Fixed points of k-nonexpansive mappings

Fixed points for multivalued mappings defined on unbounded sets in Banach spaces

In this paper, we give a necessary and sufficient condition under which a variational inequality defined on unbounded sets in a Banach space has a solution. Furthermore, we establish a necessary and sufficient condition under which the minimax equality on unbounded sets is true.

A "purely geometric" proof of the Lindenstrauss-Troyanski result ([2], [6]) on strongly exposed points of weakly compact sets in Banach spaces is given.I would like to thank the referee for his suggestions.We consider a Banach-space X, \\ ||.If G X and if e > 0, then B(,e) = {x G X; \\x -|| < e}.The closed convex hull of a set A G X, is denoted c(A).Let C be a convex subset of X, then Ce is the set of the extremal points of C. If F is a convex, weakly compact, F a convex, closed, bounded set of X and J a closed subinterval of [0,1 ], (K, B,J) denotes the closed, convex setWe have to introduce a few geometrical definitions, referring to [4].Suppose that C is a nonempty, bounded, closed and convex subset of X.Let"strongly exposed" if there exists / G X* (\\f\\ = 1) such that Ve > 0, 3a > 0 with G S(fa, C) G B(e).Let S be the set of all/ G X* such that 11/11 = 1 and/strongly exposes some point of C. Proposition I.If K is a nonempty, convex, weakly-compact subset of X and if B is convex, closed and bounded with K fl B = 0, then the set D = c(K U F) = (K,B, [0,1]) has the following property: Ve > 0, 3 G K such that c(D\B(,e)).First we observe that it is sufficient to prove the proposition for X separable.Indeed, suppose D does not have the required property.Then there exists e > 0 with Vx G K: x G c(D\B(x,e)) and hence, Vx G K, 3A(x) G D with the properties: x G c(A(x)), A(x) n B(x, e) = 0 and A(x) is countable.By induction we construct a sequence (Kn,Bn)n where Kn is countable in K and Bn is countable in F. Let (K0,B0) = ({x},0), where x is some element of K. Suppose we already found (Kn,Bn).Consider Vy G UxeKnA(x) an element ky in K and by in B, with y G c({ky,by)).Let

Iterative approximation of fixed points of nonexpansive mapping is a very active theme in many aspects of mathematical and engineering areas, in particular, in image recovery and signal processing. Because the errors usually occur in few places, it is necessary to show that whether the iterative algorithm is robust or not. In the present work, we prove that Krasnoselski-Mann's algorithm is robust for asymptotically nonexpansive mapping in a Banach space setting. Our results generalize the corresponding results existing in the literature.

In this paper, we prove some fixed point theorems for condensing operators in the setting of Banach spaces via measure of non-compactness, without using regularity. Our results improve and generalize many known results in the literature.

On a new approach of enriched operators

By a metric fixed point theorem we mean an existence result for a fixed point of a mapping f under conditions which depend on a metric d, and which are not invariant when we replace d by an equivalent metric. The best known metric fixed point theorem is the Banach theorem, also called the contractive mapping principle: “every contraction from a complete metric space into itself has a (unique) fixed point”. It is clear that a contractive mapping can lose this property if d is replaced by an equivalent metric. The Banach theorem is a basic tool in functional analysis, nonlinear analysis and differential equations. If we relax the contractive condition, requiring only that the mapping be nonexpansive, that is, d(f (x), f (y)) ≤ d(x, y), then trivial examples show that the Banach theorem need no longer hold. This failure may have been the reason why no significant result about the existence of fixed points for nonexpansive mappings was obtained for many years. However, in 1965, Browder [Br1 and Br2], Göhde [Go] and Kirk [Ki1] proved the following results: “let X be a Banach space, C a closed, convex and bounded subset of X and T : C → C a nonexpansive mapping. If X is either a Hilbert space, or a uniformly convex Banach space or a reflexive Banach space with normal structure, then T has a fixed point”. This result is, in some sense, surprising because it uses convexity hypotheses (more usual in topological fixed point theory) and geometric properties of the Banach spaces (commonly used in linear functional analysis, but rarely considered in nonlinear analysis prior to this time). The above results were the starting point for a new mathematical field: the application of the geometric theory of Banach spaces to fixed point theory. The texts [GK1], [AK], [KZ] and [Z] constitute excellent surveys of this theory.KeywordsBanach SpaceFixed Point TheoremNonexpansive MappingFixed Point TheoryConvex Banach SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper is concerned with a class of problem of best simultaneous approximations from $RS$-sets in Banach spaces $X$. It is shown that the best simultaneous approximations from an $RS$-set is strongly unique in the case when $X$ is a real Banach space, and strongly unique of order $\alpha \geq 2$ in the case when $X$ is a complex Banach space.

In this paper, we first introduce a Bregman prox-convexity notion as the generalization of the convexity for Bregman prox-convexity in Banach spaces. We verify the single-valudeness and Bregman firm nonexpansiveness of the Bregman proximity operator of the functions and provide examples of the notion. Using this notion, we show that the Bregman proximal point scheme remains convergent under appropriate conditions on the functions to be minimized in the setting of Banach spaces.

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R n . In this context, we establish a subdifferential characterization and show that qualified convexly C 1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C 1 in the context of certain uniformly convex spaces.

ABSTRACTFor a bounded closed convex subset C of a Banach space X with the origin as an interior point, we study the convexity issue of proximinal sets G in Banach spaces in the sense of generalized best approximation determined by the Minkowski functional generated by C. Some sufficient conditions such as the smoothness and compactly locally uniform convexity of C, or the weakly uniform convexity of are provided for a proximinal set G to be convex.

We study the notion of (I)-generating introduced by V. Fonf and J. Lindendstrauss and a related notion of (I)-envelope. As a consequence of our results we get an easy proof of the James characterization of weak* compactness in Banach spaces with weak angelic dual unit ball and an easy proof of the James characterization of reflexivity within a large class of spaces. We also show by an example that the general James theorem cannot be proved by this method.

We introduce a new iterative scheme for nding a common elementof the solutions set of a generalized mixed equilibrium problem and the xedpoints set of an in nitely countable family of nonexpansive mappings in a Banachspace setting. Strong convergence theorems of the proposed iterative scheme arealso established by the generalized projection method. Our results generalize thecorresponding results in the literature.

Weak convergence of random sets in Banach spaces

Introduction. S. Mazur [10] was the first to consider the following smoothness property in normed linear spaces, called the Mazur Intersection Property (MIP