A Common Element Of The Set Of Solutions For Mixed Equilibrium Problem, Variational Inequality And Operator Semi-groups

This thesis focuses on the iterative algorithm for a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed points for nonexpansive semi-groups and Optimization problem in Hilbert space, a hybrid algorithm for the variational inequalities and mixed equilibrium problems over the common fixed points set of nonexpansive semi-groups in Hilbert space, and a appro-ximation method for a common element of the set of solutions for mixed equilibrium problems and relatively quasi-nonexpansive semi-groups in Banach space. The main research methods of this three parts prove that the corresponding iterative sequence are Cauchy column, so first we prove the boundedness of the sequence, second we prove that the norm of the difference between two number in the sequence strong converges to zero, and then we prove that when the sequence under the action of semi-group, its norm strong converges to zero, then we prove some inequalities are established to show that the public solution belongs to the solution of optimization problem or a variational inequality problem set, finally, according to a lemma to prove that the sequence of strong convergence to seek the solution.The combination of mixed equilibrium problem with fixed point of operator problems has made the rich research achievements, people tried to study the tentacles extend to mixed equilibrium problem with common fixed point of semi-group. Accor-ding to this kind of problem, I did some system research of operator semi-group and mixed equilibrium problems during the period of graduate study, and then I will divide this paper into five chapters:the first and second chapter are the introduction and prepare knowledge. In the third chapter of this thesis, we introduce a new hybrid iterative algorithm for finding a common element of the set of solutions for a system of mixed equilibrium problems and the set of common fixed points for nonexpansive semi-groups, which solves Optimization problem in Hilbert space. Its innovation is that we generalize the incorporation of the optimization problem.the problem of mixed equilibrium and the finite family of nonexpansive mapping into the combina-tion of a family of mixed equilibrium problems, optimization problems and infinite families of semigroup problems in Hilbert space.In the fourth chapter, we use the hybrid iterative algorithm of nonexpansive semi-groups in Hilbert space, and introdu-ce a mapping F which is η-strongly monotone and κ-Lipschitz continuous to discuss the iterative sequence generated by hybrid iterative algorithm,when mixed equilibri-um auxiliary problem is satisfied and under the limit of some relevant parameters,we got the public solution of the mixed equilibrium problem and the fixed point set of nonexpansive semi-groups, which solves a class of variational inequalities. And under some appropriate conditions we prove the strong convergence theorem of the iterative algorithm. Its feature is that we structure the mixed iterative algorithm for the public solutions of mixed equilibrium problems, variational inequality and nonexpansive semi-groups in Hilbert space.In the fifth chapter, through the generalized projection operator, we build a iterative algorithm for finding a common element of the set of solutions for mixed equilibrium problems and the set of common fixed points for relatively quasi-nonexpansive semi-groups in Reflexive and strictly convex Banach space. Under suitable conditions some strong convergence theorem for the iterative algorithm are proved. The differences of the former two parts lies in the research fra-mework of extend from Hilbert space to reflexive and strictly convex Banach space, as well as the research object from the nonexpansive semi-groups to relatively quasi-nonexpansive semi-groups.