A variational inequality index for condensing maps in Hilbert spaces and applications to semilinear elliptic inequalities

A variational inequality index for γ-condensing maps is established in Hilbert spaces. New results on existence of nonzero positive solutions of variational inequalities for such maps are proved by using the theory of variational inequality index. Applications of such a theory are given to existence of nonzero positive weak solutions for semilinear second order elliptic inequalities, where previous results of variational inequalities for S-contractive maps cannot be applied.     

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.     

A Variational Inequality Theory with Applications to $$-Laplacian Elliptic Inequalities

The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous \(\psi_{p}\)-dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for \(p\)-Laplacian elliptic inequalities are obtained under suitable conditions.     

Existence results of semilinear differential variational inequalities without compactness

ABSTRACT

The purpose of this paper is to study a class of semilinear differential variational systems with nonlocal boundary conditions, which are obtained by mixing semilinear evolution equations and generalized variational inequalities. First we prove essential properties of the solution set for generalized variational inequalities. Then without requiring any compactness condition for the evolution operator or for the nonlinear term, two existence results for mild solutions are established by applying a weak topology technique combined with a fixed point theorem.     

kth Order convergence for a semilinear elliptic boundary value problem in the divergence form

The convergence problem of approximate solutions for a semilinear elliptic boundary value problem in the divergence form is studied. By employing the method of quasilinearization, a sequence of approximate solutions converging with the kth (k ? 2) order convergence to a weak solution for a semilinear elliptic problem is obtained via the variational approach.     

Multiplicity of solutions for semilinear variational inequalities via linking and ∇-theorems

We prove the existence of three distinct nontrivial solutions for a class of semilinear elliptic variational inequalities involving a superlinear nonlinearity. The approach is variational and is based on linking and ∇-theorems. Some nonstandard adaptations are required to overcome the lack of the Palais-Smale condition.     

THE ELLIPTIC VARIATIONAL INEQUALITIES WITH DOUBLE DEGENERATE AND GENERAL GROWTH CONDITIONS

A class of quasilinear elliptic variatlonal inequalities with double degenerate is discussed in this paper. We extend the Keldys-Fichera boundary value problem and the first boundary problem of degenerate elliptic equation to the variatlonal inequalities. We establish the existence and uniquenees of the weak solution of conespondlng problem under monstandard growth conditions.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

Prox-regularization and solution of ill-posed elliptic variational inequalities

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem.In particular, regularization on the kernel of the differential operator and regularization with respect to a weak norm of the space are studied. These approaches are illustrated by two nonlinear problems in elasticity theory.     

Hardy-Sobolev critical singular elliptic equations with mixed Dirichlet-Neumann boundary conditions

Three important inequalities (the Poincaré, Hardy and generalized Poincaré inequalities) on the mixed boundary conditions are firstly established by some analytical techniques. Then the existence and multiplicity of positive solutions are studied for a class of semilinear elliptic equations with mixed Dirichlet-Neumann boundary conditions involving Hardy terms and Hardy-Sobolev critical exponents by using the variational methods.     

Stability for semilinear elliptic variational inequalities depending on the gradient

In this work we give a result concerning the continuous dependence on the data for weak solutions of a class of semilinear elliptic variational inequalities (P_n) with a nonlinear term depending on the gradient of the solution. This paper can be seen as the second part of the work Matzeu and Servadei (2010) [9], in the sense that here we give a stability result for the C^1,α-weak solutions of problem (P_n) found in Matzeu and Servadei (2010) [9] through variational techniques. To be precise, we show that the solutions of (P_n), found with the arguments of Matzeu and Servadei (2010) [9], converge to a solution of the limiting problem (P), under suitable convergence assumptions on the data.     

On One of Matérn's Hard-core Point Process Models

By means of time discretization, we approximate evolution variational inequalities by the corresponding elliptic variational inequalities. Using ROTHE'S method (method of lines), an approximate solution is constructed by means of direct variational methods. Existence, uniqueness and regularity of solutions as well as convergence of the approximate solutions are proved.     

Functional a posteriori estimates for elliptic variational inequalities

The paper is concerned with a new way of deriving computable estimates for the difference between the exact solutions of elliptic variational inequalities and arbitrary functions in the corresponding energy space that satisfy the main (Dirichlét) boundary conditions. Unlike the method derived earlier, the estimates are obtained by certain transformations of variational inequalities without using duality arguments. For linear elliptic and parabolic problems, this method was suggested by the author in previous papers. The present paper deals with two different types of variational inequalities (also called variational inequalities of the first and second kind). The techniques discussed can be applied to other nonlinear problems related to variational inequalities. Bibliography: 20 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 147–164.     

On the existence of a solution to a class of variational inequalities

In this note we consider a class of semilinear elliptic variational inequalities on H ¹(Ω) space. With the aid of the mountain-pass principle and the Ekeland variational principle we prove the existence of solutions.     

Parabolic variational inequalities: The Lagrange multiplier approach

Parabolic variational inequalities are discussed and existence and uniqueness of strong as well as weak solutions are established. Our approach is based on a Lagrange multiplier treatment. Existence is obtained as the unique asymptotic limit of solutions to a family of appropriately regularized nonlinear parabolic equations. Two regularization techniques are presented resulting in feasible and unfeasible approximations respectively. Monotonicity results of the regularized solutions and convergence rate estimate are established. The results are applied to the Black–Scholes model for American options. The case of the bilateral constraints is also treated. Numerical results for the Black–Scholes model are presented and prove the practical efficiency of our results.     

Differential quasi-variational inequalities in finite dimensional spaces

In this paper, we introduce and study a new class of differential quasi-variational inequalities in finite dimensional Euclidean spaces. First, we prove existence theorems for Carathéodory weak solutions of the differential quasi-variational inequalities under various conditions. Furthermore, we establish a convergence result on Euler time-dependent procedure for solving the initial-value differential set-valued variational inequalities.     

Weak sharp solutions for generalized variational inequalities

We consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued, and not necessarily monotone. We extend the concept of weak sharpness to this more general framework, and establish some of its characterizations. We establish connections between weak sharpness and (1) gap functions for variational inequalities, and (2) global error bound. When the solution set is weak sharp, we prove finite convergence of the sequence generated by an arbitrary algorithm, for the monotone set-valued case, as well as for the case in which the underlying set-valued map is either Lipschitz continuous in the set-valued sense, for infinite dimensional spaces, or inner-semicontinuous when the space is finite dimensional.     

Differential mixed variational inequalities in finite dimensional spaces

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4]. In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Carathéodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.     

General algorithm of random solutions for random nonlinear variational inequalities in banach Spaces *

In this paper, we study a class of random nonlinear variational inequalities in Banach spaces. By applying a random minimax inequahty obtained by Tarafdar and Yuan, some existence uniqueness theorems of random solutions for the random nonhnear variational inequalities are proved. Next, by applying the random auxiliary problem technique, we suggest an innovative iterative algorithm to compute the random approximate solutions of the random nonlinear variational inequahty. Finally, the convergence criteria is also discussed     

半线性椭圆方程正解的等周不等式

证明了一类半线性椭圆方程正解满足等周不等式,并得到了此解的最佳上界估计.