Furthermore, we give some inclusion relations concerning the space ℓpλ(B) and we construct the basis for the space ℓpλ(B), where 1 ⩽ p < ∞. Furthermore, we determine the alpha-, beta- and gamma-duals of the space ℓpλ(B) for 1 ⩽ p ⩽ ∞. Finally, we investigate some geometric properties concerning Banach-Saks type p and give Gurarii's modulus of convexity for the normed space ℓpλ(B).We introduce the sequence space  From the summability theory perspective, the role played by the algebraical, geometrical, and topological properties of the new Banach spaces which are defined by the matrix domain of triangle matrices in sequence spaces is well-known.w, we denote the space of all real or complex valued sequences. Any vector subspace of w is called a sequence space.By μ with a linear topology is called a K-space provided that each of the maps pi : μ → ℂ defined by pi(x) = xi is continuous for all i ∈ ℕ, where ℂ denotes the complex field and ℕ = {0,1, 2,…}. A K-space is called an FK-space provided μ is a complete linear metric space. An FK-space whose topology is normable is called a BK-space  be an infinite matrix of real or complex numbers ank, where n, k ∈ ℕ. Then, we say that A defines a matrix transformation from μ into γ and we denote it by writing A : μ → γ, if for every sequence x = (xk) ∈ μ the sequence Ax = {(Ax)n}, the A-transform of x is in γ, whereLet μ : γ) denotes the class of all matrices A such that A : μ → γ. Thus, A ∈ (μ : γ) if and only if the series on the right hand side of n}n∈ℕ ∈ γ for all x ∈ μ. The matrix domain μA of an infinite matrix A in a sequence space μ is defined byA = (ank) is said to be a triangle if ann ≠ 0 for all n ∈ ℕ and ank = 0 for k > n. The study of matrix domains of triangles has a special importance due to the various properties which they have. For example, if A is a triangle and μ is a BK-space, then μA is also a BK-space with the norm given by ||x||μA = ||Ax||μ for all x ∈ μA.The notation Nq and cNq in =∑k=0ed as in .x ∈ ℓpλ(B) is given by (x = ∑kμk(x)bk)(. Since the transformation T defined from ℓpλ(B) to ℓp by k ∈ ℕ. That is to say that the representation (x ∈ ℓpλ(B) is unique. Let us show that the uniqueness of representation for given by . Supposeentation  of x ∈ ℓℓpλ(B) and ℓ∞λ(B). We start with the definition of the alpha-, beta- and gamma-duals of a sequence space.In this section, we give some theorems determining the alpha-, beta- and gamma-duals of the spaces x and y are sequences and X and Y are subsets of ω, then we write x · y = (xkyk)k=0∞, x−1∗Y = {a ∈ ω : a · x ∈ Y}, andX and Y. One can easily observe for a sequence space Z with Y ⊂ Z and Z ⊂ X that the inclusions M ⊂ M and M ⊂ M hold. The alpha-, beta- and gamma-duals of a sequence space, which are, respectively, denoted by Xα, Xβ, and Xγ are defined byXα ⊂ Xβ ⊂ Xγ. Also, it can be easily seen that the inclusions Xα ⊂ Yα, Xβ ⊂ Yβ, and Xγ ⊂ Yγ hold, whenever Y ⊂ X. Now, we may begin with quoting the following lemmas α = t∞λ and [ℓpλ(B)]α = tqλ for 1 < p ⩽ ∞, where the matrix T = (tnkλ) is defined via the sequence a = (an) ∈ w byn, k ∈ ℕ. Define the sets a = (an) ∈ w and 1 < p < ∞. Then, by using (n ∈ ℕ thatax = (anxn) ∈ ℓ1 whenever x = (xk) ∈ ℓpλ(B) if and only if Ty ∈ ℓ1 whenever y = (yk) ∈ ℓp. This means that a = (ak)∈[ℓpλ(B)]α if and only if T ∈ (ℓp : ℓ1). Therefore, we get by T instead of A that a = (ak)∈[ℓpλ(B)]α if and only ifℓpλ(B)]α = tqλ, for 1 < p ⩽ ∞. Similarly, we get from (a = (ak)∈[ℓ1λ(B)]α if and only if T ∈ (ℓ1 : ℓ1) which is equivalent to  is defined for all n, k ∈ ℕ byax = (anxn) ∈ cs whenever x = (xk) ∈ ℓpλ(B) if and only if Dy ∈ c whenever y = (yk) ∈ ℓp. This means that a = (ak)∈[ℓpλ(B)]β if and only if D ∈ (ℓp : c), where 1 ⩽ p ⩽ ∞. Therefore, we derive from (ℓpλ(B)]β = eqλ∩d1λ∩d4λ for 1 < p < ∞. Since beta-dual of the space ℓpλ(B) for the cases p = 1 and p = ∞ can be similarly computed, we omit the details. This completes the proof. ∑k=0nakxkive from  and 52)(63)∑k=0np ⩽ ∞. Then, Let 1 < This may be obtained in the similar way used in the proof of ℓpλ(B) : ℓ∞), (ℓpλ(B) : c0), (ℓpλ(B) : c), (ℓpλ(B) : ℓ1), (ℓ1λ(B) : ℓp), (ℓ∞λ(B) : ℓp), where 1 ⩽ p ⩽ ∞. Also, by means of a given basic lemma, we derive the characterizations of certain other classes. Since the characterization of matrix mapping on the space ℓpλ(B) can be proved in a similar way, we omit the proof for the cases p = 1 and p = ∞ and consider only the case 1 < p < ∞ in the proofs of theorems given in this section.In this section, we characterize the matrix classes (A = (ank), we write for brevity thatFor an infinite matrix mas (see ) which aA = (ank) be an infinite matrix. Then, the following statements hold. (i)A ∈ (ℓ1 : c0) if and only if supn,k∈ℕ | ank | <∞ and(ii)p < ∞. Then, A ∈ (ℓp : c0) if and only if (k∈ℕ∑n | ank|q < ∞. Let 1 <  only if  holds an(iii)A ∈ (ℓ∞ : c0) if and only if ∈(ℓ1 : ℓp) if and only if supn∈ℕ∑k | ank|q < ∞. Let 1 ⩽ p < ∞. Then, A = (ank)∈(ℓ∞ : ℓp) if and only if supK∈ℱ∑n|∑k∈Kank|p < ∞.Let 1 < A = (ank) be an infinite matrix. Then, the following statements hold.(i)p < ∞. Then, A ∈ (ℓpλ(B) : ℓ∞) if and only ifLet 1 < (ii)A ∈ (ℓ1λ(B) : ℓ∞) if and only if ( only if  and 69)A ∈ (ℓ1(iii)A ∈ (ℓ∞λ(B) : ℓ∞) if and only if ( only if  and 69)A ∈ (ℓ∞Let x ∈ ℓpλ(B), where 1 < p < ∞. Then, we have by ank)k∈ℕ ∈ [ℓpλ(B)]β for all n ∈ ℕ and this implies the existence of Ax. Also, it is clear that the associated sequence y = (yk) is in the space ℓp ⊂ c0.(i) Assume that the conditions –71) hol hol71) hmth partial sum of the series ∑kankxk as follows:m → ∞ thatℓp : ℓ∞) by n in (Ax ∈ ℓ∞ and hence A ∈ (ℓpλ(B) : ℓ∞).Let us now consider the following equality derived by using relation  from mthby using –70), we, wemth pfollows:∑k=0mankx→ ∞ that∑kankxk=∑A = (ank)∈(ℓpλ(B) : ℓ∞), where 1 < p < ∞. Then, since (ank)k∈ℕ ∈ [ℓpλ(B)]β for all n ∈ ℕ by the hypothesis, the necessity of (ank)k∈ℕ ∈ [ℓpλ(B)]β, (x ∈ ℓpλ(B) and y ∈ ℓp which are connected by relation (fn on ℓpλ(B) byℓpλ(B) and ℓp are norm isomorphic, it should follow with  are pointwise bounded. Thus, we deduce by Banach-Steinhaus theorem that these functionals are uniformly bounded, which yields that there exists a constant K > 0 such that ||fn|| ⩽ K for all n ∈ ℕ. This shows the necessity of the condition ]β,  holds forelation . Let us low with  that(79ondition  which coA = (ank) be an infinite matrix. Then, the following statements hold. (i)A ∈ (ℓ1λ(B) : c) if and only if ( only if  and 69)A ∈ (ℓ1(ii)p < ∞. Then, A ∈ (ℓpλ(B) : c) if and only if (Let 1 <  only if –71) hol holp < ∞ only if  also hol(iii)A ∈ (ℓ∞λ(B) : c) if and only if , a, aA ∈ , where 1 < p < ∞. Then, Ax exists and by using (k ∈ ℕ that n → ∞ which leads us with (k ∈ ℕ. This shows that (αk) ∈ ℓq. Since x ∈ ℓpλ(B), we have y ∈ ℓp. Therefore, we derive by applying Hölder's inequality that (αkyk) ∈ ℓ1 for each y ∈ ℓp.We consider only part (ii). Assume that nditions –71) and andA satnditions , and x ∈by using , we have us with  to the fϵ > 0, choose a fixed k0 ∈ ℕ such thatm0 ∈ ℕ such thatm⩾m0. Thus, by using n → ∑kαkyk as n → ∞ which means that Ax ∈ c; that is, A = (ank)∈(ℓpλ(B) : c).Now, for any given  follows  that theby using , we get A ∈ (ℓpλ(B) : c), where 1 < p < ∞. Then, since c ⊂ ℓ∞, A ∈ (ℓpλ(B) : ℓ∞). Thus, the necessity of (x ∈ ℓpλ(B). Since Ax ∈ c by our assumption, we derive by  is  is A ∈ n = xn+1 for all n ∈ ℕ. A Banach limit L is defined on ℓ∞ such that L(x)⩾0 for x = (xk), where xk⩾0 for all k ∈ ℕ, L(Px) = L(x), L(e) = 1, where e = . A sequence x = (xk) ∈ ℓ∞ is said to be almost convergent to the generalized limit α if all Banach limits of x is α β for all n ∈ ℕ and this implies that Ax exists. Besides, it follows by combining (Ax ∈ ℓ1 and so we have A ∈ (ℓpλ(B) : ℓ1).Suppose that nditions –71),  thaive from  holds whA ∈ (ℓpλ(B) : ℓ1), where 1 < p < ∞. Since ℓ1 ⊂ ℓ∞, A ∈ (ℓpλ(B) : ℓ∞). Thus, Ax ∈ ℓ1 by the hypothesis, we deduce by ∈(ℓ1λ(B) : ℓp) if and only if (Let 1 ⩽  only if  and 69)p < ∞. Thx ∈ ℓ1λ(B). Then, y ∈ ℓ1. We have by ank)k∈ℕ ∈ [ℓ1λ(B)]β for each n ∈ ℕ and this implies that Ax exists. Furthermore, by  : ℓp).Suppose that the conditions , 69), a, a69), amore, by , one cane; since  and (69)e; since  as m → ∞relation  holds. Trelation  and 95)x ∈ ℓ1λA ∈ (ℓ1λ(B) : ℓp), where 1 ⩽ p < ∞. Since ℓp ⊂ ℓ∞, then A ∈ (ℓ1λ(B) : ℓ∞). Thus, Ax ∈ ℓp by our assumption, we deduce by ∈(ℓ∞λ(B) : ℓp) if and only if (Let 1 <  only if  and 69)p < ∞. Thλ and μ be any two sequence spaces, let A be an infinite matrix and B a triangle matrix. Then, A ∈ (λ : μB) if and only if BA ∈ (λ : μ). Let It is trivial that A = (ank) be an infinite matrix and u = (un) and v = (vn) be sequences of non-zero numbers, and define the matrix C = (cnk) by cnk = un∑j=0nvjajk for all n, k ∈ ℕ. Then, the necessary and sufficient conditions in order A belongs to any of the classes (ℓpλ(B) : ℓ∞), (ℓpλ(B) : ℓp), and (ℓpλ(B) : c) are obtained from respective ones in Theorems A by those of the matrix C. The spaces ℓ∞, ℓp, and c are defined in  characterizes the uniformly convex spaces. In [Gurarii's modulus of convexity is defined byε ⩽ 2. It is easily shown that δX(ε) ⩽ βX(ε) ⩽ 2δX(ε) for any 0 ⩽ ε ⩽ 2. Further, if 0 < βX(ε) < 1, then X is uniformly convex, and if βX(ε) < 1, then X is strictly convex.In the present section, we investigate some geometric properties of the space ity see , 25) def defℓpλaces. In , GurariiX is said to have the Banach-Saks property if every bounded sequence (xn) in X admits a sequence (zn) such that the sequence {tk(z)} is convergent in the norm in X [A Banach space orm in X , where(X is said to have the weak Banach-Saks property whenever given any weakly null sequence (xn) in X and there exists a subsequence (zn) of (xn) such that the sequence {tk(z)} is strongly convergent to zero.A Banach space In , García-X with R(X) < 2 has a weak fixed point property. A Banach space ℓpλ(B) has Banach-Saks type p. The space εn) be a sequence of positive numbers for which ∑n=1∞εn ⩽ 1/2. Let (xn) be a weakly null sequence in B(ℓpλ(B)). Set u0 = x0 and u1 = xn1 = x1. Then, there exists t1 ∈ ℕ such thatxn) is a weakly null sequence” implies that xn → 0 with respect to the coordinatewise, there exists n2 ∈ ℕ such thatn⩾n2. Set u2 = xn2. Then, there exists t2 > t1 such thatxn → 0 with respect to the coordinatewise, there exists n3 > n2 such thatn⩾n3. If we continue this process, we can find two increasing sequences (ti) and (ni) of natural numbers such thatn⩾nj+1 anduj = xnj. Hence,x||ℓpλ(B) < 1. Thus, ||x||ℓpλ(B)p < 1, and we haven + 1)p1/ for all n ∈ ℕ and 1 ⩽ p < ∞, we haveℓpλ(B) has Banach-Saks type p. Let (R(ℓpλ(B)) = R(ℓp) = 2p1/, since ℓpλ(B) is linearly isomorphic to ℓp. Note that  Thus, by Remarks p < ∞. Then, the sequence space ℓpλ(B) has the weak fixed point property. Let 1 < ℓpλ(B) isε ⩽ 2. Gurarii's modulus of convexity for the normed space x ∈ ℓpλ(B). Then, we haveε ⩽ 2 and consider the following sequences:z = (zn) and t = (tn), we obtain the following equalities:α ⩽ 1Let p < ∞, we haveTherefore, for 1 ⩽ ε > 2, βℓpλ(B)(ε) = 1. Thus, ℓpλ(B) is strictly convex. For ε ⩽ 2, βℓpλ(B)(ε) ⩽ 1. Thus, ℓpλ(B) is uniformly convex. For 0 < The following statements hold. α = 1/2, βℓpλ(B)(ε) = δℓpλ(B)(ε). For Er of order r, the method Ar, and the generalized difference matrix B in the sequence spaces ℓp and ℓ∞ investigated by Altay et al. [B, our corresponding results are much more general than the results given by Kirişçi and Başar [ℓpλ(B). It is obvious that the matrix Er, Ar, or B. So, the present results are new. As a natural continuation of this paper, one can study the domain of the matrix ℓ∞(p), c(p), c0(p), and ℓ(p).The domain of Euler means y et al. , Aydın ay et al. , and Kiry et al. , respectnd Başar . Additio


