limyk = L.Let The proof of the following theorem is obtained by using the same techniques of Savas and Das [α ∈ ∩ℓ∞(Δm) is a closed subset of ℓ∞(Δm).Let Sα-statistically convergent sequences and strong wα-summable sequences.In the following theorem we investigate the relationship between α and β be fixed real numbers such that 0 < α ≤ β ≤ 1, and let p be a positive real number; then wα ⊂ Sβ, and the inclusion is strict.Let ε > 0 and wα − limxk = L; then we can writeδ > 0, we haveLet This completes the proof.α = β, we show the strictness of the inclusion wα ⊂ Sβ for a special case. For this, consider the sequence x = (xk) defined byTaking ε > 0 and α ∈ ) for α ∈ , since I is admissible. So xk↛wα.and for any m-bounded and Sα-convergent, but need not to be wα-summable. For this, consider a sequence x = (xk) defined by (x ∈ ℓ∞(Δm) and x ∈ Sα for α ∈  for α ∈ . Therefore, x ∈ Sα∖wα for α ∈ .The converse of fined by . It can The following result is a consequence of wα-convergent to L, then it is Sα-convergent to L.If a sequence is α ≤ β ≤ 1, and let p be a positive real number; then wα ⊂ wβ and the inclusion is strict.Let 0 < p = 1, we show the strictness of the inclusion wα ⊂ wβ for a special case. Define the sequence x = (xk) such thatThe inclusion part of proof is trivial. Taking x ∈ wβ for 1/2 < β < 1 but x ∉ wα for 0 < α < 1/2.So The following result is a consequence of α ≤ β ≤ 1 be a positive real number. Thenα = β, then wα = wβ,if wα ⊂ w for each α ∈  ⊂ Sβ, and the inclusion is strict.Let x ∈ Sα. Then given α and β such that 0 < α ≤ β ≤ 1, we may writeSα ⊂ Sβ.Let Sα ⊂ Sβ for a special case. Define the sequence x = (xk) such thatx ∈ Sβ for 1/2 < β ≤ 1, but x ∉ Sα for 0 < α ≤ 1/2.We show the strictness of the inclusion The following result is a consequence of α ≤ 1 be a real number; then Sα ⊂ S.Let 0 <


Also, we give some results about role of apoint in the existence of endpoints.We introduce  X, d) be a metric space, CB(X) the collection of all nonempty bounded and closed subsets of X, and H the Hausdorff metric with respect to d; that is, H = max{supx∈Ad, supy∈Bd} for all A, B∈CB(X), where d = infy∈Bd. Let T : X → 2X be a multifunction. An element x ∈ X is said to be a fixed point of T whenever x ∈ Tx. Also, an element x ∈ X is said to be an endpoint of T whenever Tx = {x}  for all x ∈ X. Suppose that ψ :  and let d = |x − y|. Define T : X → CB(X) byx = 1 and y = 3/2, thenψ :  and β = 0 otherwise. First suppose that x ∉  or that y ∉ . If x, y ∈  = 1. But, H = 0, and so βH ≤ ψ). If x ∈  = 0. Hence, βH ≤ ψ). Now, suppose that x, y ∈ . In this case, we have β ≥ 1, H = H = (1/2)d, and N = max{d, x/2, y/2,  + d)/2}. Thus, d ≤ N, and soT is a β-generalized weak contractive multifunction. Now, we show that T is β-admissible. If β ≥ 1, then A, B ⊂ , and so Tx = {x/2}∈ and Ty = {y/2}∈ for all x ∈ A and y ∈ B. Thus, β ≥ 1 for all x ∈ A and y ∈ B. Now, suppose A =  and x0 = 1/4. Then, Tx0 = {1/8}∈ and ⊂. Hence, β ≥ 1. Now, we show that T satisfies the condition (H). First note that, for each ɛ > 0, there exists z ∈ X such that supa∈Tzd < ɛ. Now, we show that for each x ∈ X there exists y ∈ Tx such that H = supb∈Tyd. If 0 ≤ x ≤ 1, then Tx = {x/2}, T(x/2) = {x/4}, andx ≤ 3/2 we have 5/2 < 4x − (3/2) ≤ 9/2, T(4x − (3/2)) = {0}. Thus,x ≤ 9/2, then Tx = {0} and T(0) = {0}. Hence,T satisfies the conditions (R) and (K). Note that, 0 is the endpoint of T.Let X be a set and β : 2X × 2X → [0, ∞) a map. We say that the set X has the property (Gβ) whenever β ≥ 1 for all subsets A and B of X with A⊈B or B⊈A.Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. Let X, d) be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a β-admissible, β-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x0 ∈ A such that β ≥ 1. If T has the approximate endpoint property and X has the property (Gβ), then T has a unique endpoint.Let  = β ≥ 1 because X has the property (Gβ). Hence,T has a unique endpoint.By using T has a unique endpoint, while X does has not the property (Gβ). Also, T has the property (R), while T is not lower semicontinuous. To see this, consider the sequence {xn} defined byk ≥ 1 and put y = 1/2 and x0 = 1. Then xn → 1 and y ∈ Tx0 = {1/2}. Let {yn} be an arbitrary sequence in X such that yn ∈ Txn for all n ≥ 1. Then, yk−12 ∈ Txk−12 and yk2 ∈ Txk2 for all k. But, yk−12 = 4xk−12 − (3/2) for sufficiently large k and yk2 = xk2/2 for all k since yk−12 → 5/2, yn↛1/2. This implies that T is not lower semicontinuous.In X, d) be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (Gβ), and there exist a subset A of X, x0 ∈ A and k ∈ [0,1) such that β ≥ 1 and βH ≤ kN for all x, y ∈ X. Then T has a unique endpoint if and only if T has the approximate endpoint property.Let (ψ(t) = kt for all t ≥ 0. Then, It is sufficient that we define T and the property (R) are independent conditions [R) to obtain the next result. Its proof is similar to the proof of It has been proved that lower semicontinuity of the multifunction nditions . We can X, d) be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0 ∈ A such that β ≥ 1. Then T has the approximate endpoint property if and only if T has an endpoint.Let  be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x0 ∈ A such that β ≥ 1. If T has the approximate endpoint property and X has the property (Gβ), then T has a unique endpoint.Let  be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (Gβ), and there exist a subset A of X, x0 ∈ A and k ∈ [0,1) such that β ≥ 1 and βH ≤ kN for all x, y ∈ X. If T has the approximate endpoint property, then Fix(T) = End(T) = {x}.Let (ψ(t) = kt, then, by using Theorem 2.10 in [T has a fixed point. Since T has the approximate endpoint property, by using T has a unique endpoint such x. Let y ∈ Fix(T). If Tx = Ty, then y = x. If Tx ≠ Ty, then β ≥ 1 because X has the property (Gβ). Also, we haveN = max{d, d, d,  + d)/2} = d. Thus, d = 0, and so Fix(T) = End(T) = {x}.If we put  2.10 in , T has aNext corollary shows us the role of a point in the existence of endpoints.X, d) be a complete metric space, x* ∈ X a fixed element, and T : X → CB(X) a multifunction such that T has the property (H) and x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B and all x ∈ A and y ∈ B. Assume that H ≤ ψ) for all x, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that x* ∈ A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn → x and x* ∈ Txn−1∩Txn, for all n ≥ 1, one has x* ∈ Txn∩Tx. Also, for each sequence {xn} in X with x* ∈ Txn−1∩Txn for all n ≥ 1, there exists a natural number k such that x* ∈ Txm∩Txn for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.Let  by β = 1 whenever x* ∈ A∩B and β = 0 otherwise, and then we use It is sufficient we define X, d) be a complete metric space, x* ∈ X a fixed element and T : X → CB(X) a lower semicontinuous multifunction such that T has the property (H) and x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B and all x ∈ A and y ∈ B. Assume thatx, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that x* ∈ A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn → x and x* ∈ Txn−1∩Txn for all n ≥ 1, we have x* ∈ Txn∩Tx. Then T has an endpoint if and only if T has the approximate endpoint property.Let  by β = 1 whenever x* ∈ A∩B and β = 0 otherwise, and then we use It is sufficient to define X, d, ≤) be an ordered metric space. Define the order ⪯ on arbitrary subsets A and B of X by A⪯B if and only if for each a ∈ A there exists b ∈ B such that a ≤ b. It is easy to check that (CB(X), ⪯) is a partially ordered set.Let  be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B and all x ∈ A and y ∈ B. Assume that H ≤ ψ) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that A0⪯Tx0. Assume that for each convergent sequence {xn} in X with xn → x and Txn−1⪯Txn, for all n ≥ 1, one has Txn⪯Tx. Also, for each sequence {xn} in X with Txn−1⪯Txn for all n ≥ 1, there exists a natural number k such that Txm⪯Txn for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.Let  = 1 whenever A⪯B and β = 0 otherwise, and then we use Define X, d, ≤) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B, all x ∈ A, and y ∈ B. Assume that H ≤ ψ) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that A0⪯Tx0. Assume that for each convergent sequence {xn} in X with xn → x and Txn−1⪯Txn, for all n ≥ 1, one has Txn⪯Tx. Also, for each sequence {xn} in X with Txn−1⪯Txn for all n ≥ 1, there exists a natural number k such that Txm⪯Txn for all m > n ≥ k. If T has the approximate endpoint property and A⪯B for all subsets A and B of X with A⊈B or B⊈A, then T has a unique endpoint.Let  = 1 whenever A⪯B and β = 0 otherwise, and then we use Define X, d) be a metric space and T : X → 2X a multifunction. We say that T is an HS-multifunction whenever for each x ∈ X there exists y ∈ Tx such that H = supb∈Tyd. It is obvious that each HS-multifunction is an multifunction which has the property (H). Thus, one can conclude similar results to above ones for HS-multifunctions. Here, we provide some ones. Although by considering HS-multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.Let  be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a β-admissible, β-generalized weak contractive HS-multifunction which has the properties (R) and (K). Suppose that there exist a subset A of X and x0 ∈ A such that β ≥ 1. