Let $(A, B)$ be a nonempty bounded closed convex proximal parallel pair in a nearly uniformly convex Banach space and $T: A\cup B \rightarrow A\cup B$ be a continuous and asymptotically relatively nonexpansive map. We prove that there exists $x \in A\cup B$ such that $\|x - Tx\| = \emph{dist}(A, B)$ whenever $T(A) \subseteq B$, $T(B) \subseteq A$. Also, we establish that if $T(A) \subseteq A$ and $T(B) \subseteq B$, then there exist $x \in A$ and $y\in B$ such that $Tx = x$, $Ty = y$ and $\|x - y\| = \emph{dist}(A, B)$. We prove the aforesaid results when the pair $(A, B)$ has the rectangle property and property $UC$. In case of $A = B$, we obtain, as a particular case of our results, the basic fixed point theorem for asymptotically nonexpansive maps by Goebel and Kirk.
Best proximity pair results for relatively nonexpansive mappings in geodesic spaces
给定度量空间中的两个非空子集$A$和$B$，映射$T : A \cup B \rightarrow A \cup B$是相对非扩张的，如果$d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$这样的映射的最佳邻近点是点$x \in A \cup B$，使得$d(x,Tx)=\text{dist}(A,B)$。本文推广了[A.A.西弗吉尼亚州埃尔德雷德Kirk，P. Veeramani，Proximal normal structure and relatively nonexpansive mappings，Studia Math.，171（2005），283-293]对于巴拿赫空间到更一般度量空间的相对非扩张映射。即在Busemann凸自反度量空间中，给出了循环和非循环相对非扩张映象最佳邻近点的存在性结果。在CAT（0）和一致凸测地空间的背景下，证明了一些特殊的结果.最后，我们表明，邻近的正常结构是一个充分的，但不是必要条件存在于$A \times B$的一对最佳邻近点。
In this work, a new concept of nonself total asymptotically nonexpansive mapping is introduced and an iterative process is considered for two nonself totally asymptotically nonexpansive mappings. Weak and strong convergence theorems for computing common fixed points of two nonself total asymptotically nonexpansive mappings are established in the framework of Banach spaces. Finally, we give an example which show that our theorems are applicable.
We prove the existence of a fixed point for mappings which satisfy some asymptotic nonexpansive conditions in Banach spaces which are either nearly uniformly convex or they satisfy that asymptotic centers of bounded sequences are compact. Nominally, we consider pointwise eventually nonexpansive mappings, pointwise asymptotically nonexpansive mappings and asymptotically type nonexpansive. We do not assume the existence of a continuous iterated, solving some long-standing open questions about existence of a fixed point for these mappings in absence of continuity.
The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of non…
In this paper, we use the concept of proximal quasi-normal structure (P. Q-N. S) to study the existence of best proximity points for cyclic mappings, cyclic contractions, relatively Kannan nonexpansive mappings, as well as for orbitally nonexpansive mappings. In this way, we generalize several recent results obtained by others.
We have derived that on certain Banach spaces having a graph structure $G$, the iterations for asymptotically $G$-nonexpansive map will converge weakly towards a fixed point. This result unifies and extends several theorems on fixed points proved by various authors for class of nonexpansive and asymptotically nonexpansive maps. As an application of this result, we derive that for maps satisfying the nonexpansive condition locally on special Banach spaces, the successive approximations converge weakly towards a fixed point.
Let LUC$(S)$ be the space of left uniformly continuous functions on a semitopological semigroup $S$. Suppose that $S$ is right reversible and $\operatorname{LUC}(S)$ has a left invariant mean. Let $(X,d)$ be a Fréchet space. Let $τ$ be a locally convex topology of $X$ weaker than the $d$-topology such that the metric $d$ is $τ$-lower semicontinuous. Let $K$ be a $d$--separable and $τ$--compact convex subset of $X$. We show that every jointly $τ$-continuous and super asymptotically $d$-nonexpansive action $S\times K\mapsto K$ of $S$ has a common fixed point. Similar results in the locally conve…
In this note, at first we prove that the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that a main result of the paper "Proximal normal structure and relatively nonexpansive mappings", Studia Math., (171~(2005) 283--293) immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of it…
By using the Ishikawa iterative algorithm, we approximate the fixed points and the best proximity points of a relatively non expansive mapping. Also, we use the von Neumann sequence to prove the convergence result in a Hilbert space setting. A comparison table is prepared using a numerical example which shows that the Ishikawa iterative algorithm is faster than some known iterative algorithms such as Picard and Mann iteration.
Let $S$ be a semitopological semigroup, and let $C$ be a nonempty closed convex subset of a reflexive Banach space. Under some amenability conditions on $S$, we provide existence results of fixed points for several Bregman nonexpansive type actions $S\times C\to C$, $(s,x)\mapsto T_s x$, of $S$ on $C$. The mappings $T_s$ we discuss include those being Bregman generalized hybrid, Bregman nonspreading, and Bregman left asymptotically nonexpansive.
Weak and strong convergence of an implicit iterative process with errors for a finite family of asymptotically quasi $I-$nonexpansive mappings in Banach space
International Islamic University Malaysia（马来西亚国际伊斯兰大学）
Let $S$ be a right reversible semitopological semigroup, and let $\operatorname{LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that $\operatorname{LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact convex subset of a Banach space. We show that there always exists a common fixed point for any jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$. Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.
For two nonempty, closed, bounded and convex subsets $A$ and $B$ of a uniformly convex Banach space $X$ consider a mapping $T:(A \times B) \cup (B \times A) \rightarrow A \cup B$ satisfying $T(A,B) \subset B$ and $T(B, A) \subset A$. In this paper the existence of a coupled best proximity point is established when $T$ is considered to be a p-cyclic contraction mapping and a p-cyclic nonexpansive mapping. The Ulam-Hyers stability of the best proximity point problem is also studied.
Maximally monotone operators and firmly nonexpansive mappings play key roles in modern optimization and nonlinear analysis. Five years ago, it was shown that if finitely many firmly nonexpansive operators are all asymptotically regular (i.e., the have or "almost have" fixed points), then the same is true for compositions and convex combinations. In this paper, we derive bounds on the magnitude of the minimal displacement vectors of compositions and of convex combinations in terms of the displacement vectors of the underlying operators. Our results completely generalize earlier works. Moreover,…
We show that in the framework of CAT(0) spaces, any convex combination of two mappings which are firmly nonexpansive -- or which satisfy the more general property $(P_2)$ -- is asymptotically regular, conditional on its fixed point set being nonempty, and, in addition, also $Δ$-convergent to such a fixed point. These results are established by the construction and study of a convex combination metric on the Cartesian square of a CAT(0) space. We also derive a uniform rate of asymptotic regularity in the sense of proof mining. All these results are then interpreted in the special case of the ma…
We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many o…
We show that there are no nontrivial surjective uniformly asymptotically regular mappings acting on a metric space and derive some consequences of this fact. In particular, we prove that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings on a bounded $τ$-compact subset of a Banach space has a common fixed point, and give a qualitative complement to the Markov-Kakutani theorem.
The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are analyzed by employing suitable conditions on the control sequences of parameters and the structural properties of the under lying space. We aim to establish strong and del-convergence results of the proposed iterative algorithm and compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family…
