A class of boundary value problems in the theory of surface waves

Boundary value problems are formulated concerning characteristic oscillations relative to capillary-sound equilibrium forms and theorems are established concerning properties of spectra of these problems; theorems are also presented concerning stability of the indicated forms of equilibrium.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 359–364, March, 1991.     

Variational relation problems and equivalent forms of generalized Fan-Browder fixed point theorem with applications to Stampacchia equilibrium problems

In this paper, we study the existence theorems of solution for variational relation problems. From the existence theorems of solution for variational relation problems, we study equivalent forms of generalized Fan-Browder fixed point theorem, existence theorems of solutions for Stampacchia vector equilibrium problems and generalized Stampacchia vector equilibrium problems. Our results contains many orginal results and have many applications in Nonlinear Analysis.     

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.     

Coincidence and maximal element theorems and their applications to generalized equilibrium problems and minimax inequalities

In this paper we obtain first a very general coincidence theorem. From this we derive a new coincidence theorem and two alternative theorems concerning existence of maximal elements. Applications of these results to generalized equilibrium problems and minimax inequalities are given in the last sections.     

Upper order-preservation of the solution correspondence to vector equilibrium problems

In this paper, several order-theoretic fixed point theorems are proved on partially ordered topological spaces. Applying these fixed point theorems, we explore the existence and upper order-preservation for parametric vector equilibrium problems. In contrast to the previous results on vector equilibrium problems, the upper order-preservation of solutions is a new subject, which would be useful for predicting the changing trend of solutions to vector equilibrium problems. In addition, neither topological continuity nor convexity of the considered vector-valued bifunction F is required in our results.     

The Study of KKM Theorems With Applications to Vector Equilibrium Problems and Implict Vector Variational Inequalities Problems

In this paper, we establish some equivalence relations between coincidence theorems and KKM-type theorems. We also obtain some new coincidence theorems and fixed point theorems. Applying the KKM-type theorems we obtain the existence theorems of generalized vector equilibrium problems. From these results, some existence theorems of generalized vector implicit variational inequality problems are established in this paper.     

Generalized Vector Equilibrium Problems for Pseudomonotone Multivalued Bifunctions<Superscript><Emphasis Type="Bold">1</Emphasis></Superscript>

In this paper, we apply a new version of the Brézis, Nirenberg, and Stampacchia theorem; we use pseudomonotonicity and some coercivity conditions to establish some existence result for a solution of generalized vector equilibrium problems for multivalued bifunctions. The proper quasiconvexity of multivalued bifunctions is introduced and existence theorems for generalized vector equilibrium problems related to multivalued mappings with the KKM property are obtained. The new results extend and modify various existence theorems for similar problems.     

System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings

In this paper, we establish the existence theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence theorems of system of mixed generalized vector variational-like inequalities and existence theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.     

Ekeland’s variational principle for vectorial multivalued mappings in a uniform space

In this paper, we establish Ekeland’s variational principle and an equilibrium version of Ekeland’s variational principle for vectorial multivalued mappings in the setting of separated, sequentially complete uniform spaces. Our approaches and results are different from those in Chen et al. (2008), Hamel (2005), and Lin and Chuang (2010) [13], [14] and [15]. As applications of our results, we study vectorial Caristi’s fixed point theorems and Takahashi’s nonconvex minimization theorems for multivalued mappings and their equivalent forms in a separated, sequentially complete uniform space. We also apply our results to study maximal element theorems, which are unified methods of several variational inclusion problems. Our results contain many known results in the literature Fang (1996) [21], and will have many applications in nonlinear analysis.     

乘积拓扑矢量空间中的不动点定理和广义矢量平衡问题组

利用零调映象的一个不动点定理,在乘积拓扑矢量空间内得到了某些新的不动点定理,作为应用,在乘积拓扑矢量空间内,对一类广义矢量平衡问题组证明了一些平衡存在性定理,这些定理推广了近期文献中的一些重要的已知结果.     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

Existence Theorems of Simultaneous Equilibrium Problems and Generalized Vector Quasi-Saddle Points

In this paper, we establish the existence theorems of simultaneous equilibrium problems. As consequences of our results, we establish the existence theorem of simultaneous mathematical programs and equilibrium problems and the existence theorems of generalized vector quasi-saddle point problems.     

Existence Theorems for Systems of Generalized Vector Quasiequilibrium Problems and Optimization Problems

In this paper, we use existence theorems for the equilibria of generalized abstract economies proved recently to establish existence theorems for systems of generalized vector quasiequilibrium problems. Then, these existence theorems for equilibrium problems are used to derive existence theorems for systems of generalized vector quasivariational-like inequality problems, and vector quasioptimization problems.This research was supported by the National Science Council of the Republic of China. The authors express their gratitude to the referees for valuable suggestions.     

GFC-空间中的KKM定理及其对一般拟平衡问题系统的应用

在GFC-空间中建立了新的KKM定理和Browder型不动点定理,作为应用,狭得了GFC-空间中的一般拟平衡问题系统和拟平衡问题系统的平衡存在定理．     

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.     

非紧L-凸度量空间中的一般拟平衡问题组(英文)

本文建立了非紧完备L-凸度量空间中新的不动点定理.作为应用,获得了非紧完备L-凸度量空间中的一般拟平衡问题组和拟平衡问题组的平衡存在定理.     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

Existence Results for Primal and Dual Generalized Vector Equilibrium Problems With Applications to Generalized Semi-Infinite Programming

In this paper, we introduce several kinds of maximal pseudomonotonicity and establish existence theorems of maximal pseudomonotonicity. From these results we establish the existence theorems of generalized vector equilibrium problems. We establish existence theorems of generalized vector semi-infinite programming, as applications of generalized vector equilibrium problems. This research was supported by the National Science Council of the Republic of China.     

Modified block iterative algorithm for Quasi-?-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

The purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi-?-asymptotically nonexpansive mappings and the set of an equilibrium problem. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. As an application, at the end of the paper a numerical example is given. The results presented in the paper improve and extend the corresponding results in Qin et al. [Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces, J. Comput. Appl. Math., 225, 2009, 20-30], Zhou et al. [Convergence theorems of a modified hybrid algorithm for a family of quasi-?-asymptotically nonexpansive mappings, J. Appl. Math. Compt., 17 March, 2009, doi:10.1007/s12190-009-0263-4], Takahashi and Zembayshi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 2009, 45-57], Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 2009, 11-20] and Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 2005, 257-266] and others.