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete metric space, β : 2X × 2X → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, and β-generalized weak contractive HS-multifunction which has the property (K). Suppose that there exist a subset A of X and x0 ∈ A such that β ≥ 1. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete metric space, x* ∈ X a fixed element, and T : X → CB(X) an HS-multifunction such that x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B, all x ∈ A, and y ∈ B. Assume that H ≤ ψ) for all x, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that x* ∈ A0∩Tx0. Assume that for each convergent sequence {xn} in X with xn → x and x* ∈ Txn−1∩Txn for all n ≥ 1 one has x* ∈ Txn∩Tx. Also, for each sequence {xn} in X with x* ∈ Txn−1∩Txn for all n ≥ 1, there exists a natural number k such that x* ∈ Txm∩Txn for all m > n ≥ k. Then T has an endpoint, and so T has the approximate endpoint property.Let  be a complete ordered metric space and T a closed and bounded valued lower semicontinuous HS-multifunction on X such that Tx⪯Ty for all subsets A and B of X with A⪯B, all x ∈ A, and y ∈ B. Assume that H ≤ ψ) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A0 of X and x0 ∈ A0 such that A0⪯Tx0. Assume that for each sequence {xn} in X with Txn−1⪯Txn for all n ≥ 1, there exists a natural number k such that Txm⪯Txn for all m > n ≥ k. Then T has an endpoint, and so T has the approximate endpoint property.Let (


Mp (1 < p < ∞) of holomorphic functions on the open unit disk 𝔻 in the complex plane. These classes are in fact generalizations of the class M introduced by Kim (1986). The space Mp equipped with the topology given by the metric ρp defined by ρp = ||f − g||p = (∫0π2logp(1 + M(f − g)(θ))(θd/2π))p1/, with f, g∈Mp and Mf(θ) = sup⩽r<10|f(reθi)|, becomes an F-space. By a result of Stoll (1977), the Privalov space Np (1 < p < ∞) with the topology given by the Stoll metric dp is an F-algebra. By using these two facts, we prove that the spaces Mp and Np coincide and have the same topological structure. Consequently, we describe a general form of continuous linear functionals on Mp (with respect to the metric ρp). Furthermore, we give a characterization of bounded subsets of the spaces Mp. Moreover, we give the examples of bounded subsets of Mp that are not relatively compact.We consider the classes  𝔻 denote the open unit disk in the complex plane and let 𝕋 denote the boundary of 𝔻. Let Lq(𝕋)  (0 < q ≤ ∞) be the familiar Lebesgue spaces on the unit circle 𝕋.Let M consists of all holomorphic functions f on 𝔻 for which+|a| = max{loga, 0} andf. The Privalov class  Np  (1 < p < ∞) consists of all holomorphic functions f on 𝔻 for whichNp is denoted as Aq.Following Kim , 2), the, the2]),or whichsup0<r<1p = 1, the condition  the classical Hardy space on 𝔻. It is known =lNp  (1 < p < ∞) was continued in 1977 by Stoll . Hence, f*(eiθ) = g*(eiθ) for almost every eiθ ∈ 𝕋, and, by Riesz uniqueness theorem, we infer that f(z) = g(z) for all z ∈ 𝔻. As, by . For a function f holomorphic in 𝔻 and for any fixed 0 ≤ ρ < 1, denote by fρ the function defined on 𝔻 as fρ(z) = f(ρz), z ∈ 𝔻. Furthermore, for a given holomorphic function f on 𝔻, letFor simplicity, here as always in the sequel, we shall write f holomorphic on the unit disk 𝔻 belongs to the class Mp if and only if it satisfiesA function f ∈ Mp. Conversely, assume that f ∈ Mp. Thenf ∈ Mp; that is, ∫0π2(log+Mf(θ))p(dθ/2π) < ∞, using (The condition  implies p. ThenMfρ(θ)⟶MfMp is closed under integration.The space f ∈ Mp, defineF(reiθ)|≤Mf(θ), and thus MF(θ) ≤ Mf(θ) for almost every θ ∈ . Therefore F ∈ Mp, as desired.For a given function f ∈ Mp, fρ → f in Mp as ρ → 1−.For each function f ∈ Mp. Since f ∈ N+, by Fatou's theorem, the radial limit f*(eiθ) = limr→1−f(reiθ) exists for almost every θ ∈ . Hence, for such a fixed θ, the function t ↦ f(teiθ) is a continuous on , and thus it is uniformly continuous on . Therefore, for such a θ, we havef ∈ Mp, we obtainfρ → f in Mp as ρ → 1−.Assume that act that  is satis we haveM(f−fρ) we will need the following lemmas.For the proof of completeness of the metric space ρ → fn in Mp as ρ → 1−, where this convergence is uniform with respect to n ∈ ℕ.If {fn} is an arbitrary Cauchy sequence in Mp. Then for a given ε > 0 there is a k ∈ ℕ such thatn ≥ k, we haveρ0 < 1  sufficiently near to 1, for whichSuppose that { we haveρp≤ρMp⊆Np is obvious, and . Letφ. DefineU ∈ Lp(𝕋) and there is a constant Ap depending only on p such thatLp is the usual norm of the space Lp(𝕋).Let 1 < We are now ready to state the following result.Mp = Np for each p > 1; that is, the spaces Mp and Np coincide.Mp⊆Np for each p > 1. For the proof of the converse of this inclusion, assume that f ∈ Np. We will show that f ∈ Mp. As noticed in f can be factorized asI(z) is the inner function and F(z) is an outer function for the class Np; that is,ω is a constant of unit modulus. Furthermore, log+ | f* | ∈Lp(𝕋). As |I(z)|≤1, for each z ∈ 𝔻, the previous factorization and the fact that F ∈ Mp immediately imply that f ∈ Mp. Sincer < 1,+ | f* | ∈Lp(𝕋), we conclude by +MF(θ) ∈ Lp(𝕋). This means that F ∈ Mp and therefore f ∈ Mp. Thus Np⊆Mp, and therefore Mp = Np. This completes the proof.By that is,F(z)=ωexpf ∈ Mp. ThenCp is a nonnegative constant depending only on p.Let F be the outer factor in the canonical factorization of f ∈ Mp. From the proof of U(θ) = log+MF(θ) and φ(θ) = log+ | f*(eiθ)| the inequality (F instead of f. Since |f(z)|≤|F(z)|, for each z ∈ 𝔻, it follows that Mf(θ) ≤ MF(θ) at almost every θ ∈ ; thus <uch thatρp((fn)r,Mp with the topology given by the metric ρp defined by  follows from By , page 51By the Lebesgue dominated convergence theorem, we havek ∈ ℕ such that |c| ≤ k. Then the triangle inequality yieldsf ↦ cf is a continuous map from Mp into Mp.Let The assertion (iv) is in fact the assertion of This concludes the proof.Mp, ρp) and  have the same topological structure.We are now ready to prove that the (metric) spaces  and  have the same topological structure.For each j : Mp → Np. Then, by the inequality  given by Stoll .(ii)⇒(iii). It follows that from the obvious inequality |Np. Choose sufficiently small ε > 0 such thatδ, 0 < δ < ε, such that (iii) holds. Choose an n ∈ ℕ for which 1/n < δ. SetEk | = 1/n < δ, and thus by (iii) we havef ∈ Np there exists a measurable set Ef ⊂ 𝕋 depending on f such thatα such that 0 < α < ε/δ. Then using the inequalityf ∈ L, we obtaindp < η, from which it follows that αL ⊂ V. Hence, L is a bounded subset of Np.(iii)⇒(i). Let we have∫02π, 0 < α0 < 1, such thatf ∈ L and |α | ≤α0. It follows thatε > 0, choose η > 0 satisfyingα0 = α0(η) satisfying ⇒(ii). Assume that uch thatows that∫02π(logows that∫02π(loguch thatδlogp1α+Mf(θ))p : f ∈ L} is uniformly integrable on 𝕋. The same assertion is also valid for the condition (iii). On the other hand, from the proof of +Mf(θ))p : f ∈ L} forms a bounded subset of the space L1(𝕋); that is, there holds+|f*(eiθ)|)p : f ∈ L} is bounded in L1(𝕋).Note that the condition (ii) from L is a subset of Mp for which the familyIf +Mf(θ))p : f ∈ L} is uniformly integrable on the circle 𝕋. This fact and the obvious inequality |f(reiθ)|≤Mf(θ), f ∈ Mp, 0 ≤ r < 1, for almost every θ ∈ , imply that the family {(log+|f(reiθ)|)p : f ∈ L, 0 ≤ r < 1} is uniformly integrable.The condition of Mp( = Np) to be bounded.The following result gives a necessary condition for a subset of L be a subset of Mp. If L is bounded in Mp, thenM∞ = maxθ<2π0≤ | f(reiθ)|, K is a positive constant, and ω(r), 0 ≤ r < 1, is a positive continuous function that does not depend on f ∈ L and for which ω(r) ↓ 0 as r → 1.Let f ∈ Np, we haveL is a bounded subset of Np, by ε > 0 there exists δ = δ(ε) > 0, such thatE ⊂ 𝕋 with the Lebesgue measure |E | <δ.By the inequqlity 5.4) from the proof of Theorem 5.2 in  asfn} ⊂ Np and for each measurable set E ⊂ 𝕋 we haveE| denotes the Lebesgue measure of E. From this and L = {fn} is bounded in Np.Define a sequence {E is relatively compact. This means that there exists a subsequence {fnk} of {fn} and a function f ∈ Np such thatf(z) ≡ e on 𝔻. On the other hand, from


