Iterative Induced Sequence on Cone Metric Spaces using Fixed Point Theorems of Generalized Lipschitzian Map

Abstract

In this paper, the concept of Lipschitzian map on cone metric spaces was examined. With this modification, some fixed point theorems of generalized Lipschitz mappings with weaker conditions on generalized Lipschitz constants were proved. Let be a complete metric space and a generalized Lipschitzian map with constant  First, defining the sequence  inductively by     and if  is the unique fixed point of , it was shown that  as  and  

Moreover, we shall assume that if  is complete cone metric space,  be a normal cone with normal constant   satisfying the generalized Lipschtizian condition where  and for any  in , then we will prove that the iterative induced defined sequence converges to the fixed point.