Two Iterative Algorithms And Their Applications For Fixed Points Of Asymptotically Non-Expansive Operators In Banach Spaces

In this thesis,we mainly study two iterative algorithms and their applications for fixed points of asymptotically non-expansive operators in Banach spaces,establish a viscosity iterative algorithm for a class of asymptotically non-expansive mappings and some new hybrid iterative schemes for the common element of the set of solutions of generalized mixed equilibrium and the set of fixed points of weakly relatively non-expansive mappings.Under certain conditions,we prove the strong convergence of the schemes.The results here improve and extend the corresponding results reported by some other authors.The content of this thesis is divided into the following three chapters:In the first chapter,we first present the development backgrounds of related issues.Then we introduce the basic concepts and lemma related to this thesis.In the second chapter,we consider a viscosity approximation scheme for a finite family of asymptotically non-expansive mappings in Banach spaces.Firstly,a new viscosity approximation scheme is proposed to study the general split common fixed point problem for a finite family of asymptotically non-expansive mappings in the framework of Banach spaces.Under some mild conditions,the strong convergence of the proposed iterative scheme is verified by some particularly complex techniques.Furthermore,the split equilibrium problem and the hierarchical variational inequality problem are solved by using our main results.In the third chapter,we aim at further enriching the content of equilibrium theory and fixed point theory,the hybrid iterative approximation method is used to study the common elements of solutions of generalized mixed equilibrium and fixed points of weak relatively non-expansive mappings under weaker conditions.A more extensive iterative approximation formula is constructed and the strong convergence of the iterative sequence generated by the formula was obtained.The results show that the new hybrid approximate algorithm converges strongly to the projection point for the more general equilibrium solutions and the fixed points of generalized non-expansive mappings in more general uniformly smooth and strictly convex Banach spaces.The results include some relevant results established by other authors as special cases.