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Nonexpansive mappings in metric and Banach spaces
Abstract
The fundamental fixed point theorem for nonexpansive mappings is reformulated in the setting of a metric space possessing
a convexity structure, and various constructive and nonconstructive proofs are compared. A new proof of the Browder-Göhde
demiclosedness principle for mappings of the formI–T, T nonexpansive, is given, and a related result of R. E. Bruck in discussed.
Si riformula, nel contesto di uno spazio metrico che possiede una struttura di convessità, il teorema fondamentale di punto
fisso per applicazioni non espansive e si confrontano fra loro varie dimostrazioni, di tipo costruttivo e non costruttivo,
di questo teorema. Viene inoltre data una nuova dimostrazione del principio di demichiusura di Browder-Göhde per applicazioni
della formaI–T (T non espansiva) e viene discusso un risultato di R. E. Bruck ad esso correlato.
... Next we discuss a constructive result discovered for Banach spaces by Kirk [13] and evolved for modular function spaces in [11]. The main ingredient in this constructive proof is a technical lemma due to Gillespie and Williams [4]. ...
... We are now ready to exhibit a modular analogue of Kirk's fixed point theorem [13]. Theorem 3.4 (see [10,Theorem 5.10] for a function modular equivalent). ...
We introduce a notion of modulated topological vector spaces, that generalises, among others, Banach and modular function spaces. As applications, we prove some results which extend Kirk’s and Browder’s fixed point theorems. The theory of modulated topological vector spaces provides a very minimalist framework, where powerful fixed point theorems are valid under a bare minimum of assumptions.
... Recently it has been shown [6,7,28,34,36,45] that many classical or nonclassical Banach spaces enjoy a uniform property with respect to a given topology. This enables the authors to prove a fixed point result via well known theorems [10,19,32]. In this work we show that there is a more general property that reduces to the main conclusion of these results. ...
... Fixed point theory for nonexpansive mappings has its origins in the 1965 existence theorems [10,19,32]. Although such mappings are natural extensions of the contraction mappings, it was clear from the outset that the study of nonexpansive mappings required techniques which go far beyond the purely metric approach. ...
In this paper we define a property that implies uniform Opial con- dition or uniform Kadec-Klee property in Banach spaces. This will unify many examples in the literature. Finally we will extend these properties into hyperbolic metric spaces.
... In this paper, we follows some ideas in [3], [4], [6], [7] to establish a generalization of Khamsis's fixed point theorem [7]. Let (X, τ ) be a topological space. ...
... In [3], Kirk proved the following lemma in metric spaces. The proof of the lemma in our setting is essentially the same as in [8]. ...
The main purpose of this paper is to define the concept of uniform normal structure and give some new fixed point theorems of nonexpansive maps in a general topological space (X,�) with a �-symmetric function.
... He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and much papers have appeared (see e. g., [19] [7] [5] [9] [12] [16] [23] [8] [20] [10] [28] [6] [11] [15] [21] [22] [27] [26]). It is worth mentioning that the results in CAT(0) spaces can be applied to any CAT(κ) space with κ ≤ 0 since any CAT(κ) space is a CAT(κ ) space for every κ ≥ κ (see [1], p. 165). ...
... Lemma 2.5 ([8] ...
Δ and strong convergence theorems are established for the Noor iterations of asymptotically nonexpansive mappings in CAT(0) spaces. Our results extend and improve the recent ones announced by S. Dhompongsa and B. Panyanak [Comput. Math. Appl. 56, No. 10, 2572–2579 (2008; Zbl 1165.65351)], B. Nanjaras and B. Panyanak [Demiclosedness principles for asymptotically nonexpansive mappings in CAT(0) spaces, submitted], B. Xu and M. Aslam Noor [J. Math. Anal. Appl. 267, No. 2, 444–453 (2002; Zbl 1011.47039)] and many others.
... There are several results on fixed point of nonexpansive mappings in the literature. One may refer to [14] and the references therein. ...
The aim of this paper is to introduce a new class of mappings called almost nonexpansive mappings in a b-metric space. Some characteristics of this class of mappings are discussed. Fixed point and common fixed point results for such mappings are obtained. An application to the Cauchy problem in a Banach space is also shown in this paper.
