Some Fixed Point Theorems for Multi-valued Maps

Abstract

notion of Picard operator was introduced by Rus. The notion of weakly Picard operators was introduced and used by Rus and his collaborators. Berinde extended the notion of weakly Picard operator to multi-valued case. Kamran introduced the notion of f-weakly Picard operators. Popescu introduced the notion of (s,r)-contractive multi-valued operator. He presented some basic problem for fixed point and strict fixed point theory for (s,r)-contractive multivalued operator. In this dissertation, we extend the notion of (s,r)-contractive multi-valued operator and proved some fixed points theorems for newly defined contraction. In chapter one, we recollect some basic definitions and results, which are needed for subsequent chapters. In chapter two, we review some fixed point theorems for (s,r)-contractive multi-valued operator. We study, in detail, some results obtained by Popescu. In chapter three, we define the notion of weakly (s,r)contractive multi-valued operator and f-(s,r)-contractive multi-valued operator and using these conditions we obtain some new fixed point theorems.