Variational Inequality And Fixed Point

It is well known that variational inequality theory and fixed point theory have many applications. Many authors have studied these topics and got lots of wonderful results. In this thesis, we consider the following problems:(1) The existence of random solution for generalized random implicit quasi-variational inequalities, and the convergence of iterative sequences generated by predictor-corrected algorithm;(2) The algorithm for generalized nonlinear mixed quasi-variational-like inequalities, and the convergence of iterative sequences generated by the algorithm;(3) The existence of fixed points of single almost asympotcally nonexpansive type mapping, and the convergence of iterative sequences generated by the three-step algorithm; the algorithm for the common fuzzy random fixed points of a family of random fuzzy mappings.This thesis is divided into five chapters.In chapter 1, we introduce the definitions of variational inequality and nonexpansive mapping. There are two basic problems of variational inequality theory and fixed point theory, the first one is the existence and uniquence of solution or fixed point; the second one is algorithms for the solution of variational inequality or the fixed point of nonexpansive mapping. Variational inequality has been extended and generalized to general variational inequality, mixed variational inequality, implicit quasi-variational inequality, and so on. The mappings involving fixed point theorey include contractive mapping, nonexpansive mapping, asympotcally nonexpansive mapping, almost asympotcally nonexpansive type mapping. There are many iterative algorithms for the solution of variational inequality and fixed point of mappings.In chapter 2, we consider the existence of solution for random implicit quasi-variational inequality with random set-valued mapping in real separable Hilbert spaces. By using the results of Huang and Cho, we construct some iterative algorithms for solving random implicit quasi-variational inequality. Under suit conditions, we can prove that the sequences generated by these iterative algorithms converge to the solutions Of random implicit quasi-variational inequalities with random set-valued mapping in real separable Hilbert spaces.Noor introduced and shudied a new class of predictor-corrector algorithm. But in his articles involving predictor-corrector algorithm, some prime results are wrong. So, in chapter 3, we consider how to use predictor-corrector algorithm to solve some variational inequalities. We introduce a new concept: generalized nonlinear mixedη-strongly monotone mapping. This concept include strongly monotonicity and Lips-chitzian continuity as especial cases. By using auxiliary principle technique, we construct some iterative algorithms to solve generalized nonlinear mixed quasi-variational-like inequalities, and give some convergence results in Hilbert spaces.In chapter 4, we combine both fuzzy theory and random theory effectively, and introduce some new concepts, such as random fuzzy accretive mapping, random fuzzy pseudocontractive mapping, random fuzzy fixed point and so on. By using Petryshyn's inequality and Himmelberg's some results, we construct some algorithms for solving the common random fuzzy fixed point, and prove the convergence of the random iterative sequences in separable real Banach spaces.In chapter 5, we introduce a new concept:p-almost asymptotically nonexpansive type mapping, which include general Lipsctzian mapping, nonexpansive mapping, asymptotically nonexpansive mapping an special cases. We construct corrected three- step iterative algorithm with error to get the fixed point of this kind of nonexpansive mapping. With suit assumptions, we prove the convergence of the sequence generated by the algorithms.