... The next result is the modular version of Kirk's celebrated fixed-point theorem [1]. The proof is constructive and was first used in the Banach-space setting by Kirk [9] and relaxes the compactness assumption in the above theorem. The main ingredient in Kirk's constructive proof is a technical lemma due to Gillespie and Williams [10]. ...
In a personal communication, Prof. Domínguez Benavides noted that a fixed-point theorem for modular nonexpansive mappings in Lp(·)(Ω) obtained under the assumptions p+<∞ and the property (R) satisfied by ρ will force p−>1. Therefore, the conclusion is well known. In this note, we establish said conclusion without the assumption p+<∞.
... where r a (X) and r (X) stand for the Chebyshev radius of X with respect to a and the Chebyshev radius of X in M respectively, and diam (X) is the diameter of X. Spaces with UNS have been deeply studied in the literature and uniformly convex Banach spaces are known to have UNS (see, for instance, [17] Initially, the previous theorem (see [21,Theorem I]) was stated with the additional requirement that the family A (M ) was compact or at least countably compact (condition (R) in [18]). It turns out that this compactness condition is superfluous as stated in the following theorem. ...
The purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences c where we study non-weakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set M=[-1,1]N has the fixed point property for d-nonexpansive mappings where d(·,·) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.
... In this paper, motivated by the above results, we prove the existence of stationary points for semicompact lower semicontinuous multifunctions in metric spaces. We also show that the semicompactness condition can be replaced by some other conditions; for instance, the condition (C λ ) which defined by Garcia-Falset et al. [8] and the convexity type condition which defined by Kirk [17]. We also provide some examples which illustrate the main results. ...
We guarantee the existence of stationary points for semicompact lower semicontinuous multifunctions in metric spaces. We also show that the semicompactness condition can be replaced by some other properties. Several examples illustrating the main results are also given.
... We conclude this section by a modular version of Kirk's fixed point theorem [20] which follows as a corollary from our former result Theorem 10 (see Corollary 11 below). ...
In this paper we introduce the notion of proximal ρ -normal structure of pair of ρ -admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.
... We conclude this section by a modular version of Kirk's fixed point theorem [17] which follows as a corollary of our Theorem 16 (see Corollary 18). ...
In this paper, we formulate best proximity pair theorems for noncyclic relatively ρ-nonexpansive mappings in modular spaces in the setting of proximal ρ-admissible sets. As a companion result, we establish a best proximity pair theorem for pointwise noncyclic contractions in modular spaces. To that end, we provide some examples throughout the paper to illustrate the validity of the obtained results. Keywords: Best proximity pair, Modular spaces, Relatively ρ-nonexpansive mappings, ρ-admissible sets, ρ-normal structure, Mathematics Subject Classification: 47H09, 41A65
... It nds many applications in nonlinear functional analysis. The existence of xed points for nonexpansive mappings in Banach and metric spaces have been investigated since the early 1960s, see, e.g., Belluce and Kirk [1, 2], Browder [3], Bruck [4], and Lim [7]. Recently, a new direction has been discovered dealing with the extension of the Banach contraction principle to metric spaces endowed with a partial order. ...
In this work we discuss the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings. In particular, we show that under suitable assumptions, this set is a monotone nonexpansive retract.
... In this context the fundamental fixed point result for nonexpansive mappings is the following (see [19], [14]). Using admissible sets it is possible to give metric space analogs of all the foregoing concepts. ...
... The original proof requires the use of Zorn's lemma which explains why the compactness of F is assumed. Other constructive proofs [10] are known which only require F to be countably compact. In this work, we will not discuss these ideas. ...
In this paper, we introduce and study one-local retract of metric spaces. In particular we prove that any commutative family of non- expansive mappings defined on a metric space with a compact and normal convexity structure has a common fixed point. This conclu- sion was known in Banach spaces with no similar result in metric spaces.
... The original proof is based on the existence of minimal elements in A(M ). Kirk [9] extended this result to metric spaces for which A(M ) is countably compact and normal. The proof is constructive and does not use Zorn's lemma. ...
In this work, we discuss the compactness of convexity structures in metric spaces. We also discuss a problem posed by Kirk on an extension of Caristi's calssical theorem.
... In this paper, we follow ideas in [6] [7] [3] [4] to establish a generalization of the well known extension of Kirk's fixed point theorem [4]. Let (X, τ ) be a topological space. ...
