Nonexpansive Mappings Research Articles

Existence of sunny nonexpansive retractions and approximation of fixed points of a representation of nonexpansive mappings

This paper presents an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth uniformly convex Banach space with respect to a left-regular sequence of means defined on a subset of l∞(S). The main results are to establish an existence theorem of a sunny nonexpansive retraction and to create an algorithm for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces.

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

Approximation of fixed points of S−generalized asymptotically nonexpansive mappings

Approximation of fixed points of S−generalized asymptotically nonexpansive mappings

Mixed Variational Inequalities and Nonconvex Analysis

In this expository paper, we provide an account of fundamental aspects of mixed variational inequalities with major emphasis on the computational properties, various generalizations, dynamical systems, nonexpansive mappings, sensitivity analysis and their applications. Mixed variational inequalities can be viewed as novel extensions and generalizations of variational principles. A wide class of unrelated problems, which arise in various branches of pure and applied sciences are being investigated in the unified framework of mixed variational inequalities. It is well known that variational inequalities are equivalent to the fixed point problems. This equivalent fixed point formulation has played not only a crucial part in studying the qualitative behavior of complicated problems, but also provide us numerical techniques for finding the approximate solution of these problems. Our main focus is to suggest some new iterative methods for solving mixed variational inequalities and related optimization problems using resolvent methods, resolvent equations, splitting methods, auxiliary principle technique, self-adaptive method and dynamical systems coupled with finite difference technique. Convergence analysis of these methods is investigated under suitable conditions. Sensitivity analysis of the mixed variational inequalities is studied using the resolvent equations method. Iterative methods for solving some new classes of mixed variational inequalities are proposed and investigated. Our methods of discussing the results are simple ones as compared with other methods and techniques. Results proved in this paper can be viewed as significant and innovative refinement of the known results.

Three Existence Results in the Fixed Point Theory

In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions.

Approximation of common fixed point of two asymptotically nonexpansive mappings in the intermediate sense for a new iteration process in CAT(0) spaces

In this paper, we establish strong and ∆−convergence for a new iteration process containing two asymptotically nonexpansive mappings in the intermediate sense which is broader than the class of asymptotically nonexpansive mappings in the context of CAT(0) spaces. Our results extend, generalize, and improve many well-known results in the literature.

Convergence analysis of Picard–SP iteration process for generalized $$\alpha $$–nonexpansive mappings

AbstractIn this manuscript, we introduce a novel hybrid iteration process called the Picard–SP iteration process. We apply this new iteration process to approximate fixed points of generalized $$\alpha $$ α –nonexpansive mappings. Convergence analysis of our newly proposed iteration process is discussed in the setting of uniformly convex Banach spaces and results are correlated with some other existing iteration processes. The dominance of the newly proposed iteration process is exhibited with the help of a new numerical example. In the end, the comparison of polynomiographs generated by other well-known iteration processes with our proposed iteration process has been presented to make a strong impression of our proposed iteration process.

Nonexpansive maps in nonlinear smooth spaces

We introduce the notion of a nonlinear smooth space generalizing both CAT ⁡ ( 0 ) \operatorname {CAT}(0) spaces as well as smooth Banach spaces. We show that this notion allows for a unified treatment of several results in functional analysis. Namely, we substantiate the usefulness of this setting by establishing a nonlinear generalization of an important result due to Reich in Banach spaces. On par with the linear context, this nonlinear version entails a convergence proof of several other methods. Here, we establish the convergence of a general form of the Halpern-type schema for resolvent-like families of functions. We furthermore prove the convergence of the viscosity generalization of Halpern’s iteration (even for families of maps) generalizing a result due to Chang. This work is set in the context of the ‘proof mining’ program, and the results are complemented with quantitative information like rates of convergence and of metastability (in the sense of T. Tao).

An inertial self-adaptive algorithm for solving split feasibility problems and fixed point problems in the class of demicontractive mappings

We propose a hybrid inertial self-adaptive algorithm for solving the split feasibility problem and fixed point problem in the class of demicontractive mappings. Our results are very general and extend several related results existing in the literature from the class of nonexpansive or quasi-nonexpansive mappings to the larger class of demicontractive mappings. Examples to illustrate numerically the effectiveness of the new analytical results are presented.

