On the existence of optimal solutions in a stochastic control model

An existence result for a stochastic control model with chance constraints, obtained by Christopeit (Ref. 1), is considerably generalized by combining a standard isometry property of Wiener integrals with a well-known lower semicontinuity result for integral functionals.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

Lower semicontinuity in domain optimization problems

In the present paper, the lower semicontinuity of certain classes of functionals is studied when the domain of integration, which defines the functionals, is not fixed. For this purpose, a certain class of domains introduced by Chenais is employed. For this class of domains, a basic lemma is proved that plays an essential role in the derivations of the lower-semicontinuity theorems. These theorems are applied to the study of the existence of the optimal domain in domain optimization problems; a boundary-value problem of Neumann type or Dirichlet type is the main constraint in these optimization problems.The author wishes to express his sincere thanks to the reviewer for his valuable comments, which made the paper more readable; the reviewer also pointed out that Lemma 2.1 in the text is a direct corollary to a lemma by Chenais (Ref. 9). He thanks Prof. Y. Sakawa of Osaka University for encouragement.     

Maximum theorems with strict quasi-concavity and applications

In this paper, we first prove the strict quasi-concavity of maximizing functions, and next, using a generalization of the KKM theorem, we prove two maximum theorems without assuming the upper semicontinuity. As an application, using a common fixed point theorem, the existence theorem of social equilibrium is obtained. Finally, we shall give two illustrative examples of systems of constrained optimization problems.     

First-Order Necessary Optimality Conditions for General Bilevel Programming Problems

In Ref. 1, bilevel programming problems have been investigated using an equivalent formulation by use of the optimal value function of the lower level problem. In this comment, it is shown that Ref. 1 contains two incorrect results: in Proposition 2.1, upper semicontinuity instead of lower semicontinuity has to be used for guaranteeing existence of optimal solutions; in Theorem 5.1, the assumption that the abnormal part of the directional derivative of the optimal value function reduces to zero has to be replaced by the demand that a nonzero abnormal Lagrange multiplier does not exist.     

Generalized Semicontinuity and Existence Theorems for Cone Saddle Points

This paper is concerned with existence theorems for generalized saddle points (cone saddle points) of vector-valued functions. A concept of lower semicontinuity for vector-valued functions is introduced and its properties are investigated. Previous results of the author are extended by using the lower semicontinuity. Accepted 26 April 1996     

Approximate selections of almost lower semicontinuous multimaps in C-spaces

We study the relations of almost lower semicontinuity, lower semicontinuity and other generalized lower semicontinuity; then we establish a new approximate selection theorem for almost lower semicontinuous multimaps with the generalized Zima type condition in C$C$-spaces. Our result unify and extend the approximate selection theorems in many published works.     

A selector theorem in Banach spaces

In the present paper, we prove a selector theorem in Banach spaces similar to that of Filippov (Ref. 1), but where the measurability of the selected function follows from a weak upper semicontinuity property.     

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.     

Existence Theorems for Periodic Differential Inclusions in IR^N

We study the periodic problem for differential inclusions in R~N.First we look for extremal periodicsolutions.Using techniques from multivalued analysis and a fixed point argument we establish an existencetheorem under some general hypotheses.We also consider the“nonconvex periodic problem”under lowersemicontinuity hypotheses,and the“convex periodic problem”under general upper semicontinuity hypotheseson the multivalued vector field.For both problems,we prove existence theorems under very general hypotheses.Our approach extends existing results in the literature and appear to be the most general results on the nonconvexperiodic problem.     

Finiteness theorems for the Picard objects of an algebraic stack

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud?s relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron–Severi groups or of the Picard group itself. We give some examples and applications. In Appendix A, we prove the semicontinuity theorem for a (non-necessarily tame) algebraic stack.     

An induction theorem and general open mapping theorems

A generalized induction theorem and some general open mapping theorems for multifunctions are straightforwardly proved based on a unique convergence procedure. Many well-known open mapping theorems, closed graph theorems, theorems of the Lusternik type, and results on approximation and semicontinuity or their refinements are obtained as consequences.     

Nemitsky's operators and lower closure theorems

This paper focuses on certain analytic criteria given by the authors in earlier works, for the geometric property of upper semicontinuity of set-valued functions, used in the proofs of lower closure theorems, and hence in existence theory. In particular, it is observed that, under Filippov-type condition (namely, when the set of controls is bounded in measure or in norm), mere Carathéodory-type continuity of the relevant functionsf is sufficient to guarantee a weak form of property (Q), and in turn the lower closure theorems.This work has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.     

Set-valued equilibrium problems with applications to Browder variational inclusions and to fixed point theory

In this paper, we deal with set-valued equilibrium problems under mild conditions of continuity and convexity on subsets recently introduced in the literature. We obtain that neither semicontinuity nor convexity are needed on the whole domain when solving set-valued and single-valued equilibrium problems. As applications, we derive some existence results for Browder variational inclusions, and we extend the well-known Berge maximum theorem in order to obtain two versions of Kakutani and Schauder fixed point theorems.     

The Study of Minimax Inequalities, Abstract Economics and Applications to Variational Inequalities and Nash Equilibria

In this survey, a new minimax inequality and one equivalent geometricform are proved. Next, a theorem concerning the existence of maximalelements for an LC-majorized correspondence is obtained.By the maximal element theorem, existence theorems of equilibrium point fora noncompact one-person game and for a noncompact qualitative game withLC-majorized correspondences are given. Using the lastresult and employing 'approximation approach', we prove theexistence of equilibria for abstract economies in which the constraintcorrespondence is lower (upper) semicontinuous instead of having lower(upper) open sections or open graphs in infinite-dimensional topologicalspaces. Then, as the applications, the existence theorems of solutions forthe quasi-variational inequalities and generalized quasi-variationalinequalities for noncompact cases are also proven. Finally, with theapplications of quasi-variational inequalities, the existence theorems ofNash equilibrium of constrained games with noncompact are given. Our resultsinclude many results in the literature as special cases.     

Optimal control of systems governed by a class of nonlinear evolution equations in a reflexive Banach space

The paper presents a closure theorem for the attainable trajectories of a class of control systems governed by a large class of nonlinear evolution equations in reflexive Banach spaces. Several existence theorems for optimal controls are proven that include a terminal control problem, a time-optimal control problem, and a special Bolza problem. Some results of independent interest are also presented.This work was supported in part by the National Research Council of Canada under Grant No. 7109.The authors would like to thank Professor L. Cesari for pointing out that joint continuity off is required for the setsG andR to satisfy the upper semicontinuity property (Theorems 5.1 and 5.2).     

A survey of credibility theory

This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.     

On supremal functionals defined in <Emphasis Type="Italic">BV</Emphasis>

This paper is devoted to the study of the lower semicontinuity properties of suitable supremal functionals defined on the space of the functions of bounded variation. The main tool in the proofs is a lower semicontinuity theorem for supremal functionals defined on measures that is inspired to the classical Reshetnyak theorem in which integral functionals are involved.      

Concepts of lower semicontinuity and continuous selections for convex valued multifunctions

We discuss several concepts of continuity, weaker than lower semicontinuity, but still implying the existence of a continuous selection for a closed convex valued multifunction from a paracompact Hausdorff topological space into a Banach space. In this way, an extension of Michael's celebrated selection theorem is given. The behavior of -envelop approximations, as well as the localization of continuous selections, is also discussed.