The main purpose of this paper is to dene the concept of p-normal structure and give some new xed point theorems of nonexpansive maps in general topological space (X; ) by introducing the notion of a -symmetric function p: X X!R+. An application to symmetrizable topological spaces has been made.
... If, furthermore, K is compact, then every chain in a partially ordered set f. D) has a supremum—the intersection of all its members. Since T is progressive on f, 3), Zermelo's theorem yields the following result (see also Kirk [6]) However, we are more interested in the following question. When does the family Fix T of all fixed points of T contain a singleton? ...
Using the Zermelo Principle, we establish a common fixed point theorem for two progressive mappings on a partially ordered set. This result yields the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings.
... The following lemma will be very helpful for the proof of Theorem 4. Its proof can be found in [6,8]. Lemma. ...
In this work we define the so-called modulus of noncompact con-vexity in modular spaces. We extend the results obtained in Banach spaces by Goebel and Sekowski while their methods can not be repro-duced as.
... The proof closely follows the proof of Proposition 1 in [11] and Theorem 21 of [12]. For the convenience of the reader we include the details. ...
A CAT(0) space is a geodesic space for which each geodesic triangle is at least as ‘thin’ as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently several years ago by Lim and Kuczumow is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular many Banach space results involving weak convergence have precise analogues in this setting. At the same time, many questions remain open.
... He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed and many of papers have appeared (see e. g., [5] [6] [7] [8] [9] [10] [11] [14] [17] [18] [20] [22] [23]). ...
Suppose that K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T:K→X be a nonexpansive nonself mapping with F(T):={x∈K:Tx=x}≠∅. Suppose that {xn} is generated iteratively by x1∈K, xn+1=P((1−αn)xn⊕αnTP[(1−βn)xn⊕βnTxn]), n≥1, where {αn} and {βn} are real sequences in [ε,1−ε] for some ε∈(0,1). Then {xn}Δ-converges to some point x∗ in F(T). This is an analog of a result in Banach spaces of Shahzad (2005) and extends a result of Dhompongsa and Panyanak (2008) to the case of nonself mappings.
In this work, we prove that metric spaces with uniform normal structure have a kind of intersection property, which is equivalent to reflexivity in Banach spaces.
We discuss Panayanak’s results on the existence of an endpoint of a multivalued nonexpansive mapping. We show that all of his results can be extended and some can be established in a wider class of mappings. Out of his three open questions, two of them are solved in affirmative.
In this work, we introduce a density property in ordered sets that is weaker than the order density. Then, we prove a strong version of a result proved by T. Büber and W. A. Kirk [in: World congress of nonlinear analysts ’92. Proceedings of the first world congress, Tampa, FL, USA, August 1992. Berlin: de Gruyter. 2115–2125 (1996; Zbl 0844.47031)], which is a special case of the Brouwer Reduction Theorem, in metric spaces relating completeness and density of ordered sets.
Mappings which are defined on metric spaces and which do not increase distances between pairs of points and their images are called nonexpansive. Thus an abstract metric space is all that is needed to define the concept. At the same time, the more interesting results seem to require some notion of topology; more specifically a topology which assures that closed metric balls are compact. This is not a serious limitation, however, because many spaces which arise naturally in functional analysis possess such topologies; most notably the weak and weak* topologies in Banach spaces.
A fixed point theorem about a “nonexpansive generalized Kannan map” is given in an abstract form, using the notion of convexity structure. This contains and extends various known theorems.
In this paper, we will consider the problem of existence of common fixed points for two
mappings T and S on a CAT(0) space X. We will suppose that T and S belong to the class of
mappings satisfying a generalization of Suzuki’s condition (C). Our result improves a number of
very recent results of A. Abkar, M. Eslamian in [2] and, as well as, those of B. Nanjaras et al. in [3].
In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a CAT(0) space. Also, we obtain the strong and Δ-convergence theorems of new multi-step and S-iteration processes in a CAT(0) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.
MSC:
47H09, 47H10, 54E40, 58C30.
