Continuous Selections and Fixed Points forφ-maps with Their Applications to Section Problems

The concept of finitely continuous topological space is introduced and the basic properties of the space are given. Several continuous selection theorems and fixed point theorems for Ф-maps are established, and as applications of the above fixed point theorems, some section problems are discussed. The results generalize and improve many corresponding conclusions.

CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES

In this paper,we prove that if X is an almost convex and 2-strictly convex space,linear operator T:X→Y is bounded,N(T)is an approximative compact Chebyshev subspace of X and R(T)is a 3-Chebyshev hyperplane,then there exists a homogeneous selection T^(σ)of T^(■)such that continuous points of T^(σ)and T^(■)are dense on Y.

Continuous selections for pseudo-lower semicontinuous set-valued mappings on convex structure spaces and hyperspace contractibility

SINCE 1956, Michael’s continuous selection theory has been applied to functional analysis,topology, approximation theory and other mathematical fields. In this letter, the concept ofthe pseudo-lower semicontinuity is introduced, and a convex structure of metric space is de-fined. A continuous selection theorem for pseudo-lower semicontinuity is given. This

The Continuous Selections for the Metric Projection

Some results concerning weakly continuous selection for set-valued mappingare given and, applied to metric projection. Let Y be a subspace of a Banach space X.If Y is a separable reflexive Banach space,reinoved a first category subset, the metricprojection is weakly lower semicontinnous and admits a weakly continuous selection.

On s-Reflexive Spaces and Continuous Selections

This paper deals with the s-reflexive spaces introduced by Yang and Zhao. The authors prove that every s-reflexive Hausdorff space is zero-dimensional, and indicate a close relationship between the theory of s-reflexive spaces and that of continuous selections.Several examples relating to s-reflexivity are given.

CONTINUOUS SELECTION THEOREMS FOR FAN-BROWDER MAPPINGS IN TOPOLOGICAL SPACES AND THEIR APPLICATIONS

The concept of Fan-Browder mappings was first introduced in topological spaces without any convex structure. Then a new continuous selection theorem was obtained for the Fan-Browder mapping with range in a topological space without any convex structure and noncompact domain. As applications, some fixed point theorems, coincidence theorems and a nonempty intersection theorem were given. Both the new concepts and results unify and extend many known results in recent literature.

Continuous Homogeneous Selections of Set-Valued Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T~ are given. The results are an answer to the problem posed by Nashed and Votruba.

INTERSECTION THEOREMS CONCERNING NONCOMPACT SETS WITH H-CONVEX SECTIONS AND THEIR APPLICATIONS

In this paper, we prove some intersection theorems concerning noncompact sets with H-convex sections which generalize the corresponding results of Ma, Fan, Tarafdar, Lassonde and Shin-Tan to H-spaces without the linear structure and to noncompact setting. An application to von Neumann type minimax theorems is given.

SYSTEM OF COINCIDENCE THEOREMS IN PRODUCT TOPOLOGICAL SPACES AND APPLICATIONS (Ⅰ)

A new notion of finite continuous topological space（in short, FC-space） with out convexity structure was introduced. A new continuous selection theorem was established in FC-spaces. By applying the continuous selection theorem, some new coincidence theorems for two families of set-valued mappings defined on product space of noncompact FC-spaces are proved under much weak assumptions. These results generalize many known results in recent literature. Some applications will be given in a follow-up paper.

Generalized Quasi-equilibrium Problems without Open Fibers

In this paper, we first prove some new selection and fixed point theorems in generalized convex spaces. Then, we establish some existence theorems of quasi-equilibrium and generalized quasi-equilibrium without the conditions of open fibers, by applying our selection and fixed point theorems.

Continuous Selection,Collectively Fixed Points and System of Coincidence Theorems in Product Topological Spaces

Some new continuous selection theorems are first proved in noncompact topological spaces. As applications, some new collectively fixed point theorems and coincidence theorems for two families of set-valued mappings defined on product space of noncompact topological spaces are obtained under very weak assumptions. These results generalize many known results in recent literature.

Aerobic granules cultivated and operated in continuous-flow bioreactor under particle-size selective pressure

A novel method based on the selective pressure of particle size （particle-size cultivation method, PSCM） was developed for the cultivation and operation of aerobic granular sludge in a continuous-flow reactor, and compared with the conventional method based on the selective pressure of settling velocity （settling-velocity cultivation method, SVCM）. Results indicated that aerobic granules could be cultivated in continuous operation mode by this developed method within 14 days. Although in the granulation process, under particle-size selective pressure, mixed liquor suspended solids （MLSS） in the reactor fluctuated greatly and filamentous bacteria dominated the sludge system during the initial operation days, no obvious difference in profile was found between the aerobic granules cultivated by PSCM and SVCM. Moreover, aerobic granules cultivated by PSCM presented larger diameter, lower water content and higher specific rates of nitrification, denitrifieation and phosphorus removal, but lower settling velocity. Under long term operation of more than 30 days, aerobic granules in the continuous-flow reactor could remain stable and obtain good chemical oxygen demand （COD）, NH4^＋-N, total nitrogen （TN） and total phosphorus （TP） removal. The results indicate that PSCM was dependent on the cultivation and maintenance of the stability of aerobic granules in continuous-flow bioreactors.

A New Fixed Point Theorem in Noncompact Hyperconvex Metric Spaces and Its Application to Saddle Point Problems

In this paper, a new fixed point theorem is established in noncompact hyperconvex metric spaces. As applications, a continuous selection and its fixed point theorem, an existence theorem for maximal elements, a Ky Fan minimax inequality and an existence theorem for saddle points are obtained.

A Fixed Point Theorem for Set-Valued Mapping in Abstract Convex Space with Application

A new fixed point theorem and the selection property for upper semi-continuous set-valued mappings in abstract convexity space are established. As their applications the existence of Nash equilibrium for n-person non-cooperative generalized games is proved.