A shorter proof on recent iterative algorithms constructed by the relaxed (u,v)(u, v)-cocoercive mappings and a similar case for inverse-strongly monotone mappings

In this short note, using the class of the relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings, we prove that if an important condition holds then we can prove the convergence of the proposed algorithm, more shorter than the original proof.Comment: 8 page

Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem for a sequene of nearly nonexpansive mappings with respect to a nonexpansive mapping. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence converges strongly to a common element of the set of the common fixed points of nearly nonexpansive mappings.Also, this point solves some variational inequality. As a special case, this projection method can be used to find the minimum norm solution of the given variational inequality. The results here improve and extend some recent corresponding results of other authors.Comment: 11 page

The Split Common Null Point Problem

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301--323]. The SCNPP with only two set-valued mappings entails finding a zero of a maximal monotone mapping in one space, the image of which under a given bounded linear transformation is a zero of another maximal monotone mapping. We present four iterative algorithms that solve such problems in Hilbert spaces, and establish weak convergence for one and strong convergence for the other three.Comment: Journal of Nonlinear and Convex Analysis, accepted for publicatio

A Solution of variational inequality problem for a finite family of nonexpansive mappings in Hilbert spaces

In this paper we prove the strong convergence of the explicit iterative process to a common fixed point of the finite family of nonexpansive mappings defined on Hilbert space, which solves the the variational inequality on the fixed points set.Comment: 9 page

A note on the hybrid steepest descent methods

The aim of this paper is to prove that, in an appropriate setting, every iterative sequence generated by the hybrid steepest descent method is convergent whenever so is every iterative sequence generated by the Halpern type iterative method

Solution of variational inequality problems on fixed point sets of nonexpansive mappings using iterative methods

In this paper, we introduce new implicit and explicit iterative schemes which converge strongly to a unique solution of variational inequality problems for strongly accretive operators over a common fixed point set of finite family of nonexpansive mappings in $q$-uniformly smooth real Banach spaces. As an application, we introduce an iteration process which converges strongly to a solution of the variational inequality which is a common fixed point of finite family of strictly pseudocontractive mappings. Our theorems extend, generalize, improve and unify the corresponding results of Xu \cite{27} and Yamada \cite{Yamada} and that of a host of other authors. Our corollaries and our method of proof are of independent interest

A New Iterative Projection Method for Approximating Fixed Point Problems and Variational Inequality Problems

In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1403.320

Golden ratio algorithms for solving equilibrium problems in Hilbert spaces

In this paper, we design a new iterative algorithm for solving pseudomonotone equilibrium problems in real Hilbert spaces. The advantage of our algorithm is that it requires only one strongly convex programming problem at each iteration. Under suitable conditions we establish the strong and weak convergence of the proposed algorithm. The results presented in the paper extend and improve some recent results in the literature. The performances and comparisons with some existing methods are presented through numerical examples.Comment: 25 pages, 5 figure

Fixed point problems in Banach spaces

We prove strong convergence theorems of some iterative algorithms in a real uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors.Comment: 24 pages. http://www.fixedpointtheoryandapplications.com/content/2013/1/20

Split common fixed point problems and its variant forms

The split common fixed point problems has found its applications in various branches of mathematics both pure and applied. It provides us a unified structure to study a large number of nonlinear mappings. Our interest here is to apply these mappings and propose some iterative methods for solving the split common fixed point problems and its variant forms, and we prove the convergence results of these algorithms. As a special case of the split common fixed problems, we consider the split common fixed point equality problems for the class of finite family of quasi-nonexpansive mappings. Furthermore, we consider another problem namely split feasibility and fixed point equality problems and suggest some new iterative methods and prove their convergence results for the class of quasi-nonexpansive mappings. Finally, as a special case of the split feasibility and fixed point equality problems, we consider the split feasibility and fixed point problems and propose Ishikawa-type extra-gradients algorithms for solving these split feasibility and fixed point problems for the class of quasi-nonexpansive mappings in Hilbert spaces. In the end, we prove the convergence results of the proposed algorithms. Results proved in this chapter continue to hold for different type of problems, such as; convex feasibility problem, split feasibility problem and multiple-set split feasibility problems.Comment: 61 page