Genome rearrangements are studied on the basis of genome-wide analysis of gene orders and important in the evolution of species. In the last two decades, a variety of rearrangement operations, such as reversals, transpositions, block-interchanges, translocations, fusions and fissions, have been proposed to evaluate the differences between gene orders in two or more genomes. Usually, the computational studies of genome rearrangements are formulated as problems of sorting permutations by rearrangement operations.O(δn) time algorithm for solving the weighted sorting problem by CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2, where n is the number of genes in the given chromosome and δ is the number of needed CCLP operations.In this article, we study a sorting problem by cut-circularize-linearize-and-paste (CCLP) operations, which aims to find a minimum number of CCLP operations to sort a signed permutation representing a chromosome. The CCLP is a genome rearrangement operation that cuts a segment out of a chromosome, circularizes the segment into a temporary circle, linearizes the temporary circle as a linear segment, and possibly inverts the linearized segment and pastes it into the remaining chromosome. The CCLP operation can model many well-known rearrangements, such as reversals, transpositions and block-interchanges, and others not reported in the biological literature. In addition, it really occurs in the immune response of higher animals. To distinguish those CCLP operations from the reversal, we call them as non-reversal CCLP operations. In this study, we use permutation groups in algebra to design an The algorithm we propose in this study is very simple so that it can be easily implemented with 1-dimensional arrays and useful in the studies of phylogenetic tree reconstruction and human immune response to tumors. Reversals, often called inversions in the biological literature, rearrange a segment of continuous integers on the chromosome by reversing the order of the integers and changing their signs [Transpositions act on two adjacent and non-overlapping segments on the chromosome by exchanging their locations [Block-interchanges function as a generalized transposition that exchanges two non-overlapping but not necessarily adjacent segments on the chromosome [Translocations affect two chromosomes by exchanging their end segments [Fusions merge two chromosomes into one chromosome and fissions split a chromosome into two chromosomes [Genome rearrangements are studied on the basis of genome-wide analysis of gene orders and important in the evolution of species -6. Sinceir signs ,7-11. Trocations ,12-15. Bromosome . Translosegments -22. Fusiomosomes ,18.inverted transposition  [Recently, great attention has been paid to the study of genome rearrangement using block-interchanges, since block-interchanges contain transpositions as a special case and, currently, the computational models involving block-interchanges are more tractable than those involving transpositions. More recently, Yancopoulos et al. defined a double cut and join (DCJ) operation that can model all the rearrangement operations described previously . The DCJsversal)  and othe• Case I – reversal:a site in Figure e site in Figure As illustrated in Figure • Case II – transposition:b site in Figure The temporary circle is cut in a new place  time algorithm for solving the problem, where n is the number of genes in the given chromosome and δ is the number of needed CCLP operations.All these seven rearrangements described above are simply called E = {1, 2, …, n} be a set of n positive integers. Then a permutation of E is defined as a one-to-one function from E into itself and can simply be denoted by a product of some cycles. For example, a permutation expressed as α =   means that α(1) = 6, α(6) = 4, α(4) = 1, α(2) = 5, α(5) = 3 and α(3) = 2. Basically, a cycle is cyclic and hence it does not matter which element in the cycle is written as the first. If the cycles in a permutation are all disjoint , then their product is called the cycle decomposition. If a cycle has k elements, then it is called a k-cycle. The element in a 1-cycle is usually called fixed. It is a convention that the 1-cycles in a permutation are not written explicitly. If all the elements in E are fixed in a permutation, then this permutation is called an identity permutation and simply denoted by 1 = (1)(2)⋯(n).Below, we introduce some definitions about the basics of permutation groups, as well as a couple of lemmas from Huang and Lu , that arcomposition (or product) of two given permutations α and β of E is a permutation, denoted by αβ, such that αβ(e) = α(β(e)) for all e ∈ E. For example, suppose that α =  and β =  are two given permutations of E = {1,2, …,6}. Then αβ = . It is not hard to see that if α and β are disjoint, then αβ = βα. The inverse of α, denoted by α–1, is a permutation such that αα–1 = α–1α = 1. The conjugation of β by α, denoted by α ⋅ β, is the permutation αβα–1.The α =  be a 2-cycle and β be an any permutation of E. If both a1 and a2 belong to the same cycle of β, then the effect of αβ (or βα) is equivalent to a fission acting on β and hence α is called a split operation of β. For instance, suppose that α =  and β =  . Then αβ =  and βα = . On the other hand, if a1 and a2 belong to two different cycles of β, then the effect of αβ (or βα) equals to a fusion acting on β and α is called a join operation of β. For instance, if α =  and β = , then αβ =  and βα = .As demonstrated in ,17,18, tα of E can be written as a composition of 2-cycles in many ways [norm of α, denoted by ||α||, is the minimum number k such that α can be expressed by a composition of k 2-cycles. The number of disjoint cycles in the cycle decomposition of α is denoted by cn(α), which needs to count those non-expressed 1-cycles in α. For instance, if α =  and E = {1, 2, …,6}, then cn(α) = 3, rather than cn(α) = 2, because α = (4). For any permutation α of E, it can be shown that ||α|| = |E| – cn(α) [α and β of E, α divides β, denoted by α|β, if and only if ||βα–1|| = ||β|| – ||α||. Actually, whether α divides β or not can be easily determined using the following lemma from [In fact, any permutation any ways . The normma from .Lemma 1[Let e1, e2, …, ke ∈ E and β be any permutation of E. Then e1, e2, …,k appear in the same cycle of β in the order of ee1, e2, …, k if and only ife |β.Lemma 1. Let e1,E as E = {±1, ±2, …, ±n} for properly modeling reversals using the permutation groups, as described in Lemma 3 below. Let Γ =  ··· . It is not difficult to verify that Γ2 = 1 and Γ–1 = Γ. If a cycle contains no e and –e at the same time, where e ∈ E, then it is called admissible and can be used to denote a DNA strand. Let π+ denote a strand of a DNA molecule π. Then π– = Γ · (π+)–1 is the reverse complement of π+, representing another strand of π. Note that π+ and π– are disjoint. For the purpose of modeling reversals using the permutation groups, the DNA molecule π is represented by the composition of its two strands π+ and π– , as demonstrated in [It is required to further extend the definition of rated in .Lemma 2 [Let π and σ be two different chromosomes. Suppose that α is a cycle in σπ–1. Then (πΓ) · α–1is also a cycle in σπ–1.Lemma 2 . Let π aα and (πΓ) · α–1 are mate cycles for each other in σπ–1 according to Lemma 2.Actually, Lemma 3[Let u and v be in the different strands of a chromosome π, that is,  ł π. Then γ = (πΓ(v), πΓ(u))  affects π as a reversal.Lemma 3. Let u au, v) acts on π as a join operation and (πΓ(v),πΓ(u)) acts on π as a split operation, indicating that a reversal acting on π can be implemented using the product of two 2-cycles and π. Actually, other non-reversal CCLP operations can be implemented by multiplying four 2-cycles (πΓ(x),πΓ(w))(πΓ(v),πΓ(u)) with the given chromosome π if the following conditions are satisfied: (1) |π, (2)  ł π (3) w ≠ Γ(x) or Γ(w) ≠ x and (4) ) ł π or (Γ(w), x) ł π. The first condition is to make sure that  and (πΓ(v),πΓ(u)) respectively act on the two strands of π as splits, which lead to two temporary circles excised from π. Note that these two temporary circles are complement to each other. The second condition is to make sure that  and (πΓ(x), πΓ(w)) respectively act on the two temporary circles and the cycles of the remaining π as joins, which paste back the two temporary circles into the remaining π. It is worth mentioning that the joins also fulfill the process of linearization with possible inversion. The inversion is performed when the temporary circles are reinserted into the chromosome strands different from the ones they come from. The third and fourth conditions are to make sure that the resulting π are admissible . Therefore, we have the following lemma.Note that in Lemma 3, (Lemma 4. Let π be a chromosome and β = (πΓ(x), πΓ(w))(πΓ(v),πΓ(u)). Suppose that the following four conditions are satisfied: (1) |π, (2)  ł π (3) w ≠ Γ(x) or Γ(w) ≠ x and (4) ) ł π or (Γ(w), x) ł π. Then β affects π as a non-reversal CCLP operation.