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

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.