I, x(t) ∈ S, ∀t ∈ I, x(0) = x0 ∈ S, (∗), where S is a closed subset in a Banach space 𝕏, I = , (T > 0), F : I × S → 𝕏, is an upper semicontinuous set-valued mapping with convex values satisfying F ⊂ c(t)(||x|| + ||x||p)𝒦, ∀ ∈ I × S, where p ∈ ℝ, with p ≠ 1, and c ∈ C. The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is,  G is included in the subdifferential of convex lower semicontinuous (l.s.c) function g : ℝn → ℝ. This result has been extended in many ways by many authors  function g : ℝn → ℝ, S is closed subset in ℝn, and F :  × ℝn is a continuous set-valued mapping. The infinite dimensional case of (S at C(t) = S + v(t), where S is a fixed nonconvex closed set and v is a mapping with finite variation. Another important study of the inclusion (C(t), has been realized by the author in  × ℍ → ℍ is an upper semi-continuous set-valued mapping with closed convex values satisfying F ⊂ c(t)(||x|| + ||x||p)𝒦, where p ∈ ℝ, with p ≠ 1, and c ∈ C and 𝒦 is a convex compact set. The paper is organized as follows. After recalling the needed concepts in C in separable Hilbert spaces.To the best of our knowledge no existing works studied the existence of solutions for differential inclusions with nonlinear growth. The main purpose of the paper is to prove the existence of solutions for (∗) in Banach spaces and for  in separThis section is devoted to recall some notations and concepts needed in the paper.𝕏 be a Banach space, let S ⊂ 𝕏 be a nonempty closed subset of 𝕏, and let dS(x) = inf{||x − s|| : s ∈ S} is the usual distance function associated with S.Let r-prox-regular sets in Hilbert spaces as the class of all closed sets S satisfying the following definition. Many equivalent definitions of this class have been used for different applications; see, for example,  and p ≥ 0 (with p ≠ 1) is a constant. Thent ∈  we gett ∈ , and hence the proof is finished.Multiplying  by v−p) yield. Let γ(t) = ∫ath(s)ds, for all t ∈  and let b1 ∈  satisfying the inequalityt ∈ J1 : = . Thent ∈ .Let tisfying  with p ∈F be a set-valued mapping satisfying the following nonlinear growth:I = , (T > 0), D is a closed nonempty set in 𝕏, c ∈ C, and p ∈ ℝ with p ≠ 1. Clearly, when p = 0, this assumption coincides with the well known linear growth; that is,x0 ∈ D. To ensure the viability of the solution on the set D, we need the following classical tangential condition:K is the Bouligand tangent cone to D at x.Let  growth:F⊂c and I1 = , and G : I × 𝕏⇉𝕏 satisfying the following:G is u.s.c. with closed convex values;G ⊂ c(t)𝒦 on J × D, for some 𝒦 convex compact set in 𝕏 and c ∈ C;G∩K ≠ ∅ on I × D.Then for every x0 ∈ D, there exists an absolutely continuous mapping x : I → D such thatLet D be a closed subset in 𝕏 and let F : D⇉𝕏 be an upper semicontinuous set-valued mapping with closed convex values and let r1, r2 > 0 be such that r1 < r2, and let ψ :  be a continuous function such that ψ(s) = 1 for s ≤ r1 and ψ(s) = 0 for s ≥ r2. Let G be a set-valued mapping defined on D as follows:F satisfies the nonlinear growth on I × D; that is, F ⊂ c(t)(||x|| + ||x||p)𝒦 on I × D, for some c ∈ C, p ∈ ℝ with p ≠ 1, and 𝒦 is a convex compact set in 𝕏, then G is upper semicontinuous on I × D with closed convex values.Let G has closed convex values. Let 𝒦0 : = (r1 + r1p)𝒦 ∪ {0}. For any t ∈ I and any x ∈ D with ||x|| < r2, we have by the convexity of 𝒦 the following:t ∈ I and any x ∈ D with x ∉ r2𝔹, we have G is closed. To do that, we fix , yn) ∈ gph G with G we havexs(n))n of (xn)n such that ψ(||xs(n)||) → 0. In this case we have for n large enough ||xs(n)|| ≤ r2 and so zs(n) is bounded which ensures that ys(n) → 0 and so ψ and the convergence of xn to α > 0 and n0 ∈ ℕ such that ψ(||xn||) > α > 0 for all n ≥ n0. Then by continuity of ψ we have F, we get G and hence the proof is achieved.Clearly, Now, we are ready to prove our main existence result under the nonlinear growth condition in Banach spaces.𝕏 be a Banach space, D ⊂ 𝕏 a nonempty closed set, and F : 𝕏⇉𝕏 satisfying the following: F is u.s.c. on I × D with F being closed convex for all  ∈ I × D;F ⊂ c(t)(||x|| + ||x||p)𝒦 on I × D, for some c ∈ C, and p ∈ ℝ with p ≠ 1, and for some convex compact set 𝒦 in 𝕏;F∩K ≠ ∅ on I × D.Then for every x0 ∈ D, there exists an absolutely continuous mapping x : I1 → D such thatI1 =  when p ∈  and I1 =  be a continuous function such that ψ(s) = 1 for s ≤ M and ψ(s) = 0 for s ≥ r. Define now the set-valued mapping G on I × D as follows:G inherits the convexity and the upper semicontinuity of the set-valued mapping F with G ⊂ c(t)𝒦0, where 𝒦0 : = r𝒦 ∪ {0}. We have to check that G satisfies the tangential condition on D. Let t ∈ I and let x ∈ D∩M𝔹. Then G = F and so the tangential condition is satisfied from (c). Assume now that t ∈ I and x ∈ D with ||x|| > M. Then by (c) there exists some z ∈ F such that z ∈ K. Let y = ψ(||x||)z. Clearly, y ∈ G and y ∈ ψ(||x||)K ⊂ K, since K is a cone. So, G∩K ≠ ∅. Therefore, the tangential condition is satisfied for G on all I × D. Consequently, all the assumptions (a), (b), and (c) in 𝒦0 instead of 𝒦 are satisfied and hence for every x0 ∈ D there exists a solution x on I of  = x0, and x(t) ∈ D on I. Let us prove that x is the desired solution for (p ≠ 1. Using x(t)|| ≤ M a.e. on I1 and so by the definition of G and ψ we get G) = ψ(||x(t)||)F) = F) that is, x is a solution of , and let C : I⇉ℍ be a set-valued mapping satisfying the following Lipschitz condition for any y ∈ ℍ and any t, t′ ∈ I:Our purpose, in this section, is to use the techniques developed previously to extend some existing results, in separable Hilbert spaces, of nonconvex sweeping processes with perturbations from the case of perturbation with linear growth to the case of perturbation with nonlinear growth. For this end let m 4.1 in .ℍ be a separable Hilbert space and let r ∈  is r-prox-regular for every t ∈ I and that the assumption  ⊂ 𝒦1 ⊂ α𝔹 for all  ∈ I × ℍ, for some compact set 𝒦1 in ℍ. Then, for any x0 ∈ C(0), the sweeping process (SPP) with the perturbation F has at least one Lipschitz continuous solution; that is, there exists an Lipschitz continuous mapping x : I → ℍ such thatI.Let sumption  holds. LUsing the techniques from the previous section and r ∈  is r-prox-regular for every t ∈ I and that the assumption , a convex compact set 𝒦, and k > 0 such thatL, k, p, and T are satisfied:u0 ∈ C(0), there exists a Lipschitz continuous mapping u : I → ℍ satisfying the following sweeping process with a perturbation:Let sumption  holds. L𝒦. By our assumptions on the constants L, T we have after simple computationsβ > 0 such thatα = βp−1)1/→ to be a continuous function such that ψ(s) = 1 for s ≤ α/2 and ψ(s) = 0 for s ≥ α and define the set-valued mapping G on ℍ as follows:G inherits the convexity of the values from F. Also, for any x ∈ α𝔹, we havex ∉ α𝔹, we have G ⊂ 𝒦0, for any  ∈ I × ℍ with G follows from the u.s.c. of F and C and G we get a Lipschitz continuous mapping x : I → ℍ such thatx is a solution of (SPP) with F. Clearly, we haveα and the assumptions on the constants L, k to deduce from (x(t)|| ≤ α/2 which yields that ψ(||x(t)||) = 1 and so G) = F). This means that x is a solution of . Thenα>4(LT+||ution of  and henc


Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions.By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of  With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation.In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in real-world applications.Therefore, by using the research methods and theoretical results of qualitative and bifurcation problems of high-dimensional systems, especially the results of homoclinic and heteroclinic orbits and their bifurcations for the systems, to study the high-dimensional infectious disease dynamics and population ecology systems to reveal the complex dynamical behavior of the nonlinear dynamical systems and the corresponding reality systems is essential.About the study of bifurcation problems of homoclinic and heteroclinic loops for two-dimensional systems, a large number of papers were obtained and achieved many good results (for some results see –6); but red with , paper .Using the expressions of z = h(t) into (Substitute (t) into , and uset), and using Φ*(t)Z(t) = I, we have Multiplying both sides by Φ* ↦ N(T) which has the following formMi = ∫∞−∞+ϕi*(t)gu(r(t), 0)dt, i = 1,3, 4, is said to the Melnikov vectors [Thus, from the flow of , we defi vectors , 14, 15.F0 : S0 ↦ S1, q0 ↦ q1 defined by the orbit of ) > ρ0 > λ1(μ).Now, we consider the map orbit of . For conτ be the flying time from q0 to q1, and s = eλ1(μ)τ− be called Silnikov time. By  is called Silnikov coordinate.(F1(q1) = q2, by , a, a12), aG =  = F(q0) − q0 = * − * as follows:So, from the above and , we haves ≥ 0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of  of the following equation:Thus, for orbit of  and the Equation  is calle(H4)λ1 < ρ1. (Nonresonant condition) μ small enough, we may assume ρ1(μ) > λ1(μ).Obviously, for μ| is small enough, the system (Suppose that hypotheses (H1)~(H4) are valid, then if |e system  exists ngij) are all zero except g31 = w14δ. Therefore, Q, μ) = , (s(0) = 0, u1(0) = 0, v0(0) = 0.Consider the solution of . Let G~Q. From , we have= ,  exists as = 0, then the solution -dimensional surface L ⊂ Rl in the small neighbourhood of μ = 0, such that when μ ∈ L and |μ | ≪1, the system , v0 = v0. Substituting into G1 = 0, we getFrom , for s ≥M1 ≠ 0, then, according to the implicit function theorem, in the neighbourhood of μ = 0, the equation M1μ + h.o.t. = 0 defines a unique (l − 1)-dimensional surface L ⊂ Rl, such that if μ ∈ L and |μ | ≪1, τ1−. Suppose F0(q2) = q3, F1(q3) = q4 = q0, and let τ2 be the time from q2 to q3, s2 = eλ1(μ)τ2−.We rewrite the time from G2 =  as follows:Similar to the previous discussion, we can get its associated successor function Q2 = , gij) are all zero except g41 = w14δ, g62 = w14δ.Denote Q2, μ) = , the bifurcation equations1(0) = 0, u1(0) = 0, v0(0) = 0, s2(0) = 0, and u3(0) = 0, v2(0) = 0.Hence, s1 = s2 = 0, then the homoclinic loop of the system ~(H4) are fulfilled, e system  does notλ1 = ρ1. For convenience, we assume the resonant condition has the following form.(H5)λ1 = ρ1 = λ, λ1(μ) = λ, ρ1(μ) = λ + α(μ)λ, where α(μ) ∈ R1, |α(μ)|≪1, and α(0) = 0. (Resonant condition) We say that the homoclinic loop is Resonance if s ≥ 0, there is a one to one correspondence between the 1-homoclinic loop and 1-periodic orbit of  of the bifurcation equation , v0 = v0. Substituting it into G1 = 0, we getAt first, we discuss the bifurcations of 1-homoclinic loop and 1-periodic orbit. Now, the bifurcation equation has the following form:G1=δ(w12G1=δ(w12N(s) = sα(μ)1+, V(s) = w12(s − δ−1M1μ) + h.o.t., we have the following conclusion.Denote s0, |μ | ≪1, the necessary condition of N(s) and V(s) are tangent at s0 is that α(μ)M1μ > 0. Meanwhile, if Δ = 1 and 0 < s0 ≪ 1, N(s), and V(s) are tangent at s0 if and only if α(μ)(1 − w12) > 0, andSuppose (H1)~(H3) and (H5) are fulfilled, then, for 0 < N(s) and V(s) are tangent at s = s0 if and only if N(s0) = V(s0), s0 = ((1 + α(μ))/α(μ))δ−1M1μ + h.o.t.. Substituting it into  is iss0 = The proof is complete.1 is the surface defined by .Suppose Σfined by , Σ¯2(s)fined by , Σ2=Σ¯2 μ ∈ Σ2,  is turneα(μ) > 0, 0 < w12 < 1. If M1 ≠ 0, then, in the neighborhood of μ = 0, there exists two (l − 1)-dimensional surfaces Σ1 and Σ2, such that, for sufficiently small |μ|, we have the following.μ ∈ Σ1. The system  has a unμ satisfies M1μ > β1(μ). The system  has no 1μ satisfies β2(μ) < M1μ < β1(μ). The system  has exacμ ∈ Σ2. The system  has exacμ satisfies M1μ < β2(μ). The system  has exacSuppose (H1)~(H3) and (H5) are fulfilled, s = |μ|2, we can treat s as a small parameter in -dimensional surface s1 = 0 and s2 = (w12)α(μ)1/ + h.o.t. > 0.Through taking a proper scale transformation, such as meter in . Accordir μ = 0,  can defiΣ¯2(0),  has exacM1μ < β2(μ), then (s1 > 0.If −1 ≪ μ), then  has exacM1μ = β1(μ), that is μ ∈ Σ1, then , then (If μ), then  has no nβ2(μ) < M1μ < β1(μ), then , then  has exacThe proof is complete.