Existence Theorems For Variational Inequalities Research Articles

Some existence results for variational inequalities with nonlocal fractional operators

In this paper we consider the following nonlocal fractional variational inequality u∈X0s(Ω),u⩽ψa.e. in Ω,〈u,v−u〉X0s(Ω)−λ〈u,v−u〉2⩾∫Ωfx,u(x),(−Δ)βu(x)(v(x)−u(x))dxaaaaaaaaaaaaaaaaaaaaaaaaaaaaafor anyv∈X0s(Ω),v⩽ψa.e. in Ω, where Ω⊂RN is a smooth bounded open set with continuous boundary ∂Ω, s∈(0,1), N>2s, λ is a real parameter, f is function with subcritical growth, β∈(0,s∕2) and ψ is the obstacle function. As it is well-known, the dependence of the nonlinearity f on the term (−Δ)βu makes non-variational the nature of this problem. Using an iterative technique and a penalization method, we get the existence of a nontrivial nonnegative solution for the problem under consideration, performing the Mountain Pass Theorem. This result can be seen as the extension of known existence theorem for variational inequalities driven by the Laplace operator (or more general uniformly elliptic operators) to the nonlocal fractional setting.

On the existence of solutions of variational inequalities in nonreflexive Banach spaces

We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.

Existence Theorems for Variational Inequalities in Banach Spaces

In this paper, by employing the notion of generalized projection operators and the well-known Fan’s lemma, we establish some existence results for the variational inequality problem and the quasivariational inequality problem in reflexive, strictly convex, and smooth Banach spaces. We propose also an iterative method for approximate solutions of the variational inequality problem and we establish some convergence results for this iterative method.

Variational Inequalities: an Elementary Approach

We will deal with the following problem: Let M be an n × n matrix with real entries. Under which conditions the family of inequalities:{\bf x}\in {\open {R}}^n; {\bf x} \ge {\bf 0};{\bf M}\cdot{\bf x}\ge {\bf 0}has non–trivial solutions? We will prove that a sufficient condition is given bym_{i,j}+m_{j,i}\ge0\quad(1\le i,j\le n);from this result we will derive an elementary proof of the existence theorem for Variational Inequalities in the framework of Monotone Operators.

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known More's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.

Functions without exceptional family of elements and the solvability of variational inequalities on unbounded sets

In this paper we prove an alternative existence theorem for variational inequalities defined on an unbounded set in a Hilbert space. This theorem is based on the concept of exceptional family of elements (EFE) for a mapping and on the concept of $(0, k)$-epi mapping which is similar to the topological degree. We show that when a k-set field is without (EFE) then the variational inequality has a solution. Based on this result we present several classes of mappings without (EFE).