Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

Sufficient conditions for the existence of a common fixed point of generalized -weakly contractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature. 1. Introduction and Preliminaries It is well known that Banach’s fixed point theorem for contraction mappings is one of the pivotal result of analysis. Let be a metric space. A mapping is said to be contraction if there exists such that for all , If the metric space is complete, then the mapping satisfying (1.1) has a unique fixed point. A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity. Kannan [1, 2] proved the following result, giving an affirmative answer to the above question. Theorem 1.1. If , where is a complete metric space, satisfies where and , then has a unique fixed point. The mappings satisfying (1.2) are called Kannan type mappings. A similar type of contractive condition has been studied by Chatterjea [3] and he proved the following result. Theorem 1.2. If , where is a complete metric space, satisfies where and , then has a unique fixed point. In Theorems 1.1 and 1.2 there is no requirement of continuity of . A map is called a weakly contractive (see [4–6]) if for each , where is continuous and nondecreasing, if and only if and . If we take , , then a weakly contractive mapping is called contraction. A map is called -weakly contractive (see [7]) if for each , where is a self-mapping, is continuous and nondecreasing, if and only if and . If we take , , then a -weakly contractive mapping is called -contraction. Further, if identity mapping and , , then a -weakly contractive mapping is called contraction. A map is called a generalized weakly contractive (see [5]) if for each , where is continuous such that if and only if . If we take , , then inequality (1.6) reduces to (1.3). Choudhury [5] shows that generalized weakly contractive mappings are generalizations of contractive mappings given by Chatterjea (1.3), and it constitutes a strictly larger class of mappings than given by Chatterjea. A map is called a generalized -weakly contractive [8] if for each , where is a self-mapping, is continuous such that if and only if . If identity mapping, then generalized -weakly contractive mapping is generalized weakly contractive. For a nonempty subset of a metric space and , an element is said to be a best approximant to