ITERATIVE ALGORITHMS FOR VARIATIONAL INCLUSIONS IN BANACH SPACES

The present paper is in two folds. In the first fold, we prove the Lipschitz continuity of the proximal mapping associated with a general strongly H-monotone mapping and compute an estimate of its Lipschitz constant under some mild assumptions imposed on the mapping H involved in the proximal mapping. We provide two examples to show that a maximal monotone mapping need not be a general H-monotone for a single-valued mapping H from a Banach space to its dual space. A class of multi-valued nonlinear variational inclusion problems is considered, and by using the notion of proximal mapping and Nadler's technique, an iterative algorithm with mixed errors is suggested to compute its solutions. Under some appropriate hypotheses imposed on the mappings and parameters involved in the multi-valued nonlinear variational inclusion problem, the strong convergence of the sequences generated by the proposed algorithm to a solution of the aforesaid problem is verified. The second fold of this paper investigates and analyzes the notion of Ca-monotone mappings defined and studied in [S.Z. Nazemi, A new class of monotone mappings and a new class of variational inclusions in Banach spaces, J. Optim. Theory Appl. 155(3)(2012) 785-795]. Several comments related to the results and algorithm appeared in the above mentioned paper are given.