β1(μ) and β2(μ), and the corresponding Σ1 and Σ2, to obtain the following theorem.Similarly, we can define the corresponding α(μ) < 0, w12 > 1. If M1 ≠ 0, then, in the neighborhood of μ = 0, there exists two (l − 1)-dimensional surfaces Σ1 and Σ2 such that for sufficiently small |μ|, the following conclusions hold.μ ∈ Σ1. The system  has a unμ satisfies M1μ < β1(μ). The system  has no 1μ satisfies β2(μ) > M1μ > β1(μ). The system  has exacμ ∈ Σ2. The system  has exacμ satisfies M1μ > β2(μ). The system  has exacSuppose (H1)~(H3) and (H5) are fulfilled, 1 is called 2-fold 1-periodic orbit bifurcation surface, Σ2 is called 1-homoclinic loop bifurcation surface Σk-homoclinic loop and k-periodic orbit, where k > 1. We may assume that k = 2.Now, we consider the nonexistence of q0 and q1 as τ1, s1 = eλτ1−. Suppose F0(q2) = q3, F1(q3) = q4 = q0, let τ2 be the time from q2 to q3, s2 = eλτ2−. Similar to the previous discussion, we can get its associated successor function G2 =  as follows:We rewrite the time from Q2 = . The equation  = 0 always has a unique solution u1 = u1,  v0 = v0,  u3 = u3,  v2 = v2, substituting it into  = 0, we gets1 = s2 = 0, then (Denote  we getδ[(w12)− we getδ[(w12)−M1 ≠ 0, the above formula defines a (l − 1)-dimensional surface L, now the 2-homoclinic loop is the 1-homoclinic loop.If s1 > 0, s2 = 0, then -dimensional surface L1, such that when |μ | ≪1 and μ ∈ L1, the system  and V(s2) be the left and right of the above formula, respectively, then N(s2) and V(s2) are tangent at some point if and only if α(μ) > 0, that is, Γ is twisted. At present, s2 satisfies 0 < s2 < δ−1M1μ ≪ 1. Substituting  δ−1M1μ = β*(μ). We notice that every 1-periodic orbit also corresponds to a solution s1 = s2 > 0 of ~H3) and (H5) are fulfilled, if Δ = 1,  and (H5)Now, we consider the stability of homoclinic loop Γ.F(q0) = F* = q2 = *, if μ = 0, and ρ1 ≠ λ1, we can get the following by (According to owing by (40)n2s ≈ δ−1x0, u1 ≈ sB1(0)/λ1−u0, substituting into , x2 ≈ n2,1, u2 = w33n2,3, v2 = O. And by Reσ(−B1(0)) < 0, ReσB2(0) > 0, s ≪ 1, we can get u2 = O(sB1(0)/λ1−u0) ≫ u0, v2 = O(sB2(0)/λ1v0) ≪ v0. Meanwhile, we make Poincaré map F restrict at the half transversal section S0+ = { ∈ S0, 0 ≤ x < δ1 < δ}, F makes the transversal line Lx = {0 ≤ x < δ1, y = δ, u = 0, v = 0} maps to the segment Lx′ = {0 ≤ x < γδ1, y = δ, u = 0, v = 0} approximately, where γ is called the shrinkage (expansion) rate. So, when ρ1/λ1 > 1(<1), we can get γ = w12−1(δ−1x0)ρ1/λ1−1 < 1(>1) for δ ≪ 1, x0 ≪ 1. Hence, we have the following.According to , we knowρ1/λ1 > 1, homoclinic loop Γ is weak stability, and Γ has a (m + 1)-dimensional stable manifold and a n-dimensional instable manifold, If ρ1/λ1 < 1, homoclinic loop Γ is weak instability, and Γ has a m-dimensional stable manifold and a (n + 1)-dimensional instable manifold.If μ = 0 and ρ1 = λ1, similar to above, we can get γ = w12−1 not difficulty. Thus, we have the following.If w12−1 < 1, homoclinic loop Γ is weak stable, and Γ has a (m + 1)-dimensional stable manifold and a n-dimensional instable manifold. If w12−1 > 1, homoclinic loop Γ is weak instable, and Γ has a m-dimensional stable manifold and a (n + 1)-dimensional instable manifold.If Besides, by the above discussion, we can get the following.The homoclinic loop or the periodic orbit of the perturbed system have the same stability with the homoclinic loop of the unperturbed system.Cr system  ∈ R2, r ≥ 5. We consider the stability of the homoclinic orbit Γ under the resonance case. Denote f(z) = (f1(z), f2(z))*, σ = exp{∫∞−∞+(∂f1/∂x + ∂f2/∂y)(r(t))dt}, Now, suppose the  system  is 2-dimσ is convergent, and Cr transformation coordinates, such that the system , H2 is Cr−3. So, in U, we have Γ∩W0s = { : x = 0}, Γ∩W0u = { : y = 0}. Thus, in U, if  ∈ Γ, (∂f1/∂x) + (∂f2/∂y) = xyH = 0.In fact, from –3, thereThe proof is complete.C2 transformation (refer [f(·) and r(t) of the divergence integration σ can be thought of the original forms of , so, soC2 tforms of .w21 = −1, w12 = 1/σ. w21 = −1. And by the Liouville formula, we have w12w21σ = 1, and by w21 = −1, we get w12 = 1/σ.According to The proof is complete.Combined with the σ < 1, the homoclinic orbit Γ is stable; If σ > 1, the homoclinic orbit Γ is instable.If


A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system. For the last two decades, there have been growing interests in studying the complex dynamics of financial systems in both micro- and macroeconomics , 2. It iOn the other hand, delays are ubiquitous in life, so it is in the social and economic activities. There are at least two ways that time delays emerge in the dynamics of economic variables. One is the time lag between the time economic decisions are made and the time the decisions bear fruit . The othThe aim of this paper is to investigate the dynamics of a financial system by considering the effect of washout filters with time delay. By analyzing the characteristic equation of linearization of the system, we theoretically prove that the Hopf bifurcations occur in the model with delay. Furthermore, by using the theory of functional differential equation and Hassard's method , we alsoThe rest of the paper is organized as follows. In x, the investment demand y, and the price index z. The model is represented by three-dimensional ODEs:a > 0 is the saving amount, b > 0 is the cost per investment, and c > 0 is the elasticity of demand of commercial market. This model is well studied in τ=τij−1 and h′(ζi) have the same sign. Thus, from Lemmas Without loss of generality, suppose that roots of . Then, ω root of . From  into  and taki we have[dλdτ]−1=i = 1,2,  are posiac − (c/b) > 0 and c + a − (1/b) > 0; then, one has the following: r⩾0 and Δ = p2 − 3q ⩽ 0, then the equilibrium P0 is stable for all τ⩾0;if r < 0 (or r⩾0 and Δ = p2 − 3q > 0) and h′(ζi) ≠ 0, then, when τ ∈ . For ψ ∈ ℂ1*), defineη(θ) = η; let A = A(0); then, A and A* are adjoint operators. By the discussion in iωiτi are eigenvalues of A. Thus, they are also eigenvalues of A*. We first need to compute the eigenvector of A and A* corresponding to iωiτi and −iωiτi, respectively.Obviously, e system , where uq(θ) = Teiωiτiθ is the eigenvector of A corresponding to iωiτi. Then, Aq(θ) = iωiτiq(θ). It follows from the definition of A, Lμϕ, and η thatα = 2x*/(iωi + b), β = iωi + a − y* − αx*, γ = 1 + (iωi + c)β, and q(0) = T.Suppose that q*(s) = Deiωiτis be the eigenvector of A* corresponding to −iωiτi. By the definition of A*, we can compute α* = −x*/(iωi − b), β* = iωi − a + y* + 2x*α*, and γ* = −(β*/(iωi − d)).Similarly, let q*(s), q(θ)〉 = 1, we need to determine the value of D. From (q*(s), q(θ)〉 = 1, In order to assure that 〈 D. From , we haveC0 at μ = 0. DefineC0, we havez and C0 in the direction of q and W is real if ut is real. We consider only real solutions. For the solution ut ∈ C0, since μ = 0, we haveq(θ) = Teiωiτiθ; then,In the following, we first compute the coordinates to describe the center manifold . Define    z(t)= we haveW=W, we haveut1(0), ut2(0), ut3(0), and ut4(0) into the above equation and comparing the coefficients with  and W11(θ). From , we haveA, we can getq(θ) = q(0)eiωiτiθ, and we haveE1 = (E1(1), E1(2), E1(3), E1(4))T ∈ ℝ4 is a constant vector. In the same way, we can also obtainE2 = (E2(1), E2(2), E2(3), E2(4))T ∈ ℝ4 is also a constant vector.In order to assure the value of θ). From  and 43)g21, we ). From (W˙=u˙t−z˙). From (W˙=u˙t−z˙ we have(A−2iωkτkts with (H20(θ)=−gE1 and E2. From the definition of A and 52), we htituting  and 62)(62)H20 and W11(θ). Furthermore, we can determine each gij. Therefore, each gij is determined by the parameters and delay in , then the Hopf bifurcation is supercritical  and the bifurcation exists for τ > τi (<τi); β2 determines the stability of the bifurcation periodic solutions; the bifurcating periodic solutions are stable (unstable) if β2 < 0 (>0); and T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2 > 0 (<0).Similarly, substituting  and 63)63) into delay in . Thus, wIn this section, we present some numerical results to verify the analytical predictions obtained in the previous section. These numerical simulation results constitute excellent validations of our theoretical analysis; it is shown that the chaotic orbit can be controlled to a periodic orbit by using washout-filter-aided controller with time delay.a = 4, b = 0.1, and c = 1.a = 4, b = 0.4, and c = 1; then, system . From the algorithm of r = (1.5d − 1.5k)2 − 2.25k2; thus, if k > d/2, then r < 0. From d = 1 and k = 1. From the algorithm of τ0≐5.5932. Thus, from P0 is asymptotically stable when τ < τ0, and, as τ crosses τ0, there are periodic orbits bifurcating from P0  .a = 4, b = 0.1, and c = 4; then, from P0 is unstable and P± are stable (k = 2, and d = 1, from the algorithm of τ is τh = 2.3542. When τ pass through τh = 2.3542, a family of periodic orbits will bifurcate from equilibria P±, respectively (In this subsection, we choose e stable . If we cectively .a = 4, b = 0.1, and c = 1. If we choose k = d = 1, a family of periodic orbits bifurcate from the equilibria of system (τ. This can be verified by From τ as bifurcating parameter, we discussed the conditions at which periodic orbits bifurcate from the equilibria P0 and P±, respectively. The stability and direction of bifurcated periodic solutions have been also investigated in detail. And the obtained results can be applied to control the chaos of this financial system.In this paper, we have investigated a financial system with time-delayed washout-filters-aided controller. Taking the time delay k is also applied to influence the dynamical behaviors of this financial system; it will be investigated in the near future.From a financial sense, the obtained results show that the delay on price index has great influence on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system, so as to make the economic system run well. On the other hand, the control gain


We study the properties of almost periodic solutions for a general discrete system of plankton allelopathy with feedback controls and establish a theorem on the uniformly asymptotic stability of almost periodic solutions. Meanwhile, {ri(n)}, {aij(n)}, {bi(n)}, {ci(n)}, {di(n)}, and {ei(n)} are bounded nonnegative sequences such thati, j = 1,2. Here, we use the notations fu = supn∈ℤ+f(n) and  fl = infn∈ℤ+f(n) for any bounded sequence {f(n)}. For the simplicity and convenience of exposition, throughout this paper we let ℤ, ℤ+, and ℝ+ denote the sets of all integers, and nonnegative integers, nonnegative real numbers, respectively.Allelopathy is first used by Hans (see ) in 1937x1(n), x2(n), u1(n), u2(n)} are positive; that is,From the point of view of biology, we only consider {For convenience, we give the following definition of persistence of system .xi*, ui*, xi*, and ui* which are independent of the solutions of system (x1(n), x2(n), u1(n), u2(n)} of system  > 0 such that each discrete interval of length l(ε) contains a τ = τ(ε) ∈ E{ε, x} such thatn ∈ ℤ. τ is called the ε-translation number of x(n).A sequence f : ℤ × 𝔻 → ℝk, where 𝔻 is an open set in ℝk. f is said to be almost periodic in n uniformly for x ∈ 𝔻 or uniformly almost periodic for short if, for any ε > 0 and any compact set S in 𝔻, there exists a positive integer l such that any interval of length l contains a τ for whichn ∈ ℤ and all x ∈ S. τ is called the ε-translation number of f.Let Next, we will introduce the following Lemmas.x(n)} is an almost periodic sequence if and only if for any sequence {hk}⊂{hk′} such that x(n + hk) converges uniformly on n ∈ ℤ as k → ∞. Furthermore, the limit sequence is also an almost periodic sequence.