π into I =  using the CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2. The basic idea behind this algorithm is as follows. As mentioned before, any permutation can be written as a product of 2-cycles and the effect of a reversal  acting on π can be simulated by multiplying two  2-cycles with π. Moreover, the product of Iπ–1 and π equals to I. All these facts indicate that one can derive a product of 2-cycles from Iπ–1 such that these 2-cycles perform as a sequence of CCLP operations to optimally transform π into I. Below, for simplicity of describing our algorithm, x and y are said to be adjacent in a permutation α if α(x) = y or α(y) = x.In this section, we design an efficient algorithm on the basis of the permutation groups that sorts a given chromosome Lemma 5. Let π = π+π–be a chromosome. Suppose that |Iπ–1and |π, that is, there are two elements x and y in a cycle of Iπ–1such that  acts on π as a split. Let β = (πΓ(y), πΓ(x)). Then there are two adjacent elements x′ and y′ in a cycle of I(βπ)–1such that  and (βπΓ(y′),βπΓ(x′)) act on βπ as joins. Moreover, the cycles in β′βπ are admissible, where β′ = (βπΓ(y′), βπΓ(x′)).Proof. For convenience, let π = π+–π = . The assumption |π indicates that x and y are in the same cycle of π, say in π+, and hence πΓ(x) and πΓ(y) are in π–. Hence, both  and (πΓ(y),πΓ(x)) act on π as splits and β = (πΓ(y), πΓ(x)) divides π into four cycles. Let ia <ia+1 <n for 1 ≤ i ≤ k – 2. This indicates that ka–1 is the maximum in ka–1 + 1 is not in I(βπ)–1(a1) = I(ka–1) = ka–1 + 1, meaning that a1 and ka–1 + 1 are adjacent in I(βπ)–1. In other words, there are two adjacent elements a1 and k–a1 + 1 in I(βπ)–1 such that , as well as (βπΓ(ka–1 + l), βπΓ((a1)), acts on βπ as a join. If the two cycles in (βπΓ(ka–1 + 1),βπΓ(a1))βπ are admissible , then we have completed the proof of this lemma based on Lemma 4. Now, suppose that the two cycles in (βπΓ(ka–1 + 1),βπΓ(a1))βπ are not admissible . We then show below that we can still find two other adjacent elements x′ and y′ in a cycle of I(βπ)–1 such that  and (βπΓ(y′),βπΓ(x′)) can join βπ into two admissible cycles. First of all, ka–1 + 1 must be in βπΓ(ka–1 + 1),βπΓ(a1))βπ is an admissible chromosome), leading to that the cycle created by joining a1, ka–1 + 1) is not admissible. Further suppose that ja is the minimum in ja) = –ja, which is the maximum in ja ≥ ka–1 + 1  are adjacent in I(βπ)–1 because I(βπ)–1(j–a–1) = I(–ja). In the following, we consider five possibilities.Case 1.j a ≠ –n and ja ≠ 1. Then I(–ja) = –ja + 1, which is not in ja is the maximum in ja + 1 is in ka–1 cannot be the maximum in ja ≥ ka–1 + 1 and hence –ja + 1 >ka–1 which contradicts to our assumption that ka–1 is the maximum in I(–ja) belongs to either ja–1, I(–ja)) acts on βπ as a join and the cycles in (βπΓI(–ja),βπΓ(–ja–1)))βπ are admissible.Case 2.j a = –n and both 1 and –1 are not in I(–ja) = 1 (instead of I(–ja) = –ja + 1 = n + 1). Because I(–ja) belongs to either ja–1,I(–ja)) acts on βπ as a join and (βπΓI(–ja),βπΓ(–j–a1)))βπ contains only admissible cycles.Case 3. ja = 1 and both n and –n are not in I(–ja) = –n (instead of I(–ja) = –ja + 1 = 0 ). Clearly, I(–ja) belongs to either ja–1,I(–ja)) acts on βπ as a join and (βπΓI(–ja), βπΓ(–ja–1)))βπ have two admissible cycles.Case 4.j a = –n and 1 or –1 is in n, 1 and –1. Then we can exchange the roles of x′ and y′ in a cycle of I(βπ)–1 such that  and (βπΓ(y′), βπΓ(x′)) can join the four cycles of βπ into two admissible cycles.Case 5. ja = 1 and n or –n is in n and among them, ja is the smallest.According to the above discussion, we have completed the proof of this lemma.Theorem 1. Let Φ denote a minimum weighted sequence of CCLP operations required to transform π into I. Then the weight of Φ is great than or equal toProof. Let Φ contain a reversals and b non-reversal CCLP operations. It is not hard to see that a + 2b is the weight of Φ. Recall that the effect of a reversal can be simulated using two 2-cycles and a non-reversal CCLP operation using four 2-cycles. It indicates that Φ can be written by a composition of 2a + 4b 2-cycles such that Φπ = I, which equals to that Iπ–1 can be expressed as a composition of 2a + 4b 2-cycles. In other words, ||Iπ–1|| ≤ 2a + 4b. As mentioned before, we also have ||Iπ–1|| = |E| – cn(Iπ–1), which bases on the lemma proposed in [E| – cn(Iπ–1) ≤ 2a + 4b and, as a result, the weight of Φ is great than or equal to posed in ,17. Therx and y in a cycle of Iπ–1 such that |π. Then, according to Lemma 5, we can always find a non-reversal CCLP operation β′β from Iπ–1 to rearrange π into β′βπ, where β = (πΓ(y), πΓ(x)) and β′ = (βπΓ(y′),βπΓ(x′)). Assume that there are no any two adjacent elements x and y in a cycle of Iπ–1 such that |π, which implies that  ł π. Then based on Lemma 3, (πΓ(y), πΓ(x)) can serve as a reversal to transform π into (πΓ(y), πΓ(x))π. Using these properties, we design Algorithm 1 to sort π into I by CCLP operations. It is not hard to see that a non-reversal CCLP operation derived in Algorithm 1 decreases the norm of Iπ–1 by 4 and a reversal by 2. Since non-reversal CCLP operations are weighted 2 and reversals are weighted 1, Algorithm 1 decreases the norm of Iπ–1 by 1 at the weight of Assume that there are at least two adjacent elements Theorem 2. Given a chromosome π, the weighted sorting problem by CCLP operations can be solved in O(δn) time when with weight ratio between reversals and non-reversal CCLP operations is 1:2, where δ is the number of CCLP operations needed to transform π into I. Moreover, the weight of the optimal solution isthat can be calculated in O(n) time in advance.Proof. As discussed before, Algorithm 1 transforms π into I by a minimum weighted sequence of δ CCLP operations, whose total weight is O(n) time. Below, the time-complexity of Algorithm 1 is analyzed. Basically, the computation in steps 1 and 2 can be done in O(n) time. As for step 3, there are δ iterations to perform. For each such iteration, it takes O(n) time to find  and  by determining every pair of adjacent elements in all the cycles of Iπ–1 and Iπ–1β, respectively, and a constant time to perform other operations in step 3.1, and also takes O(n) time to perform step 3.2. Therefore, the cost of step 3 is O(δn). Step 4 is executed in constant time. Totally, the time-complexity of Algorithm 1 is O(δn).x on a circular chromosome, a CCLP operation acting on x has an equivalent one without acting on x. Based on this property, one can further prove that the problem of sorting by CCLP operations is equivalent for circular and linear chromosomes.It is worth mentioning here that our algorithm is applicable to both circular and linear chromosomes. Actually, using similar discussion as in , one canO(δn) time algorithm for solving the weighted sorting problem by CCLP operations when the weight ratio between reversals and non-reversal CCLP operations is 1:2, where n is the number of genes and δ is the number of needed CLLP operations. As described in this article, this algorithm is very simple so that it can be easily implemented using 1-dimensional arrays and useful in the studies of phylogenetic tree reconstruction and human immune response to tumors. It would be an interesting future work to design efficient algorithms for solving the problem of sorting by CCLP operations when all the CCLP operations are weighted equally.In this article, we have introduced and studied the sorting problem by CCLP operations, where CCLP is a cut-circularize-linearize-and-paste operation that can model several known and unknown rearrangements. In addition, we have proposed an The authors declare that they have no competing interests.CLL conceived of this study, designed and analyzed its algorithm and drafted the manuscript. KHH and KTC participated in the design and analysis of the algorithm and the draft of the manuscript. All authors read and approved the final manuscript.


