A Hybrid Proximal Point Algorthm And The Approximate Bundle Methods For Solving A Class Of Variational Inclusion Problems

The contents of this paper can be divided into three parts.In the first part, for a class of nonlinear variational inclusion problems in Banach spaces, we introduce the notion of ( A,η)-maximal accretive, which is a generalization of the A -maximal monotonicity mapping in Hilbert spaces. We extend the resolvent operator technique by studying the properties of ( A,η)-maximal accretive operators and generalize the concept of resolvent operators associated with the A -maximal monotonicity mappings to the one associated with ( A,η)-maximal accretive operators. By using the new resolvent operator technique, we prove the existence and uniqueness of the solution to a class of nonlinear vatiational inclusion problems. At the end of this part, we discuss the properties of this new resolvent operator.Next, in the second part, a general framework for a hybrid proximal point algorithm using the notion of ( A,η)-maximal accretive is proposed. Convergence analysis for this algorithm in the context of solving a class of nonlinear varitional inclusion problems is also explored. The obtained results generalize the conclusions of nonlinear variational inclusions in [1] to the one associated with ( A,η)-maximal accretive operators.In the final part, for solving a class of generalized variational inequalities problems, auxiliary principle is generalized. By combining the bundle methods with auxiliary principles, we propose an approximate bundle-type auxiliary principle method for the considered problem which is to find the zero point of the sum of two operators, one is monotonic single-valued, and the other is the subdifferential of a low semi continuous, proper, convex function. During the construction of the algorithm, the condition imposed on the auxiliary function is weakened, not necessarily strongly convex, convex is enough. Finally, we prove the weak convergence of the proposed algorithm.