On stably quasimonotone hemivariational inequalities

We establish some existence results for variational-hemivariational inequalities of the Hartman-Stampacchia type involving stably quasimonotone set-valued mappings on bounded, closed and convex subsets in reflexive Banach spaces. We also derive a sufficient condition for the existence and boundedness of solutions.     

Existence results for evolution hemivariational inequalities of second order

In this paper, we consider evolution hemivariational inequalities of second order with a time-dependent pseudomonotone operator and nonmonotone multivalued perturbations. We present the existence of solutions for such inequality. The proof profits from a result on the surjectivity of operators of pseudomonotone type. We discuss some examples which indicate the practical importance of our theoretical findings.     

Existence of anti-periodic solutions for hemivariational inequalities

J. Y. Park and T. G. Ha [Nonlinear Anal., 2008, 68: 747-767; 2009, 71: 3203-3217] investigated the existence of anti-periodic solutions for hemivariational inequalities with a pseudomonotone operator. In this note, we point out that the methods used there are not suitable for the proof of the existence of anti-periodic solutions for hemivariational inequalities and we shall give a straightforward approach to handle these problems. The main tools in our study are the maximal monotone property of the derivative operator with antiperiodic conditions and the surjectivity result for L-pseudomonotone operators.     

Quasilinear elliptic inclusions of Clarke's gradient type under local growth conditions

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient that satisfies only a one-sided local growth condition. An appropriate modification of the associated potential function along with truncation techniques will allow us to apply the theory of multivalued pseudomonotone operators for obtaining existence and comparison results under only one-sided local growth conditions on Clarke's generalized gradient.     

Existence of solutions for hyperbolic hemivariational inequalities

In this paper we study a hyperbolic hemivariational inequality with a nonlinear, pseudomonotone operator depending on the derivative of an unknown function and a linear, monotone operator depending on an unknown function. Using the surjectivity result for L-pseudomonotone operators, an existence result for such inequalities is proved.     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.     

Existence of a nontrivial solution for a class of hemivariational inequality problems at double resonance

In this paper, we discuss a class of semilinear elliptic hemivariational inequality problems. By using the nonsmooth minimax principle for locally Lipschitz functions, we establish the existence of a nontrivial solution for the semilinear elliptic hemivariational inequality problem, where incomplete double resonance occurs at infinity between two distinct consecutive eigenvalues.     

Existence and uniqueness results for a class of quasi-hemivariational inequalities

In this paper we are concerned with the study of a nonstandard quasi-hemivariational inequality. Using a fixed point theorem for set-valued mappings the existence of at least one solution in bounded closed and convex subsets is established. We also provide sufficient conditions for which our inequality possesses solutions in the case of unbounded sets. Finally, the uniqueness and the stability of the solution are analyzed in a particular case.     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ₁ of (−Δ,H ₀ ¹ (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.     

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.     

A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities

A priori bounds for solutions of a wide class of quasilinear degenerate elliptic inequalities are proved. As an outcome we deduce sharp Liouville theorems. Our investigation includes inequalities associated to p-Laplacian and the mean curvature operators in Carnot groups setting. No hypotheses on the solutions at infinity are assumed. General results on the sign of solutions for quasilinear coercive/noncoercive inequalities are considered. Related applications to population biology and chemical reaction theory are also studied.     

Nonexistence results for some nonlinear nonlocal elliptic inequalities with variable exponents

We provide sufficient conditions for the nonexistence of nontrivial nonnegative solutions for some nonlinear elliptic inequalities involving the fractional Laplace operator and variable exponents. The used techniques are based on the test function method. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential

We prove nonexistence results for higher-order semilinear evolution equations and inequalities of the form

     

Existence results for Stampacchia and Minty type vector variational inequalities

In this article, we consider the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for set-valued maps and prove the existence of their solutions in the setting of Banach spaces as well as topological vector spaces. We point out that our vector variational inequalities extend and generalize several vector variational inequalities that appeared in the literature. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities.     

Unique solvability and exponential stability of differential hemivariational inequalities

ABSTRACT

In this paper, we study a differential hemivariational inequality (DHVI, for short) in the framework of reflexive Banach spaces. Our aim is three fold. The first one is to investigate the existence and the uniqueness of mild solution, by applying a general fixed-point principle. The second one is to study its exponential stability, by employing the formula for the variation of parameters and inequality techniques. Finally, the third aim is to illustrate an application of our abstract results in the study of an initial and boundary value problem which describes the contact of an elastic rod with an obstacle.     

Existence and multiplicity results for Dirichlet problems with p-Laplacian

In this paper we apply Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems.     

Global classical solutions for a class of quasilinear hyperbolic systems

In this paper we investigate the mechanism of singularity formation to the Cauchy problem of quasilinear hyperbolic system.Moreover, we present some examples to show some significant problems.     

Gradient estimates for solutions to quasilinear elliptic equations with critical sobolev growth and hardy potential

This note is a continuation of the work[17].We study the following quasilinear elliptic equations(■)where 1 p N,0 ≤μ ((N-p)/p)~p and Q ∈ L~∞(R~N).Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.