BASIC THEORY AND APPLICATIONS OF PROBABILISTIC METRIC SPACES(Ⅰ)

This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces. Menger PM-spaces and probabilistic normed linear spaces (PN-space ) are given and the characterizations of various probabilistically bounded sets are presented. As applications we utilize these results obtained in this paper to study the linear operator theory and fixed point theory on PM-spaces.     

(Φ,△)-TYPE PROBABILISTIC CONTRACTOR AND SOLUTIONS FOR A NONLINEAR OPERATOR EQUATIONS IN MENGER PN-SPACES

The purpose of this paper is to introduce the coneept of (Φ,△)-type probabilistic contractor in Menger PN-spaces and to study the existence and uniqueness of solutions for the nonlinear operator equations with such probabilistic contractor in Menger PN-spaces.The results presented in this paper improve and extend the corresponding results in [1] and [4-8].     

ON THE PROBABILISTIC INNER PRODUCT SPACES

In this paper a new defintion for probabilistic inner product spaces is given.By virtueof this definition,some convergence theorems,Schwarz inequality and the orthogonalconcept for probabilistic inner product spaces are established and introduced.Moreover,the relationship between this kind of spaces and probabilistic normed spaces is consideredalso.     

FIXED POINT THEOREMS IN PROBABILISTIC INNER PRODUCT SPACES

The purpose of this paper is to Introduce a new moditying detinition for probabilistisinner product space,and to establish several new fixed point theorems for mappings on suchkind of spaces.As an example of applications.weutilize the results of this paper to study theexistence and uniqueness of solution of Uryson’s integral equation in IV.     

TOPOLOGICAL DEGREE THEORY AND FIXED POINT THEOREMS IN PROBABILISTIC METRIC SPACES

The Leray-Schauder topological degree theory is established in the probabilistic linearnormed spaces.Based.on this theory,some fixed point theorems for mappings in theprobabilistic linear normed spaces are shown.     

Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semi-groups in Banach spaces

The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces. The results presented in this paper extend and improve the corresponding results of some people.     

ON THE EXISTENCE OF SOLUTIONS FOR EQUATIONS WITH ACCRETIVE MAPPINGS IN PROBABILISTIC NORMED SPACES

The purpose of this paper is to introduce the concept of accretive mapping inprobabilistic normed space(PN space,in short)and to study the existence problem ofsolutions for equations with accretive mappings in PN space and some existence theoremsare shown.     

Mann and Ishikawa iterative approximation of solutions form-accretive operator equations

The purpose of this paper is to study the Mann and Ishikawa iterative approximation of solutions for m-accretive operator equations in Banach spaces. The results presented in this paper extend and improve some authors’ recent results.     

THE THEORY AND APPLICATIONS OF GENERALIZED H-SPACE

This paper brings forward the concept of generalized H-spaces which extends the concepts of H-spaces and almost probabilistic metric spaces.In this paper,the uniformity and properties far generalized H-space are considered.The conditions of metrization and the form of metric functions for generalized H-spaces,H-spaces and Menger PM-spaces are given and the characteristics of completeness and compactness for generalized H-spaces are presented.The results of this paper generalize and unify some recent results of[1-2,8,10].     

ON UNIFORM CONVERGENCE OF LINEAR OPERATORS ON THE PROBABILISTIC NORMED SPACE

In this paper,we introduced the notion of uniform convergence of the linear operatorson the probabilistic normed space,and the notion of probabilistic distance between theoperators,which describes the above convergence completely.In terms of these notions,weobtained the essential features of the continuity of operators,and of the uniformconvergence of operator sequences,and we also obtained the closure of continuity andcomplete continuity under the operation of the limit of uniform convergence.     