Convergence of Fibonacci–Ishikawa iteration procedure for monotone asymptotically nonexpansive mappings

In uniformly convex Banach spaces, we study within this research Fibonacci–Ishikawa iteration for monotone asymptotically nonexpansive mappings. In addition to demonstrating strong convergence, we establish weak convergence result of the Fibonacci–Ishikawa sequence that generalizes many results in the literature. If the norm of the space is monotone, our consequent result demonstrates the convergence type to the weak limit of the sequence of minimizing sequence of a function. One of our results characterizes a family of Banach spaces that meet the weak Opial condition. Finally, using our iterative procedure, we approximate the solution of the Caputo-type nonlinear fractional differential equation.

On Fixed Point for Nonexpansive Mappings in Partial Metric Spaces

On Fixed Point for Nonexpansive Mappings in Partial Metric Spaces

Proof-Theoretical Aspects of Nonlinear and Set-Valued Analysis

AbstractThis thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.The abstract was taken directly from the thesis.Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, GermanyE-mail: pischke@mathematik.tu-darmstadt.de.URL: https://doi.org/10.26083/tuprints-00026584.

Halpern inertial subgradient extragradient algorithm for solving equilibrium problems in Banach spaces

ABSTRACT In this work, we study an equilibrium problem involving a pseudomonotone bifunction in the context of uniformly smooth and 2-uniformly convex real Banach spaces. For the purpose of solving the fixed point problem of a finite family of multi-valued relatively nonexpansive mappings and the pseudomonotone equilibrium problem, we introduce an inertial subgradient extragradient method and establish its strong convergence. To increase the step size and enhance the performance of our iterative method, we use a new parameter that is independent of the inertial extrapolation method. We point out that our step size is chosen in a self-adaptive manner, making it simple to calculate our iterative algorithm without having to know the Lipschitz constants beforehand. Finally, we give some numerical results for the proposed algorithm and comparison with some other known algorithms.

Demiclosed principle and some fixed-point theorems for generalized nonexpansive mappings in Banach spaces

The aim of this paper is to discuss some results concerning the demiclosedness principle of generalized, nonexpansive mappings in uniformly convex spaces. Further, we present some new fixed-point theorems for generalized nonexpansive mappings in different settings of Banach spaces.

On Fixed Point of Monotone \((\alpha,\beta)\)-Nonexpansive Mappings in Ordered Hyperbolic Metric Spaces

On Fixed Point of Monotone \((\alpha,\beta)\)-Nonexpansive Mappings in Ordered Hyperbolic Metric Spaces

On some results related to the Karlsson–Nussbaum conjecture in geodesic spaces

We show a Wolff–Denjoy type theorem in the case of a one-parameter continuous semigroup of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor, we are also able to prove some specific properties directly related to the Karlsson–Nussbaum conjecture.

Fixed points of continuous pseudocontractive mappings on convex, closed unbounded domains

The purpose in this article is to discuss under what conditions a continuous pseudocontractive mapping, in particular a nonexpansive map, has a fixed point when its domain is a closed, convex and unbounded subset of a Banach space. In fact, we will give a characterization of this property. The Halpern iterative algorithm for nonexpansive mappings is also investigated.

Some convergence results for G− mean nonexpansive mappings in uniformly convex Banach space endowed with graph

Some convergence results for G− mean nonexpansive mappings in uniformly convex Banach space endowed with graph

Modified inertial subgradient extragradient with auxiliary parameters and parallel viscosity algorithm for minimization problem induced by bounded linear operator over common solution of fixed points of nonexpansive mappings and pseudomonotone equilibrium problems

Modified inertial subgradient extragradient with auxiliary parameters and parallel viscosity algorithm for minimization problem induced by bounded linear operator over common solution of fixed points of nonexpansive mappings and pseudomonotone equilibrium problems

An Algorithm for Nonlinear Problems Based on Fixed Point Methodologies With Applications

This research presents a highly efficient fixed point algorithm for the computation of fixed points for a very general class of nonexpansive mappings called generalized (α, β)-nonexpansive mappings within the context of uniformly convex Banach space. Our research establishes both weak and strong convergence theorems of the scheme. Furthermore, we demonstrate that the class of generalized (α, β)-nonexpansive mappings contain many classes of nonlinear mappings of the classical literature. Then, we perform various numerical computations to prove the efficiency of the proposed approach. We also study the convergence analysis of the scheme for two dimensional space with taxicab norm. Moreover, we show that our new result gives an alternative approach for solving Caputo fractional differential equation in a novel mappings setting.