Surveys in Mathematics and its Applications (Apr 2018)
Some fixed point theorems involving rational type contractive operators in complete metric spaces
Abstract
Let (X, d) be a complete metric space and T from X to X a mapping of X. In 1975 Dass and Gupta introduced the following rational type contractive condition to prove a generalization of Banach's Fixed Point Theorem: For α,β∈[0,1), such that α + β < 1, we have for all x, y ∈ X, d(Tx,Ty)≤ α * d(y,Ty)*(1+d(x,Tx))⁄(1+d(x,y))+β*d(x,y), where T is continuous. There are several generalization and extension of Dass and Gupta's result under the hypothesis that T is continuous and α + β<1. In this paper, we prove some fixed point theorems in a complete metric space setting by employing more general rational type contractive conditions than the above one. We show in our results that the continuity of the above operator T is unnecessary and the restrictive condition that α + β < 1 is also removed. Our results generalize and extend those of Das and Gupta and several known results in the literature.
