Strict Pseudocontractive Mappings Research Articles

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

A modified algorithm for solving split feasibility problem for countable family of multivalued κ- demi- contractive mappings and total asymptotically strict pseudo-contractive mapping.

A modified algorithm for solving split feasibility problem for countable family of multivalued κ- demi- contractive mappings and total asymptotically strict pseudo-contractive mapping.

Mann–Dotson’s algorithm for a countable family of non-self strict pseudo-contractive mappings

The aim of this paper to present some weak and strong convergence results for countable family of non-self mappings. More precisely, we employ the Mann–Dotson’s algorithm to approximate common fixed points of a countable family of non-self k-strict Pseudocontractive mappings in q-uniformly smooth Banach spaces.

Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems

In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point problem (FPP) in Hilbert spaces. Our proposed methods involve a projection onto a half-space and self-adaptive step size. We prove that the sequence generated by our proposed methods converges strongly to a common solution of the VIP and FPP of an infinite family of strict pseudo-contractive mappings in Hilbert spaces under some mild assumptions when the underlying operator is monotone and Lipschitz continuous. Furthermore, we apply our results to find a common solution of VIP and zero-point problem (ZPP) for an infinite family of maximal monotone operators. Finally, we provide some numerical experiments of the proposed methods in comparison with other existing methods in the literature.

Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strict pseudocontractive mappings

Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strict pseudocontractive mappings

Common solutions to a finite family of inclusion problems and an infinite family of fixed point problems by a generalized viscosity implicit scheme including applications

This manuscript deals with two problems: the first one is a variational inclusion problem involving an m-accretive mapping and a finite family of inverse strongly accretive mappings, and the other one is a fixed point problem having an infinite family of strict pseudo-contraction mappings in Banach spaces. To approximate the common solution of these problems, we design a generalized viscosity implicit iterative scheme with Meir–Keeler contraction. A strong convergence result for the proposed iterative scheme is established. Applications based on convex minimization problem, linear inverse problem, variational inequality problem and equilibrium problem are derived from the main result. The numerical applicability of the main result and some applications are demonstrated by three examples. Our result extends, generalizes and unifies the previously known results given in literature.

The modified viscosity implicit rules for variational inequality problems and fixed point problems of nonexpansive mappings in Hilbert spaces

The purpose of this paper is to introduce a modified viscosity implicit rules for finding a common element of the set of solutions of variational inequality problems for two inverse-strongly monotone operators and the set of fixed points of one nonexpansive mapping in Hilbert spaces. Under some suitable assumptions imposed on the parameters, we obtain some strong convergence theorems. We also apply our main results to solve fixed point problems for strict pseudocontractive mappings and equilibrium problems in Hilbert spaces. A numerical example is given for supporting our main results.

Systems of Variational Inequalities with Nonlinear Operators

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces.

Boundary Point Method and Mann-Dotson’s Algorithm for Nonself Strict Pseudo-contractive Mappings in Uniformly Smooth Banach Spaces

Let C be a closed, convex and nonempty subset of a q-uniformly smooth Banach space X. Let be a strict pseducontractive inward mapping. We consider the boundary point map depending on C and T defined by , for all . Then for a suitable step-by-step construction of the control coefficients by using the function , we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if and weak convergence if .

Strong convergence theorems for fixed point problems for strict pseudo-contractions and variational inequalities for inverse-strongly accretive mappings in uniformly smooth Banach spaces

In this paper, we first study a simple viscosity iterative algorithm for finding a fixed point of a nonexpansive mapping in uniformly smooth Banach space and obtain a strong convergence theorem under suitable conditions. Then, we apply our iterative algorithm to find a common element of the set of fixed points for a strict pseudo-contraction and the set of fixed points for a nonexansive mapping in uniformly smooth Banach space. We also apply our main results to find a common element of the set of fixed points for strict pseudo-contraction and nonexpansive mapping, the set of solutions of general variational inequalities for two inverse-strongly accretive mappings and equilibrium problems in uniformly smooth Banach spaces or Hilbert spaces. Finally, two numerical examples are given in support of our main results.

