Convergence and stability of an iterative algorithm for a system of generalized implicit variational-like inclusions in Banach spaces

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• 作者：K.R. Kazmi^a ; ¹ ; ^{krkazmi@gmail.com} ; Naeem Ahmad^b ; Mohammad Shahzad^c
• 关键词：System of generalized implicit variational-like inclusions ; M-畏-proximal mapping ; System of implicit equations ; Iterative algorithm ; Convergence and stability analysis
• 刊名：Applied Mathematics and Computation
• 出版年：2012
• 期刊代码：7_00963003
• 类别：cp
• 出版时间：15 May, 2012
• 卷：218
• 期：18
• 页码：9208-9219
• 文件大小：294 K

摘要

In this paper, we give the notion of M-畏-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space, which is an extension of proximal mappings studied in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369-383; K.R. Kazmi, M.I. Bhat, Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces, Appl. Math. Comput. 113 (2005) 153-165; K.R. Kazmi, M.I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, J. Comput. Appl. Math. 233 (2009) 361-371]. We prove its existence and Lipschitz continuity in reflexive Banach space. Further, we consider a system of generalized implicit variational-like inclusions in Banach spaces and show its equivalence with a system of implicit equations using the concept of M-畏-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational-like inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational-like inclusions and discuss the convergence and stability analysis of the iterative algorithm in the setting of uniformly smooth Banach spaces.