The Common Element Of Equilibrium Problems And The Fixed Points Of Operator Semi-groups

As well known, equilibrium problems and operator semi-groups are two hot issues in the field of nonlinear analysis recently. Equilibrium problems can provide us with a unified structure to study a class of related issues arising in economics, physics, transportation, game theory and optimization. Therefore, it has very important theory foundation and broad application value in the field of nonlinear analysis. Semi-group of operator theory is a branch of functional analysis, it has a specific application background in formal languages, automaton, linear dynamic system and others; In particular, it plays a vital role in abstract space of impulsive differential systems and the controllability of impulsive delay differential systems research. Therefore, it has become one of the focus research topics for finding some effective algorithms.In resent years, many researchers have considered a lot of iterative algorithms about equilibrium problems, and many significant research results have been obtained. The convergence of iterative algorithms about the fixed point of operators is one of research topics which has been attracted much attention in the field of nonlinear analysis. In 2005, P.L.Combettes and S.A.Hirstoaga firstly tied equilibrium problems and the iterative algorithms of the fixed points together, and set up an iterative algorithm approaching the common element of equilibrium problems and the fixed points of operator using the fixed point iteration algorithm principle. Since then, many important results were obtained in this field. However, how to combine operator semi-groups with equilibrium problems, and establish some iterative algorithms to approach the common element of equilibrium problems and the fixed points of operator semi-groups is a new research subject. The results play an important role in nonlinear theory and application. Inspired and motivated by these results, in this paper we mainly study the following three problems:Firstly, we construct a Mann iterative algorithm to find the common element of the set of solutions for equilibrium problems and the fixed points of a strongly continuous semi-group of non-expansive mapping in Hilbert space. Meanwhile we get two weak and one strong convergence theorems. The results extend the related results announced by L.-C.Ceng, S. Al-Homidan e t al. (2009)Secondly, we propose a hybrid iterative algorithm for finding the common element of the solutions of an equilibrium problem and the fixed points of asymptotically non-expansive semi-groups in Hilbert space. Under some appropriate conditions, we establish a strong convergence theorem of the sequence generated by our proposed scheme. The results extend and improve the corresponding results announced by Tae-Hwa Kima and Xu (2006) and Tada and Takahash (2007).Lastly, we introduce an iterative algorithm for finding the common element of the solutions of a generalized equilibrium problem, a maximal monotone operator and total quasi-φ-asymptotically non-expansive semi-groups in a reflexive, strictly convex, and smooth Banach space. And we also obtain some strong convergence theorems of the sequences. The results improve and extend the corresponding results announced by L.-C.Ceng. et al. (2012) and Shih-Sen Chang, Lin Wang e t al. (2012).