COMMON COUPLED FIXED POINT FOR HYBRID PAIR OF MAPPINGS UNDER GENERALIZED NONLINEAR CONTRACTION

Abstract

Abstract. We establish a coupled coincidence and common coupled xed point theorem for hybrid pair of mappings under generalized non-linear contraction. An example supporting to our result has also beencited. We improve, extend and generalize several known results. 1. Introduction and PreliminariesLet (X;d) be a metric space and CB(X) be the set of all nonempty closedbounded subsets of X:Let D(x;A) denote the distance from xto AˆXandHdenote the Hausdor metric induced by d;that is,D(x; A) = inf a2A d(x; a)and H(A; B) = maxˆsup a2A D(a; B); sup b2B D(b; A)˙; for all A; B2CB(X):The study of xed points for multivalued contractions and non-expansive map-pings using the Hausdor metric was initiated by Markin [20]. The existenceof xed points for various multivalued contractive mappings has been studiedby many authors under di erent conditions. For details, we refer the readerto ([2], [7], [8], [9], [10], [11], [13], [14], [15], [16], [21], [25], [26], [27]) and thereference therein. The theory of multivalued mappings has applications in con-trol theory, convex optimization, di erential inclusions and economics. Nadler[21] extended the famous Banach Contraction Principle [3] from single-valuedmapping to multivalued mapping.Bhaskar and Lakshmikantham [5] introduced the notion of coupled xedpoint and mixed monotone mappings for single valued mappings. Bhaskarand Lakshmikantham [5] established some coupled xed point theorems and