{x(n)} satisfies x(n) > 0 andn ∈ ℤ+, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. ThenAssume that {x(n)} satisfiesn→+∞x(n) ≤ x*, bux* > al, x(n0) > 0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and n0 ∈ ℤ+. ThenAssume that {In this section, we will establish sufficient conditions for the persistence of system .x1(n), x2(n), u1(n), u2(n)} of system (Assume that  and 3)  3) hold;x1(n), x2(n), u1(n), u2(n)} be any positive solution of system xi(p)/∏q=0n−11 − di(q)), i = 1,2. Since 0 < d1l < 1, we can find a g ∈ ℤ+ such that 1 − d1l = eg−; by Stolz's theorem, we haveε → 0 and substituting (Let {f system . By assuApplying limsupn→uch thatxi(n)≤xi∗i, j = 1,2, i ≠ j, are satisfied, where ui* and xi* are the same as those in Assume that  and 3);;3); furtε > 0 and n > n2, according to n2 ∈ ℤ+ such thatε → 0 in the above inequality leads toε > 0, there exists a large enough integer n0 such thatvi = ∑p=0n−1(ei(p)xi(p)/∏q=0n−11 − di(q)), i = 1,2. As 0 < diu < 1, similar to the analysis in the proof of For leads toliminfn→Now, we are in a position to state If the inequalities in , 3), an, an3), aBased on f : ℤ × SB → ℝk, SB = {x ∈ ℝk : ||x|| < B}, and f is almost periodic in n uniformly for x ∈ SB and is continuous in x. The product system of system =f defined for n ∈ ℕ, ||x|| < B, ||y|| < B satisfying thata(||x − y||) ≤ V ≤ b(||x − y||), where a, b ∈ K, with K = {a ∈ C : a(0) = 0}, and a is increasing;V − V| ≤ L(||x1 − x2|| + ||y1 − y2||), where L > 0 is a constant;|V(10)≤−aV, where 0 < a < 1 is a constant and ΔV(10) = V, f) − V.ΔMoreover, if there exists a solution φ(n) of system (φ(n)||≤B* < B for n ∈ ℤ+, then there exists a unique uniformly asymptotically stable almost periodic solution p(n) of system  is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of  = {x1(n), x2(n), u1(n), u2(n)} of system (xi* ≤ xi(n) ≤ xi*, ui* ≤ ui(n) ≤ ui*.According to Ω ≠ Ø.Assume that , 3), an, an3), ai, j = 1,2, i ≠ j. From X(n) = {x1(n), x2(n), u1(n), u2(n)} of system  → xi*(n) uniformly in n on any finite subset B of ℤ as α → ∞, wherem is a finite number.It is now possible to show by an inductive argument that the system  leads to3xi(n)=xi, ui(k + τα)} are uniformly bounded for large enough α.In fact, for any finite subset a1 ∈ B, we can choose a subsequence τα(1) of {τα} such that {xi(a1 + τα(1)), ui(a1 + τα(1))} uniformly converges on ℤ+ for α large enough.Now, for a2 ∈ B, we can choose a subsequence τα(2) of {τα} such that {xi(a2 + τα(2)), ui(a2 + τα(2))} uniformly converges on ℤ+ for α large enough.Similarly, for am ∈ B, we can choose a subsequence {ταm−1)(} of {ταm)(} such that {xi(am + ταm), ui(am + ταm)} uniformly converges on ℤ+ for α large enough.Repeating this procedure, for ταm)}, {aij(n)}, {bi(n)}, {ci(n)}, {di(n)}, and {ei(n)} are almost periodic sequence, for the above sequence τα, τα → ∞ as α → ∞, there exists a subsequence which we still denote it by {τα} , such thatα → ∞ uniformly on ℤ+.Since {δ ∈ ℤ, we can assume that τα + δ ≥ n0 for δ large enough. Letting  n ≥ 0 and n ∈ ℤ, by an inductive argument of  = {x1*(n), x2*(n), u1*(n), u2*(n)} is a solution of system (ℤ+. It is clear that 0 ≤ xi* ≤ xi(n) ≤ xi*; 0 ≤ ui* ≤ ui(n) ≤ ui*, n ∈ ℤ+.By the arbitrariness of Ω ≠ Φ. So Before stating β < 1, whereX(n) = {x1(n), x2(n), u1(n), u2(n)} of system  = exppi(n); from system  = {p1(n), p2(n), u1(n), u2(n)} satisfyingX, U) ∈ ℝ2+2, we define a normZ = {p1(n), p2(n), u1(n), u2(n)} and W = {q1(n), q2(n), v1(n), v2(n)} be any two solutions of system =p system (pi(n+1)=pf system  defined a ∈ C, a(x) = C12x2, b ∈ C, and b(x) = C22x2; thus condition (i) in ThenL = 4max{Ai, Bi}, i = 1,2. And In addition,V along the solutions of (epi(n) − eqi(n) = ξ(n)Mi(n), where ξ(n) lies between epi(n) and eqi(n).Finally, calculating the Δtions of  leads toj = 1,2 thatβ = min{rij, rij*}, i, j = 1,2, i ≠ j. That is, there exists a positive constant 0 < β < 1 such thatβ < 1, condition (iii) of X(n) = {x1(n), x2(n), u1(n), u2(n)} of system  = {x1(n), x2(n), u1(n), u2(n)} of system , o, o54), of system  which is(55)ΔV2(n)=∑i=1


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Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated. The problem of stability of solutions is one of the major problems in the theory of differential equations. Lyapunov function and the Lyapunov direct method allow us to obtain sufficient conditions for the stability of a system without explicitly solving the differential equations , 2. The Only a few decades ago, it was realized that fractional calculus provides an attractive tool for modelling the real world problems. The differentiation and integration of arbitrary orders have found applications in diverse fields of science and engineering like viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos, and fractals –5. RecenIn practical situations, it is possible to have not only a change in initial position but also in initial time because of all kinds of disturbed factors. When we do consider such a deviation in initial time, it causes measuring the difference between any two different solutions starting with different initial times. From this point of view, several studies have been made on this problem to explore the stability and boundedness, criteria for differential systems relative to initial time difference (ITD) by using variation of parameters and differential inequalities technique –13. In tf :  → ℝ be a function.Fractional calculus generalizes the derivative and the integral of a function to a noninteger order , 4, 14. q > 0 is defined asThe fractional integral (or the Riemann-Liouville (RL) type integral) of order q ∈  = sqX(s) − ∑k=0n−1sk[Dq−k−1f(t)]t=0, where Dq represents 0Dtq.ℒ[cDqf(t)] = sqX(s) − ∑k=0n−1sq−k−1fk)((0), where cDq represents 0cDtq.cDqIqf(t) = f(t) and DqIqf(t) = f(t).IqDqf(t) = f(t) − ∑k=1n−1[Dq−kf(t)]t=a((t − a)q−k/Γ(k − q + 1)) and IqcDqf(t) = f(t)‒∑k=0n−1((t − a)k/k!)fk)((a).cDqf(t) = Dq(f(t) − ∑k=0n−1((t − a)k/k!)fk)((a)).DqC = C(t − a)q−/Γ(1 − q) and cDqC = 0, where C is arbitrary constant.q < 1, f ∈ C and J = . Then IVP's are equivalent to the following Volterra fractional integral equations:After giving definition and properties of fractional integral and derivatives, we consider fractional order IVP with (RL) and Caputo derivative, respectively;vely. In  authors t0 ∈ ℝ+, f ∈ C and 0 < q < 1. Suppose that the function f is smooth enough to guarantee existence, uniqueness, and continuous dependence of solutions of IVP (x(t) = x the solution of (τ0 ∈ ℝ+ and y(t) = y be the solution of the system (x(t) = x is the solution on which we shall study stability and boundedness criteria with respect to it. Set η = τ0 − t0 > 0 and denote S(ρ) = {x ∈ ℝn : ||x|| < ρ}.Consider the IVP for the system of nonlinear fractional differential equationDqcx(t)=fDqcx(t)=fDqcx(t)=fBefore giving our comparison theorem, stability, boundedness criteria, and Lagrange stability for FDE we need to introduce the following definitions.x(t) = x of ((S1)ϵ > 0 and t0 ∈ ℝ+; there exist δ = δ > 0 and stable with ITD, if given (S2)S1) holds with δ and t0.uniformly stable with ITD, if (The solution  x0) of  is said (B1)α > 0 and t0 ∈ ℝ+; there exist β = β > 0 such thatequibounded with ITD, if given (B2)B1) holds with β independent of t0;uniformly bounded with ITD, if ((A1)ϵ > 0 and each α > 0 there exists T = T such thatattractive in the large with ITD, if for every (L1)Lagrange stable if (B1) and (A1) hold.The system  is said a is said to belong to the class 𝒦 such thatA function x = x(t) = x is the solution of  is the solution of  = x be a solution of (x(t). Consider the solution Y(t) = Y of  = Y(t) − x(t) is a solution of the IVP (Let ution of . Study t y0) of . Set they0) of (Dqcz=f~(tx(t) of (The IVP  has a zex(t) of  is reducx(t) of .x(t). Consider the solution of (y(t) = y. Set the IVP asz = y(t + η) − x(t) is a solution of (x(t) could not be reduced to the study of stability of the zero solution of an appropriate fractional differential system.Study the stability with initial time difference of ution of  with dife IVP asDqcz=f~, if and only if the Caputo derivative of cDqm(t) exists and satisfiesm ∈ Cq. Suppose that for any t1 ∈  = 0 and m(t) < 0 for t0 ≤ t < t1; then it follows thatLet m(t0) < 0 we have Dqm(t0) < 0. Therefore, we obtainFrom the relation between Riemann-Liouville and Caputo fractional derivatives we writeive from  and 18)(17)Dqcm, f ∈ C andv(t0) < w(t0) impliesLet v(t0) < w(t0) and v(t), w(t) are continuous, there exists a t1 ∈  = w(t1) and v(t) < w(t) for t0 ≤ t < t1. Set m(t) = v(t) − w(t). Then m(t1) = 0 and m(t) < 0 for t1 ∈  such that m(t0) ≤ u0. Then one hasAssume that r(t), it is enough to prove that m(t) < u, t0 ≤ t ≤ T, where u is any solution of the IVPm(t) < u. And from limϵ→0u = r(t) uniformly on each compact set t0 ≤ t ≤ T0 < T, we get the desired estimate m ∈ Cq, g ∈ C and(ii)r(t) = r of the IVPt ≥ τ0;the maximal solution (iii)g is nondecreasing in t for each w and τ0 > t0.Then (a) m(t) ≤ r(t + η), t ≥ t0 and (b) m(t − η) ≤ r(t), t ≥ τ0.Assume thatw is any solution ofϵ→0w = r on every compact set .(a) It is well known that if w0 = w, we have w0 = w = w = w0 + ϵ > w0 ≥ m(t0). Thus we get m(t0) < w0. On the other hand, using (iii)m(t) < w0,  t ≥ t0 and hence it follows that m(t) ≤ r(t + η),  t ≥ t0.Setting m0(t) = m(t − η) so that m0(τ0) = m(t0) ≤ w0 < w0 + ϵ andm0(t) < w, t ≥ τ0. The conclusion follows taking the limit as ϵ → 0. The proof of theorem is complete.(b) We set In the following theorem, we obtain a comparison result in terms of Lyapunov-like functions with ITD.(i)V ∈ C, V is locally Lipschitzian in x ∈ ℝn, g ∈ C and(ii)r(t) = r of the fractional scalar differential equationt ≥ τ0;the maximal solution (iii)g is nondecreasing in t for each u.Then V ≤ u0 impliesAssume thatm(t) = V − x) so thatz = y − x so thatV is locally Lipschitzian in x and L > 0 is the Lipschitz constant and ϵ(hq)/hq → 0 as h → 0, we haveDefine A comparison principle is obtained by employing the notion of Lyapunov function together with the theory of differential inequalities. In this part, one can see Lyapunov-like function as a transformation which reduces the study of stability, boundedness, and Lagrange stability properties relative to ITD of a given complicated system to a relatively simpler scalar equation.(i)V ∈ C, V is locally Lipschitzian in x ∈ ℝn, g ∈ C and(ii)r(t) = r of  of  exists f(iii)a, b ∈ 𝒦 such that b(||x||) ≤ V ≤ a(||x||) for  ∈ ℝ+ × S(ρ);there exists (iv)g is nondecreasing in t for each u and g = 0;Then the stability properties of the null solution of .Assume thatution of  imply thϵ < ρ be given. Then by definition of equistability given b(ϵ) > 0, τ0 ∈ ℝ+, there exist a δ1 = δ1 such thatu is any solution of the  > 0 as 0 < a(δ2) < δ1. Obviously limτ0,y0)→ − x|| = 0. Then given ϵ > 0 and τ0 ∈ ℝ+, δ3 = δ3 > 0 such thatu0 = a(||y0 − x0||) and choose δ = min. Then we claim thatt1 > τ0 and a solution y with ||y0 − x0|| < δ and y0 − x0|| < δ, by (iii) we havex of (Assume that the null solution of  is equisn of the . Choose uch that||y0−x0||uch that||y(t1+η)uch thatu(t)<b(ϵ)uch that||y(t1+η), (ii), In this section, Lagrange stability, which includes boundedness criteria, of fractional dynamic systems are discussed by employing comparison method.Let the assumption of α ≥ 0 be given, and let ||y0 − x0|| ≤ α. In view of (iii) V ≤ a(α) = α1. Assume that  such thatb(u) → ∞ as u → ∞; we can choose a β = β verifying the relationβ > 0 and τ0 ∈ ℝ+, δ1 = δ1 > 0 such thatu0 = V. Then we claim thatt* > τ0 and a solution y of (We need to prove (B1) and (S3) for . Let α ≥ume that  is Lagraume that  is boundhen from , given β y0) of  such thaelations , 51), ≥β1.olds for .ϵ > 0, α ≥ 0, τ0 ∈ ℝ+ be given and ||y0 − x0|| ≤ α. In view of (iii), V ≤ a(α) = α1. Since  and τ0 ∈ ℝ+ there exist a T = T such thattk} ∈ ℝ+, tk → ∞ as k → ∞, tk > τ0 + T and a solution y of  of  such thas (iii), , 56), a, aϵ > 0,y0) of (||y(tk+η)Firstly, differential inequalities and a new comparison principle for fractional differential equations relative to initial time difference have been developed and then stability, boundedness criteria, and Lagrange stability relative to initial time difference have been proved by employing comparison method.


