Projection Method For Finding Research Articles

Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems

In this paper, we introduce a new inertial self-adaptive projection method for finding a common element in the set of solution of pseudomonotone variational inequality problem and set of fixed point of a pseudocontractive mapping in real Hilbert spaces. The self-adaptive technique ensures the convergence of the algorithm without any prior estimate of the Lipschitz constant. With the aid of Moudafi’s viscosity approximation method, we prove a strong convergence result for the sequence generated by our algorithm under some mild conditions. We also provide some numerical examples to illustrate the accuracy and efficiency of the algorithm by comparing with other recent methods in the literature.

A hybrid shrinking projection method for common fixed points of a finite family of demicontractive mappings with variational inequality problems

In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of variational inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

A hybrid projection method for solving a common solution of a system of equilibrium problems and fixed point problems for asymptotically strict pseudocontractions in the intermediate sense in Hilbert spaces

Abstract In this paper, we introduce a new iterative algorithm which is constructed by using the hybrid projection method for finding a common solution of a system of equilibrium problems of bifunctions satisfying certain conditions and a common solution of fixed point problems of a family of uniformly Lipschitz continuous and asymptotically λ i -strict pseudocontractive mappings in the intermediate sense. We prove the strong convergence theorem for a new iterative algorithm under some mild conditions in Hilbert spaces. Finally, we also give a numerical example which supports our results. MSC:47H05, 47H09, 47H10.

Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set‐valued maximal monotone mappings and inverse‐strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez‐Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

The Shrinking Projection Method for Common Solutions of Generalized Mixed Equilibrium Problems and Fixed Point Problems for Strictly Pseudocontractive Mappings

We introduce the shrinking hybrid projection method for finding a common element of the set of fixed points of strictly pseudocontractive mappings, the set of common solutions of the variational inequalities with inverse-strongly monotone mappings, and the set of common solutions of generalized mixed equilibrium problems in Hilbert spaces. Furthermore, we prove strong convergence theorems for a new shrinking hybrid projection method under some mild conditions. Finally, we apply our results to Convex Feasibility Problems (CFP). The results obtained in this paper improve and extend the corresponding results announced by Kim et al. (2010) and the previously known results.

Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi‐nonexpansive mappings, the set of the variational inequality for an α‐inverse‐strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2‐uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well‐known strong convergence results in the literature.

A Shrinking Projection Method for Generalized Mixed Equilibrium Problems, Variational Inclusion Problems and a Finite Family of Quasi-Nonexpansive Mappings

The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S.Takahashi and W. Takahashi (2008), and many others.

Existence and approximation of solutions for Fredholm equations of the first kind with applications to a linear moment problem

The Cimmino algorithm is an interative projection method for finding almost common points of measurable families of closed convex sets in a Hilbert space. When applied to Fredholm equations of the first kind, the Cimmino algorithm produces weak approximations of solutions provided that solutions exist. We show that for consistent Fredholm equations of the first kind whose data satisfy some spectral conditions, the sequences produced by the Cimmino algorithm converge not only weakly but also in norm. Using this fact, we obtain an existence criterion for solutions to a class of moment problems and show that if problems in this class have solutions, then the Cimmino algorithm generate norm approximations of such solutions.