Generic existence of Lipschitzian solutions of optimal control 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 this paper we consider a large class of optimal control problems which is identified with a complete metric space of integrands without convexity assumptions and show that for a generic integrand the corresponding optimal control problem possesses a unique solution and this solution is Lipschitzian.     

The structure of approximate solutions of variational problems without convexity

In this work we study the structure of approximate solutions of variational problems with continuous integrands f:[0,∞)×Rn×Rn→R1 which belong to a complete metric space of functions. We do not impose any convexity assumption. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.     

Minima in elliptic variational problems without convexity assumptions

We prove an abstract existence theorem for the minimum of the functional

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.      

A turnpike result for autonomous variational problems

In this paper we examine the structure of extremals of variational problems with continuous integrands f:R ⁿ ?×?R ⁿ ?→?R ¹ which belong to a complete metric space of functions. Our results deal with the turnpike properties of variational problems. To have this property means that the solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.     

Existence of solutions in variational relation problems without convexity

Two main existence conditions for solutions of variational relation problems are established without convexity. The first one is based on a finite solvability property and the second one on generalized KKM mappings. These conditions unify and strengthen several existing results in the literature on the topic. A model of satisficing process by rejection is considered which gives an economic interpretation of the introduced concepts.     

Existence of Solutions of a Vector Optimization Problem with a Generic Lower Semicontinuous Objective Function

We study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of lower semicontinuous bounded-from-below objective functions . We establish the existence of a G _δ everywhere dense subset ℱ of such that, for any objective function belonging to ℱ, the corresponding minimization problem possesses a solution.     

Convergence of Krasnoselskii-Mann iterations of nonexpansive operators

In this paper, we show that a generic nonexpansive operator on a closed and convex, but not necessarily bounded, subset of a hyperbolic space has a unique fixed point which attracts the Krasnoselskii-Mann iterations of this operator.     

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.     

Generic properties of compact metric spaces

We prove that there is a residual subset of the Gromov-Hausdorff space (i.e. the space of all compact metric spaces up to isometry endowed with the Gromov-Hausdorff distance) whose elements enjoy several unexpected properties. In particular, they have zero lower box dimension and infinite upper box dimension.     

A nonintersection property for extremals of variational problems with vector-valued functions

In this work we study the structure of extremals of variational problems with vector-valued functions on [0,∞). We show that if an extremal is not periodic, then the corresponding curve in the phase space does not intersect itself.     

Existence theorems in optimal control problems without convexity assumptions

We give existence theorems of solutions for Lagrange and Bolza problems of optimal control. These results are obtained without convexity assumptions on the cost function with respect to the control variable. We extend a Cesari's theorem to cost functions which are nonlinear with respect to the space variable and to problems which are governed by a differential inclusion. Moreover, we consider the case where the control variable belongs to a space of measurable functions and the case where this variable belongs to a Lebesgue space.     

Existence theorems in optimal control problems without convexity assumptions

A correction in the statement of Proposition 4.1 of Ref. 1 is given.     

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.     

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”.     

Nondifferentiability of an objective function at its point of minimum for most minimax problems

In this paper we study a class of minimax problems where . We show that the subclass of all problems for which there exists a point of minimum such that and is small. The paper is dedicated to the memory of Alexander M. Rubinov.     

Well-posedness in the generalized sense for variational inclusion and disclusion problems and well-posedness for optimization problems with constraint

In this paper, we study the well-posedness in the generalized sense for variational inclusion problems and variational disclusion problems, the well-posedness for optimization problems with variational inclusion problems, variational disclusion problems and scalar equilibrium problems as constraint.     

Estimates of approximate solutions and well-posedness for variational inequalities

The purpose of this paper is to estimate the approximate solutions for variational inequalities. In terms of estimate functions, we establish some estimates of the sizes of the approximate solutions from outside and inside respectively. By considering the behaviors of estimate functions, we give some characterizations of the well-posedness for variational inequalities. This work was partially supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China (10671135).     

Generic Existence and Approximation of Fixed Points for Nonexpansive Set-valued Maps

We study nonexpansive set-valued maps in Banach and metric spaces. We are concerned, in particular, with the generic existence and approximation of fixed points, as well as with the structure of fixed point sets.