In this paper, we study the strong convergence of Moudafi’s viscosity approximation methods for approximating a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings in
spaces. We prove that the proposed iterative scheme converges strongly to a common fixed point of a one-parameter continuous semigroup of nonexpansive mappings which is also a unique solution of the variational inequality. The results presented in this paper extend and enrich the existing literature.
An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to
spaces.
AMS Subject Classification:
54E40, 54H25, 47H10, 37C25.
It is our purpose in this paper first to introduce the class of total asymptotically nonexpansive nonself mappings and to prove the demiclosed principle for such mappings in
spaces. Then, a new mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of two total asymptotically nonexpansive mappings and two total asymptotically nonexpansive nonself mappings is constructed. Under suitable conditions, some strong convergence theorems and Δ-convergence theorems are proved in a
space. Our results improve and extend the corresponding results of Agarwal, O’Regan and Sahu (J. Nonlinear Convex Anal. 8(1):61-79, 2007), Guo et al. (Fixed Point Theory Appl. 2012:224, 2012. doi:10.1186/1687-1812-2012-224), Sahin et al. (Fixed Point Theory Appl. 2013:12, 2013. doi:10.1186/1687-1812-2013-12), Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011), Khan et al. (Nonlinear Anal. 74:783-791, 2011), Xu (Nonlinear Anal., Theory Methods Appl. 16(12):1139-1146, 1991), Chidume et al. (J. Math. Anal. Appl. 280:364-374, 2003) and others.
MSC:
47J05, 47H09, 49J25.
Suppose K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T : K → X be a nonself mapping, satisfying Condition (E) with F(T): = {x ∈ K : Tx = x} ≠ ∅. Suppose {x
n
} is generated iteratively by x
1 ∈ K, x
n
+1 = P ((1 - α
n
)x
n
⊕ α
n
TP [(1 - β
n
)x
n
⊕ β
n
Tx
n
]),n ≥ 1, where {α
n
} and {β
n
} are real sequences in [ε, 1 - ε] for some ε ∈ (0, 1). Then, {x
n
} Δ-converges to some point x
⋆ in F(T). This extends a result of Laowang and Panyanak [Fixed Point Theory Appl. 367274, 11 (2010)] for nonself mappings satisfying Condition (E).
In this paper, we introduce generalized hybrid mapping on CAT(0) spaces. The class of generalized hybrid mappings contains the class of nonexpansive mappings, nonspreading mappings, and hybrid mappings. We study the fixed point theorems of generalized hybrid mappings on CAT(0) spaces. We also consider some iteration processes for generalized hybrid mappings on CAT(0) spaces, and our results generalize some results of fixed point theorems on CAT(0) spaces and Hilbert spaces.
Uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity are a natural generalization of both uniformly convexnormed spaces and CAT(0) spaces. In this article, we discuss the existence of fixed points and demiclosed principle for mappings of asymptotically non-expansive type in uniformly convex W-hyperbolic spaces with monotone modulus of uniform convexity. We also obtain a Δ-convergence theorem of Krasnoselski-Mann iteration for continuous mappings of asymptotically nonexpansive type in CAT(0) spaces.
MSC: 47H09; 47H10; 54E40
In this paper, we consider some iteration processes for one-parameter continuous semigroups of nonexpansive mappings in a nonempty compact convex subset C of a complete CAT(0) space X and prove that the proposed sequence converges to a common fixed point for these semigroups of nonexpansive mappings. Note that our results generalize Cho et al. result (Nonlinear Anal. 74:6050-6059, 2011) and related results.
In this paper, we study the convergence of implicit Mann iteration processes with bounded perturbations for approximating a common fixed point of nonexpansive semigroup in CAT(0) spaces. We obtain the △-convergence results of implicit Mann iteration schemes with bounded perturbations for a family of nonexpansive mappings in CAT(0) spaces. Under certain and different conditions, we also get the strong convergence theorems of implicit Mann iteration schemes with bounded perturbations for nonexpansive semigroups in CAT(0) spaces. The results presented in this paper extend and enrich the existing literature.
MSC:
47H05, 47H10, 47J25.
In this paper, we introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in
spaces. As a consequence, we obtain a Δ-convergence theorem of total asymptotically nonexpansive nonself mappings in
spaces. Our results extend and improve the corresponding recent results announced by many authors.
MSC: 47H09, 47J25.
