Regularized gap functions for variational problems

We consider variational problems in Banach spaces. Well-posedness concepts for such problems are introduced and investigated by means of two gap functions and their Moreau-Yosida regularizations.     

The studies of systems of variational inclusions problems and variational disclusions problems with applications

In this paper, we study existence theorems of solutions for systems of variational inclusions problems and systems of variational disclusions problems. From these existence results, we establish existence theorems of solutions for systems of generalized vector quasiequilibrium problems and systems of quasioptimization problems.     

A variational technique for optimal boundary control in a hyperbolic problem

We investigate the problem of controlling the boundary functions in a one dimensional hyperbolic problem by minimizing the functional including the final state. After proving the existence and uniqueness of the solution to the given optimal control problem, we get the Frechet differential of the functional and give the necessary condition to the optimal solution in the form of the variational inequality via the solution of the adjoint problem. We constitute a minimizing sequence by the method of projection of the gradient and prove its convergence to the optimal solution.     

The existence results for optimal control problems governed by a variational inequality

In this paper, we establish the existence of the optimal control for an optimal control problem where the state of the system is defined by a variational inequality problem with monotone type mappings. Moreover, as an application, we get several existence results of an optimal control for the optimal control problem where the system is defined by a quasilinear elliptic variational inequality problem with an obstacle.     

Equivalence of variational inequality problems to unconstrained minimization

In this paper we propose a class of merit functions for variational inequality problems (VI). Through these merit functions, the variational inequality problem is cast as unconstrained minimization problem. We estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of VI. This work was supported by the state key project “Scientific and Engineering Computing”.     

Generalized Nash equilibrium problem, variational inequality and quasiconvexity

It is well known that the generalized Nash equilibrium problem, a model for multi-leader–follower games, can be reformulated as a quasivariational inequality. We show that, in fact, a reformulation in terms of a variational inequality can be obtained in the general setting of quasiconvex nondifferentiable decision functions. An existence result is deduced.     

Stability of welding problem with perturbations

This paper discusses the stability of the welding problem of the infinite plane with a hole and a plate which are welded together. Using the complex variable methods, error estimates of stresses and displacement are obtained.     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Generic well-posedness of variational problems without convexity assumptions

The Tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions. In our previous work a generic well-posedness result (with respect to variations of the integrand of the integral functional) without the convexity condition was established for a class of optimal control problems satisfying the Cesari growth condition. In this paper we extend this generic well-posedness result to two classes of variational problems in which the values at the end points are also subject to variations. The main results of the paper are obtained as realizations of a general variational principle.     

New existence criteria of the solutions for a variational relation problem

Many quasivariational inclusions or quasiequilibrium problems, encountered in the literature, are special cases of a variational relation problem proposed in a recent paper by Agarwal et al. (J. Optim. Theory Appl. 2012;155:417–429). The purpose of this paper is to establish new existence results for the solutions of this problem. The main ingredients in the proofs are some continuous selection and fixed point theorems, and an interesting section result. In the last section, we prove that, applied for some concrete relations, our results are different from those obtained by other authors.     

A non-interior-point smoothing method for variational inequality problem

In this paper, we focus on the variational inequality problem. Based on the Fischer-Burmeister function with smoothing parameters, the variational inequality problem can be reformulated as a system of parameterized smooth equations, a non-interior-point smoothing method is presented for solving the problem. The proposed algorithm not only has no restriction on the initial point, but also has global convergence and local quadratic convergence, moreover, the local quadratic convergence is established without a strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems

In this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters.     

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.     

A one-dimensional variational problem with gradient constraint

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.     

A theorem of variational inclusion problems and various nonlinear mappings

In this paper, we prove a strong convergence theorem for finding a common element of the set of fixed points of a finite family of nonexpansive mappings and the set of solutions of a finite family of variational inclusion problems in Hilbert spaces. Moreover, we utilize our main result to fixed point problems of various nonlinear mappings and the set of solutions of variational inclusion problems.     

Nonoccurrence of the Lavrentiev phenomenon for many nonconvex constrained variational problems

In this paper we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex nonautonomous constrained variational problems. A state variable belongs to a convex subset of a Banach space with nonempty interior. Integrands belong to a complete metric space of functions which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. In our previous work Zaslavski (Ann. Inst. H. Poincare, Anal. non lineare, 2006) we considered a class of nonconstrained variational problems with integrands belonging to a subset and showed that for any such integrand the infimum on the full admissible class is equal to the infimum on a subclass of Lipschitzian functions with the same Lipschitzian constant. In the present paper we show that if an integrand f belongs to , then this property also holds for any integrand which is contained in a certain neighborhood of f in . Using this result we establish nonoccurrence of the Lavrentiev phenomenon for most elements of in the sense of Baire category.      

On variational inequalities for auction market problems

We give an equivalent variational inequality formulation for a general class of equilibrium problems based upon auction decision rules. We show that a general relaxation iterative process with conditional gradient extrapolation ensures convergence to a solution under rather mild assumptions.      

The critical Neumann problem for semilinear elliptic equations with concave perturbations

We consider the semilinear Neumann problem involving the critical Sobolev exponent with an indefinite weight function and a concave purturbation. We prove the existence of two distinct solutions.