The Convergence Of Generalized Equilibrium Problems And The Split Generalized Equilibrium Problem

The generalized equilibrium problem and split feasibility problem are current hot issues in the field of nonlinear analysis. The research of generalized equilibrium problem is very meaningful on the theory and application of nonlinear analysis. It includes, as special cases, optimization problems, variational inequalities problems, mini-max problems, the Nash equilibrium problem in noncooperative games and others. The split feasibility problem is closely related to the convex feasibility problem. It is a basic problem in mathematics, physics and many other subjects. The problem has been applied to image restoration, radiation treatment planning and crystallography in actual life. The split generalized equilibrium problem is a combination of generalized equilibrium problem and split feasibility problem. In this thesis, we will study in the following two parts:Firstly, we propose a shrinking projection method for finding a common element of the set EP of the solutions of a generalized equilibrium problem, the set F(S) of fixed points of a quasi-φ-asymptotically nonexpansive mapping and the set T-1O of zeros of a maximal monotone operator T in uniformly smooth and uniformly convex Banach spaces. It is proven that under appropriate conditions, the iterative sequence generated by the method converges strongly to some point in EP∩T-10∩F(S).Secondly, we introduce a split generalized equilibrium problem in Hilbert spaces. We consider five iterative algorithms to solve this problem, and we prove the convergence of iterative sequences.