NONLINEAR SEMIGROUPS AND DIFFERENTIAL INCLUSIONSIN PROBABILISTIC NORMED SPACES

I.IntroductionTheconceptofaccretivemappingsisoffundamentalimportanceinthetheoryofsetvaluednonlinearoperators,differentialequationsandpartialdifferentialequationsinBanachspaces,whichwasintroducedindependentlybyF.E.Browdert31andT.Katolll].Ontheotherhand,manyauthorshavedoneconsiderableworkonsemigroupsofnonlinearcontractions,differentialequationsandevolutionequationsinBanachspacesandHilbertspaceal"='4'?.8.l:,13j.Recently,theauthorsintroducedtheconceptofaccretivemappingst"andsomeelementaryprope…     

Probabilistic norm of operators and resonance theorems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｓｔ,ａｓｔｈｅｄｅｆｉｎｉｔｉｏｎｓａｂｏｕｔｓｏｍｅｃｏｎｃｅｐｔｓｉｎｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｌｉｎｅａｒｓｐａｃｅ（ｂｒｉｅｆｌｙ,ＰＮｓｐａｃｅ）ｃｏｐｉｅｄｔｈｅｃｏｒｅｓｐｏｎｄｉｎｇ...     

Solution and stability of mixed type functional equation in non-Archimedean random normed spaces

The generalized stability of the Euler-Lagrange quadratic mappings in the framework of non-Archimedean random normed spaces is proved. The interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean spaces, and the theory of functional equations is presented.     

On the probabilistic inner product spaces

In this paper a new definition for probabilistic inner product spaces is given. By virtue of this definition, some convergence theorems, Schwarz inequality and the orthogonal concept for probabilistic inner product spaces are established and introduced. Moreover, the relationship between this kind of spaces and probabilistic normed spaces is considered also.Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Topological degree theory and fixed point theorems in probabilistic metric spaces

The Leray-Schauder topological degree theory is established in the probabilistic linear normed spaces. Based on this theory, some fixed point theorems for mappings in the probabilistic linear normed spaces are shown.The projects supported by National Natural Science Foundation of China.     

NEW VERSION OF KKMTHEOREM IN PROBABILISTIC METRIC SPACES WITH APPLICATIONS

ＮＥＷＶＥＲＳＩＯＮＯＦＫＫＭＴＨＥＯＲＥＭＩＮＰＲＯＢＡＢＩＬＩＳＴＩＣＭＥＴＲＩＣＳＰＡＣＥＳＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＺｈａｎｇＳｈｉｓｈｅｎｇ（张石生）ＹｅｏｌＪｅＣｈｏ（赵烈济）ＷｕＸｉａｎｇ（吴鲜）（ＲｅｃｅｉｖｅｄＭａｙ７，１９９...     

BASIC THEORY AND APPLICATIONS OF PROBABILISTIC METRIC SPACES (Ⅰ)

This paper is devoted to the study of the basic theory and applications of probabilistic metric spaces (PM-space). In this paper the topological structure and properties for PM-space are considered. The conditions of metrization and the form of metric functions for PM-spaces. Menger PM-spaces and probabilistic normed linear spaces (PN-space) are given and the characterizations of various probabilistically bounded sets are presented. As applications we utilize these results obtained in this paper to study the linear operator theory and fixed point theory on PM-spaces. Projects supported by the Science Fund of the Chinese Academy of Sciences.     

On probabilistic norm of a linear operators and space of operators

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１］,ＸｉａｏｉｎｔｒｏｄｕｃｅｄｔｈｅｎｏｔｉｏｎｏｆｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｏｆａｌｉｎｅａｒｏｐｅｒａｔｏｒｏｎＭｅｎｇｅｒｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｓｐａｃｅ（ｂｒｉｅｆｙ,ＭｅｎｇｅｒＰＮ＿ｓｐａｃｅ）．Ａｎｄｂｙｕｓｉｎｇｔｈｅｎｏｔｉｏｎ,ｕｎｄｅｒｔｈｅｃｏｎｄｉｔｉｏｎｏｆｔ＿ｎｏｒｍΔｓａｔｉｓｆｙｉｎｇΔ（ｔ,ｔ）≥ｔ（ｔ∈（０,１］）,ｈｅｇａｖｅａｄｅｓｃｒｉｐｔｉｏｎｏｆｂｏｕｎｄｅｄｎｅｓｓｏｆｌｉｎｅａｒｏｐｅ…