The Existence Of Nonzero Solutions For Variational Inequalities By Using The Fixed Point Index Approach

Variational inequality theory is a important branch of nonlinear analysis. It has been applied intensively to a large areas such as mechanics, differential equation, economic mathematics, oper-ations research, optimization and controllable theory and nonlinear programming. The existence of nonzero solutions for variational inequalities is an important aspect of variational inequality theory, it has been extensively discussed by many authors in recent years. In this dissertation, we mainly focus on the existence of nonzero solution to singe-valued variational inequalities and bilinear variational inequalities. The dissertation is organized as follows:In chapterⅠ, we make a brief introduce background and present situation of research for variational inequalities. Moreover, we also introduce some basic conceptions which are used in this dissertation.In chapterⅡ, we will consider the existence of nonzero solutions for the following single-variational inequality in reflexive Banach space:which consists in finding u∈K such thatWe convert the variational inequality problem into a fixed point problem. By using fixed point index approach of generalized projection operator in Banach spaces, we obtain the main conclusions as follows:Theorem 2.3.2 Let X be a real smooth reflexive and strictly convex Banach space, let K be a nonempty closed convex subset of X with 0∈K. Suppose that A:K→X* is a single-valued completely continuous mapping. If the following assumptions hold(a) for any sequence{xn} (?) K with||xn||→+∞, we have(b) there exist x0∈rcK\{0} and a neighborhood V(0) of zero point such that for all x∈K∩V(0), it holds that > . Then the variational inequality (2.1.1) has a nonzero solution.Theorem 2.3.3 Let X be a real smooth reflexive and strictly convex Banach space, let K be a nonempty closed convex subset of X with 0∈K. Suppose that A:K→X* is a single-valued completely continuous mapping. If the following assumptions hold(a) for any sequence{xn} (?) K with ||xn||→0, we have(b) there exist x0∈rcK\{0} and a constantρ> 0 such that for all x∈K with||x||>ρ, we have > . Then the variational inequality (2.1.1) has a nonzero solution.We obtain some results about existence of nonzero solutions for the semilinear second-order elliptic equation by using Theorem 2.3.2 and Theorem2.3.3.In chapterⅢ, we will consider the existence of nonzero solutions for the following bilinear inequality in reflexive Banach space:which consists in finding u∈K such thatWe convert the variational inequality problem into a fixed point problem. By using fixed point index approach of compacting mapping in Banach spaces, we obtain the main conclusions as follows:Theorem 3.3.1 Let X be a real reflexive Banach space and f∈X*, K be a nonempty closed convex subset of X with 0∈K. Suppose that g:K→X* is weak to norm continuous functional. Let the following assumptions be satisfied(a) there exists u0∈rcK\{0} such that < 0;(b) there exist two constants C> 0 andα≥0 such that for large enough||u||> b;Then the bilinear variational inequality (3.1.1) has a nonzero solution.Theorem 3.3.2 Let X be a real reflexive Banach space and f∈X*, K be a nonempty closed convex subset of X with 0∈K. Suppose that g:K→X* is weak to norm continuous functional. Let the following assumptions be satisfied(a) for any sequence{un} (?) K with||un||→+∞, we have (b) there exist u0∈rcK\{0} and an neighborhood V(0) of zero point such that for any given u∈K∩V(0), it holds that a(u, u0)+b(u, u0)< (g(u)+f, u0>. Then the bilinear variational inequality (3.1.1) has a nonzero solution.Theorem 3.3.3 Let X be a real reflexive Banach space and f∈X*, K be a nonempty closed convex subset of X with 0∈K. Suppose that g:K→X* is weak to norm continuous functional. Let the following assumptions be satisfied(a) for any sequence{un} (?) K,||un||→0, we have(b) there exist u0∈rcK\{0} and a constant p> 0, such that for any given u∈K with ||u||>ρ, it holds that Then the bilinear variational inequality (3.1.1) has a nonzero solution.