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We also characterize ideals, nilradicals, and nilpotent elements of such algebras.We describe weak-BCC-algebras  in which the condition ( It holds in BCK-algebras and in some generalizations of BCK-algebras, but not in BCC-algebras. BCC-algebras satisfying this identity are BCK-algebras (cf. [One of very important identities is the identity (ras (cf.  or 7]).).x∗y)∗z ras (cf. .x∗y)∗z = (x∗z)∗y is satisfied only in the case when elements x, y belong to the same branch. We describe some endomorphisms of such algebras, ideals, nilradicals, and nilpotent elements.In this paper, we will study weak-BCC-algebras in which the condition  of type  satisfying the following axioms: x∗y)∗(z∗y))∗(x∗z) = 0, )BZ-algebrea  x = 00∗is called a BCC-algebra. A BCC-algebra with the condition (vi) x∗(x∗y))∗y = 0 x∗y)∗z = (x∗z)∗y. (One can prove (see  or 7]) ) 7]) thaG; ∗, 0) of type  satisfying the axioms (i), (ii), (iii), (iv), and (vi) is called a BCI-algebra. A BCI-algebra satisfies also (vii). A weak-BCC-algebra is a BCI-algebra if and only if it satisfies (vii).An algebra  of type  with a relation ⩽ defined by x∗y)∗(z∗y) ⩽ x∗z,((ii′)x ⩽ x,(iii′)x∗0 = x,(iv′)x ⩽ y  and  y ⩽ x  imply  x = y.An algebra , it follows that in weak-BCC-algebras, implicationsproper. Proper weak-BCC-algebras have at least four elements . But thThey are proper, because in both cases (3∗2)∗1 ≠ (3∗1)∗2.Since two nonisomorphic weak-BCC-algebras may have the same partial order, they cannot be investigated as algebras with the operation induced by partial order. For example, weak-BCC-algebras defined by  and 5)  5) have n⩾4, there exist at least two proper weak-BCC-algebras of order n which are not isomorphic.The methods of construction of weak-BCC-algebras proposed in  show tha⩽) elements of G is denoted by I(G). Elements belonging to I(G) are called initial.The set of all minimal  = 0∗x. The main properties of this map in the case of weak-BCC-algebras are collected in the following theorem proved in , we mean the set of all elements of G such that 0∗xk ∈ A; that is,Let G defined in A = {0, a} and any natural k, we have  =  = B(0),  = G. But for B = {a, e}, we get  = {d},  = B(c). The set  is empty.In the weak-BCC-algebra G defined byk-nilradical of A = {0,5} is equal to G; each k-nilradical of B = {1,4} is empty.The solid weak-BCC-algebra A its k-nilradical is also nonempty? The answer is given in the following proposition.The first question is when for a given nonempty set k-nilradical  of a nonempty subset A of a weak-BCC-algebra G is nonempty if and only if A contains at least one element a ∈ I(G).A xk = 0∗ak for every x ∈ B(a) and any positive k. So, x ∈  if and only if 0∗ak ∈ A. The last means that 0∗ak ∈ A∩I(G) because I(G) is a subalgebra of G.From the proof of I(G); k] = G for every k..Indeed, 0∗I(G) has n elements, then  = G for any subset A of G containing 0, and  = ∅ if 0 ∉ A.If xk = 0∗ak for every x ∈ B(a) and any k. Since 0∗ak ∈ I(G) and I(G) is a group-like subalgebra of G, 0∗xk = ak− in the group (I(G); ·, −1, 0) (I(G) has n elements, then obviously 0∗xn = an− = 0 ∈ A. Hence, x ∈ . This completes the proof.Similarly, as in previous proofs, we have 0∗, −1, 0) . If I(G)x ∈ B(a). Then x ∈  if and only if B(a)⊂.Let xk = 0∗ak, we have x ∈ ⇔a ∈ .Since 0∗A; k] = ⋃{B(a) : 0∗ak ∈ A}. is a subalgebra of G such that A⊆.Let x, y ∈ . Then 0∗xk, 0∗yk ∈ A and 0∗(x∗y)k = (0∗xk)∗(0∗yk) ∈ A, by x∗y ∈ . Clearly A⊆.Let k-nilradical of an ideal is also an ideal.In a solid weak-BCC-algebra, a A be a BCC-ideal of G. If y ∈  and (x∗y)∗z ∈ , then 0∗yk ∈ A and A∋0∗((x∗y)∗z)k = ((0∗xk)∗(0∗yk))∗(0∗zk), by A∋(0∗xk)∗(0∗zk) = 0∗(x∗z)k. Thus, x∗z ∈ .Let Note that the last two propositions are not true for weak-BCC-algebras which are not solid.G induced by the symmetric group S3 is not solid because S3 is not an abelian group -fold p-ideal is also an -fold p-ideal.In a solid weak-BCC-algebra, a k-nilradical of an -fold p-ideal A of G is an ideal of G. If y, (x∗zm)∗(y∗zn)∈, then 0∗yk, 0∗((x∗zm)∗(y∗zn)k) ∈ A. Hence, applying xk ∈ A. So, x ∈ .By k-nilradical  of an ideal A does not save all properties of an ideal A. For example, if an ideal A is a horizontal ideal, that is, x ∈ A∩B(0)⇔x = 0, then a k-nilradical  may not be a horizontal ideal. Such situation takes place in a weak-BCC-algebra defined by  means that 0∗x3 ∈ A and x ∈ B(0) which is also true for x ≠ 0.Note that in general, a fined by . In thisk-nilradicals. Below, we present the list of the main types of ideals considered in BCI-algebras and weak-BCC-algebras.Nevertheless, properties of many main types of ideals are saved by their A of a weak-BCC-algebra G is called (i)antigrouped, if(ii)associative, if(iii)quasiassociative if(iv)closed, if(v)commutative, if(vi)subcommutative, if(vii)implicative if(viii)subimplicative if(ix)weakly implicative if(x)obstinate, if(xi)regular, if(xii)strong, iffor all x, y, z ∈ G.An ideal A of a weak-BCC-algebra G has the property 𝒫 if it is one of the above types, that is, if it satisfies one of implications mentioned in the above definition.We say that an ideal A of a solid weak-BCC-algebra G has the property 𝒫, then its k-nilradical  also has this property.If an ideal A is antigrouped. Let φ2(x)∈. Then 0∗(φ2(x))k ∈ A. Since, by φ2 is an endomorphism of each weak-BCC-algebra, we haveφ2(0∗xk) ∈ A, which according to the definition implies 0∗xk ∈ A. Hence, x ∈ .(1)  A is associative. If (x∗y)∗z,  y∗z ∈ , then 0∗((x∗y)∗z)k ∈ A and 0∗(y∗z)k ∈ A which, in view of xk)∗(0∗yk))∗(0∗zk) ∈ A and (0∗yk)∗(0∗zk) ∈ A. Since an ideal A is associative, this implies 0∗xk ∈ A; that is, x ∈ .(2)  A is quasiassociative. Similarly as in the previous case x∗(y∗z),  y ∈  means that 0∗(x∗(y∗z))k ∈ A and 0∗yk ∈ A. Hence, (0∗xk)∗((0∗yk)∗(0∗zk)) ∈ A. This implies 0∗(x∗z)k = (0∗xk)∗(0∗zk) ∈ A. Consequently, x∗z ∈ .(3)  A is closed. Let x ∈ . Then, 0∗xk ∈ A. Thus,x ∈ .(4)  A is commutative. Let x∗y ∈ . Then, 0∗(x∗y)k ∈ A. From this, we obtain (0∗xk)∗(0∗yk) ∈ A, which gives 0∗(x∗(y∗(y∗x)))k = (0∗xk)∗((0∗yk)∗((0∗yk)∗(0∗xk))) ∈ A. Hence, x∗(y∗(y∗x))∈.(5)  For other types of ideals, the proof is very similar.


