Generalized G-KKM Theorems in Generalized Convex Spaces and Their Applications

Some new generalized G-KKM and generalized S-KKM theorems are proved under the noncompact setting of generalized convex spaces. As applications, some new minimax inequalities, saddle point theorems, a coincidence theorem, and a fixed point theorem are given in generalized convex spaces. These theorems improve and generalize many important known results in recent literature.     

KKM type theorems and generalized vector equilibrium problems in noncompact FC-spaces

A new KKM type theorem is first proved under noncompact setting of FC-spaces. As applications of the KKM type theorem, we establish some new existence theorems of solutions for generalized vector equilibrium problems under noncompact setting of FC-spaces. These theorems improve and generalize many known results in literature.     

Generalized 2-KKM theorems and their applications in hyperconvex metric spaces

In this work, we first define the 2-KKM mapping and the generalized 2-KKM mapping on a metric space, and then we apply the property of the hyperconvex metric space to get a KKM theorem and a fixed point theorem without a compactness assumption. Next, by using this KKM theorem, we get some variational inequality theorems and minimax inequality theorems.     

Common fixed point theorems for three maps in discontinuous Gb metric spaces

In [Aghajani A, Abbas M, Roshan JR. Common fixed point of generalized weak contractive mappings in partially ordered G_b-metric spaces. Filomat, 2013, in press], using the concepts of G-metric and b-metric Aghajani et al. defined a new type of metric which is called generalized b-metric or G_b-metric. In this paper, we prove a common fixed point theorem for three mappings in G_b-metric space which is not continuous. An example is presented to verify the effectiveness and applicability of our main result.     

Backward Φ-shifts and universality

In this paper we consider spaces of sequences which are valued in a topological space E and study generalized backward shifts associated to certain selfmappings of E. We characterize their universality in terms of dynamical properties of the underlying selfmappings. Applications to hypercyclicity theory are given. In particular, Rolewicz's theorem on hypercyclicity of scalar multiples of the classical backward shift is extended.     

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.     

W-G-F-KKM mapping, intersection theorems and minimax inequalities in FC-space

By using basic KKM theorem, a new matching theorem and some minimax inequalities for set-valued mappings defined on the FC-spaces are proved under very weak assumptions. These results generalized many known results from the recent literature.     

Solutions of system of generalized vector quasi-equilibrium problems in locally G-convex uniform spaces

In this paper we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally G-convex uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally G-convex uniform spaces. These theorems are new and generalize some known results in the literature.     

The matching theorems and coincidence theorems for generalized R-KKM mapping in topological spaces

In this paper we present some new matching theorems with open cover and closed cover by using the generalized R-KKM theorems [L. Deng, X. Xia, Generalized R-KKM theorem in topological space and their applications, J. Math. Anal. Appl. 285 (2003) 679-690] in the topological spaces with property (H). As applications, some coincidence theorems are established in topological spaces. Our results extend and generalize some known results.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

L-凸空间中一个新的GLKKM定理及其对带上下界的广义平衡问题的应用(英文)

在非紧L-凸空间中建立了一个新的GLKKM定理。作为应用,获得了带上下界的广义平衡问题的解的存在定理.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

An intersection theorem in L-convex spaces with applications

A new intersection theorem is obtained in L-convex spaces without linear structure. As its applications, a fixed point theorem, a maximal element theorem, a coincidence theorem, some new minimax inequalities and a saddle point theorem are given in L-convex spaces. Our results generalize many known theorems in the literature.     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

KKM property and fixed point theorems in metric spaces

In this paper, first part, we establish some fixed point theorems for a k-set contraction map on the nearly-subadmissible subsets of a complete metric space. The second part, we deduce a generalization of the approximate fixed point theorem for the lower semicontinuous mappings on a metric space.     

Some fixed-point theorems and minimax inequalities in FC-space

By applying the existence theorem of maximal elements, some new collectively fixed-point theorems for a family of set-valued mappings defined on the product space of noncompact FC-space are proved and some new theorems about minimax inequality involving two functions are given to show the relations of fixed-point theorem and minimax inequality in FC-spaces. These results improve and generalize many important results in the recent literature.     

Approximation of fixed points of uniformly R-subweakly commuting mappings

We introduce a new class of uniformly R-subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically S-nonexpansive mappings in a Banach space with uniformly Gâteaux differentiable norm.     

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.     

On the topological classification of dynamic equations on time scales

This paper considers the topological classification of non-autonomous dynamic equations on time scales. In this paper we show, by a counterexample, that the trivial solutions of two topologically conjugated systems may not have the same uniform stability. This is contrary to the expectation that two topologically conjugated systems should have the same topological structure and asymptotic behaviors. To counter this mismatch in expectation, we propose a new definition of strong topological conjugacy that guarantees the same topological structure, and in particular the same uniform stability, for the corresponding solutions of two strongly topologically conjugated systems. Based on the new definition, a new version of the generalized Hartman–Grobman theorem is developed. We also include some examples to illustrate the feasibility and effectiveness of the new generalized Hartman–Grobman theorem.     

拓扑空间上的KKM定理,不动点定理,广义变分不等式

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型定理以及若干个变形结果,然后作为应用给出了非紧的拓扑空间上不动点定理和广义变分不等式解的存在性定理.