Algorithmic and Analytical Approach for a System of Generalized Multi-valued Resolvent Equations-Part I: Basic Theory

The concept of resolvent operator associated with a $P$-$\eta$-accretive mapping is used in constructing of a new iterative algorithm for solving a new system of generalized multi-valued resolvent equations in the framework of Banach spaces. Some definitions along with some new concrete examples are provided. The main result of this paper is to prove the Lipschitz continuity of the resolvent operator associated with a $P$-$\eta$-accretive mapping and to compute an estimate of its Lipschitz constant under some new appropriate conditions imposed on the parameters and mappings involved in it. In part II, the convergence analysis of the sequences generated by our proposed iterative algorithm under some appropriate conditions is studied. The results presented in this paper are new, and improve and generalize many known corresponding results.