The approach used in this paperis the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Passtheorem are used to prove the existence of at least two nontrivial solutions.A class of nonlinear Neumann problems driven by  Of the existing works in the literature, the majority deal with problems in which the potential function is smooth  ∈ C1(ℝ)). We mention the works of Mihailescu p(x)-Laplhailescu  and Cammhailescu . Problemhailescu , 7, who n et al.  studied f Clarke . The autV(x) ≡ 1, then problem (If  problem  as folloproblem (−div(|∇uu|p(x)−2∇u) is said to be p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous and in general, it has not the first eigenvalue. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years. These problems are interesting in applications to modeling electrorheological fluids  and andu|p(xids (see ).Lp(x)(Ω) and the generalized Lebesgue-Sobolev space Wp(x)1,(Ω). In the third section, we give the assumptions on the nonsmooth potentials F,  G and prove the multiplicity results for problem (Ω) and Wp(x)1,(Ω). For the details, see φ(γ(t)),  𝒯 = {γ ∈ C:γ(0) = x0, γ(1) = x1}, then c is a critical value of φ and c ≥ infU∂φ.Let In this section, we will discuss the existence of weak solution of .F and G are given as follows. H(F) : F : Ω × ℝ → ℝ is a function such that F = 0 almost everywhere on Ω and satisfies the following facts: t ∈ ℝ, x ↦ F is measurable;(1) for all  x ∈ Ω, t ↦ F is locally Lipschitz.(2) for almost all  H(G) : G : ∂Ω × ℝ → ℝ is a function such that G = 0 almost everywhere on ∂Ω and satisfies the following facts: t ∈ ℝ, x ↦ G is measurable;(1) for all  x ∈ ∂Ω, t ↦ G is locally Lipschitz.(2) for almost all Our hypotheses on nonsmooth potential φ : X → ℝ for problem , H(G), and the following conditions hold:f1) ≤ α+ < p− such thatthere exist  x ∈ Ω, all t ∈ ℝ and w ∈ ∂F;for almost all g1) with 1 < β(x) ≤ β+ < p− < p∗(x) such thatthere exist  x ∈ ∂Ω, all t ∈ ℝ and w ∈ ∂G.for almost all Suppose that φ is locally Lipschitz in X.Then, J ∈ C1, we have t ∈ .By Br = {x ∈ X : ||u−u0||X ≤ r}.Let Br is w-compact. Then, we obtain that there exists a positive constant M, such that r.Note that u1, u2 ∈ Br, we haveTherefore, for any f1) and Lebourg mean value theorem, we haveOn the other hand, by (α′(x))+(1/α(x)) = 1.Hence,Obviously, it is verified thatWp(x)1,(Ω)↪Lα(x)(Ω) is a compact imbedding.So,c4, such thatWp(x)1,(Ω)↪Lβ(x)(∂Ω) is a compact imbedding.As above, there is a positive constant φ is locally Lipschitz.Therefore, f1) and (g1) in α(x) ≤ α+ < p− < p*(x) such thatthere exist  x ∈ Ω, all t ∈ ℝ and w ∈ ∂F;for almost all β ∈ C(∂Ω) with 1 < β(x) ≤ β+ < p− < p∗(x), such thatthere exist  x ∈ ∂Ω, all t ∈ ℝ and w ∈ ∂G, then the result of for almost all If assumptions (H(F), H(G), (f1), (g1), and the following conditions (f2)-(f3), (g2) hold:f2);where 1 < f3) and 0 < |t | ≤δ;g2) and 0 < |t | ≤δ, whereLet Then, the problem  has at lThe proof is divided into four steps as follows.Step  1. We will show that φ is coercive in this step.x ∈ Ω, by H(F) (2), t ↦ F is differentiable almost everywhere on ℝ and we haveFirstly, for almost all f1), we havex ∈ Ω and t ∈ ℝ.Moreover, from (α(x) ≤ α+ < p− < p*(x) and 1 < β(x) ≤ β+ < p− < p∗(x); then by Wp(x)1,(Ω)↪Lα(x)(Ω) and Wp(x)1,(Ω)↪Lβ(x)(∂Ω) (compact imbedding).Note that 1 < Furthermore, there exist u|Lα(x)(Ω) > 1, |u|Lβ(x)(∂Ω) > 1 and ||u|| > 1, we haveSo, for any |Hence,Step  2. We will show that the φ is weakly lower semicontinuous.un⇀u weakly in Wp(x)1,(Ω); by Let By Fatou's lemma, we haveThus,u0 ∈ Wp(x)1,(Ω) such thatHence, by Weierstrass theorem, we deduce that there exists a global minimizer Step  3. In this step, we prove that φ(u0) < 0.f2), we haveBy (x0 ∈ Ω and Br2(x0)⊆Ω with 2r < 1. Let η ∈ C0∞(Br2(x0)) such that η = 1, x ∈ Br(x0); 0 ≤ η(x) ≤ 1 and |∇η | ≤2/r. Denote s = 2δ; thenSuppose that Step  4. We will show that there exists another nontrivial weak solution of problem ( problem .p+ < ρ− ≤ ρ+ < p∗(x).Let u ∈ X with ||u|| < 1, by c7 and For f3), we haveBy (f3), we obtainλ ∈ , 0 < |t | ≤δ.From Lebourg's mean value theorem and |≤δ} and Ω2 = {x ∈ Ω : |u(x)|≥δ}.Divide u ∈ X such that ||u|| = d, we haveFor any φ is coercive; hence, it satisfies the nonsmooth C-condition. So by the nonsmooth Mountain Pass theorem (consequence of u1 ∈ Wp(x)1,(Ω) such thatu1 is another nontrivial critical point of φ.Note that Using the similar and simpler arguments, we can prove the following theorems.H(F), H(G), (f1), (f2), (g1), and the following conditions (f3′), (g2′) hold:f3′) = 0, for all x ∈ Ω, 0 < |t | ≤δ;g2′) = 0, for  all  x ∈ ∂Ω, 0 < |t | ≤δ, whereLet Then, the problem  has at lφ(u0) < 0 under the assumptions of The steps are similar to those of f2), we haveStep 3′. By (x0 ∈ Ω and Br2(x0)⊆Ω with 2r < 1.Suppose that η ∈ C0∞(Br2(x0)) such that η = 1, x ∈ Br(x0); 0 ≤ η(x) ≤ 1 and |∇η | ≤(2/r). Denote s = 2δ; thenLet H(F), H(G), (f2),  Let  H(G), f,  (f¯1),φ has a critical point, which is just the solution of problem  − θ(x)) > 0, infx∈∂Ω(β(x) − θ(x)) > 0 and θ− > 1. In the following, we will show that F satisfies hypotheses H(F) and (f1) − (f3), and G satisfies hypotheses H(G) and (g1)-(g2).Let t ↦ |t − δ|θ(x) and t ↦ |t − δ|α(x) are convex functions; thus, F is also convex. Since t ↦ |t − δ|θ(x), t → |t − δ|α(x) are locally Lipschitz, so t ↦ F is locally Lipschitz. Thus, t ↦ F is regular. ThenNote that w ∈ ∂F, we haveHence, for any Thus, we haveH(F) and (f1) − (f3) hold. In a similar fashion, we have that conditions H(G) and (g1)-(g2) hold.Therefore, conditions


FWC-spaces, in short) which have no linear, convex, and topological structure. Using the maximal element theorem, we develop new existence theorems of solutions to variational relation problem, generalized equilibrium problem, equilibrium problem with lower and upper bounds, and minimax problem in FWC-spaces. The results represented in this paper unify and extend some known results in the literature.A maximal element theorem is proved in finite weakly convex spaces ( In 1983, by using fixed point theorems for set-valued mappings, Yannelis and Prabhakar  improvedH-spaces and obtained some minimax inequalities, variational inequalities, and quasivariational inequalities by using this maximal element theorem. Subsequently, Wu  and  = , where τ is a family of subsets of X. We can verify that  is not a topological space. For simplicity, we write X instead of . Let Y =  =  for each z = ∑i=0nλiei ∈ Δn. It is easy to see that  forms an FWC-space. Now we define a set-valued mapping T : X → 2Y byN = {u0, u1,…, un}∈〈D〉, we have T(φN(Δn)) = . Therefore, the composition T|φN∘φN : Δn → 2T(φN(Δn)) is an upper semicontinuous set-valued mapping with nonempty compact contractible values. By Lemma 1 of  and  = , where τ is a family of subsets of X. It is easy to check that  is not a topological space. For simplicity, we will write X instead of . Define a set-valued mappings H : Y → 2D such thatY. Therefore, H−1 is compactly open-valued, and hence, (ii) of P : X → 2Y such thatT : X → 2Y be defined byY. Let 1) of N = {u0, u1,…, un}∈〈D〉, we define a set-valued mapping φN : Δn → 2X by φN(z) = {0,4} for all z ∈ Δn. Then  forms an FWC-space. For each y ∈ , we havey ∈ Y, X∖P−1(y) is an FWC-subspace of  relative to H(y). Therefore, by x ∈ X, T(x)⊆P(x) and T, we can obtain T−1 : Y → 2X as follows:T−1(y)⊆P−1(y) for each y ∈ Y. Hence, for each x ∈ X, T(x)⊆P(x). By using the same method as in Let T = P in If X, D; φN) be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let H : Y → 2D, Q : D → 2X, and (i)u ∈ D, H−1(u) is compactly open;for each (ii)N = {u0, u1,…, un}∈〈D〉 and each {ui0, ui1,…, uik}⊆N,for each (iii)one of the following conditions holds:1) is an FWC-subspace of  relative to LN, Y, andfor each Let (Then there exists X = D and Q(x) = {x} in Taking X; φN) be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let H : Y → 2X and (i)x ∈ X, H−1(x) is compactly open;for each (ii)N = {x0, x1,…, xn}∈〈X〉 and each {xi0, xi1,…, xik}⊆N,for each (iii)one of the following conditions holds:1), Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and FWC-spaces.R be a relation linking y ∈ Y and x ∈ X. Find x ∈ X.Let Z be a nonempty set, Q : X → 2Z a set-valued mapping, and R a relation linking y ∈ Y and z ∈ Z. Find x ∈ X and each z ∈ Q(x).Let Let  be an FWC-space, Y a Hausdorff topological space, and K a nonempty compact subset of Y. Let R be a relation linking elements x ∈ X, y ∈ Y such that(i)x ∈ X, the set {y ∈ Y : R  does  not  hold} is compactly open;for each (ii)N = {x0, x1,…, xn}∈〈X〉, each {xi0, xi1,…, xik}⊆N and each y ∈ T(φN(Δk)), there exists for each (iii)one of the following conditions holds:1), Y, andfor each Let  is compactly open. By (ii), for each N = {x0, x1,…, xn}∈〈X〉 and each {xi0, xi1,…, xik}⊆N, we have(a)N0 ∈ 〈X〉 such that N = {x0, x1,…, xn}∈〈X〉, Y;there exists (b)N ∈ 〈X〉, there exists a subset LN of X containing N such that LN is an FWC-subspace of , Y, andfor each Therefore, by x ∈ X.Define a set-valued mapping X = Y and T(x) = {x} for every x ∈ X in By taking X; φN) be an FWC-space and K a nonempty compact subset of X, where X is a Hausdorff topological space. Let I is the identity mapping on X. Let R be a relation linking elements x ∈ X, y ∈ X such that(i)x ∈ X, the set {y ∈ X : R  does  not  hold} is compactly open;for each (ii)N = {x0, x1,…, xn}∈〈X〉, each {xi0, xi1,…, xik}⊆N and each y ∈ φN(Δk), there exists for each (iii)one of the following conditions holds:1)  does  not  hold} and φN : Δn → 2X is a compact set-valued mapping for each N = {u0, u1,…, un}∈〈D〉;there exists 2) andfor each Let  be an FWC-space, where X is a Hausdorff compact topological space. Let I is the identity mapping on X. Let R be a relation linking elements x ∈ X, y ∈ X such that the following conditions hold:x ∈ X, the set {y ∈ X : R  does  not  hold} is compactly open;for each N = {x0, x1,…, xn}∈〈X〉, each {xi0, xi1,…, xik}⊆N and each y ∈ φN(Δk), there exists for each Let (x ∈ X.Then there exists K = X. Then (iii1) of Let x ∈ X, the set {y ∈ X : R  holds} is closed; (2) (ii) of Theorem 2.1 of Pu and Yang [x0, x1,…, xn}∈〈X〉, there exists a continuous mapping φN : Δn → X such that, for each λ = {λ0, λ1,…, λn} ∈ Δn, there exists i ∈ J(λ) such that R(φN(λ), xi) holds, where J(λ) = {i ∈ {0,1,…, n} : λi > 0}. By (ii) of Theorem 2.1 of Pu and Yang [X, φN) in Theorem 2.1 of Pu and Yang [FWC-space; (3) in X needs not to have the fixed point property, but X in Theorem 2.1 of Pu and Yang [I on X in Theorem 2.1 of Pu and Yang [N = {x0, x1,…, xn}∈〈X〉 and for every continuous mapping ψ : φN(Δn) → Δn, the composition ψ∘φN : Δn → Δn is continuous, where φN coincides with the one in (ii) of Theorem 2.1 of Pu and Yang [z0 ∈ Δn such that z0 = ψ∘φN(z0), which implies that It is interesting to compare and Yang  can be sand Yang , we knowand Yang  forms anand Yang  needs toand Yang , we mustand Yang . Then byX; φN) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let Q : X → 2Z be set-valued mappings and R a relation linking elements y ∈ Y, z ∈ Z. Assume that(i)x ∈ X, the set {y ∈ Y : P(y)⋂Q(x) ≠ ∅} is compactly open, where P : Y → 2Z is defined by P(y) = {z ∈ Z : R  does  not  hold} for each y ∈ Y;for each (ii)N = {x0, x1,…, xn}∈〈X〉, each {xi0, xi1,…, xik}⊆N and each y ∈ T(φN(Δk)), there exists R holds for each for each (iii)one of the following conditions holds:1), Y, andfor each Let (x ∈ X and each z ∈ Q(x).Then there exists Y and X be defined by P(y)⋂Q(x) = ∅. Then by (i), for each x ∈ X, the set N = {x0, x1,…, xn}∈〈X〉, each {xi0, xi1,…, xik}⊆N and each y ∈ T(φN(Δk)), there exists (a)N0 ∈ 〈X〉 such that N = {x0, x1,…, xn}∈〈X〉, Y;there exists (b)N ∈ 〈X〉, there exists a subset LN of X containing N such that LN is an FWC-subspace of  andfor each where Y. Therefore, by x ∈ X; that is, x ∈ X and each z ∈ Q(x).Let the relation FWC-spaces and convex spaces, respectively. It follows from the previous analysis that FWC-spaces include convex spaces as special cases; (b) The class of better admissible mappings in ℬ, respectively. By ℬ is contained in Z in T in T in 1) of N = {x0, x1,…, xn}∈〈X〉, each x ∈ co{x0, x1,…, xn} and each y ∈ T(x), there exists Rholds for all z ∈ Q(xj). (ii) of N = {x0, x1,…, xn}∈〈X〉, we can define a continuous mapping φN : Δn → co(N)⊆X byX; φN) forms an FWC-space. On the basis of this fact, we can see that (ii) of Theorem 3.1 of Balaj and Lin [(1) and Lin  can be s and Lin  implies X = Z and Q(x) = {x} for each x ∈ X in (2) P : Y → 2Z be defined by P(y) = {z ∈ Z : R  does  not  hold} for each y ∈ Y. (i)′z ∈ Q(X), the set {y ∈ Y : R  holds} is compactly closed; for each (ii)′Z is a topological space, the set-valued mapping P is lower semicontinuous, and Q has open values.Let z∈Q(x){y ∈ Y : R  holds} is compactly closed for each x ∈ X. Thus, {y ∈ Y : P(y)⋂Q(x) ≠ ∅} = Y∖⋃z∈Q(x){R  does  not  hold} is compactly open. If (ii)′ holds, then by the definition of a lower semicontinuous set-valued mapping, for each x ∈ X, the set {y ∈ Y : P(y)⋂Q(x) ≠ ∅} = Y∖{y ∈ Y : P(y)⋂Q(x) = ∅} is open and thus, compactly open.Suppose that (i)′ is satisfied. Then ⋂In recent years, many authors  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y × X → 2Z, L : Y × D → 2Z, F, W : Y → 2Z, Q : D → 2X, and (i)x ∈ X and each y ∈ T(x), J⊆F(y);for each (ii)u ∈ D, the set {y ∈ Y : L⊆W(y)} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : J⊈F(y)} is an FWC-subspace of  relative to the set {u ∈ D : L⊈W(y)};for each (iv)one of the following conditions holds:1)⊈W(y)} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let (u ∈ D.Then there exists P : X → 2Y and H : Y → 2D byT(x)⊆P(x) for each x ∈ X. By (ii), for each u ∈ D, H−1(u) is compactly open. Now, we show that (iii) of N = {u0, u1,…, un}∈〈D〉 and {ui0, ui1,…, uik}⊆N such thaty* ∈ P(φN(Δk)) such that y* ∈ H−1(uij) for each j ∈ {0,1,…, k}; that is, uij ∈ {u ∈ D : L⊈  W(y*)}. By (iii), we havey* ∈ P(φN(Δk)), it follows that there exists 1) of 1) and the definition of H, we know that there exists N0 ∈ 〈D〉 such that Y∖K⊆⋃d∈N0H−1(d) and for each N = {u0, u1,…, un}∈〈D〉, Y. Therefore, (iv1) of 2) of 2) and the definition of H again, we know that for each N ∈ 〈D〉, there exists a subset LN of D containing N such that Q(LN) is an FWC-subspace of  relative to LN andY. Therefore, (iv2) of u ∈ D. This completes the proof.Define ies thatJ⊈FFWC-spaces and FC-spaces, respectively. By the previous analysis, we know that FWC-spaces include FC-spaces as special cases; (b) The class of better admissible mappings in ℬ, respectively. By ℬ is contained in 2) of By using the same argument as in X, D; φN) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y × X → 2Z, L : Y × D → 2Z, F, W : Y → 2Z, Q : D → 2X, and (i)x ∈ X and each y ∈ T(x), J⊈F(y);for each (ii)u ∈ D, the set {y ∈ Y : L⊈W(y)} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : J⊆F(y)} is an FWC-subspace of  relative to the set {u ∈ D : L⊆W(y)};for each (iv)one of the following conditions holds:1)⊆W(y)} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  The class of better admissible mappings in ℬ, respectively. It follows from ℬ is contained in 2) of X, D; φN) be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y × X → 2Z, L : Y × D → 2Z, F, W : Y → 2Z, Q : D → 2X, and (i)x ∈ X and each y ∈ T(x), J⋂F(y) ≠ ∅;for each (ii)u ∈ D, the set {y ∈ Y : L⋂W(y) ≠ ∅} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : J⋂F(y) = ∅} is an FWC-subspace of  relative to the set {u ∈ D : L⋂W(y) = ∅};for each (iv)one of the following conditions holds:1)⋂W(y) = ∅} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let J : Y × X → 2Z, L : Y × D → 2Z, F, W : Y → 2Z, Q : D → 2X, and (i)x ∈ X and each y ∈ T(x), J⋂F(y) = ∅;for each (ii)u ∈ D, the set {y ∈ Y : L⋂W(y) = ∅} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : J⋂F(y) ≠ ∅} is an FWC-subspace of  relative to the set {u ∈ D : L⋂W(y) ≠ ∅};for each (iv)one of the following conditions holds:1)⋂W(y) ≠ ∅} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space, Y a Hausdorff topological space, K a nonempty compact subset of Y, and Z a nonempty set. Let F, W : Y → 2Z, Q : D → 2X, and E be a nonempty set and G : Y → 2E a set-valued mapping. Let ζ : E × Y × X → Z and ξ : E × Y × D → Z be two single-valued mappings. Assume that(i)x ∈ X and each y ∈ T(x), ζ(G(y), y, x)⊈F(y);for each (ii)u ∈ D, the set {y ∈ Y : ξ(G(y), y, u)⊈W(y)} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : ζ(G(y), y, x)⊆F(y)} is an FWC-subspace of  relative to the set {u ∈ D : ξ(G(y), y, u)⊆W(y)};for each (iv)one of the following conditions holds:1)(ivY∖K⊆⋃u∈N0{y ∈ Y : ξ(G(y), y, u)⊆W(y)} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  the class ℬ in Theorem 4.4 of Fang and Huang [2) of FWC-spaces. We emphasis that X, D, Z, and E in (1)nd Huang  as a spend Huang . In factnd Huang , we can nd Huang  imply (iZ and E in (ii)′W is open in Y × Z;the graph of (ii)′′G is upper semicontinuous on each compact subset of Y with nonempty compact values and for each u ∈ D, ξ is continuous on each compact subset of E × Y.(2) Let H−1 : D → 2Y is a set-valued mapping with compactly open values. Thus, these solution sets are compactly closed subsets of the compact set The solution sets of generalized equilibrium problems considered in Theorems C be a nonempty closed subset of a locally convex semireflexive topological vector space X, and let F be a real-valued function on C × C. In 1999, Isac et al. [y ∈ C, where c1, c2 are two real numbers with c1 ≤ c2. Later, Li [G-convex spaces.Let c et al.  first raater, Li  introducater, Li  obtainedFWC-spaces.In this section, we apply X, D; φN) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ and ν be real-valued functions on Y × X and Y × D, respectively. Let g and h be real-valued functions on Y such that g(y) ≤ h(y) for each y ∈ Y. Assume that(i)x ∈ X and each y ∈ T(x), g(y) ≤ μ ≤ h(y);for each (ii)u ∈ D, the set {y ∈ Y : g(y) ≤ ν ≤ h(y)} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : μ > h(y)  or  μ < g(y)} is an FWC-subspace of  relative to the set {u ∈ D : ν > h(y)  or  ν < g(y)};for each (iv)one of the following conditions holds:1) > h(y)  or  ν < g(y)} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  The condition that there are four functions in g(y) = c1 and h(y) = c2 for all y ∈ Y, where c1 and c2 are real numbers such that c1 ≤ c2. In this case, Let X, D; φN) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ and ν be real-valued functions on Y × X and Y × D, respectively. Let c1 and c2 be two real numbers such that c1 ≤ c2. Assume that(i)x ∈ X and each y ∈ T(x), c1 ≤ μ ≤ c2;for each (ii)u ∈ D, the set {y ∈ Y : c1 ≤ ν ≤ c2} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : μ > c2  or  μ < c1} is an FWC-subspace of  relative to the set {u ∈ D : ν > c2  or  ν < c1};for each (iv)one of the following conditions holds:1) > c2  or  ν < c1} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ and ν be real-valued functions on Y × X and Y × D, respectively. Let c be a real number. Assume that(i)x ∈ X and each y ∈ T(x), μ = c;for each (ii)u ∈ D, the set {y ∈ Y : ν = c} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : μ > c  or  μ < c} is an FWC-subspace of  relative to the set {u ∈ D : ν > c  or  ν < c};for each (iv)one of the following conditions holds:1) > c  or  ν < c} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let  in Corollary 3.1 of Li [It is interesting to compare .1 of Li  as speci.1 of Li ; (3)    be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ1 and ν1 be real-valued functions on Y × X and Y × D, respectively. Let c be a real number. Assume that(i)x ∈ X and each y ∈ T(x), μ1 ≤ c;for each (ii)u ∈ D, the set {y ∈ Y : ν1 ≤ c} is compactly closed;for each (iii)y ∈ Y, the set {x ∈ X : μ1 > c} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > c};for each (iv)one of the following conditions holds:1) > c} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let ′u ∈ D, ν1 is lower semicontmuous on each nonempty compact subset of Y.For every (iii)′N = {u0, u1,…, un}∈〈D〉, every {ui0, ui1,…, uik}⊆N, and every y ∈ Y, we have μ1 ≥ minj≤k0≤ν1 for all x ∈ φN(Δk).For every (ii)-(iii) of y ∈ Y, N = {u0, u1,…, un}∈〈D〉, and {ui0, ui1,…, uik}⊆N∩{u ∈ D : ν1 > c} such that φN(Δk)⊈{x ∈ X : μ1 > c}. Hence, there exists x ∈ φN(Δk) such that μ1 ≤ c. Since {ui0, ui1,…, uik}⊆N∩{u ∈ D : ν1 > c}, we have ν1 > c for each j ∈ {0,1,…, k}. By (iii)′, we obtain the following contradiction:It is clear that (ii)′ implies (ii) of P : X → 2Y and H : Y → 2D byu ∈ D.c ∈ ℝ be given. Let us define two real-valued functions μ1 : Y × X → ℝ and ν1 : Y × D → ℝ byμ1 and ν1 satisfy all conditions of u ∈ D; that is, u ∈ D, which implies that Conversely, let X, D; φN), Y be as in μ1 and ν1 be real-valued functions on Y × X and Y × D, respectively. Let c be a real number. Assume thatx ∈ X and each y ∈ T(x), μ1 ≤ c;for each u ∈ D, the set {y ∈ Y : ν1 ≤ c} is compactly closed;for each y ∈ Y, the set {x ∈ X : μ1 > c} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > c}.for each Let (u ∈ D.Then there exists Q : D → 2X by Q(u) = X for each u ∈ D. For each N ∈ 〈D〉, let LN = D. Let 2) of Define a set-valued mapping FWC-spaces which contain G-convex spaces adopted in Theorem 3.3 of Tan [G-co(A) is compact in Theorem 3.3 of Tan [FWC-spaces without any linear, convex, and topological structure. The comparison details between 3 of Tan ; (2) The3 of Tan ; (3) the3 of Tan  is dropp3 of Tan . In fact3 of Tan  and Rema3 of Tan . Corolla3 of Tan  from topX, D; φN) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ1 and ν1 be real-valued functions on Y × X and Y × D, respectively. Let c be a real number. Assume that(i)u ∈ D, the set {y ∈ Y : ν1 ≤ c} is compactly closed;for each (ii)y ∈ Y, the set {x ∈ X : μ1 > c} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > c};for each (iii)one of the following conditions holds:1) > c} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y;2) is an FWC-subspace of  relative to LN, Y, andfor each Let , μ1 ≤ c. Hence, by u ∈ D. This completes the proof.If (a) is false, then it follows that for each X, D; φN) be an FWC-space and K a nonempty compact subset of a Hausdorff topological space Y. Let Q : D → 2X and μ1 and ν1 be real-valued functions on Y × X and Y × D, respectively. Assume that supx∈X,y∈T(x)μ1<+∞ and the following conditions hold:(i)u ∈ D, the set {y ∈ Y : ν1 ≤ supx∈X,y∈T(x)μ1} is compactly closed;for each (ii)y ∈ Y, the set {x ∈ X : μ1 > supx∈X,y∈T(x)μ1} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > supx∈X,y∈T(x)μ1};for each (iii)either1) > supx∈X,y∈T(x)μ1} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y or2) is an FWC-subspace of  relative to LN, Y, andfor each Let μ1. By the definition of c, (a) of u ∈ D. In particular, we have Let μ1′ = −μ1 and ν1′ = −ν1 and adjusting the corresponding conditions of u ∈ D. In particular, we have FWC-spaces.By setting T in If X, D; φN), K, Y, and Q be as in μ1 and ν1 be real-valued functions on Y × X and Y × D, respectively. Assume that supx∈Xμ1(T(x), x)<+∞ and the following conditions hold:(i)u ∈ D, the set {y ∈ Y : ν1 ≤ supx∈Xμ1(T(x), x)} is compactly closed;for each (ii)y ∈ Y, the set {x ∈ X : μ1 > supx∈Xμ1(T(x), x)} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > supx∈Xμ1(T(x), x)};for each (iii)either1) > supx∈Xμ1(T(x), x)} for some N0 ∈ 〈D〉 and for each N = {u0, u1,…, un}∈〈D〉, Y or2) is an FWC-subspace of  relative to LN, Y, andfor each Let  = {x} for all x ∈ X, we can obtain the following result from By taking X, D; φN) be an FWC-space, where X is a Hausdorff topological space. Let K be a nonempty compact subset of X. Let Q : D → 2X be a set-valued mapping and let IX is the identity mapping on X. Let μ1 and ν1 be real-valued functions on X × X and X × D, respectively. Assume that supx∈Xμ1<+∞ and the following conditions hold:(i)u ∈ D, the set {y ∈ X : ν1 ≤ supx∈Xμ1} is compactly closed;for each (ii)y ∈ X, the set {x ∈ X : μ1 > supx∈Xμ1} is an FWC-subspace of  relative to the set {u ∈ D : ν1 > supx∈Xμ1};for each (iii)either1) > supx∈Xμ1} for some N0 ∈ 〈D〉 and φN : Δn → 2X is a compact set-valued mapping for each N = {u0, u1,…, un}∈〈D〉 or2) relative to LN andfor each Let  ≤ supx∈Xμ1.Then there exists FWC-spaces which include L-convex spaces adopted in Corollary 5 of Jin and Cheng [H(A) in Corollary 5 of Jin and Cheng [X implies (i) of nd Cheng  as specind Cheng  is compand Cheng . In factX, D; φN) be an FWC-space, where X is a Hausdorff topological space. Let K be a nonempty compact subset of X. Let Q : D → 2X be a set-valued mapping and let IX is the identity mapping on X. Let μ1 and ν1 be real-valued functions on X × X and X × D, respectively. Assume that infx∈Xμ1>−∞ and the following conditions hold:(i)u ∈ D, the set {y ∈ X : ν1 ≥ infx∈Xμ1} is compactly closed;for each (ii)y ∈ X, the set {x ∈ X : μ1 < infx∈Xμ1} is an FWC-subspace of  relative to the set {u ∈ D : ν1 < infx∈Xμ1};for each (iii)either1) < infx∈Xμ1} for some N0 ∈ 〈D〉 and φN : Δn → 2X is a compact set-valued mapping for each N = {u0, u1,…, un}∈〈D〉 or2) relative to LN andfor each Let  ≥ infx∈Xμ1.Then there exists μ1′ = −μ1 and ν1′ = −ν1, we can see that the conclusion of By setting Y; φN11) and  be two FWC-spaces, where X and Y are two Hausdorff topological spaces. Let Y × X; φN11 × φN22) is an FWC-space defined as in μ1 be a real-valued function on Y × X. Assume thatz, w) ∈ Y × X, each  ∈ T, and each α ∈ ℝ, μ1 ≤ α or μ1 ≥ α;for each  ∈ Y × X and each α ∈ ℝ, the sets {x ∈ X : μ1 ≤ α} and {y ∈ Y : μ1 ≥ α} are compactly closed;for each  > α} is an FWC-subspace of ;for each y ∈ Y and each α ∈ ℝ, the set {w ∈ X : μ1 < α} is an FWC-subspace of .for each Let  = supy∈Yinfx∈Xμ1.Then infα ∈ ℝ such thaty, x) ∈ Y × X, there exists υ : (Y × X)×(Y × X) → ℝ byυ, for each  ∈ Y × X, each  ∈ T, we have υ, ) ≤ 0. For each  ∈ Y × X, we havez, w) ∈ Y × X, the set { ∈ Y × X : υ, ) ≤ 0} is compactly closed.It is clear that the following inequality4 of Tan , we defiy, x) ∈ Y × X, the set { ∈ Y × X : υ, ) > 0} is an FWC-space of . In fact, for each  ∈ Y × X, we have the following:N = N1 × N2 = {, ,…, }∈〈Y × X〉 and each {, ,…, }⊆{ ∈ Y × X : υ, ) > 0}∩N, we haveφN(Δk) = φN11(Δk) × φN22(Δk), it follows from (φN(Δk)⊆{ ∈ Y × X : υ, ) > 0}, which implies that for each  ∈ Y × X, the set { ∈ Y × X : υ, ) > 0} is an FWC-subspace of . Thus, by X = D and μ1 = ν1, there exists z, w) ∈ Y × X. Hence, for each  ∈ Y × X, either x∈Xsupy∈Yμ1 = supy∈Yinfx∈Xμ1. This completes the proof.Now, we show that for each ( we haveφN11(Δk)⊆tradicts . TherefoFWC-spaces which contain G-convex spaces adopted in Theorem 4.4 of Tan [G-co(A) and each G-co(B) in Theorem 4.4 of Tan [4 of Tan ; (b) the4 of Tan  are comp4 of Tan ; (d) (ii4 of Tan .


