Contractive Set-valued Mappings Research Articles

Fixed point theorems and applications in p-vector spaces

The goal of this paper is to develop new fixed points for quasi upper semicontinuous set-valued mappings and compact continuous (single-valued) mappings, and related applications for useful tools in nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex and p-vector spaces for p in (0, 1]. In particular, we first develop general fixed point theorems for quasi upper semicontinuous set-valued and single-valued condensing mappings, which provide answers to the Schauder conjecture in the affirmative way under the setting of locally p-convex spaces and topological vector spaces for p in (0, 1]; then the best approximation results for quasi upper semicontinuous and 1-set contractive set-valued mappings are established, which are used as tools to establish some new fixed points for nonself quasi upper semicontinuous set-valued mappings with either inward or outward set conditions under various boundary situations. The results established in this paper unify or improve corresponding results in the existing literature for nonlinear analysis, and they would be regarded as the continuation of the related work by Yuan (Fixed Point Theory Algorithms Sci. Eng. 2022:20, 2022)–(Fixed Point Theory Algorithms Sci. Eng. 2022:26, 2022) recently.

Fixed Point Theorem and Related Nonlinear Analysis by the Best Approximation Method in p-Vector Spaces

The goal of this paper is to develop some new and useful tools for nonlinear analysis by applying the best approximation approach for classes of semiclosed 1-set contractive set-valued mappings in locally p-convex or p-vector spaces for In particular, we first develop general fixed point theorems for both set-valued and single-valued condensing mappings which provide answers to the Schauder conjecture in the affirmative way under the setting of (locally p-convex) p-vector spaces, then the best approximation results for upper semi-continuous and 1-set contractive set-valued are established, which are used as tools to establish some new fixed points for non-self set-valued mappings with either inward or outward set conditions under various situations. These results unify or improve corresponding results in the existing literature for nonlinear analysis.

On fixed points of locally and pointwise contractive set-valued maps with an application to the existence of Nash equilibrium in games

We establish the existence of fixed points for set-valued maps defined on metric spaces and satisfying a pointwise or a local version of Banach’s contraction property. As an application, we demonstrate the existence of Nash equilibrium in a general class of strategic games played on metric spaces of strategies.

Nonlinear analysis by applying best approximation method in p-vector spaces

It is known that the class of p-vector spaces (0 < p leq 1) is an important generalization of the usual norm spaces with rich topological and geometrical structure, but most tools and general principles with nature in nonlinearity have not been developed yet. The goal of this paper is to develop some useful tools in nonlinear analysis by applying the best approximation approach for the classes of 1-set contractive set-valued mappings in p-vector spaces. In particular, we first develop general fixed point theorems of compact (single-valued) continuous mappings for closed p-convex subsets, which also provide an answer to Schauder’s conjecture of 1930s in the affirmative way under the setting of topological vector spaces for 0 < p leq 1. Then one best approximation result for upper semicontinuous and 1-set contractive set-valued mappings is established, which is used as a useful tool to establish fixed points of nonself set-valued mappings with either inward or outward set conditions and related various boundary conditions under the framework of locally p-convex spaces for 0 < p leq 1. In addition, based on the framework for the study of nonlinear analysis obtained for set-valued mappings with closed p-convex values in this paper, we conclude that development of nonlinear analysis and related tools for singe-valued mappings in locally p-convex spaces for 0 < p leq 1 seems even more important, and can be done by the approach established in this paper.

Analyzing Approximate Value Iteration Algorithms

In this paper, we consider the stochastic iterative counterpart of the value iteration scheme wherein only noisy and possibly biased approximations of the Bellman operator are available. We call this counterpart the approximate value iteration (AVI) scheme. Neural networks are often used as function approximators, in order to counter Bellman’s curse of dimensionality. In this paper, they are used to approximate the Bellman operator. Because neural networks are typically trained using sample data, errors and biases may be introduced. The design of AVI accounts for implementations with biased approximations of the Bellman operator and sampling errors. We present verifiable sufficient conditions under which AVI is stable (almost surely bounded) and converges to a fixed point of the approximate Bellman operator. To ensure the stability of AVI, we present three different yet related sets of sufficient conditions that are based on the existence of an appropriate Lyapunov function. These Lyapunov function–based conditions are easily verifiable and new to the literature. The verifiability is enhanced by the fact that a recipe for the construction of the necessary Lyapunov function is also provided. We also show that the stability analysis of AVI can be readily extended to the general case of set-valued stochastic approximations. Finally, we show that AVI can also be used in more general circumstances, that is, for finding fixed points of contractive set-valued maps.

An extended local principle of fixed points for weakly contractive set-valued mappings

In this paper, we establish an extension of the well-known Contraction Mapping Principle for Set-Valued Mappings. We consider a larger class of weakly contractive mappings, which allows us to cover, extend and unify many assertions of fixed point theory. As an application, we derive two quantitative stability results for fixed points of weakly contractive maps.

Fixed points of locally fuzzy contractive set-valued mappings in fuzzy metric spaces

Fixed points of locally fuzzy contractive set-valued mappings in fuzzy metric spaces

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

ABSTRACTIn this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].

