The Convergence For Iterative Approximate Solution On Two Class Problems In General Spaces

In this paper, we mainly study strong convergence problems of asymptotically quasi-Φ-nonexpansive mappings in the intermediate sense and mixed equilibrium problems with relaxed 77-a-monotone mappings by using the hybrid method. Our results extend and improve the existing relevant results. Specifically stated as follows:In the Chapter 1, we discuss the issues of domestic and foreign scholars about this research, some scholars research results in recent years, some explored backgrounds and advantages of the article, which thus highlights the application value and the practical significance of this topic.In the Chapter 2, we introduce some relevant knowledge, some basic concepts and notations.In the Chapter 3, we study the convergence for iterative approximate solution on two class problems in General Spaces, by constructing the first new iterative algorithms, we prove the strong convergence of the sequences generated by the algorithms to a fixed point of infinitely asymptotically quasi-^-nonexpansive mappings in the intermediate sense which is also a common solution of a mixed equilibrium problem with relaxed η-α-monotone mapping. The result improves the corresponding ones of Yang L. et al. in 2012 from asymptotically quasi-Φ-nonexpansive mappings to asymptotically quasi-Φ-nonexpansive mappings in the intermediate sense, from a generalized mixed equilibrium problem to a mixed equilibrium problem with relaxed η-α-monotone mapping, and those of Kong D.Z. in 2013 from a uniformly convex and uniformly smooth Banach space to a strictly convex and uniformly smooth Banach space with Kadec-Klee property, from quasi-Φ-nonexpansive mappings to asymptotically quasi-Φ-nonexpansive mappings in the intermediate sense, from a generalized equilibrium problem to a mixed equilibrium prob-lem with relaxed η-α-monotone mapping. Meanwhile, the result improves and generalizes many other recent corresponding results. More details please read the Chapter 3.In the Chapter 4, we study the convergence for iterative approximate solution on two class problems in General Spaces, by constructing the second new iterative algo-rithms, we prove the strong convergence of the sequences generated by the algorithms to a fixed point of infinitely asymptotically quasi-Φ-nonexpansive mappings in the inter- mediate sense which is also a common solution of infinitely mixed equilibrium problems with relaxed η-α-monotone mappings. The result improves the corresponding ones of Chen M.J. et al. in 2014 from a uniformly convex and uniformly smooth Banach space to a strictly convex and uniformly smooth Banach space with Kadec-Klee property, from quasi-Φ-nonexpansive mappings to asymptotically quasi-^-nonexpansive mappings in the intermediate sense, from a mixed equilibrium problem with relaxed η-α-monotone map-ping to infinitely mixed equilibrium problems with relaxed η-α-monotone mappings, and those of Huang C.Y. et al. in 2014 from an equilibrium problem to infinitely mixed equi-librium problems with relaxed η-α-monotone mappings. Meanwhile, the result improves and generalizes many other recent corresponding results. More details please read the Chapter 4.