As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.We prove a general result on Ulam's type stability of the functional equation  Its A be a nonempty set,  be a metric space, 𝒞 ⊂ ℝ+A2 be nonempty, 𝒯 be an operator mapping 𝒞 into ℝ+A, and ℱ1, ℱ2 be operators mapping nonempty 𝒟 ⊂ XA into XA2. We say that the equation𝒯-stable provided for every ε ∈ 𝒞 and φ0 ∈ 𝒟 withφ ∈ 𝒟 of  Roughly speaking, 𝒯-stability of  solutioility of  is alwayility of ) to an eility of . The nexE1 and E2 be two normed spaces and let c ≥ 0 and p ≠ 1 be fixed real numbers. Let f : E1 → E2 be an operator such thatp ≥ 0 and E2 is complete, then there is a unique operator T : E1 → E2 that is additive  = T(x) + T(y) for x, y ∈ E1) and such thatp < 0, then f is additive.Let p = 0 yields the result of Hyers )of Hyers  and it iof Hyers ; cf. alsof Hyers , 19) thashown in  that estp < 0, can be described as the φ-hyperstability of the additive Cauchy equation for φ ≡ c(||x||p + ||y||p) . The subsequent remark shows this.The second statement of e, e.g., , 21, 22;e, e.g., –25). It p < 0, a ≥ 0, I = , and f, T : I → ℝ be given by T(x) = 0 and f(x) = xp for x ∈ I. Then clearlyx ≤ y. Then (x + y)p ≤ (2x)p = 2pxp ≤ xp ≤ xp + yp, whence |f(x + y) − f(x) − f(y)| = xp + yp − (x+y)p ≤ xp + yp.Let In this paper we prove a quite general result that allows us to generalize and extend In the proof of the main theorem in this paper, we use the following fixed point result that can be easily derived from ). For a Z is a nonempty set,  is a complete metric space, f1, f2 : Z → Z, 𝒯 : YZ → YZ is an operator satisfying the inequality+Z → ℝ+Z is an operator defined byε : Z → ℝ+ and φ : Z → Y such thatn denotes the nth iterate of Λ . Then there exists a unique fixed point ψ of 𝒯 withAssume that X, +), we denote by Aut  X the family of all automorphisms of X. Moreover, for each u ∈ XX we write ux : = u(x) for x ∈ X and we define u′ ∈ XX by u′x : = x − ux.Given a group  and  be commutative groups, d be a complete metric in E that is invariant  = d for x, y, z ∈ X), H : (X∖{0})2 → ℝ+, andu ∈ Aut  X. Assume that f : X → E satisfies the inequality𝒰 ⊂ l(X) such thatT : X → E fulfilling the inequalityLet (𝒰 ⊂ l(X) be nonempty and let  = u′x, and f2(x) = ux. Moreover,  Andξ, μ ∈ EX0, x ∈ X0, and u ∈ 𝒰. Consequently, for each u ∈ 𝒰, also ,n thatx ∈ X0, n ∈ ℕ0 (nonnegative integers), and u ∈ 𝒰. Hence,Z = X0, Y = E, ε : = εu, and φ = f. According to it, the limitx ∈ X0 and u ∈ 𝒰,Tu : X → E defined byTu′ is a fixed point of 𝒯u.Note that, in view of the definition of x, y ∈ X0, x + y ≠ 0, n ∈ ℕ0, and u ∈ 𝒰.Now we show thatn = 0 is just d. Then, by (Tu(x) + Tu(−x) = 0 and consequently Tu(x − x) = Tu(0) = 0 = Tu(x) + Tu(−x).In view of , it is oThen, by ,(34)Tu,tisfying  and 35)T : X → Ytions to  for all j ∈ ℕ0,j = 0 is exactly ,Tw view of ,(38)d,Twj ∈ ℕ0,d(T(x),TwTu = Tw for each u ∈ 𝒰 , which  implies  with T :equality  means thThus we have completed the proof of X, H, E, and d be as in 𝒰 ⊂ l(X) such that (f : X → E satisfying (Let uch that  holds antisfying  is additf : X → E satisfies (T : X → E such that (H𝒰(x) = 0 for x ∈ X∖{0}, this means that f(x) = T(x) for x ∈ X∖{0}, whencef is additive  proof of ).The next corollary corresponds to the results on the inhomogeneous Cauchy equation  in 30–3–330–35.X, H, E, and d be as in F : X2 → E. Suppose thatF ≠ 0 for some x0, y0 ∈ X∖{0}, and there exists a nonempty 𝒰 ⊂ l(X) such that (f : X → E.Let uch that  and 41)X, H, E, f : X → E is a solution to  = f(x0 + y0) − f(x0) − f(y0) = 0, which is a contradiction.Suppose that ution to . Then, with d = ||x − y||, and H : (X∖{0})2 → ℝ+. For each n ∈ ℤ define μn : X → X by μnx : = nx for x ∈ X. Let H be given byp < 0, q < 0, c ≥ 0, and d ≥ 0. Thenx, y ∈ X∖{0}, k, n ∈ ℤ, and kn ≠ 0. Hence,M > 1 such thatf : X → E satisfying =c|H : (X∖{0})2 → ℝ+ has the following a bit more involved formp < 0, q < 0, c ≥ 0, and d ≥ 0 and additive injections η, χ : X → E1 . So, we have the following corollary corresponding to the hyperstability results in [Clearly, the above reasoning also works (after an easy modification) when the function sults in , 21, 24 sults in , 23, 25)H be given by . Then every f : X → E satisfying , becausex, y ∈ X∖{0}, k, n ∈ ℤ, and kn ≠ 0. So we have the following hyperstability result, as well .H be given by . Then every f : X → E satisfying , becausex, y ∈ X∖{0}, k, n ∈ ℤ, and kn ≠ 0. So, we have yet the following.It is easily seen that another example of the function tisfying  is givenH be given by . Then every f : X → E satisfying . Then, for every f : X → E satisfying  ≤ 2p−. Consequently, by 𝒰 = {u0}, there is a unique additive T : X → E such thatLet  satisfy  and u0


Characterizations in a uniform setting are proved, using techniques from thedomain of nonautonomous evolution equations with unbounded coefficients, and connections with the classic notion of trichotomy are given. The statements are sustained by several examples.We construct a framework for the study of dynamical systems that describe phenomena from physics and engineering in infinite dimensions and whose state evolution is set out by skew-evolution semiflows. Therefore, we introduce the concept of  The possibility of reducing the nonautonomous case in the study of associated evolution operators to the autonomous case of evolution semigroups on various Banach function spaces can be considered as an important way towards applications issued from the real world. Of great importance in the study the solution of differential equations is the approach by evolution families, as the techniques from the domain of non autonomous equations with unbounded coefficients in infinite dimensions can be extended in this direction.Appropriate for the study of evolution equations in infinite dimensions are also the skew-evolution semiflows, introduced by us in , as geneω-trichotomy for skew-evolution semiflows in Banach spaces and to give several conditions in order to describe the behavior related to the third subspace.The techniques used in the investigation of exponential stability and exponential instability were generalized for the case of exponential dichotomy in , 8 and fX, d) be a metric space, V a Banach space, and V* its topological dual. Let ℬ(V) be the space of all V-valued bounded operators defined on V. The norm of vectors on V and on V* and of operators on ℬ(V) is denoted by ||·||. Let us consider that Y = X × V and T = { ∈ ℝ+2 : t ≥ t0}. I is the identity operator.Let  is called evolution cocycle over an evolution semiflow φ if it satisfies the following properties:A mapping Φ : C : T × Y → Y defined byφ, is called skew-evolution semiflow on Y.The mapping f : ℝ+ → ℝ+* be a decreasing function with the property that there exists limt→∞f(t) = a > 0. We denote by 𝒞 = 𝒞 the set of all continuous functions x : ℝ+ → ℝ+, endowed with the topology of uniform convergence on compact subsets of ℝ+, metrizable by means of the distanceLet x ∈ 𝒞, then, for all t ∈ ℝ+, we denote that xt(s) = x(t + s), xt ∈ 𝒞. Let X be the closure in 𝒞 of the set {ft, t ∈ ℝ+}. It follows that  is a metric space, and the mapping φ : Δ × X → X, φ = 𝓍t−s is an evolution semiflow on X.If V = ℝ2, with the