یک الگوریتم تصویری پیش‌رو – پس‌رو برای تقریب ریشۀ مجموع دو عملگر

یک الگوریتم تصویری پیش‌روپس‌رو برای یافتن ریشۀ مجموع دو عملگر غیرخطی در فضای هیلبرت را در نظر می‌گیریم. دنبالۀ تولید شده به‌وسیلۀ الگوریتم به‌صورت قوی همگرا به ریشۀ مجموع دو عملگر به‌طور قوی یکنوای معکوس و یکنوای ماکسیمال است. نتیجه به‌دست آمده را برای حل مسئلۀ نامساوی تغییراتی، مسئلۀ نقطه ثابت و مسئلۀ تعادل به‌کار می‌بریم.

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

Introduction lrm;One of the most important classes of mappings is the class of lrm; lrm;monotone mappings due to its various applications lrm;. lrm;For solving many lrm; lrm;important problems lrm;, lrm;it is required to solve monotone inclusion lrm; lrm;problems lrm;, lrm;for instance lrm;, lrm;evolution equations lrm;, lrm;convex optimization lrm; lrm;problems lrm;, complementarity problems and variational inequalities lrm; lrm;problems.The first algorithm for approximating the zero points of the lrm; lrm;monotone operator introduced by Martinet. lrm;In the past decades lrm;, lrm;many authors prepared various algorithms and investigated the existence and convergence of zero points for maximal monotone mappings in Hilbert spaces lrm;. lrm;A generalization of finding zero points of nonlinear operator is to find zero points of the sum of an lrm; lrm;inverse strongly monotone operator and a maximal monotone operator lrm;. lrm;Passty introduced lrm; lrm;an iterative methods so called forwardbackward method for finding zero points of the sum of two operators lrm;. lrm;There are various applications of the problem of finding zero points of the sum of two operators.Recently lrm;, lrm;some authors introduced and studied some algorithms for lrm; lrm;finding zero points of the sum of a inverse strongly lrm; lrm;monotone operator and a maximal monotone operator under different lrm; lrm;conditions.In this paper lrm;, lrm;motivated and inspired in above lrm;, lrm;a shrinking projection algorithm is introduced for finding zero points of the sum of an inverse strongly monotone operator and a maximal monotone operator lrm;. lrm;We prove the strong convergence theorem lrm; lrm;under mild restrictions imposed on the control sequences lrm;.Material and methodsIn this scheme, first we gather some lrm;definitions and lemmas of geometry of Banach spaces and monotone lrm; lrm;operators lrm;, lrm;which will be needed in the remaining sections lrm;. lrm;In lrm; the next section lrm;, lrm;a shrinking projection algorithm is proposed and a lrm; lrm;strong convergence theorem is established for finding a zero point lrm; lrm;of the sum of an inverse strongly monotone operator and a maximal lrm; lrm;monotone operator lrm;.Results and discussion lrm;The generated sequence by the presented algorithm converges strongly to a zero point of the sum of an inverse strongly lrm; lrm;monotone operator and a maximal monotone operator lrm; lrm;in Hilbert spaces. lrm;ConclusionIn this paper lrm;, lrm;we present an iterative algorithm lrm;for approximating a zero point of the sum of an inverse strongly lrm; lrm;monotone operator and a maximal monotone operator lrm; lrm;in Hilbert spaces. lrm;Under some mild conditions lrm;, lrm;we show the convergence theorem of the mentioned algorithm lrm;. lrm;Subsequently lrm;, lrm;some corollaries and applications of those main result is provided lrm;. lrm;This observation may lead to the future works that are to analyze and discuss the rate of convergence of these suggested algorithms lrm;.We obtain some applications of main theorem for solving variational inequality problems and finding fixed points of strict pseudocontractions lrm;../files/site1/files/62/7Abstract.pdf