The Existence Of Nonzero Solutions For Set-valued Variational Inequalities

Variational inequality theory is an important part of nonlinear analysis, and has been applied intensively to a large areas such as mechanics, cybernetics, differential equation, quantitative economics, optimization theory and nonlinear programming. The existence of nonzero solutions for variational inequalities, as an important aspect of variational inequality theory, has been extensively discussed by many authors in recent years. In this dissertation, we mainly focus on the existence of nonzero solution to set-valued variational inequalities. The dissertation is organized as below:In chapterâ… , we make a brief review of related work on existence of nonzero solution to set-valued variational inequalities. Moreover, we introduce some basic conceptions and lemmas which are used in this dissertation.In chapterâ…¡, we will consider the existence of nonzero solutions forγ-strongly monotone variational inequalities.we convert the variational inequality problem into a fixed point problem. By using fixed point index approach of strict set-contractive mappings in Banach spaces, we obtain the main conclusions in chapterâ…¡as follows:Theorem 2.3.1 Let X be a real reflexive Banach space and f∈X^*, if be a nonempty closed convex subset of X with 0∈K. Suppose that j : X→R is a proper lower-semicontinuous and convex functional with j(0) = 0 and j(K) (?) [0, +∞]. A : K→2^X* isγ-strongly monotone and upper hemicontinuous with nonempty compact convex values with 0∈A(0), g : K→X^* is aβ-set-contractive mapping, whereβ<γ, if the following assumptions hold:(a) for any sequence {u_n} (?) K with‖u_n‖→+∞, we have (b) there exist u₀∈rcK\{0} and an neighborhood V(0) of zero point such that for all u∈K∩V(0), all u^*∈A(u), it holds that*,u₀> +j_∞(u₀) < 0>.Then the set-valued variational inequality (2.1.1) has a nonzero solution.Theorem 2.3.2 Let X be a real reflexive Banach space and f∈X^*, if be a nonempty closed convex subset of X with 0∈K. Suppose that j : X→R is a proper lower-semicontinuous and convex functional with j(0) = 0 and j(K) (?) [0,+∞]. A : K→2^X* isγ-strongly monotone and upper hemicontinuous with nonempty compact convex values, A is bounded with 0∈A(0), g : K→X^* is boundedβ-set-contractive, whereβ<γ, if the following assumptions hold:(a) for any sequence {u_n} (?) K with‖u_n‖→0, we have(b) there exist u₀∈rcK\{0} and a constantρ> 0 such that for all u∈K with‖u‖>ρ, all u^*∈A(u), we have*,u₀> +j_∞(u₀) < 0>.Then the set-valued variational inequality (2.1.1) has a nonzero solution.In chapterâ…¢, we will consider the existence of nonzero solutions for monotone variational inequalities.we convert the variational inequality problem into a fixed point problem . By using fixed point index approach of compact mappings in locally convex spaces, we obtain the main conclusions in chapterâ…¢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 j : X→R is a proper lower semicontinuous and convex functional with j(0) = 0, j(K) (?) [0, +∞] and(?)j(u) = +∞. A : X→2^X* is monotone and hemicontinuous with nonemptycompact convex values, 0∈A(0). g : K→X^* is continuous in the weak-strong topology. If the following assumptions hold:(a) for any sequence {u_n} (?) K with‖u_n‖→+∞, we have (b) there exist u₀∈rcK\{0} and an neighborhood V(0) of zero point such that for all u∈K∩V(0), all u^*∈A(u), it holds that*,u⁰> +j_∞(u₀) < 0>.Then the set-valued 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 anonempty closed convex subset of X with 0∈K. Suppose that j : X→R is aproper lower semicontinuous and convex functional with j(0) = 0, j(K) (?) [0,+∞] and(?) j(u) = +∞. A : K→2^X* is monotone and hemicontinuous with nonemptycompact convex values, 0∈A(0), A is bounded with 0∈A(0). g : K→X^* is continuous in the weak-strong topology. If the following assumptions hold:(a) for any sequence {u_n} (?) K,‖u_n‖→0, we have(b) there exist u₀∈rcK\{0} and a constantρ> 0, such that for all u∈K with‖u‖>ρand all u^*∈A(u), it holds that*,u₀> +j_∞(u₀) < 0>.Then the set-valued 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 j : X→(-∞, +∞] is a proper lower semicontinuous and convex functional, A is bounded with j(0) = 0 and j(K) (?) [0, +∞] A : X→2^X* is monotone and hemicontinuous with nonempty compact convex values, 0∈A(0). g : K→X^* is continuous in the weak-strong topology. If the following assumptions hold:(a)(b) there exists u₀∈rcK\{0} such that 0>≠0;(c) there exist constants b, c > 0, for all‖u‖> b, it holds that‖g(u)‖/‖u‖≤c.Then the set-valued variational inequality (3.1.1) has a nonzero solution.Theorem 3.3.4 Let X be a real Banach space and a point f∈X^*, K be anonempty closed convex subset of X with 0∈K. Suppose that A : K→2^X* is monotone and hemicontinuous with nonempty compact convex values, 0∈A(0). Suppose that j : X→R is a proper lower semicontinuous and convex functional, j(0) = 0, j(K) (?) [0, +∞]. g : K→X^* is continuous in the weak-strong topology. If the following assumptions hold:(a)(b) there exist a neighbourhood V(0) of zero with respect to K and a constant C > 0 such that for any given u∈V(0),‖g(u)‖< C‖u‖;(c) there exist a point u₀∈rcK\{0} such that j_∞(u₀) < +∞, 0> < 0 andThen the set-valued variational inequality (3.1.1) has a nonzero solution.