fa یک الگوریتم تصویری پیش‌رو – پس‌رو برای تقریب ریشۀ مجموع دو عملگر جبر alg علمی پژوهشی بنیادی S <span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">یک الگوریتم تصویری پیش‌رو</span></span><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">-</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">پس‌رو 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</span></span> <span style="position:relative;top:4.0pt;"><span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"> <img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > </span></span></span><span dir="LTR"><span style="font-size:10.0pt;">-</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">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</span></span><span dir="LTR"><span style="font-size:10.0pt;">.</span></span><span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;"></span></span></span><br> <strong><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"></span></span></strong><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"></span></span><br> <span dir="LTR"><span style="font-family:Times New Roman,serif;"><span style="font-size:10.0pt;">&nbsp;</span></span></span> <strong>Introduction</strong><br> &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.<br> 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;.<br> &lrm;A generalization of finding zero points of nonlinear operator is to find zero points of the sum of an&lrm; &lrm;<img alt="" height="14" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-inverse strongly monotone operator and a maximal monotone operator&lrm;. &lrm;Passty introduced&lrm; &lrm;an iterative methods so called forward-backward 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.<br> Recently&lrm;, &lrm;some authors introduced and studied some algorithms for&lrm; &lrm;finding zero points of the sum of a <img alt="" height="14" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-inverse strongly&lrm; &lrm;monotone operator and a maximal monotone operator under different&lrm; &lrm;conditions.<br> 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;.<br> <strong>Material and methods</strong><br> In 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;<span dir="RTL"> the next </span>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;.<br> <strong>Results and discussion</strong><br> &lrm;The generated sequence by&nbsp; the presented algorithm converges strongly to a zero point of the sum of an <img alt="" height="14" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-inverse strongly&lrm; &lrm;monotone operator and a maximal monotone operator&lrm; &lrm;in Hilbert spaces. &lrm;<br> <strong>Conclusion</strong><br> In this paper&lrm;, &lrm;we present an iterative algorithm &lrm;for approximating a zero point of the sum of an <img alt="" height="14" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="16" >-inverse strongly&lrm;<span dir="RTL"> &lrm;</span>monotone operator and a maximal monotone operator&lrm; &lrm;in Hilbert spaces. <ul> <li>&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&nbsp; provided&lrm;.</li> <li>&lrm;This observation may lead to the future works that are to analyze and discuss the rate of convergence of these suggested algorithms&lrm;.</li> <li>We obtain some applications of main theorem for solving variational inequality problems and finding fixed points of strict pseudocontractions&lrm;.<a href="./files/site1/files/62/7Abstract.pdf">./files/site1/files/62/7Abstract.pdf</a></li> </ul> عملگر یکنوای ماکسیمال, عملگر حلال, الگوریتم تصویری پیش‌رو – پس‌رو. Maximal monotone operator, Resolvent operator, Forward-backward projection algorithm. 215 224 http://mmr.khu.ac.ir/browse.php?a_code=A-10-403-1&slc_lang=fa&sid=1 Vahid Dadashi وحید داداشی v.dadashi@gmail.com 10031947532846003757 10031947532846003757 Yes دانشگاه آزاد اسلامی، واحد ساری، گروه ریاضی