Research On Several Algorithms Of Variational Inequality And Fixed Point Problem

In this dissertation, we study the variational inequalities problems, the fixed point problems and split feasibility problems in the setting of infinite-dimensional Hilbert spaces. In order to solve these problems, we modify others’ relaxed viscosity approximation method, steepestdescent method and extragradient method, and prove the convergence of these modified algorithms. The results in this paper can be viewed as the improvement, extension and supplementation of the corresponding results announced by many others. This dissertation consists of six chapters.1. In Chapter 1, we state that the research background and present situation of variational inequality theory and fixed point theory, and we also briefly introduce the main work and the structure arrangement of this dissertation.2. In Chapter 2, we recall some basic concepts and theories.3. In Chapter 3, we present a new relaxed viscosity approximation method, and prove the strong convergence of the method to a common fixed point of finitely many nonexpansive mappings and a strict pseudocontraction that also solves a suitable equilibrium problem and a general system of variational inequalities.4. In Chapter 4, we introduce one hybrid implicit steepest-descent scheme and another hybrid explicit steepest-descent scheme for finding a solution of the general system of variational inequalities(in short, GSVI) with the constraints of finitely many variational inclusions for maximal monotone and inverse-strongly monotone mappings and a minimization problem for a convex and continuously Fr′echet differentiable functional(in short, CMP) in a real Hilbert space. We establish the strong convergence of these two hybrid steepest-descent schemes to the same solution of the GSVI, which is also a common solution of these finitely many variational inclusions and the CMP. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the hybrid implicit and explicit steepest-descent schemes for finding a common fixed point of finitely many strictly pseudocontractive mappings, and then derive their strong convergence to the unique solution of some hierarchical fixed point problem. Our results extend, improve, complement and develop the corresponding ones given by some authors recently in this area.5. In Chapter 5, we consider a triple hierarchical inequality problem(THVIP),that is,a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem.Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and the present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities(SHVI),that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem.Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.6. In Chapter 6, we consider and study the modified extragradient methods for finding a common element of the solution set Γ of a split feasibility problem(SFP) and the fixed point set Fix(S) of a strictly pseudocontractive mapping S in the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element of Fix(S) ∩Γ where S is strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S) ∩Γ. We also propose another extragradient-like algorithm for finding an element of Fix(S) ∩Γ where S : C → C is nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element of Fix(S) ∩Γ.