Iterative Approximation Methods For Variational Inequalities And Fixed Point Problems

In this PhD thesis, it is investigated that the algorithms of iterative approximate find points for solutions of some kinds of generalized variational inequalities (inclusions). In the real iterative approximate process, combing the theories of Banach Geometry,Critical Point Theory,Variational Methods, Nonlinear Approximate Theory in Banach Spaces and Fixed Point Theory,we study the existence and approximate problems of solutions for some kinds of variational inequalities(inclusions) by using the tools of metric projective operator, sunny non-expansive retraction,resolvent operator equation etc. The results in this paper can be viewed as the improvement, development and supplementation of the corresponding results in some references.The main contents of this thesis are in the following:l.The historic background of variational inequality theory is recalled briefly. Then the summary of the work in this paper are given.2.We recall some basic concepts and theories.3.We use the metric projective operator to study iterative approximations for finding a common element of the fixed points of a non-expansive mapping and the set of a solutions of a variational inequality for an inverse-strongly monotone mappings in Hilbert spaces. And we prove the convergence of iterative methods.4. We propose a new set of a general system of variational inclusions in a q-uniformly smooth Banach space. We also study the properties of this system of variational inclusions by using sunny non-expansive retraction, resolvent operator, demi-closedness principle and the hybrid method in mathematical programming. Meanwhile, we analyze the convergence for a common fixed point of a infinite family of strict pesudo-contractions and establish a strong convergence theorem for a common fixed point of a infinite family of strict pesudo-contractions and the solution of the general system of variational inclusions in a Banach space.5. We propose a new set of a general system of variational inequalities in a q-uniformly smooth Banach space. We also study the properties of this system of variational inclusions by using sunny non-expansive retraction, resolvent operator, demi-closedness principle and the hybrid method in mathematical programming. Meanwhile, we analyze the convergence for a common fixed point of a infinite family of strict pesudo-contractions and establish a strong convergence theorem for a common fixed point of a infinite family of strict pesudo-contractions and the solution of the general system of variational inequalities in a Banach space.6.We present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points of non-expansive mappings in q-uniformly smooth Banach spaces. And we establish weak and strong convergence convergence7. We present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem and the set of fixed points of strict pesudo-contractions in q-uniformly smooth Banach spaces. And we establish weak and strong convergence convergence theorem for strictly pseudo-contractions. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a strict pesudo-contraction.