In this survey we present an exposition of the development during the last decade of metric fixed point theory on hyperconvex metric spaces. Therefore we mainly cover results where the conditions on the mappings are metric. We will recall results about proximinal nonexpansive retractions and their impact into the theory of best approximation and best proximity pairs. A central role in this survey will be also played by some recent developments on
-trees. Finally, some considerations and new results on the extension of compact mappings will be shown.
This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and others. In most cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural and can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. The material is presented in a systematic and rigorous manner that allows readers to grasp the key ideas and to gain a working knowledge of the theory. Despite the fact that the work is largely self-contained, extensive bibliographic references are included, and open problems and further development directions are suggested when applicable. The monograph is targeted mainly at the mathematical research community but it is also accessible to graduate students interested in functional analysis and its applications. It could also serve as a text for an advanced course in fixed point theory of mappings acting in modular function spaces.
In this paper, we give strong convergence theorems for the modified S-iteration process of asymptotically quasi-nonexpansive mappings on a CAT(0) space which extend and improve many results in the literature.
MSC:
47H09, 47H10.
In this paper, we study the strong and △-convergence theorems of SP-iteration for nonexpansive mappings on a CAT(0) space. Our results extend and improve many results in the literature.
MSC:
47H09, 47H10.
Let X1 and X2 be two complex normed linear spaces and let D1 be a domain (that is, a nonempty open connected subset) in X1. A mapping f : D1 → X2 is said to be holomorphic in D1 if it is Fréchet differentiable at each point of D1. If D1 and D2 are domains in X1 and X2, respectively, then H(D1, D2) will denote the family of all holomorphic mappings from D1 into D2.
We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some xed point theorems proved recently are consequences of a classical theorem of Kirk.
Let (G, ∑, μ) be a finite, atomless measure space and let Lφ be an Orlicz space of measurable functions on G. We consider some geometrical properties of the functional , called the Orlicz modular. These properties, like strict convexity, uniform convexity or uniform convexity in every direction, can be equivalently expressed in terms of the properties of the corresponding Orlicz function φ. We use these properties in order to prove some fixed point results for mappings T: B → B, B ⊂ Lφ, that are nonexpansive with respect to the Orlicz modular ρ, i.e., ρ(Tƒ − Tg) ⩽ ρ(ƒ − g) for all ƒ and g in B. We prove also existence and uniqueness in Lφ of the best approximant with respect to ρ and some convex subsets of Lφ. Our results are valid also in the case when the Orlicz function φ does not satisfy the Δ2-condition. This demonstrates the advantage of our method because, in the latter case, both Luxemburg's and Orlicz's norms cannot possess suitable convexity properties.
By using the concept of △-convergence introduced by Lim [T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182], we are able to give the CAT(0) space analogs of results on weak convergence of the Picard, Mann and Ishikawa iterates proved in uniformly convex Banach spaces by Opial [Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597], Ishikawa [S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976) 65–71] and Tan and Xu [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301–308].
We propose a new concept of set convergence in a Hadamard space and obtain its
equivalent condition by using the notion of metric projections. Applying this
result, we also prove a convergence theorem for an iterative scheme by the
shrinking projection method in a real Hilbert ball.
We prove the demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. As a consequence, we obtain a Δ-convergence theorem of the Krasnosel'skii-Mann iteration for asymptotically nonexpansive mappings in this setting. Our results extend and improve many results in the literature.
A new characterization of normal structure is given, which allows to prove permanence properties of normal structure such as preservation under finite direct-sum-operations - e.g., the lpN-direct sums, 1 < ≤ ∞ - as well as under certain infinite direct-sum-operations - e.g., the lp-direct sums, 1 < p < ∞. Furthermore, it is shown that a normed space has isonormal structure - i.e., it is isomorphic to a normally structured space - if and only if it can be mapped by a continuous linear one-to-one operator into some normally structured space. Finally, some problems are discussed, such as preservation of normal structure under the l12-direct-sum-operation. To solve the latter at least partially, a sum-property is introduced which implies normal structure. This sum-property is implied by all known sufficient conditions for normal structure, and it is preserved under all finite direct-sum-operations.
In this paper, we consider an iteration process for approximating common fixed points of two nonexpansive mappings and prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.