Fixed Point Theorems for Set-Valued Mappings under Contractive Condition in Quasi-Metric Spaces

Abstract In this paper, we introduce the concept of a generalized weak contraction for set-valued mappings defined on quasi-metric spaces. We show the existence of fixed points for generalized weakly contractive set-valued mappings. Indeed, we have a generalization of Nadler’s fixed point theorem and Banach’s fixed point theorem in quasi-metric spaces and, further, investigate the convergence of iterate scheme of the form xn+1 ∈ Fxn with error estimates.

Fixed points of fuzzy contractive set-valued mappings and fuzzy metric completeness

In this paper, we introduce a new fuzzy contraction via a new concept of the fuzzy sets called fw-distances initiated in the paper, which is a generalization of a fuzzy contractive mapping initiated in the article (Fuzzy Sets Syst. 159:739-744, 2008). A fixed point theorem is established by using this type of contraction of set-valued mappings in fuzzy metric spaces which are complete in the sense of George and Veeramani. As an application of our results, we give characterizations of fuzzy metric completeness. The results are supported by examples.

Fixed point theorems for weak \alpha*-(\phi,L)-contractive set-valued maps in cone metric spaces

The notion of weak α∗-(φ, L)-contractive set-valued maps in cone metric spaces is introduced. A new fixed point theorem for these setvalued maps is established.

Generalized Contractive Set-Valued Maps on Complete Preordered Quasi-Metric Spaces

By using a suitable modification of the notion of a w-distance we obtain some fixed point results for generalized contractive set-valued maps on complete preordered quasi-metric spaces. We also show that several distinguished examples of non-metrizable quasi-metric spaces and of cones of asymmetric normed spaces admit w-distances of this type. Our results extend and generalize some well-known fixed point theorems.

Fixed points and endpoints of contractive set-valued maps in cone uniform spaces with generalized pseudodistances

We introduce the concept of contractive set-valued maps in cone uniform spaces with generalized pseudodistances and we show how in these spaces our fixed point and endpoint existence theorem of Caristi type yields the fixed point and endpoint existence theorem for these contractive maps.MSC:47H10, 54C60, 47H09, 54E15, 46A03, 54E50, 46B40.

Some results on set-valued contractions in abstract metric spaces

In this paper, the concept of contractive set-valued maps in the frame of abstract metric spaces is studied and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, several known fixed point results are either generalized or extended, including the corresponding recent fixed point results of Wardowski [D. Wardowski, Endpoints and fixed points of set-valued contractions in cone metric spaces, Nonlinear Anal. 71 (2009) 512–516] as well as Klim and Wardowski [D. Klim, D. Wardowski, Dynamic process and fixed points of set-valued nonlinear contractions in cone metric spaces, Nonlinear Anal. 71 (2009) 5170–5175]. Examples are given to show that our results are distinct from the existing ones.

FIXED POINT THEOREMS FOR GENERALIZED NONEXPANSIVE SET-VALUED MAPPINGS IN CONE METRIC SPACES

Abstract. In 2007, Huang and Zhang [1] introduced a cone metric spacewith a cone metric generalizing the usual metric space by replacing thereal numbers with Banach space ordered by the cone. They consideredsome xed point theorems for contractive mappings in cone metric spaces.Since then, the xed point theory for mappings in cone metric spaces hasbecome a subject of interest in [1-6] and references therein. In this paper,we consider some xed point theorems for generalized nonexpansive set-valued mappings under suitable conditions in sequentially compact conemetric spaces and complete cone metric spaces. 1. Introduction and PreliminariesIn 2007, Huang and Zhang [1] introduced a cone metric space with a conemetric generalizing the usual metric space by replacing the real numbers withBanach space ordered by the cone. They considered some xed point theoremsfor contractive mappings in cone metric spaces. Since then, the xed pointtheory for mappings in cone metric spaces has become a subject of interest in[1-6] and references therein. Especially, for single-valued mappings, Choudhuryand Metiya [6] considered some xed point theorems for weak contraction incone metric spaces in 2010. In 2011, Wardowski [2] introduced H-cone metricin the collection of subsets of a given cone metric space. And he consideredthe concept of set-valued contraction of Nadler-type and proved a xed pointtheorem for contractive set-valued mappings in H-cone metric spaces. Inspiredand encouraged by the previous works, in this paper we consider some xedpoint theorems for generalized nonexpansive set-valued mappings under suit-able conditions in sequentially compact cone metric spaces and complete conemetric spaces.Let Ebe a real Banach space and Pbe a subset of E.Pis called a cone if and only if

Some generalizations of Caristi’s fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem

In this paper, we first prove some generalizations of Caristi’s fixed point theorem. Then we give some applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem.

A Common End Point Theorem for Set-Valued Generalized -Weak Contraction

We introduce the class of generalized -weak contractive set-valued mappings on a metric space. We establish that such mappings have a unique common end point under certain weak conditions. The theorem obtained generalizes several recent results on single-valued as well as certain set-valued mappings.

An Order on Subsets of Cone Metric Spaces and Fixed Points of Set-Valued Contractions

AbstractIn this paper at first we introduce a new order on the subsets of cone metric spaces then, using this definition, we simplify the proof of fixed point theorems for contractive set-valued maps, omit the assumption of normality, and obtain some generalization of results.