A generalized forward–backward splitting method for solving a system of quasi variational inclusions in Banach spaces

The purpose of this paper is by using the generalized forward–backward splitting method and implicit midpoint rule to propose an iterative algorithm for finding a common element of the set of solutions to a system of quasi variational inclusions with accretive mappings and the set of fixed points for a $$\lambda $$ -strict pseudo-contractive mapping in Banach spaces. Some strong convergence theorems of the sequence generated by the algorithm are proved. The results presented in the paper extend and improve some recent results. At the end of the paper, some applications to a system of variational inequalities problem, monotone variational inequalities, convex minimization problem and convexly constrained linear inverse problem are presented.

A projection-type method for solving multi-valued variational inequalities and fixed point problems

In this paper, we introduce a projection algorithm for finding a common element of the set Fix(T) of fixed points of a -strict pseudocontraction mapping and the set S of solutions of a multi-valued variational inequality problem without monotonicity. We show that the sequence generated by our method is globally convergent to some common element of Fix(T) and S, provided that the mapping F is continuous with nonempty compact convex values. Preliminary numerical experiments are reported.

Modified viscosity implicit rules for nonexpansive mappings in Hilbert spaces

In this paper, we study a modified viscosity implicit rule of a nonexpansive mapping in Hilbert spaces. Strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As applications, we obtain some strong convergence theorems for solving fixed-point problems of strict pseudocontractive mappings and finite equilibrium problems in Hilbert spaces. We also give some numerical examples to support our main results.

The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces

The aim of this paper is to introduce a viscosity iterative algorithm for the implicit midpoint rule of nonexpansive mappings in uniformly smooth spaces. Under some appropriate conditions on the parameters, we prove some strong convergence theorems. As applications, we apply our main results to solving fixed point problems of strict pseudocontractive mappings, variational inequality problems in Banach spaces and equilibrium problems in Hilbert spaces. Finally, we give some numerical examples for supporting our main results.

Split feasibility and fixed point problems for asymptotically <i>k</i>-strict pseudo-contractive mappings in intermediate sense

Split feasibility and fixed point problems for asymptotically <i>k</i>-strict pseudo-contractive mappings in intermediate sense

Convergence theorems for strict pseudo-contractions in CAT(0) metric spaces

In this paper, we introduce the notion of strict pseudo-contractive mappings in the framework of CAT(0) metric spaces. Some properties of such mappings including demiclosed principle are investigated. Also, strong convergence and ?-convergence of the well-known Mann iterative algorithm is established for strict pseudo-contractive mappings.

A new general iterative scheme for split variational inclusion and fixed point problems of [formula omitted]-strict pseudo-contraction mappings with convergence analysis

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k-strictly pseudo-contractive nonself mappings. Strong convergence theorem is established under some suitable conditions in a real Hilbert space, which also solves some variational inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors. Finally, some examples to study the rate of convergence and some illustrative numerical examples are presented.

Strong convergence theorem for strict pseudo-contractions in Hilbert spaces

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, 2015), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, 2015) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof.

Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems

In this paper, we consider the split feasibility problem (SFP) in infinite‐dimensional Hilbert spaces and propose some subgradient extragradient‐type algorithms for finding a common element of the fixed‐point set of a strict pseudocontraction mapping and the solution set of a split feasibility problem by adopting Armijo‐like stepsize rule. We derive convergence results under mild assumptions. Our results improve some known results from the literature. Copyright © 2016 John Wiley & Sons, Ltd.

Iterative methods for convex proximal split feasibility problems and fixed point problems

In this paper we prove strong convergence result for a problem of finding a point which minimizes a proper convex lower-semicontinuous function f which is also a fixed point of a total asymptotically strict pseudocontractive mapping such that its image under a bounded linear operator A minimizes another proper convex lower-semicontinuous function g in real Hilbert spaces. In our result in this work, our iterative scheme is proposed with a way of selecting the step-size such that its implementation does not need any prior information about the operator norm ||A|| because the calculation or at least an estimate of the operator norm ||A|| is very difficult, if it is not an impossible task. Our result complements many recent and important results in this direction.