Generalizations Of Caristi's Fixed Point Theorem And Existence Of Positive Periodic Solutions Of Retarded Functional Differential Equations

This paper is divided into two parts. The first part is concerned with Caristi's fixed point theorem, while the existence of positive periodic solutions of retarded functional differential equations is discussed in the second part.Fixed point theory has been great attention to mathematicians (especially, applied mathematicians) and engineers. Recently, many mathematicians have studied and tried to find the relation among all kinds of fixed point theorems. Many new fixed point theorems have appeared, the most famous one of which is Caristi's fixed point theorem (1976). The first part of this paper deals with set-valued Caristi's fixed point theorem on basis of Caristi's work. A new set-valued Caristi's fixed point theorem, as well as its detail proof, is presented by using partial order theory, which generalized some related results .The theory of monotone dynamical systems has given rise to great attention. The second part of this paper is concerned with the theory of retarded functional differential equations (RFDEs) and applications to the existence of positive periodic solutions of RFDEs. By combining the theory of monotone semiflows generated by RFDEs and the fixed point theorem of abstract operators, the existence of positive periodic solutions of RFDEs is established under the suitable conditions which are useful and easy to verify. Nontrivial applications of our results to some periodic ecological systems are also presented.