Studies on common fixed points of single and multi valued mappings

Abstract

Fixed point theory is a beautiful mixture of analysis, topology and geometry. Over the past five decades or so, the theory of fixed points has been revealed as one of the most powerful tools in the study of nonlinear phenomena. Fixed point theorems give the conditions under which single or multi-valued self-mappings have a solution. Since a fixed point of a self-mapping can be viewed as a common fixed point of the mapping paired with the identity mapping, common fixed point theorems provide a wider class of fixed point theorems. In this thesis, we study common fixed point theorems in classical, fuzzy and intuitionistic fuzzy metric spaces and discuss generalization of some known results in the classical metric space. Further we extend some established results in classical metric space to fuzzy and intuitionistic fuzzy metric spaces. We define a modified weakly contractive condition for a hybrid pair of self mappings in classical metric space and prove common fixed point theorem thereby generalizing the results of Peter Z. Daffer and Hideaki Kaneko and I.Beg and M.Abbas. We present the application of these results to invariant approximations. We prove the fuzzy versions of the results of Aamir and El Moutawakil in fuzzy metric space, employing the property (E.A) of pair of single valued self mappings. We replace the completeness by some sufficient conditions to obtain fixed point theorems for multi-valued self-mappings in fuzzy metric space. A common fixed point theorem for sequence of mappings is derived. The notion of Common property (E.A) is extended to fuzzy metric space and as a sequel the existence of coincident and common fixed points of hybrid pairs of mappings is established. We extend the common property (E.A) to intuitionistic fuzzy metric space and obtain some common fixed point theorems. Also, we prove common fixed point theorems for weakly contractive mappings. newline newline newline