Fixed Point Solutions Of Nonexpansive Mappings For Several Classes Of Optimization Problems

Theory of variational inequality is an important part of the theory of nonlinear func-tional analysis. The main work is to use some iterative algorithms research fixed point solutions of nonexpansive mappings for several classes of optimiziation problems, which also solve a certain variational inequality.This paper includes four chapters. Now we describe them one by one. In Chapter1, we recall the simple history of the nonlinear sciences and introduce the main work of this paper. In Chapter2, we introduce general iterative algorithms for finding common solutions of a mixed equilibrium problem, general system of variational inequalites, and the fixed point problem of an infinite family of nonexpansive self-mappings in a real Hilbert space, and derive its strong convergence. In Chapter3, we study Ishikawa’s iterative algorithm for finding a common solution of a monotone inclusion problem, a fixed point problem and a equilibrium problem in a real Hilbert space, and derive its strong convergence. In Chapter4, on the basis of Yamada’s hybrid steepest-decent method, we propose a general iterative method for solving the constrained convex minimization problem. It is proved that the sequences generated by proposed implicit and explicit schemes converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. The results presented in this paper improve, extend and develop some recent corresponding results in the literature.