A class of subsets of a bounded metric space is said to be normal if each member of contains a nondiametral point. An induction proof is given for the following. Suppose M is a nonempty bounded metric space which contains a class of subsets which is countably compact, normal, stable under arbitrary intersections, and which contains the closed balls in M. Then every nonexpansive self-mapping of M has a fixed point.
A fixed point theorem for nonexpansive mappings in dual Banach spaces is proved. Applications in certain Banach lattices are given.
Starting with a decreasing map φ: X → X on a partially ordered set X we construct a map It φ which intuitively can be understood as the iteration (countable or transfinite) of φ. The main properties which It φ inherits from φ are investigated. As application of the main result some fixpoint theorems are proved. Besides, our method yields constructive proofs for results which are usually demonstrated with the help of the axiom of choice.
This is a brief survey of those aspects of fixed point theory for nonexpansive mappings which require assumptions weaker than the axiom of choice in its most general form. In 1977 B. Fuchssteiner observed that the basic existence theorem for such mappings could be derived constructively from an early result essentially due to Zermelo. This paper begins with a reformulation of Fuchssteiner’s idea in a more abstract framework. It is then shown that within this framework, but with the additional assumption of separability on the space, one can deduce that minimal sets exist using only a countable version of the axiom of choice (dependent choices). The existence of minimal sets is crucial to the application of the well-known Goebel-Karlovity lemma which exploits pathological behavior of nontrivial weakly compact sets which are minimal with respect to being nonempty, weakly compact, convex, and T-invariant for nonexpansive T. Section 4 contains a refinement of this lemma and Section 5 contains an application of this refinement.
The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of a nonexpansive mapping T and a finite family of nonexpansive mappings {T} i=1 N , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H. K. Xu and R. G. Ori [Numer. Funct. Anal. Optimization 22, No. 5–6, 767–773 (2001; Zbl 0999.47043)], Z. Opial [Bull. Am. Math. Soc. 73, 591–597 (1967; Zbl 0179.19902)], and others.
The concept of uniform convexity in a normed linear space is based on the geometric condition that if two members of the unit ball are far apart, then their midpoint is well inside the unit ball. We consider here a generalization of this concept whose geometric significance is that the collection of all chords of the unit ball that are parallel to a fixed direction and whose lengths are bounded below by a positive number has the property that the midpoints of the chords lie uniformly deep inside the unit ball. This notion, called uniform convexity in every direction ( UCED ), was first used by A. L. Garkavi [ 5; 6 ] to characterize normed linear spaces for which every bounded subset has at most one Čebyŝev center. We discuss questions of renorming spaces so as to be UCED and forming products of spaces that are uniformly convex in every direction.
We prove that ifC is a bounded closed convex subset of a uniformly convex Banach space,T:C→C is a nonlinear contraction, andS
n
=(I+T+…+T
n−1
)/n, then limn
‖S
n
(x)−TS
n
(x)‖=0 uniformly inx inC. T also satisfies an inequality analogous to Zarantonello’s Hilbert space inequality. which permits the study of the structure of the weak ω-limit set of an orbit. These results are valid forB-convex spaces if some additional condition is imposed on the mapping.
The following theorem is proven:if E is a uniformly rotund Banach space with a Fréchet differentiable norm, C is a bounded nonempty closed convex subset of
E, and T: C→C is a contraction, then the iterates {T
nx} are weakly almost-convergent to a fixed-point of T.
Interation and fixpoints
- B Fuchssteiner
Asymptotic center and nonexpancive mappings in some conjugate spaces
- C Lim T
LIM T. C., Asymptotic center and nonexpancive mappings in some conjugate spaces. Pacific J. Math. 90 (1980), 135-143.
Interation and ]ixpoin~s
- Fuchsstniner B
FUCHSSTNINER B., Interation and ]ixpoin~s. Pacific J. Math. 67 (1977),
73-80.
- N Dunford
- Schwartz J
DUNFORD N. and SCHWARTZ J., Linear Operators. Vol. 1, Interscience, New
York, 1958.
Fixed point theory for nonexpansivc mappings
- W A Kxrk
KXRK W. A., Fixed point theory for nonexpansivc mappings. Proc. Workshop on Fixed Point Theory, Univ. de Sherbrooke, June 1980 (to appear).
On the center of a convex set
- M S Brodskii
- M. S. Brodskii