A ‎‎‎Forward-Backward Projection Algorithm for Approximating of the Zero of the Sum of Two Operators

‎One of the most important classes of mappings is the class of‎ ‎monotone mappings due to its various applications‎. ‎For solving many‎ ‎important problems‎, ‎it is required to solve monotone inclusion‎ ‎problems‎, ‎for instance‎, ‎evolution equations‎, ‎convex optimization‎ ‎problems‎, complementarity problems and variational inequalities‎ ‎problems.The first algorithm for approximating the zero points of the‎ ‎monotone operator introduced by Martinet. ‎In the past decades‎, ‎many authors prepared various algorithms and investigated the existence and convergence of zero points for maximal monotone mappings in Hilbert spaces‎.‎A generalization of finding zero points of nonlinear operator is to find zero points of the sum of an‎ ‎-inverse strongly monotone operator and a maximal monotone operator‎. ‎Passty introduced‎ ‎an iterative methods so called forward-backward method for finding zero points of the sum of two operators‎. ‎There are various applications of the problem of finding zero points of the sum of two operators.Recently‎, ‎some authors introduced and studied some algorithms for‎ ‎finding zero points of the sum of a -inverse strongly‎ ‎monotone operator and a maximal monotone operator under different‎ ‎conditions.In this paper‎, ‎motivated and inspired in above‎, ‎a shrinking projection algorithm is introduced for finding zero points of the sum of an inverse strongly monotone operator and a maximal monotone operator‎. ‎We prove the strong convergence theorem‎ ‎under mild restrictions imposed on the control sequences‎.

In this scheme, first we gather some ‎definitions and lemmas of geometry of Banach spaces and monotone‎ ‎operators‎, ‎which will be needed in the remaining sections‎. ‎In‎ the next section‎, ‎a shrinking projection algorithm is proposed and a‎ ‎strong convergence theorem is established for finding a zero point‎ ‎of the sum of an inverse strongly monotone operator and a maximal‎ ‎monotone operator‎.

‎The generated sequence by  the presented algorithm converges strongly to a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces. ‎

In this paper‎, ‎we present an iterative algorithm ‎for approximating a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces.Under some mild conditions‎, ‎we show the convergence theorem of the mentioned algorithm‎. ‎Subsequently‎, ‎some corollaries and applications of those main result is  provided‎.This observation may lead to the future works that are to analyze and discuss the rate of convergence of these suggested algorithms‎.We obtain some applications of main theorem for solving variational inequality problems and finding fixed points of strict pseudocontractions‎.