An iterative method for variational inclusions and fixed points of total uniformly $L$-Lipschitzian mappings

Abstract

"The characterizations of $m$-relaxed monotone and maximal $m$-relaxed monotone operators are presented and by defining the resolvent operator associated with a maximal $m$-relaxed monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. By using resolvent operator associated with a maximal $m$-relaxed monotone operator, an iterative algorithm is constructed for approximating a common element of the set of fixed points of a total uniformly $L$-Lipschitzian mapping and the set of solutions of a variational inclusion problem involving maximal $m$-relaxed monotone operators. By employing the concept of graph convergence for maximal $m$-relaxed monotone operators, a new equivalence relationship between the graph convergence of a sequence of maximal $m$-relaxed monotone operators and their associated resolvent operators, respectively, to a given maximal $m$-relaxed monotone operator and its associated resolvent operator is established. At the end, we study the strong convergence of the sequence generated by the proposed iterative algorithm to a common element of the above mentioned sets."