Existence results for semilinear elliptic variational inequalities with changing sign nonlinearities

In this paper we present some existence results for a class of semilinear elliptic variational inequalities, depending on a real parameter λ, with changing sign nonlinearities. The fundamental tool to prove the existence result is a penalization method combined with the Mountain Pass Theorem and the Linking Theorem, respectively in the case λ < λ ₁ and λ ≥ λ ₁, where λ₁ is the first eigenvalue of the uniformly elliptic operator A involved in the variational inequality.     

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.     

Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities

In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] and [12].     

Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces

In this paper we study the feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. We show that the strict feasibility implies solvability of vector variational inequalities with moving cones under suitable conditions. Further we show that under suitable conditions, the homogeneous vector variational inequality with a moving cone is solvable whenever it is feasible. As consequences, we obtain the solvability of vector variational inequalities with feasibility assumptions in Banach spaces.     

Semilinear elliptic variational inequalities with dependence on the gradient via Mountain Pass techniques

In this paper we consider a semilinear variational inequality with a gradient-dependent nonlinear term. Obviously the nature of this problem is non-variational. Nevertheless we study that problem associating a suitable semilinear variational inequality, variational in nature, with it, and performing an iterative technique used in De Figueiredo et al. (2004) [6] in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity. We prove the existence of a non-trivial non-negative weak solution u for our problem using essentially variational methods, a penalization technique and an iterative scheme. Via Lewy-Stampacchia’s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is α-H?lder continuous on for any α∈(0,1).     

Levitin-Polyak well-posedness of variational inequalities

In this paper we consider the Levitin-Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin-Polyak well-posedness by considering the size of Levitin-Polyak approximating solution sets of variational inequalities. We also show that the Levitin-Polyak well-posedness of variational inequalities is closely related to the Levitin-Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin-Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution.     

Existence results for semilinear elliptical hemivariational inequalities

We are interested in studying the existence of solutions to an elliptical hemivariational inequality, depending on a real parameter λ$\lambda$. The main tool in the proof of our results is a critical point theorem recently established. We obtain the existence of solution through a direct method, both with a changing sign nonlinearity of the kind p(x)f(ξ)$p\left(x\right)f\left(\xi \right)$ and in the classical one P(x,ξ)$P(x,\xi )$ too.     

Optimal control of parabolic variational inequalities with delays and state constraint

In this paper, an optimal control problem for parabolic variational inequalities with delays and state constraint is investigated and the necessary conditions for optimal controls are derived.     

Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle

This work deals with strict solutions of set-valued optimization problems under the set optimality criterion. In this context, we introduce a new approximate solution concept and we obtain several properties of these solutions when the error is fixed and also for their limit behavior when the error tends to zero. Then we prove a general existence result, which is applied to obtain approximate Ekeland variational principles.     

Multiple bifurcation branches for variational inequalities

We prove the existence of two bifurcation branches for a variational inequality in a case when the corresponding asymptotic problem is nonsymmetric. We use a nonsmooth variational framework and a blow-up argument which allows to find multiple critical points possibly at the same level. An application to plates with obstacle is presented.     

Uniqueness of solutions of the parabolic variational inequality with variable exponent of nonlinearity

In this article, new properties of variable exponent Lebesque and Sobolev spaces were examined. Using these properties we prove that the solution of some parabolic variational inequality is unique with the given conditions.     

Energy minimizing harmonic maps with an obstacle at the free boundary

We consider partial regularity for energy minimizing maps satisfying a partially free boundary condition. This condition takes the form of the requirement that a relatively open subset of the boundary of the domain manifold be mapped into a closed submanifold with non-empty boundary, contained in the target manifold. We obtain an optimal estimate on the Hausdorff dimension of the singular set of such a map. Our result can be interpreted as regularity result for a vector-valued Signorini, or thin-obstacle, problem.     

Existence results for quasilinear parabolic hemivariational inequalities

This paper is devoted to the periodic problem for quasilinear parabolic hemivariational inequalities at resonance as well as at nonresonance. By use of the theory of multi-valued pseudomonotone operators, the notion of generalized gradient of Clarke and the property of the first eigenfunction, we build a Landesman-Lazer theory in the nonsmooth framework of quasilinear parabolic hemivariational inequalities.     

Critical points for nondifferentiable functions in presence of splitting

A classical critical point theorem in presence of splitting established by Brézis-Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational-hemivariational eigenvalue problems.     

An LQP-based descent method for structured monotone variational inequalities

This paper proposes a descent method to solve a class of structured monotone variational inequalities. The descent directions are constructed from the iterates generated by a prediction-correction method [B.S. He, Y. Xu, X.M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Comput. Optim. Appl. 35 (2006) 19-46], which is based on the logarithmic-quadratic proximal method. In addition, the optimal step-sizes along these descent directions are identified to accelerate the convergence of the new method. Finally, some numerical results for solving traffic equilibrium problems are reported.     

Extragradient methods for solving nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.     

An inexact implicit method for general mixed variational inequalities

In the present paper, we present an inexact implicit method with a variable parameter for general mixed variational inequalities. We use a self-adaptive technique to adjust parameter ρ$\rho$ at each iteration. The main advantage of this technique is that the method can adjust the parameter automatically and the numbers of iteration are not very sensitive to different initial parameter _ρ0.${\rho }_0.$     

Projection algorithms for solving a system of general variational inequalities

In this paper, we introduce and consider a new system of general variational inequalities involving four different operators. Using the projection operator technique, we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of variational inequalities involving three operators, variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.