Fixed point index approach for solutions of variational inequalities

Since the fundamental theory of variational inequality was founded in the 1960s, the variational inequality theory with applications has made powerful progress and has become an important part of nonlinear analysis. It has been applied intensively to mechanics, differential equation, cybernetics, quantitative economics, optimization theory, nonlinear programming, and so forth (see [2]). In virtue of minimax theorem of Ky Fan and KKM technique, variational inequalities, generalized variational inequalities, and generalized quasivariational inequalities were studied intensively in the last 20 years with topological method, variational method, semiordering method, and fixed point method [2]. However, the existence of nonzero solutions for variational inequalities, as another important topic of variational inequality theory, has been rarely discussed. It is of theoretical and practical significance to study the existence of nonzero solutions for variational inequalities. In this paper, we will discuss the existence of nonzero solutions for a class of generalized variational inequalities for multivalued mappings by fixed point index approach in reflexive Banach space. Let Y , Z be two topological spaces. A multivalued mapping F : Y → 2Z is called upper semicontinuous at y0 ∈ Y if for each neighbourhood V ⊂ Z of F(y0), there exists a neighbourhood U of y0 such that the set F(U) ⊂ V . Suppose that E1, E2 are two real Banach spaces, D ⊆ E1. A multivalued mapping A : D → 2E2 is said to be k-set-contractive on D if there exists a constant k such that α(A(S)) ≤ kα(S) whenever α(S) = 0, S ⊆ D, where α is the Kuratowski measure of noncompactness. A mapping A is called condensing on D if α(A(S)) < α(S) whenever α(S) = 0, S ⊆ D. It is easily seen that a mapping A is condensing when k < 1. Let X be a Banach space, X∗ its dual, and (·,·) the pair between X∗