Algorithmic and Analytical Approach of Solutions of a System of Generalized Multi-valued Nonlinear Variational Inclusions

Abstract

The main contributions of this paper is twofold. First, our primary concern is to suggest a new iterative algorithm using the $P$-$\eta$-proximal-point mapping technique and Nadler's technique for finding the approximate solutions of a system of generalized multi-valued nonlinear variational-like inclusions. Under some appropriate conditions imposed on the parameters and mappings involved in the system of generalized multi-valued nonlinear variational-like inclusions, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of the aforesaid system is proved. Second, the $H(.,.)$-$\eta$-cocoercive mapping considered in [R. Ahmad, M. Dilshad, M. Akram, Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach sapces, Filomat 26(5)(2012) 897--908] is investigated and analyzed, and the fact that under the assumptions imposed on $H(.,.)$-$\eta$-cocoercive mapping, every $H(.,.)$-$\eta$-cocoercive mapping is $P$-$\eta$-accretive and is not a new one is pointed out. At the same time, some important comments on $H(.,.)$-$\eta$-cocoercive mapping and the results given in the above-mentioned paper are stated.