Two types of minimax theorems for vector-valued functions

It seems that minimax theorems for vector-valued functions found in recent papers have something in common. Taking note of this, we improve several results in the author's recent works and state two types of minimax theorems for vector-valued functions. One theorem refers to functions with some special convexity properties; the other theorem refers to separated functions of the typef(x, y)=u(x)+v(y). The proofs are based on the existence of weak cone saddle points off and on a condition about a pointed convex cone which induces a partial ordering in the image space off. We need the condition (C{0})+clC–C, which implies the Sterna-Karwat condition for a convex coneC of a Hausdorff topological vector space.The author thanks the referees for their valuable suggestions on the original draft. Also, he is grateful to S. Yoshiara for his useful suggestions on the English presentation.     

ProperD-solutions of multiobjective programming problems with set functions

A special class of solutions for multiobjective programming problems with set functions is considered. A subset of nondominated solutions, called properD-solution set, with respect to a given domination structure is characterized under two situations, with and without inequality constraints.The authors greatly appreciate valuable comments received from the referees.     

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

A type of Strassen's theorem for positive vector measures with values in dual spaces

In this paper, we extend a type of Strassen's theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.

     

Remarks on quasi-variational inequalities and fixed points in locally convex topological vector spaces

In this note, we first prove existence theorems for noncompact generalized quasivariational inequalities. As applications, two fixed point theorems for upper or lower semicontinuous multivalued mappings without compact domains are given in locally Hausdorff topological vector spaces. These results generalize or improve corresponding results in the literature.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

Random fixed point theorems for a certain class of mappings in banach spaces

Let (, ) be a measurable space and C a nonempty bounded closed convex separable subset of p-uniformly convex Banach space E for some p > 1. We prove random fixed point theorems for a class of mappings T: × C C satisfying: for each x, y C, and integer n 1,

Stability for parametric implicit vector equilibrium problems

In this paper, we consider a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces where a mapping f and a set K are perturbed by parameters and λ, respectively. We establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution set mapping S:Λ₁×Λ₂→2^X for such parametric implicit vector equilibrium problems.