The Research Of A Class Of Generalized Set-Valued Variational Inclusion Problems

Set-valued variational inclusion problems, research on which touch upon such mathematical branches as convex analysis, linear and nonlinear analysis, nonsmooth analysis, set-valued analysis, partially ordered theory and graphical convergence theory. involve mathematical economics, finance, control theory, mechanics, physics and so on. They become an important foundation and tool for studying multiobjective and multilevel programs and one of focal point problems paid close attention by scholars in the field of applied mathematics. Therefore, the research for them has important learning value and certain degree of difficulty.Just as weakening convexity of objective functions being a central issue in optimization problems, weakening monotonicity of set-valued mappings is an important research direction in set-valued variational inclusion problems. In 2000. Lee and Ding introduced the concepts of - monotonicity and - subdifferential forset-valued mappings and proper functionals, respectively, and Lee also presents an open problem: if Q: H - 2H is a maximal -monotone set-valued mapping, thenunder what conditions do we have rge(I + Q) - H ? Here H is a real Hilbert space. : Hx H - H is a single-valued mapping, > 0 is an any given constant. I and rge(I + Q) denote the identity mapping and range of the set-valued mapping I + Q, respectively.This paper is devoted to study systematically a class of generalized set-valued variational inclusion problems, which is an unity and extension of a large number of known variational inequalities, mixed variational inequalities and variational inclusions, from theory and algorithms. The research is carried on from four aspects. One is, based on answering the above open problem on a finite dimensional Euclidean space by means of partially ordered theory, to research the existence of solutions, global error bounds of proximal solutions and sensitivity of parametric unique solutions and present a class of variable-parameter three-step iterative algorithms for generalized set-valued variational inclusion problems by using - resolvent operator of set-valued mapping.Two is to consider the convexity, closedness and boundedness of the solution set of general set-valued variational inclusion problems and the sensitivity of the parametric solution set by means of graphical convergence theory. Three is to discuss directly the existence of solutions by using analytical methods for set-valued mixed quasi-variational-like inequalities and suggest a class of direct variable-parameter three-stepiterative algorithms for solving generalized set-valued variational inclusions. Finally, the relationships between generalized set-valued variational inclusion problems and non-convex programming are studied. Details are as follows. The background, present research situation and mathematical models of generalized set-valued variational inclusion problems are introduced briefly. The related references are synthesized. The notion of - resolvent operator of set-valued mappings is introduced. Thesame rank Lipschitz continuous development of single-valued mappings is proven by means of partially ordered theory on finite dimensional Euclidean spaces. The problem that under what conditions the - resolvent operator of a maximal TJ - monotone set-valued mapping is a Lipschitz continuous single-valued mapping on whole space, which also answers the open problem mentioned above, is studied on finite dimensional Euclidean spaces. The problem is researched that under what conditions the - resolvent operator of - subdifferential mapping of a proper functional is aLipschitz continuous single-valued mapping on whole space. The existence conditions of unique solutions are discussed by using -reslovent operators for set-valued mixed quasi-variational-like inequality and generalized set-valued variational inclusion problems on finite dimensional Euclidean spaces; respectively. The problem that under what conditions the solution set is nonempty (not necessarily unique solution) is...