Several Problems Of Solutions To Nonlinear Variational Inclusions

Variational inequality is an important part of nonlinear analysis theory, and varia-tional inclusions are key generalizations of variational inequalities. The purpose of thepaper is to discuss the existence, uniqueness, convergence of algorithms, and iterativeapproximation of solutions to nonlinear variational inclusion problems. The resultspresented in this paper improve and extend some corresponding results.This paper includes four chapters. Now we will describe them briefly.In chapter 1, we recall the history and introduce the backgrounds of the researchon nonlinear variational inclusions and the main work of this thesis.In chapter 2, we study a class of variational inclusions of k-subaccretive type withgeneralized Lipschitzian mappings in real reflexive Banach spaces. we prove the exis-tence and uniqueness of solutions to the class of variational inclusions and the conver-gence of Ishikawa iteration approximating process with mixed errors.In chapter 3, we introduce a new concept ofη-relaxed (γ, r)-cocoercive mappingand study a class of nonlinear generalized quasi-variational-like inclusions withη-relaxed(γ, r)-cocoercive mapping in real Banach spaces. By using resolvent operator techniquefor (A,η)-accretive mappings, we prove the existence of solutions to the nonlinear gen-eral quasi-variational-like inclusions in q-uniformly smooth Banach spaces.In chapter 4, We introduce the concept of relaxed (γ, r, s)-cocoercive mappings andstudy a new system of nonlinear implicit variational inclusions (GSIVI) in real Banach spaces. By using resolvent operator technique for (A,η)-accretive mappings, we provethe existence and uniqueness of solutions to the implicit variation inclusions (GSIVI)and the convergence of Ishikawa iterative process with mixed errors for approximatingsolutions in q-unifomly smooth Banach spaces.