**INTERNATIONAL JOURNAL OF ENGINEERING, SCIENCE AND - Volume 6, Issue 8,, December 2017 (Special Issue)**

**Pages:** 87-91

**Date of Publication:** 24-Dec-2017

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**Fixed Point Theorems in Metric Spaces and its Applications to **

**Cone Metric Spaces**

**Author:** Dr. Capt. K. Sujatha*

**Category:** Engineering, Science and Mathematics

**Abstract:**

Fixed point theorems provide conditions under which maps (single or multi-valued) have solutions. The theory itself is a beautiful mixture of analysis, topology, and geometry. In particular, fixed point techniques have been applied in such diverse fields as Biology, Chemistry, Economics, Engineering, Game Theory, and Physics. Fixed point theory plays an important role in functional analysis, approximation theory, differential equations and applications such as boundary value problems etc. The concept of a metric space was introduced in 1906 by M. Frechet [2]. One extension of metric spaces is the so called cone metric space. In cone metric spaces, the metric is no longer a positive number but a vector, in general an element of a Banach space equipped with a cone. In 2007, Huang & Xian [3] introduced the notion of a cone metric space and established some fixed point theorems in cone metric spaces, an ambient space which is obtained by replacing the real axis in the definition of the distance, by an ordered real Banach space whose order is induced by a normal cone P. In this paper we introduce the notion of quasi generalized contraction pair of self maps on a cone metric space and observe that this notion is weaker than the notion of generalized contraction pair of self maps introduced in [1]. Also we prove a common fixed point theorem for a quasi generalized contraction pair of self maps and provide examples analyzing various situations

**Keywords:** Complete metric space, cone metric space, contraction pair, fixed point, generalized contraction pair, Quasi generalized contraction pair.