We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated. Fuzzy commutative algebra, presented a fuzzy ideal theory of commutative rings, and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field. The notion of a fuzzy subset of a set was introduced by Zadeh . His semX) and obtained a semigroup structure. Fayoumi  is said to be  abelian fuzzy if μ(x∗y) = μ(y∗x) for all x, y ∈ X. Let  be a left-zero-semigroup; that is, x∗y = x for all x, y ∈ X. Let μ : X →  be an abelian fuzzy subset of X. Then, μ(x) = μ(x∗y) = μ(y∗x) = μ(y) for all x, y ∈ X. It follows that μ is a constant map. Let  ∈ ZBin(X), then every abelian fuzzy subset on  is a constant function. If . Then, μ(x∗y) = μ(y∗x) for all x, y ∈ X. Let  ∈ ZBin(X). By x ≠ y in X, then  is either a left-zero-semigroup or a right-zero-semigroup. It follows that either x∗y = x, y∗x = y or x∗y = y, y∗x = x. In either cases, we obtain μ(x) = μ(y) for all x, y ∈ X, proving that μ is a constant function. Assume that X, ∗) ∈ Bin(X), we denote the set of all abelian fuzzy subgroupoids on  by A. Given a groupoid  ∈ Bin(X). Then,  is commutative if and only if A = X. Let  is not commutative; that is, there exist x, y ∈ X such that x∗y  ≠y∗x. If we let α : = x∗y and let μ : = χα be the characteristic function of α, then μ(x∗y) = 1, μ(y∗x) = 0, proving that μ is not an abelian fuzzy subset of . The converse is straightforward. Assume that  ∈ Bin(X), we define a fuzzy subset μα : X →  by μα(x): = α for all x ∈ X, where α ∈ . Denote by C: = {μα | α ∈ }. Then, C⊆A for all groupoids  whatsoever. Thus, the extreme of non-commutativity is the situation C = A. Given . If μ is one-one, then  is commutative. Let μ ∈ A, then μ(x  ∗  y) = μ(y∗ x) for all x, y ∈ X. Since μ is one-one, we have x∗y = y∗x for all x, y ∈ X. If X, ∗) ∈ Bin(X), we define a set a : = { ∈ X × X | x∗y = y∗x}. Note that ∈a implies ∈a as well. If we let ν : = χX,∗)aa, then ν is an abelian fuzzy subgroupoid on X × X. Given aμ : = { ∈ X × X | μ(x∗y) = μ(y∗x)}, then X, ∗)a⊆aμ, a = aμ, if μ is constant, then aμ = X × X. if Let It is straightforward. X, ∗) ∈ Bin(X), then there exists a fuzzy subset μ of X such that a = aμ. If  ∈ Bin(X) such that a ≠ aμ for any fuzzy subset μ of X. Then, there exists an element ∈aμ∖a. It follows that μ(x∗y) = μ(y∗x), but x∗y ≠ y∗x for some x, y ∈ X. If we let α∶ = x∗y and let μ∶ = χα be the characteristic function of α, then μ(x∗y) = 1, but μ(y∗x) = 0, which proves that ∉aμ. Assume that there exists  ∈ Bin(X) and let · be the usual product on the set of real numbers. If μ : → is a homomorphism, then μ ∈ A and aμ = X × X. Let → is a homomorphism, then μ(x∗y) = μ(x) · μ(y) = μ(y) · μ(x) = μ(y∗x) for all x, y ∈ X; that is, μ ∈ A.If x, y) ∈ X × X, then μ(x∗y) = μ(x) · μ(y) = μ(y∗x), proving that ∈aμ. If  ∈ Bin(X). Define a binary operation “⋆” on . Let (Bin(X). The relation ≤ is a quasi order on X, ∗)aμ⊆aμ for any fuzzy subset μ : X → , we have ≤ for all  ∈ Bin(X).Since ≤ and ≤, then aμ⊆aμ and aμ⊆aμ, and hence, aμ⊆aμ, for any fuzzy subset μ : X → . It follows that ≤. If (X), since aμ = aμ for any fuzzy subset μ : X →  does not imply  = . Note that the relation ≤ described in X, ∗) ∈ Bin(X), we defineGiven (X)〉∶ =   { |  ∈ Bin(X)}. If we define a relation ≤q on 〈Bin(X)〉 byX), ≤q〉) is a partially ordered set. For partially ordered sets, we refer to [Let 〈Bin] =  in 〈Bin(X)〉, then A = A. If  = . Then, aμ = aμ for any fuzzy subset μ : X → . It follows thatA = A. Assume that , then aμ∩aν⊆aλ. Let x, y)∈aμ∩aν, then μ(x∗y) = μ(y∗x) and ν(x∗y) = ν(y∗x) for all x, y ∈ X. It follows thatx, y)∈aλ.If  is a left-zero-semigroup. Define two fuzzy subsets μ, ν on X by μ(x)∶ = x, ν(x)∶ = 1 − x for all x ∈ X. Then, μ and ν are one-one mappings. Let ∈ be a real number such that 0 < ∈<1/2, and let u∶ = (1/2)−∈, v∶ = (1/2)+∈. Then, μ(v) − μ(u) = ν(u) − ν(v) ≠ 0. Let λ∶ = (1/2)(μ + ν). Then,u, v)∈aλ.Let X, ∗) is a left-zero-semigroup and μ is one-one, we have aμ = { | μ(x∗y) = μ(y∗x)} = { | μ(x) = μ(y)} = △(X) = aν. This shows that ∉△(X) = aμ∩aν. Since . If λ∶ = tμ + (1 − t)ν, where t ∈ , and if aμ = aν, then aμ = aν⊆aλ. Let μ, ν be fuzzy subsets of . If λi∶ = tiμ + (1 − ti)ν, , where t1, t2 ∈  with t1 ≠ t2, then aλ1∩aλ2⊆aμ∩aν. Let x, y)∈aλ1∩aλ2, theni = 1,2. It follows thatμ(x∗y) − μ(y∗x) = ν(x∗y) − ν(y∗x) = 0, so that ∈aμ∩aν. This proves the theorem. If , and let λi∶ = tiμ + (1 − ti)ν, , where t1, t2 ∈  with t1 ≠ t2. If aμ = aν, then aλ1∩aλ2 = aμ = aν. Let It follows immediately from μ, ν of X and groupoids  and  such thatWe give a pause to find some examples of fuzzy subsets X : = {a, b, c} with the tables:μ, ν on X by μ(a) = 1, μ(b) = 1/2, μ(c) = 1/3 and ν(a) = 1, ν(b) = ν(c) = 1/2. Then, it is easy to see that a = {, , } = aμ, aν = aν = aν = X × X. Hence, aμ⊆aμ, aν = aν, and aμ⊆aν, aν⊆aν. Let X, ∗) ∈ Bin(X), and let μ be fuzzy subsets of X. Define a set Zμ byμ-center of . In Zμ = ∅, Zν = X. Let  ∈ Bin(X). Then, x ∈ Zμ if and only if ({x} × X)∪(X × {x})⊆aμ. Let  ∈ Bin(X), and let μ : X →  be a fuzzy subset of X. Define a set OZμ byμ-center, we obtain the following. Let ∩Zμ ≠ ∅, then μ is a constant function. If x ∈ OZμ∩Zμ, then μ(x∗y) = μ(y∗x) for all y ∈ X. Since x ∈ OZμ, we have μ(x) = μ(y) for all y ∈ X, proving that μ is a constant function. If μ is a constant function, then OZμ = X = Zμ. If μ is a constant function, then μ(x∗y) = μ(y∗x) for all x, y ∈ X. It follows that Zμ = X.x ∉ OZμ. Then, there exists an y ∈ X such that μ(x∗y) = μ(y∗x), but μ(x) ≠ μ(y). Since μ is a constant function, we obtain μ(x) = μ(y) ≠ μ(x), a contradiction. Hence, OZμ = X.Assume that


We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature. Liouville property is an interesting topic in analysis since Liouville found that a bounded holomorphic function on the complex plane must be a constant function. It has been studied by many geometers. For example, Yau  studied In this paper, we prove the following Liouville property of holomorphic maps. This is, in some sense, a generalization of Yau's Liouville property for holomorphic maps. M be a complete Kähler manifold with nonnegative holomorphic bisectional curvature and N a complete simply connected Kähler manifold with sectional curvature ≤−c/(1+r)2, where c is some positive constant and r is the distance function of N to a fixed point o. Then, any holomorphic map from M to N must be constant. Let N which is guaranteed by the classical Hessian comparison. Then, by using Ni-Tam's [Our strategy to prove this result is by first showing that there are plenty of bounded pluri-subharmonic functions on the target Ni-Tam's  LiouvillIn order to show the existence of enough bounded plurisubharmonic functions on complete simply connected Kähler manifolds with negative quadratically decayed sectional curvature, we need the following technical lemma. c, there is a bounded function f ∈ C2) such that f(0) = 0 and f(r) > 0 for r > 0;f′(r) > 0 for r ≥ 0;f′′(r) + c/(1 + r)coth(cr/(1 + r))f′(r) ≥ 0 for r ≥ 0. For any positive constant r → ∞ and ccothc > 1. There are two positive numbers R and δ, such thatr ≥ R. Leth ∈ C). Moreover,r ≥ R. Letf ∈ C2) satisfying (1) and (2). Moreover, when r ≥ R,f is bounded.Note thatFinally,Mn, g) be a complete simply connected Kähler manifold with sectional curvature ≤−c/((1+r)2), where c is some positive constant and r is the distance function of M with respect to a fixed point o ∈ M. Then, for any point p ∈ M, there is a bounded continuous plurisubharmonic function ϕ such that ϕ > 0 on M∖{p} and ϕ(p) = 0. Let 2, where c1 is some positive constant. By Hessian comparison, we havee1, e2,…, en be a parallel unitary frame along geodesic rays emanating from p. Thenrp)α = 0 for any α > 1.Let f be a bounded function in C2) such that f(0) = 0 and f(r) > 0 for r > 0; f′(r) > 0 for r ≥ 0; r ≥ 0.  Let ϕ(x) = f(rp(x)). Then, by (ϕ is a bounded continuous pluri-subharmonic function on M with ϕ > 0 on M∖{p} and ϕ(p) = 0. By Then, by , we haveWe are now ready to prove f : M → N be a nonconstant holomorphic map. Then, there are two points p, q in M, such that f(p) ≠ f(q). By ϕ be a bounded continuous pluri-subharmonic function on N such that ϕ(f(p)) = 0, ϕ(f(q)) > 0. Then, ϕ(f) is a bounded continuous pluri-subharmonic function on M that is nonconstant. This contradicts Theorem  3.2 in [We proceed by contradiction. Let   3.2 in . ℂn constructed by Seshadri [r2logr) outside a compact subset. This means that the curvature decayed rate cannot be raised to be greater than 2 in The unitary invariant metric on Seshadri  has nega