Fixed point theory and nonexpansive mappings

2015

In this paper, we present some fixed point theorems for fundamentally nonexpansive mappings in Banach spaces and give one common fixed point theorem for a commutative family of demiclosed fundamentally nonexpansive mappings on a nonempty weakly compact convex subset of a strictly convex Banach space with the Opial condition and a uniformly convex in every direction Banach space, respectively; moreover, we show that the common fixed points set of such a family of mappings is closed and convex.

2012

A Banach space X is said to satisfy property (D) if there exists α ∈ [0,1) such that for any nonempty weakly compact convex subset E of X, any sequence {xn }⊂ E which is regular relative to E, and any sequence {yn }⊂ A(E,{xn}) which is regular relative to E, we have r(E,{yn}) ≤ αr (E,{xn}). A this property is the mild modification of the (DL)-condition. Let X be a Banach space satisfying property (D) and let E be a weakly compact convex subset of X .I fT : E → E is a mapping satisfying condition (E) and (Cλ) for some λ ∈ (0,1). We study the existence of a fixed point for this mapping.

2016

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and convex, then its the fixed points set is nonempty, closed and convex.

Nonlinear Analysis-theory Methods & Applications, 1988

Applied Mathematics Letters, 2010

compact convex subsets of a Banach space X which is uniformly convex in every direction. Furthermore, if {T i } i∈I is any compatible family of strongly nonexpansive self-mappings on such a C and the graphs of T i , i ∈ I, have a nonempty intersection, then T i , i ∈ I, have a common fixed point in C.

Proceedings of the American Mathematical Society, 2005

Let K K be a compact metrizable space and let C ( K ) C(K) be the Banach space of all real continuous functions defined on K K with the maximum norm. It is known that C ( K ) C(K) fails to have the weak fixed point property for nonexpansive mappings (w-FPP) when K K contains a perfect set. However the space C ( ω n + 1 ) C(\omega ^{n}+1) , where n ∈ N n\in \mathbb {N} and ω \omega is the first infinite ordinal number, enjoys the w-FPP, and so C ( K ) C(K) also satisfies this property if K ( ω ) = ∅ K^{(\omega )}=\emptyset . It is unknown if C ( K ) C(K) has the w-FPP when K K is a scattered set such that K ( ω ) ≠ ∅ K^{(\omega )}\not =\emptyset . In this paper we prove that certain subspaces of C ( K ) C(K) , with K ( ω ) ≠ ∅ K^{(\omega )}\not = \emptyset , satisfy the w-FPP. To prove this result we introduce the notion of ω \omega -almost weak orthogonality and we prove that an ω \omega -almost weakly orthogonal closed subspace of C ( K ) C(K) enjoys the w-FPP. We show an example o...

1986

2015

In this paper, we established some weak and strong convergence theorems for common fixed points of three nonself asymptotically Banach spaces. Our results extended and improve the result announed by Wang[6] [Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006)550-557.] and WeiQiDeng, Lin Wang and Yi-Juan Chen[13] [Strong and Weak Convergence Theorems for common fixed points of two asymptotically nonexpansive mappings in Banach spaces, International Mathematical Forum, Vol. 7, 2012, no. 9, 407 – 417.] For a smooth banach space E, let us assume that K is a nonempty closed convex subset of with P as a sunny nonexpansive retraction. Let,T1, T2, T3 : K → E be three weakly inward nonself asymptotically nonexpansive mappings with respect to P with three sequences kn i ∁ [1,∞) satisfying (kn i ∞ n=1 −1) < ∞ ,(i=1,2,3) and F(T1)∩ F(T2)∩ F T3 = xεk, T1x = T2x = T3x = x respectively . For any given x1 ∈...

Abstract and Applied Analysis, 1998

LetXbe a Banach space andτa topology onX. We say thatXhas theτ-fixed point property (τ-FPP) if every nonexpansive mappingTdefined from a bounded convexτ-sequentially compact subsetCofXintoChas a fixed point. Whenτsatisfies certain regularity conditions, we show that normal structure assures theτ-FPP and Goebel-Karlovitz's Lemma still holds. We use this results to study two geometrical properties which imply theτ-FPP: theτ-GGLD andM(τ)properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of theτ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach spaceXsuch that theτ-FPP is shared by any isomorphic Banach spaceYsatisfying that the Banach-Mazur distance betweenXandYis less than some of these constants.

Fixed Point Theory and Applications, 2014

We introduce the concept of ψ-firmly nonexpansive mapping, which includes a firmly nonexpansive mapping as a special case in a uniformly convex Banach space. It is shown that every bounded closed convex subset of a reflexive Banach space has the fixed point property for ψ-firmly nonexpansive mappings, an important subclass of nonexpansive mappings. Furthermore, Picard iteration of this class of mappings weakly converges to a fixed point. MSC: 47H06; 47J05; 47J25; 47H10; 47H17