Strong convergence theorems by hybrid and shrinking projection methods for sums of two monotone operators

Abstract

In this paper, we introduce two iterative algorithms for finding the solution of the sum of two monotone operators by using hybrid projection methods and shrinking projection methods. Under some suitable conditions, we prove strong convergence theorems of such sequences to the solution of the sum of an inverse-strongly monotone and a maximal monotone operator. Finally, we present a numerical result of our algorithm which is defined by the hybrid method.

Strong convergence of two inertial projection algorithms in Hilbert spaces

“…We also give some numerical examples to illustrate the computational performance of our proposed algorithms. Our methods improve and generalize some results presented by Mainge and Moudafi [13], Malitsky and Semenov [17], Kazmi et al [10], Dong and Lu [8], Yuying and Plubtieng [24], Dong et al [7], Tan, Xu and Li [23]. This paper is organized as follows.…”

Strong Convergence of a New Generalized Viscosity Implicit Rule and Some Applications in Hilbert Space

“…This problem includes various important problems such as convex minimization problem, variational inequality problem, linear inverse problem and split feasibility problem. One of most popular methods for solving this inclusion problem is the forward-backward splitting method [24][25][26][27].…”

Strong convergence for a modified forward-backward splitting method in Banach spaces

“…Later, strong convergence by the viscosity approximation [25] which extends that by Halpern's type iteration was studied by many researchers ( [7,10,32,36] and references therein) in a real Hilbert space. On the other hand, strong convergence by the hybrid method [14] and shrinking projection method [40] were researched by several authors ( [28,29,43] and references therein) in a real Hilbert space.…”