Integral means of set-valued maps

In this paper, we consider a differential inclusion governed by a set-valued nonexpansive mapping and study the asymptotic behavior (weak and strong convergence) of its solutions with various assumptions on this mapping. Then for a set-valued nonexpansive mapping, we define the corresponding resolvent (proximal) operator as a set-valued mapping and study some of its elementary properties. Subsequently, we apply the resolvent operator to state the implicit discretization of the differential inclusion and study the asymptotic behavior of its solutions which yields similar convergence results as in the continuous case. This provides an algorithm for approximating a fixed point of a set-valued nonexpansive mapping which extends the classical proximal point algorithm. An application to variational inequalities and a numerical comparison with another iterative method for approximating a fixed point of set-valued nonexpansive mappings are also presented.

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.

Continuous and Lipschitzian selections from a measurable set-valued map

The goal of this paper is an analytical basis of a region growing method using set-valued maps. They are to provide an approach to image segmentation that does not use a priori conditions on the final segment. Seizing a suggestion of Demongeot and Leitner ([8]), we start with a compact subset of the grey-valued image, and the region growing method is based on the continuous deformation of sets for decreasing some error functional. Avoiding any further restrictions on these sets leads to describing the process as a set-valued map. The ansatz of the deforming sets utilizes reachable sets of differential inclusions that admit more than one velocity of propagation at each point. So set-valued maps underlie a mathematical segmentation problem posed and solved in the first part. Then we present a suggestion how to apply the analytical results to a very simple computer implementation.

In this note we introduce a notion of the weak contingent generalized gradient for set-valued mappings associated with the contingent epiderivative of set-valued mappings introduced in "E. Bednarczuk and W. Song (1998). Contingent epiderivative and its applications to set-valued optimization. Control and Cybernetics, 27, 376-386; G.Y. Chen and J. Jahn (1998). Optimally conditions for set-valued optimization problems. Mathematical Methods of Operations Research, 48, 187-200." and prove that, under some additional condition, it coincides with the weak subdifferential introduced in "T. Tanino (1992). Conjugate duality in vector optimization. Journal of Mathematical Analysis and Applications, 167, 84-97." when the set-valued map is cone-convex. We also study the weak contingent generalized gradient of a sum of two set-valued mappings and optimality conditions for a set-valued vector optimization problem.

Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem.

This paper deals with a convex vector optimization problem with set-valued maps. In the absence of constraint qualifications, it provides, by the scalarization theorem, sequential Lagrange multiplier conditions characterizing approximate weak Pareto optimal solutions for the problem in terms of the approximate subdifferentials of the marginal function associated with corresponding set-valued maps. The paper shows also that this result yields the approximate Lagrange multiplier condition for the problem under a new constraint qualification which is weaker than the Slater-type constraint qualification. Illustrative examples are also provided to discuss the significance of the sequential conditions.

In this paper, I introduce two kinds of ε-subgradients—ε-Pareto subgradients and ε-Benson proper subgradients—for set-valued maps. I give sufficient conditions for the existence of such ε-subgradients. A generalized ε-Moreau–Rockafellar type theorem for ε-Benson proper subdifferentials of set-valued maps is formulated and applied to establish ε-optimality conditions for ε-Benson proper efficient solutions of vector optimization problems with set-valued maps. Scalarization theorems for such a problem are also obtained.

In this paper, we focus on regularity and set-valued functions. The regularity was first introduced by S.C. Kleene (1952) into the propositional connectives of a ternary logic. Then, M. Mukaidono (1986) expanded the regularity of Kleene into n-variable ternary functions, and a ternary function which is regular is called a regular ternary logic function. Some studies expanded regular ternary logic functions into /spl tau/-valued functions, and studied properties of them. In this paper, we propose another extension of the concepts of the regularity in the sense of Kleene and Mukaidono. That is, we introduce regularity into r-valued set-valued functions. Further, we give properties of the set-valued functions with the regularity.

Conjugate duality in vector optimization

Variational characterization of the contingent epiderivative

Let X be a separable Banach space, G a LCA group, and let L1(G, P KC (X))be the space of all compact convex set-valued functions defined on a locally compact abelian group G, where P KC (X) denotes the family of compact convex sets of X. L1 is a metric space under the metric for any F1, F2 ϵ L1 (G,P KC (X))where h denotes the Hausdorff metric on P KC (X). We formulate the convolution of Ll and L1(G), and prove that any set-valued mapping F ϵ L1 (G,P KC (X)) is approximated by the convolution of an approximate identity {e a } for L1(G) and F self under a specific metric topology D1 in L1. This property will be extended to study Fourier analysis of set-valued maps.

In this paper, we propose a new perspective to discuss the N-order fixed point theory of set-valued and single-valued mappings. There are two aspects in our work: we first define a product metric space with a graph for the single-valued mapping whose conversion makes the results and proofs concise and straightforward, and then we propose an SG-contraction definition for set-valued mapping which is more general than some recent contraction’s definition. The results obtained in this paper extend and unify some recent results of other authors. Our method to discuss the N-order fixed point unifies N-order fixed point theory of set-valued and single-valued mappings.

A generalized vector variational inequality problem with a set-valued semi-monotone mapping

In hyperconvex metric spaces, we first present a coincidence point theorem for condensing set-valued self-maps. Then we consider the best approximation problem and the best proximity problem for set-valued mappings that are condensing. As an application, we derive a coincidence point theorem for nonself-condensing set-valued maps.