norm ||v|| = |v1| + |v2|, v =  ∈ V. If u : ℝ+ → ℝ+*, then the mapping Φu : Δ × X → ℬ(V) defined byφ, and C =  is a skew-evolution semiflow.We consider that A particular class of skew-evolution semiflows is emphasized in the following.C =  and a parameter λ ∈ ℝ. We define the mappingLet us consider a skew-evolution semiflow Cλ =  is also a skew-evolution semiflow, called λ-shifted skew-evolution semiflow on Y. We will call Φλ the λ-shifted evolution cocycle.We observe that X be the metric space defined in the first Example. We define the mapping φ0 : ℝ+ × X → X, φ0 = xt, where xt(τ) = x(t + τ), for all τ ≥ 0, which is a classic semiflow on X. Let us consider for every x ∈ X the parabolic system with Neumann's boundary conditions as follows:Let V = ℒ2 be a separable Hilbert space with the orthonormal basis {en}n∈ℕ, e0 = 1, y ∈ , n ∈ ℕ.Let D(A) = {v ∈ ℒ2, v(0) = v(1) = 0}, and we define the operatorWe denote that 𝒞0-semigroup S, defined byV. For every 𝓍 ∈ X, let us define an operator A(x) : D(A) ⊂ V → V, A(x) = x(0)A, which allows us to rewrite system  is a linear skew-product semiflow strongly continuous on Y. Also, for all v0 ∈ D(A), we have obtained that v(t) = Φx0, t ≥ 0, is a strongly solution of system  is a skew-product semiflow on Y, then the mapping C : T × Y → Y, C = , Φv), whereis a classic cocycle over the semiflow f system . As C0 Y. Hence, the skew-evolution semiflows generalize the notion of skew-product semiflows.is a skew-evolution semiflow on t, s, x) is the solution of the Cauchy problemMore directly, if Φ is a linear skew-evolution semiflow.then Let us recall the definition of a semigroup of linear operators, and let us give an example which shows that this is generating a skew-evolution semiflow.S : ℝ+ → ℬ(V) is called semigroup of linear operators on V if the following relations hold:A mapping S : T × X → ℬ(V), defined by ΦS = S(t − s), which is an evolution cocycle on V over evolution semiflows given, for example, by φ = xt−s v, which are considered measurable. A particular case of skew-evolution semiflows is given by the following.Other examples of skew-evolution semiflows are given in . The asyC =  is ∗-strongly measurable if, for every  ∈ T × X × V*, the mapping given by s ↦ ||Φ)*v*|| is measurable on .A skew-evolution semiflow ω-trichotomy. Some examples and connections with the classic concept of exponential trichotomy are also provided.We intend to give a new approach for the property of trichotomy for skew-evolution semiflows, the C : T × Y → Y, C = , Φv) be a skew-evolution semiflow on Y. We recall that a mapping P : X → ℬ(V) with the propertyLet projections family on V.is called P : X → ℬ(V) is said to be invariant relative to the skew-evolution semiflow C =  ifA projections family The splitting of the state space into three subspaces will be assured by the following.Pk}k∈{1,2,3} are said to be compatible with a skew-evolution semiflow C =  ifThree projections families {1) each of Pk, k ∈ {1,2, 3} is invariant relative to C;(c2)  for all x ∈ X, the projections verify the relationsAψ ≠ ∅ and Bψ ≠ ∅;(ii)μ ∈ Aψ implies the existence of a constant M ∈ ℝ+ such that(iii)λ ∈ Bh implies that there exists a constant m ∈ ℝ+ such thatFor every function Let us denote thatC =  is called ω-trichotomic if there exist three projections families {Pk}k∈{1,2,3} compatible with C and some functions ψ, ζ, χ : ℝ+ → ℝ+* with the propertiesA skew-evolution semiflow t1)P1(x)v|| ≤ ψ(t − t0)||P1(x)v||;||Φ(t2)(ζ(t − t0)||P2(x)v|| ≤ ||ΦP2(x)v||;t3)P3(x)v|| ≤ χ(t − t0)||P3(x)v|| and χ(t − t0)||ΦP3(x)v|| ≥ ||P3(x)v||,||Φ ∈ T and all  ∈ Y.for all  be a decreasing function with the property that there exists limt→∞f(t) = l > 0. Let us denote that λ > f(0). Let us consider the Banach space V = ℝ3 with the norm |||| = |v1| + |v2| + |v3|, v =  ∈ V. The mappingt ≥ t0 ≥ 0,  ∈ Y, is an evolution cocycle over the evolution semiflow given in P1, P2, P3 : X → ℬ(ℝ3) by P1(x)v = , P2(x)v = , P3(x)v = , for all x ∈ X and all v =  ∈ ℝ3. The following inequalitiesLet t, s),  ∈ T and all  ∈ Y. The mappings ψ, ζ, χ : ℝ+ → ℝ+*, defined byhold for all ,  ∈ T and x ∈ X. We remark that the following relations hold:k = Pk(x), ∀ ∈ ℝ+ × X;Φk)Φk = Φk, ∀,  ∈ T, ∀x ∈ X.Φfor every  is ω-trichotomic if and only if there exist some constants N1, N2, N3 ≥ 1, ν1, ν2, ν3 > 0 and three projections families {Pk}k∈{1,2,3} compatible with C such that A skew-evolution semiflow t, t0) ∈ T and all  ∈ Y.for all  ≤ Meμt, for all  t ≥ 0, and for λ ∈ Bψ there exists a constant m ∈ ℝ+ with the property ψ(t) ≥ meλt, for all t ≥ 0.such that relations (i)–(iii) of N1 ≥ 1 and ν1 > 0, such that ψ(t) ≤ N1eν1t−, for all t ≥ 0. Hence, relation  ≥ N2eν2t, for all t ≥ 0, which implies  is obt implies 2). From Hence, relation 3) is satSufficiency. Let us assume that there exist three projections families compatible with C and N1, N2, N3 ≥ 1, ν1, ν2, ν3 > 0 such that relations –(et3)3SufficieWe obtain the relationsC is ω-trichotomic, which ends the proof.Hence, relations (i)–(iii) of We obtain a characterization for the property of trichotomy, by means of the shifted skew-evolution semiflow.C =  is ω-trichotomic if and only if there exist three projections families {Pk}k∈{1,2,3} compatible with C and1 is exponentially stable;the evolution cocycle Φ2 is exponentially instable;the evolution cocycle Φλ > 0 such that the λ-shifted evolution cocycle Φλ3 is exponentially stable and the −λ-shifted evolution cocycle Φλ−3 is exponentially instable.there exists a constant A skew-evolution semiflow Necessity. Statements (i) and (ii) are obtained immediately from the necessity of N3 ≥ 1 and ν3 > 0 such thatλ = 2ν3 > 0. We obtain successivelyLet us consider that t, t0) ∈ T and all  ∈ Y, which shows that Φλ3 is exponentially stable. Also, we havefor all ,  ∈ T and all  ∈ Y, which proves that Φλ−3 is exponentially instable.for all (Sufficiency. Relations (i) and (ii) in λ > 0 such that the skew-evolution semiflow Φλ3 is exponentially stable, then there exists some constants N ≥ 1 and ν > 0 such thatFurther, we obtaint, s),  ∈ T and all  ∈ Y. Denotingfor all (the first relation in 3) is obtλ > 0 such that the −λ-shifted skew-evolution semiflow Cλ− is exponentially instable, then there exist some constants N ≥ 1 and ν > 0 such thatIf there exists a constant t, s),  ∈ T and all  ∈ Y. If we consider ν3 defined as in  is obtω-trichotomy is given relative to the dual space V* of the Banach space V. To this aim, let us consider three projections families {Pk}k∈{1,2,3} compatible with C such that the evolution cocycle Φ1 has exponential growth and the evolution cocycle Φ2 has exponential decay.Another characterization for the property of strongly measurable skew-evolution semiflow C =  is ω-trichotomic if the and only if the following statements hold:(i)p > 1, all  ∈ T, and all  ∈ X × V*;there exists (ii)p > 1, all  ∈ T, and all  ∈ Y;there exists (iii)α < 0 and t0 ∈ ℝ+ and all  ∈ Y and there exist β < 0 and t, t0) ∈ T and all  ∈ Y.there exist A ∗-Necessity.  (i) As C is ω-trichotomic, according to ψ : ℝ+ → ℝ+*, with the property An equivalent relation is obtained, if we consider t, t0) ∈ T and all  ∈ Y. According to the properties of function ψ, there exist N ≥ 1 and ν > 0 such that the following relations hold:for all  ∈ T and all  ∈ X × V*, where we have denoted that for all  According to ζ assures the existence of some constants N ≥ 1 and ν > 0, such that, for all p > 0, the following relations hold:The property of function t, t0) ∈ T and all  ∈ Y, where we have denoted that for all ((iii) Both relations are obtained by a similar proof as in (i) and (ii), according to Sufficiency.  (i) As the evolution cocycle Φ1 has exponential growth, there exist M ≥ 1 and ρ > 0 such thatt ≥ t0 + 1. We consider thatLet p > 1 there exists q > 1 such that 1/p + 1/q = 1. We obtainFor every Hence, we have thatt ∈  we obtainFor which implies thatFurther, the following relations hold:t0, t], it follows thatBy integrating on ,and, if x, v) ∈ Y. Hence,ζ : ℝ+ → ℝ+* defined byfor all (t2) of with the property α-shifted evolution cocycle Φα3 in (i) and for the β-shifted evolution cocycle Φβ3 in (ii) leads to the existence of two functions χ1, χ2 : ℝ+ → ℝ+*, with the properties (iii) A similar proof for the If we consider the definition of the shifted evolution cocycle, the previous relations are equivalent witht3) of We define ω-trichotomy.The property described by relation  is also


Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied. Lp-function, 1 < p < ∞, as the Walsh-Fourier series. Walsh-type wavelet packet expansion has been studied by the researchers Billard q1/, 1 < q < ∞. Let 𝔹q denote the space of measurable functions f on  with a unique expansion x = ∑j=0∞xj2j−1−  by their associated binary sequence {xi}. For two such points x, y ∈ , define We can carry over the operator ⊕ to the interval  by identifying those x, y ∈ . With this definition, we havex, y for which x ⊕ y is defined, , and define {fn}n≥2 recursively byLet n, J ∈ ℕ, 2J ≤ n < 2J+1.ThenUk = u1 + u2 + ⋯+uk for k = 1,2,…, n; it is also called Abel's transformation. ConsiderWn}n=0∞ be the Walsh system. ThenC is a finite positive constant, K ≥ 1, 2K ≤ n < 2K+1, and for all pairs x, y ∈ −log ≥ |a − b | /2. This completes the proof of Therefore, using f in the periodic Walsh-type wavelet packets,The dyadic arithmetic mean of partial sums for the expansion of a measurable (integrable) function ackets,4(σ2Nf)(x)q-block β is associated with the dyadic interval I ⊂ [0,1). If 1 < α|I|, then |I|q1−/αq ≤ 1/α, and using the fact that the operator f → 𝔾cdf(x)) is of strong type . We haveSuppose that the α|I| with I = [a, b). Put Now suppose that 1 ≥ KN denote the operator kernel associated with the projection operators C (independent of N) such that Fix  Terence ). KN, we obtainUsing the estimate  on the kβ||1 ≤ 1 and x − a | , |x − b | ≥|I | /2, therefore,I and β and hence Since ||α > 0 and a q-block β supported on the dyadic interval I ⊂ [0,1); two cases are considered.Fix α | I|, then |I|q1−/αq ≤ 1/α. Therefore, using Theorem 5.1. [If 1 < rem 5.1. , page 27α | I| with I = [a, b). LetLet 1 ≥ Notice thatx ∈ [0,1), we haveFor Eαl1,l2| withHence, it suffices to estimate |x ∈ ℝ∖I; thenx ∈ Eαl1,l2, there is an increasing sequence Jk → ∞ for whichC > 0 and for k = 1,2,…. Since Jk → ∞, thereforeFix Eαl1,l2| ≤ 1/α and consequentlyUsing that f = ∑k=1∞ckβk be a function of 𝔹q. ThenL1 convergence of the average sum defining f. SinceLet thereforeThis completes the proof of Following corollary can be deduced from our theorems.f ∈ 𝔹q, 1 < q < ∞, in Let Let us write f = ∑k=1∞ckβk ∈ 𝔹q, let gK = ∑k=1Kckβk, and observe that ||f−gK||𝔹q → 0. For each x ∈ [0,1), writeWith ThusFrom this it follows thatThis completes the proof of the corollary.


We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into anotherdifferential equations of integer order with respect to the new variable. Then, using Riccati transformation,inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results. In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades . In Dtα(·) denotes the modified Riemann-Liouville derivative with respect to the variable t, the function a ∈ Cα, R+), r ∈ Cα2, R+), p, q ∈ C, R+) and Cα denotes continuous derivative of order α.In , Jumarieα are listed as follows :(2)x(t) of  of  is callex(t) of  is calleWe organize the next as follows. In ξ = tα/Γ(1 + α), ξi = tiα/Γ(1 + α), i = 0,1, 2,3, 4,5, R+ = , h1, h2, H ∈ C, R) satisfyH has continuous partial derivatives ∂H/∂ξ and ∂H/∂s on −4/3∫ξ0ξs−4/3ds) = exp(3[Γ(4/3)]−4/3[ξ0−1/3 − ξ−1/3]), which implies 1 ≤ A(ξ) ≤ exp(3[Γ(4/3)]−4/3ξ0−1/3). On the other hand, T > ξ2 such that T, ∞). In ∫ξ0dζ=∞.In , lettingerefore,  is oscilConsider the following fractional differential equation:t0 = 2, α = 1/2, a(t) = t1/4, r(t) ≡ 1, q(t) ≡ 1, then we obtain (ξ0 = 21/2/Γ(3/2), A(ξ) = exp([Γ(3/2)]−1/2∫ξ0ξs−3/2ds) = exp([Γ(3/2)]−1/2[2ξ0−1/2 − 2ξ−1/2]), which implies 1 ≤ A(ξ) ≤ exp(2[Γ(3/2)]−1/2ξ0−1/2). On the other hand, T > ξ2 such that T, ∞).In , if we se obtain . So ξ0 λ = 2, for any sufficiently large l, we haveFrom the analysis above, one can see the  holds. Wds=∞.In ,59)∫ξ0∫ξ0(58)∫tζ=∞.So,  hold. Onζ=∞.So, , after pds=∞.So  holds, ads=∞.So  is oscil


G-metric spaces, fixed points of maps that satisfy the generalized -Chatterjea type contractive conditions are obtained. The results presented in the paper generalize and extend several well known comparable results in the literature.In the framework of ordered  P in G-metric spaces. Recently, Shatanawi   for all  x, y ∈ X;ype (see ) if therChatterjea type [k ∈  ≤ k[d + d] for all x, y ∈ X.jea type  if thereLet →[0, ∞) is continuous and nondecreasing with ψ(t) = 0  if and only if t = 0} andΨ = { φ | φ : [0, ∞)5 → [0, ∞) is lower semi-continuous with φ = 0 if and only if t1 = t2 = t3 = t4 = t5 = 0}.Φ = {We define two classes of mappings as follows:G-metric space is said to have a sequential limit comparison property if for every nondecreasing sequence (nonincreasing sequence) {xn} in  X  such that G → 0 as n → ∞ implies that xn⪯x  (x⪯xn), respectively.An ordered partial φ, ψ)-Chatterjea type contractive conditions on partially ordered complete generalized metric space. We start with the following result.In this section, we obtain fixed point results for mappings satisfying generalized   be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfyingψ ∈ Ψ, φ ∈ Φ withx, y ∈ X with x⪯y. Suppose that there exists x0 ∈ X with x0⪯fx0. If f is continuous or X a sequential limit comparison property, then f has a fixed point in X.Let  > 0 for every  n ∈ ℕ. If not, then xn = xn+1 for some n and xn becomes a fixed point of f. Using  ≥ G for some n ≥ 0, it follows that φ) = 0, a contradiction. Therefore, for all n ≥ 0,M = G. Now {G} is a decreasing sequence, so there exists L ≥ 0 such that limn→∞G = L. This gives limn→∞G  =  limn→∞M = L. By lower semicontinuity of φ,L = 0. Taking the upper limits as n → ∞ on both sides ofφ = 0 and we conclude thatIf f. Using , we obtaxn} is a G-Cauchy sequence in  X. If not, then there exist ε > 0 and integers nk and mk with mk > nk > k such thatε ≤ limk→∞G.Next, we show that {ffect of  and 14)xn} is ak→∞G ≤ ε so thatε ≤ limk→∞G, andε ≤ limk→∞G.On the other hand,ned with  and 16)(18)G(xnkplies by  and 19)(18)G ≤ ε. Thus,k → ∞ implies thatε > 0.But fromher with  and 16)(25)G(xnkε).From , we obtaxn} is a G-Cauchy sequence and by G-completeness of X, there exists u ∈ X such that {xn}G-converges to u as n → ∞. If f is continuous, then it is clear that fu = u. Next, if X has a sequential limit comparison property, then we have xn⪯u for all n ∈ ℕ. From  = G. Thus, from , G, 0) = 0  so that  G = 0  and, hence, fu = u.It follows that { ℕ. From , we haveℕ. From   be a partially ordered set and  f  be a nondecreasing self mapping on a complete G-metric space  X  satisfyingψ ∈ Ψ, φ ∈ Φ withx, y ∈ X with x⪯y with ai ≥ 0 for all i = 1,2, 3,4 with a1 + a2 + a3 + 2a4 ≤ 1. Suppose that there exists x0 ∈ X with x0⪯fx0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Let   = max{|x − y|, |y − z|, |z − x|} be a G-metric on X. Define f : X → X byψ(t) = (3/4)t and φ = (1/12)(t1 + t2 + t3 + t4 + t5) for all t, t1, t2, t3, t4, t5 ∈ [0, ∞).Let x, y ∈ X with x⪯y, we havea1 = a2 = a3 = a4 = 1/6, where a1 + a2 + a3 + 2a4 ≤ 1. Hence, the conditions of f.Now, for all ).Thus,  is satisX, ⪯) be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfyingφ ∈ Φ,x, y ∈ X with x⪯y. Suppose that there exists x0 ∈ X with x0⪯fx0. If f is continuous or X has a sequential limit comparison property, then f has a fixed point in X.Let  be a partially ordered set and f be a nondecreasing self mapping on a complete G-metric space X satisfyingφ ∈ Φ,x, y ∈ X with x⪯y, where ai > 0 for i = 1,2, 3,4 with a1 + a2 + a3 + 2a4 ≤ 1. Suppose that there exists  x0 ∈ X  with  x0⪯fx0. If  f  is continuous or  X  has a sequential limit comparison property, then f has a fixed point in  X.Let (


Lp spaces, 1 ≤ p < ∞ for univariate and multivariate functions, respectively. Furthermore, we obtain these types of approximation theorems by means of 𝒜-summability which is a stronger convergence method thanordinary convergence.Korovkin-type theorem which is one of the fundamental methods inapproximation theory to describe uniform convergence of any sequence ofpositive linear operators is discussed on weighted  Lp-spaces ..Lp-spacLp-spaces was obtained by Gadjiev and Aral  and χ2A(t) = 1 − χ1A(t) for any A ≥ 0. We can choose a sufficient large A such that for every ε > 0Let k, n, and f and the linearity of the operators Lj, we getUsing the assumption of the convergence of the series  for eachK > 0 such thatBy condition , there eHence, from , we compf ∈ Lp,ω(ℝ) the inequalityLp,ω ⊂ Lp. Since the space of continuous functions is dense in Lp, given f ∈ Lp,ω(ℝ), for each ε′ > 0, there exists a continuous function φ on  satisfying the condition φ(x) = 0 for |x| > A such thatFor every function Using the inequalities  and 18)18), we gχ2A1χ1Aφ = 0 for some A1 > A, we get the inequalityOn the other hand, since Mφ = maxt∈ℝ|φ(t)|χ1A(t), we getNow, by supposing that ω ∈ L1(ℝ), we can choose the number A1 such thatSince Using this inequality, we havenIn′:As a corollary, we get the following inequality for supφχ1A is a continuous function on , for any given ε′ > 0, there exists a δ > 0 such thatSince t ∈  and that x ∈  is as follows:bk) is a sequence of positive real numbers satisfying the conditionThe Kantorovich variant of the Szász-Mirakyan operators  by replaIt is known thatFurthermore by simple calculations, we obtainAlso,f ∈ Lp,ω(ℝ), we haveHence, conditions , 11) ar ar11) arLp,Ω(ℝm), m ∈ ℕ, instead of Lp,Ω(ℝn) to avoid any confusion about the indices of 𝒜 = {An}.Also, analogue of 𝒜 = {An} be a sequence of infinite matrices with nonnegative real entries and let {Lj} be a sequence of positive linear operators from Lp,Ω(ℝm) into Lp,Ω(ℝm). Assume thatLet f ∈ Lp,Ω(ℝm), one hasIfχ1A of the ball |x| ≤ A and χ2A(t) = 1 − χ1A(t), it is possible to choose a sufficient large A such thatConsidering the characteristic function K such that ||Akn) = 0 for |x| > A such thatOn the other hand, by condition  there exk, n, and f, and using the linearity of the operators Lj, which means the linearity of Akn)|χ1A(t). Furthermore, we can choose A1 such that ||χ2A1||p,Ω < ε′/Mθ, and for sufficiently large k, we estimate supn||Akn)((1)−1||p,Ω < ε′/Mθ. Using these estimations in  = ex−y−. Note that this selection of Ω satisfies condition  with an integral mean of f over the interval  ×  is as follows:bk) is a sequence of positive real numbers satisfying the conditionThe Kantorovich variant of the Szász-Mirakyan operators  by replaIt is known thatFurthermore we obtainAlso,f ∈ Lp,Ω(ℝ2), we haveHence, conditions  and 11)11) are p


Many known results appear as special consequences of our work.The aim of this paper is to study the problem of coefficient bounds for a newly defined subclass of  𝒜(p) denote the class of functions f(z) of the form𝕌 = {z : |z| < 1}. Also let 𝒮p* and 𝒦p denote the well-known classes of p-valent starlike functions and p-valent convex functions, respectively.Let f(z) ∈ 𝒜(p) given by (g(z) ∈ 𝒜(p) given byf(z) and g(z) is given byDδ+p−1 for p-valent analytic functions given byδ > −p,x)n is a Pochhammer symbol given byδ is any integer greater than −p,𝒱𝒟pλ of p-valent analytic functions as follows.For given by  and g(z)given byg(z)=zp+∑p-valent function f(z) of the form , if and only ifα ≥ 0,  b ∈ ℂ∖{0}, δ > −p, λ is real with |λ| < (π/2), and 0 ≤ β < 1.An analytic the form  belongs α, β, λ, p, b, and δ in 𝒱𝒟pλ, we obtain many important subclasses studied by various authors in earlier papers; see for details [𝒱𝒟1λ ≡ 𝒮λ* and 𝒱𝒟1λ ≡ 𝒦λ, studied by Spacek [y Spacek  and Robey Spacek , respecty Spacek –11;𝒱𝒟10 ≡ 𝒮𝒟 and 𝒱𝒟10 ≡ 𝒦𝒟, studied by both Owa et al. and Shams et al. [s et al. , 13;𝒱𝒟1λ ≡ 𝒰𝒮𝒫(λ) and 𝒱𝒟1λ ≡ 𝒰𝒞𝒮𝒫(λ), introduced by Ravichandran et al. [n et al. ;𝒱𝒟10 ≡ 𝒱𝒟, considered by Latha [by Latha ;𝒱𝒟10 ≡ 𝒮*(β) and 𝒱𝒟10 ≡ 𝒦(β), the well-known classes of starlike and convex functions of order β.From the above special cases we note that this class provides a continuous passage from the class of starlike functions to the class of convex functions.By giving specific values to  details –6; we liα ≥ 0, 0 ≤ β < 1, δ > −1, λ is real with |λ| < (π/2), and b ∈ ℂ∖{0}.We will assume throughout our discussion, unless otherwise stated, that f(z) ∈ 𝒱𝒟pλ with 0 ≤ α ≤ β. Then f(z) ∈ 𝒱𝒟pλ, where ζ = (β − α)/(1 − α).Let f(z) ∈ 𝒱𝒟pλ. Then we obtainα ≤ β, then we can easily obtainLet f(z) ∈ 𝒱𝒟pλ, thenφn+p−1(δ) is given by (If given by  and(15)f(z) ∈ 𝒱𝒟pλ. Then by p(z) byp(z) is analytic in E with p(0) = 1 and Rep(z) > 0, z ∈ E. Letη is given by  byeiλ(1−2b+given by . Using ( E. Let1p(z)=1+∑nThat is,2eiλ(1−α) sides,22eiλ(1−α)nctions [|an+p−1|≤ows that  is true.nd using , we haveerefore,  holds fon = k; that is,n = k + 1, and hence by using mathematical induction,  ∈ 𝒮𝒟, thenα = 0 in If obertson .λ = 0, p = 1, b = 1, and δ = 1 in By setting f(z) ∈ 𝒦𝒟, thenα = 0 in If f(z) ∈ 𝒜(p) and satisfiesφk(δ) is given by (f(z) ∈ 𝒱𝒟pλ.If given by , then f ∈ 𝒜(p) and satisfiesf(z) ∈ 𝒮p*(β), the class of p-valent starlike functions of order β.If λ = 0, δ = −p + 2, b = 1, and α = 0 in For f(z) ∈ 𝒜(p) and satisfiesf(z) ∈ 𝒦p(β), the class of p-valent convex functions of order β.If p = 1 in both the last two corollaries, one obtains the results for the classes 𝒮*(β) and 𝒦(β) which was proved by Merkes et al. [Further for s et al.  and Silvs et al. , respect