The Existence Of Solutions To Set-valued Variational Inequalities

This thesis is devoted to the existence of solution to set-valued variationalinequalities under suitable assumptions. We not only propose some convergenceanalysis of iterative algorithms for solving variational inequalities, but also establishthe equivalence between the existence of zeros for set-valued mappings and thesolvability of the variational inequality. Therefore, we only focus on the existence ofzeros for set-valued mappings under suitable assumptions. The paper is organizedas follows:In chapter 1, we introduce the backgrounds and developments of iterative algorithmsand existence of zeros of mapping for variational inequality . In addition,we recall some basic conceptions and notions which are used in this dissertation.In chapter 2, firstly we introduce a generalized f-projection operator in Banachspace and two iterative algorithms:Algorithm 2.2.1 Starting with an arbitrary initial point x0 2 B, generate asequence {xn} via the following iterative scheme:where {α_n} satisfies the conditions: Algorithm 2.2.3 Starting with an arbitrary initial point x0 2 B, generate asequence {xn} via the following iterative scheme:where {αn} satisfies the conditions:Secondly, we present the existence of solutions and approximation of the so-lutions for generalized variational inequalities GV I(K,T,f) in noncompact subsetsof Banach spaces by using above two iterative schemes, respectively. The mainconclusions are following:Theorem 2.2.1 Let B be a uniformly convex and uniformly smooth Banachspace; let K be a closed and convex subset of B and 0∈K. Let f : K→R beconvex, lower semi-continuous. Let T : K→2B? be upper semi-continuous withclosed values; suppose that there exists a positive numberβ, such that J -βT :K→2B is compact. Moreover,(1) f(x)≥0 for all x∈K and f(0) = 0;(2) for any x∈K, any u∈Tx,Let the sequence {xn} be generated by Algorithm 2.2.1. Then GV I(K,T,f) hasa solution x^*∈K and there exists a subsequence {xni} of {xn} such that xni→x?as i→∞.Theorem 2.2.3 Let B be a uniformly convex and uniformly smooth Banachspace and let K be closed convex subset of B and 0∈K. Let f : K→R be convex,lower semi-continuous and f(x)≥0 for all x∈K and f(0) = 0. Let T : K→2B?be upper semi-continuous with closed values; suppose that there exists a positivenumberβ, such that for any x∈K, any u∈Tx,and J -βT : K→2^B* is compact. LetΩbe the solution set of the variational inequality GV I(K,T,f), ifThen GV I(K,T,f) has a solution x?∈K and the sequence {xn} generated byAlgorithm 2.2.1 converges to x^*∈K.Theorem 2.2.4 Let B be a uniformly convex and uniformly smooth Banachspace; let K be a closed and convex subset of B and 0∈K. Let f : K→R beconvex, lower semi-continuous. Let T : K→2^B* be upper semi-continuous withclosed values; suppose that there exists a positive numberβ, such that J -βT :K→2^B* is compact. Moreover,(1) f(x)≥0 for all x∈K and f(0) = 0;(2) for any x∈K, any u∈Tx,Let the sequence {xn} be generated by Algorithm 2.2.3. Then GV I(K,T,f) has asolution x^*∈K and there exists a sequence {xni} of {xn} such that xni→x^* asi→∞.Theorem 2.2.5 Let B be a uniformly convex and uniformly smooth Banachspace and let K be closed convex subset of B and 0∈K. Let f : K→R be convex,lower semi-continuous and f(x)≥0 for all x∈K and f(0) = 0. Let T : K→2^B*be upper semi-continuous with closed values; suppose that there exists a positivenumberβ, such thatand J -βT : K→2^B* is compact.Let be the solution set of GV I(K,T,f), ifThen GV I(K,T,f) has a solution x^*∈K and the sequence {xn} defined by Algo-rithm 2.2.3 converges to x^*∈K.Our results not only generalize the previous known results for variational in-equalities from single-valued case to set-valued, but also extend compact subset conditions to the noncompact subset of Banach spaces and without assuming thepositive homogeneity of f. Finally, we obtain the corresponding results in Hilbertspace.In chapter 3, we establish the equivalence between the existence of zeros forset-valued mappings and the solvability of the variational inequality.Theorem 3.2.2 Let B be a reffexive, strictly convex and smooth Banach space.Let K be a nonempty closed convex subset of B and let T : K→2^B* be a set-valuedmapping. If for all x∈K and all u∈Tx with J^-1(Jx - u)∈K, we haveThen if and only if x0 solves the variational inequality V I(T,K).By using this result, we establish some existence theorem of zeros for quasi-monotone set-valued mappings, which condition is weaker than some well-knownconditions.Theorem 3.2.3 Let B be a re?exive, strictly convex and smooth Banachspace. Let K be a nonempty closed convex subset of B and let T : K→2^B*be a quasimonotone, upper hemicontinuous mapping with nonempty, convex, compact values and satisfy the following coercivity condition: there existsρ> 0, forall x∈K\Bˉ(0,ρ), there exists y∈K with y < x such that for all x^*∈T(x)Moreover, we obtain fixed point theorem of generalized inward mapping inHilbert space.