Infinite programming and theorems of the alternative

In this paper, we obtain optimal versions of the Karush–Kuhn–Tucker, Lagrange multiplier, and Fritz John theorems for a nonlinear infinite programming problem where both the number of equality and inequality constraints is arbitrary. To this end, we make use of a theorem of the alternative for a family of functions satisfying a certain type of weak convexity, the so‐called infsup‐convexity.     

FC-空间上的不动点定理和相交定理及其应用

介绍了没有凸结构和线性结构的有限连续拓扑空间(简称为FC-空间);得到了若干个非紧的FC-空间上不动点定理的开[闭]表现形式并建立了FC-空间上的连续选择定理.应用以上结果,在非常弱的假设下得出若干的相交定理和重合点定理,改进和推广了文献中的相应结果.     

Converse duality in Banach space

Two theorems on converse duality are obtained for mathematical programs in Banach spaces. The proofs are based on a Banach-space generalization of the F. John necessary condition for a constrained minimum. No use is made of Kuhn-Tucker constraint qualifications. In the second theorem, the primal program contains a nonlinear equality constraint, and a converse duality theorem is obtained, using a modified concept of convexity; this result appears new, even for finite-dimensional programs. The results are applied to a problem in optimal control.     

FC-空间上的鞍点定理及应用

在不具有任何凸性结构和线性结构的有限连续空间(简称FC-空间)中给出了一个鞍点定理.并应用此定理,在很弱的条件下证明了一些非空交定理及截口定理,推广了近期文献中的一些相关的结果.     

On a generalized sup-inf problem

In this paper, necessary and sufficient conditions for solvability of nonlinear inequality systems are given using certain generalized convexity concepts. Our results imply some theorems of Kirszbraun, Fan, Minty, Simons, Sebestyén, and Gwinner-Oettli.The authors are grateful to the referees for their helpful comments. They thank one of the referees, who emphasized the connection between the Wald minimax theorem and Theorem 2.1, and suggested an alternative proof of Theorem 2.1 in Section 4.     

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.     

Generalized KKM type theorems in FC-spaces with applications (II)

This paper is a continuum of the preceding paper of author. By applying a coincidence theorem in noncompact FC-space without any convexity structure due to author, a new KKM type theorem is first proved under noncompact setting of FC-spaces. The equivalent relation between the coincidence theorem and the KKM type theorem is also established. As applications of the KKM type theorem, we establish some new existence theorems of solutions for three classes of generalized vector equilibrium problems under noncompact setting of FC-spaces. These theorems improve and generalize many known results in literature.     

Generalized Motzkin Theorems of the Alternative and Vector Optimization Problems

In this paper, we introduce a definition of generalized convexlike functions (preconvexlike functions). Then, under the weakened convexity, we study vector optimization problems in Hausdorff topological linear spaces. We establish some generalized Motzkin theorems of the alternative. By use of these theorems of the alternative, we obtain some Lagrangian multiplier theorems. A saddle-point theorem and a scalarization theorem are also derived.Communicated by F. GiannessiThe author thank Ginndomenico Mastrocni for helpful and useful comments.     

Existence theorems without convexity assumptions for optimal control problems governed by parabolic and elliptic systems

We give existence theorems for optimal control problems governed by partial differential equations. We make no convexity assumption with respect to the control variables. The results follow from a combination of the Lyapounov convexity theorem and a condition of concavity with respect to the state variable.     

随机积分和微分方程解的存在性准则

在[1]中我们已证明了一个一般的随机不动点定理并给出了某些应用,在本文中我们将给出该结果的进一步应用.首先证明了一随机Darbo不动点定理,然后利用此定理在紧性假设下给出了非线性随机Volterra积分方程和非线性随机微分方程Cauchy问题随机解的存在性准则.我们的定理改进和推广了Lakshmikantham[3,4],Vaugham[2],De Blasi和Myjak[5]等人的结果.     

拓扑空间中新的广义L-KKM型定理及其应用

In this paper, some new generalized L-KKM type theorems with finitely open values and with finitely closed values are established without any convexity structure in topological spaces. As applications, some new matching theorem, fixed point theorem and existence the orem of equilibrium problem with lower and upper bounds are also given under some suitable conditions. These theorems presented in this paper unify and generalize some corresponding known results in recent literatures.     

Generalization of some duality theorems in nonlinear programming

Pseudoconvexity of a function on one set with respect to some other set is defined and duality theorems are proved for nonlinear programming problems by assuming a certain kind of convexity property for a particular linear combination of functions involved in the problem rather than assuming the convexity property for the individual functions as is usually done. This approach generalizes some of the well-known duality theorems and gives some additional strict converse duality theorems which are not comparable with the earlier duality results of this type. Further it is shown that the duality theory for nonlinear fractional programming problems follows as a particular case of the results established here.     

Generalized type theorems in topological spaces with applications

In this work, some new generalized type theorems for generalized mappings with compactly open values are established in topological spaces without any convexity assumptions. As applications, a Ky Fan type matching theorem, fixed point theorem and coincidence theorem are obtained in topological spaces. These results generalize some known results from the recent literature.     

一般拓扑空间上的KKM定理,匹配定理,重合点定理

引进没有任何凸结构的拓扑空间上的广义R-KKM映射的定义,并利用古典的KKM原理得到一般拓扑空间上的KKM型定理,然后利用该结果得到Lassonde型匹配定理和Klee型相交定理,最后作为应用给出非紧的拓扑空间上不动点定理和重合点存在定理.推广和改进了文献中的相应结果.     

New Generalized R-KKM Type Theorems in General Topological Spaces and Applications

In this paper, some new generalized R-KKM type theorems for generalized R-KKM mappings with finitely closed values and with finitely open values are established in noncompact topological spaces without any convexity structure under much weaker assumptions. As applications, some new minimax inequalities, saddle point theorem and equilibrium existence theorem for equilibrium problems with lower and upper bounds are established in general noncompact topological spaces. These theorems unify and generalize many known results in the literature.     

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.     

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.     

Separation theorems for nonconvex sets and application in optimization

The aim of this paper is to present separation theorems for two disjoint closed sets, without convexity condition. First, a separation theorem for a given closed cone and a point outside from this cone, is proved and then it is used to prove a separation theorem for two disjoint sets. Illustrative examples are provided to highlight the important aspects of these theorems. An application to optimization is also presented to prove optimality condition for a nonconvex optimization problem.     

SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS ON PRODUCT FC-SPACES

By applying a maximal element theorem on product FC-space due to author, some new equilibrium existence theorems for generalized games with fuzzy constraint corre- spondences are proved in FC-spaces.By using these equilibrium existence theorems,some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are established in noncompact product FC-spaces.These results improve and generalize some recent results in literature to product FC-spaces without any convexity structure.     

Two-function upward-downward minimax theorems

Two-function upward-downward minimax theorems are derived which contain Simons upward-downward minimax theorem as well as Domokos minimax theorem as special cases. Here the convexity assumptions on two functions are given by mixing up their values as means proposed by Lin and Quan.