Fixed Points and Asymptotic Behavior for Asymptotically Nonexpansive Type Semigroups in Banach Spaces

Let X be a Banach space with a weakly continuous duality map Jψ, C a non-empty weakly compact convex subset of X, and T = (T(t) : t ∈ S} an asymptotically nonexpansive type semigroup on C. In this paper, the inequality K ∩ F(T) ≠ (?) is characterized, where K is a subset of C and F(T) is the set of all common fixed points of T. Furthermore, it is shown that an almost-orbit     

Ergodic Retraction Theorem and Weak Convergence Theorem for Reversible Semigroups of Non-Lipschitzian Mappings

Let G be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banaeh space E with a Frechet differentiable norm, and T = {Tt ： t ∈ G} be a continuous representation of G as nearly asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F（T） of T in C is nonempty. It is shown that if G is right reversible, then for each almost-orbit u（.） of T, ∩s∈G ^-CO{u（t） ： t ≥ s} ∩ F（T） consists of at most one point. Furthermore, ∩s∈G ^-CO{Ttx ： t ≥ s} ∩ F（T） is nonempty for each x ∈ C if and only if there exists a nonlinear ergodic retraction P of C onto F（T） such that PTs - TsP = P for all s ∈ G and Px ∈^-CO{Ttx ： s ∈ G} for each x ∈ C. This result is applied to study the problem of weak convergence of the net {u（t） ： t ∈ G} to a common fixed point of T.     

空间中渐近非扩张型映射的渐近行为

Let X be a uniformly convex Banach space X such that its dual X^＊ has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt ： t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 ： t∈ G} and give its weak convergence theorem.     

Convergence analysis of iterative sequences for a pair of mappings in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space E. Let S ： C→ C be a quasi-nonexpansive mapping, let T ： C→C be an asymptotically demicontractive and uniformly Lipschitzian mapping, and let F ：= {x ∈C ： Sx = x and Tx = x}≠Ф Let {xn}n≥0 be the sequence generated irom an arbitrary x0∈Cby xn＋i=（1-cn）Sxn＋cnT^nxn, n≥0.We prove the necessary and sufficient conditions for the strong convergence of the iterative sequence {xn} to an element of F. These extend and improve the recent results of Moore and Nnoli.     

关于含渐近等距于$l^{\beta}(0<\beta<1)$子空间的赋$\beta$范空间

We get the characterizations of the family of all nonnegative, subadditive,β-absolutely homogeneous and continuous functionals defined on X, when the ;3-normed space X contains an asymptotically isometric copy of l^β. Moreover, it is proved that if a closed bounded β-convex subset K of a β-normed space contains an asymptotically isometric β-basis, then K contains a closed β-convex subset C which fails the fixed point property.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

Weak Convergence Theorems for Nonself Mappings

Let E be a real uniformly convex and smooth Banach space, and K be a nonempty closed convex subset of E with P as a sunny nonexpansive retraction. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with a sequence {k(i)n}  [1, ∞)(i = 1, 2), and F := F(T1)∩F(T2) = . An iterative sequence for approximation common fixed points of the two nonself asymptotically nonexpansive mappings is discussed. If E has also a Fr′echet differentiable norm or its dual E*has Kadec-Klee property, then weak convergence theorems are obtained.     

COMMON FIXED POINTS WITH APPLICATIONS TO BEST SIMULTANEOUS APPROXIMATIONS

For a subset K of a metric space (X , d) and x ∈ X ,P K (x) ={y ∈ K : d(x, y) = d(x, K) ≡ inf { d(x, k) : k ∈ K }}is called the set of best K-approximant to x. An element g。∈ K is said to be a best simultaneous approximation of the pair y 1 , y 2 ∈ X if max{d(y 1 , g。), d(y 2 , g。) } = inf g ∈ K max { d(y 1 , g), d(y 2 , g)}.In this paper, some results on the existence of common fixed points for Banach operator pairs in the framework of convex metric spaces have been proved. For self mappings T and S on K, results are proved on both T - and S- invariant points for a set of best simultaneous approximation. Some results on best K-approximant are also deduced. The results proved generalize and extend some results of I. Beg and M. Abbas [1] , S. Chandok and T.D. Narang [2] , T.D. Narang and S. Chandok [11] , S.A. Sahab, M.S. Khan and S. Sessa [14] , P. Vijayaraju [20] and P. Vijayaraju and M. Marudai [21] .     

依中间意义渐近非扩张族的几乎轨道的渐近行为

设C是具有Frechet可微范数的一致凸Banach空间E的非空子集,T={T(t):t∈S}是依中间意义渐近非扩张的一族C上的自映象,F是F(T)的子集,其中,F(T)表示族T={T(t):t∈S}的所有公共不动点之集。本文证明了,如果u:S→C是T={T(t):t∈S}的几乎轨道,并满足下列条件:(a)ω_w({u(t):t∈S}) F;(b)({u(t):t∈S}∪F) C。则(i)F=且||u(t)||=∞;或(ii)F≠且u(t)弱收敛到F的一个元。     

On Asymptotically Isometric Copies of l~β(0 < β < 1)

We get the characterizations of the family of all nonnegative,subadditive,β-absolutely homogeneous and continuous functionals defined on X,when the β-normed space X contains an asymptotically isometric copy of lβ.Moreover,it is proved that if a closed bounded β-convex subset K of a β-normed space contains an asymptotically isometric lβ-basis,then K contains a closed β-convex subset C which fails the fixed point property.