Volume 6, Issue 2 (Vol. 6, No. 2 2020)                   mmr 2020, 6(2): 215-224 | Back to browse issues page


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Dadashi V. A ‎‎‎Forward-Backward Projection Algorithm for Approximating of the Zero of the Sum of Two Operators. mmr 2020; 6 (2) :215-224
URL: http://mmr.khu.ac.ir/article-1-2710-en.html
, v.dadashi@gmail.com
Abstract:   (2651 Views)
Introduction
‎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.
Material and methods
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.
Results and discussion
‎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. ‎
Conclusion
In this paper‎, ‎we present an iterative algorithm ‎for approximating a zero point of the sum of an -inverse strongly‎monotone 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../files/site1/files/62/7Abstract.pdf
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Type of Study: S | Subject: alg
Received: 2017/11/25 | Revised: 2020/09/13 | Accepted: 2018/08/5 | Published: 2020/01/25 | ePublished: 2020/01/25

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