Nonlinear Analysis 74 (2011) 1436–1444 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Multivalued generalized nonlinear contractive maps and fixed points Abdul Latif a,∗ , Afrah A.N. Abdou b a Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b Science Faculty For Girls, King Abdulaziz University, P.O. Box 14884, Jeddah 21434, Saudi Arabia article info Article history: Received 1 August 2010 Accepted 4 October 2010 MSC: 54H25 54C60 Keywords: Metric space Fixed point Hausdorff metric Multivalued nonlinear contractive map w-distance abstract We introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps. Consequently, several known fixed point results are either improved or generalized including the corresponding recent fixed point results of Ciric [L.B. Ciric, Multivalued nonlinear contraction mappings, Nonlinear Anal. 71 (2009) 2716–2723], Klim and Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139], Feng and Liu [Y. Feng, S. Liu, Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112] and Mizoguchi and Takahashi [N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188]. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction and preliminaries Let (X , d) be a metric space, 2 X a collection of nonempty subsets of X , and CB(X ) a collection of nonempty closed bounded subsets of X , Cl(X ) a collection of nonempty closed subsets of X , K (X ) a collection of nonempty compact subsets of X . For any A, B ∈ CB(X ), let H(A, B) = max{sup x∈A d(x, B), sup y∈B d(y, A)}, where d(x, B) = inf y∈B d(x, y). H is called the Hausdorff metric induced by d. An element x ∈ X is called a fixed point of a multivalued map T : X → 2 X if x ∈ T (x). We denote Fix(T ) ={x ∈ X : x ∈ T (x)}. A sequence {x n } in X is called an orbit of T at x 0 ∈ X if x n ∈ T (x n−1 ) for all n ≥ 1. A map f : X → R is called lower semicontinuous if any sequence {x n } ⊂ X with x n → x ∈ X implies that f (x) ≤ lim inf n→∞ f (x n ). Investigations on the existence of fixed points of multivalued contractions in metric spaces were initiated by Nadler [1]. Using the Hausdorff metric, he established the following multivalued version of the well known Banach contraction principle. Theorem 1.1. Let (X , d) be a complete metric space and let T : X → CB(X ) be a map such that for a fixed constant h ∈ (0, 1) and for each x, y ∈ X, H(T (x), T (y)) ≤ hd(x, y). Then Fix(T )  =∅. ∗ Corresponding author. Tel.: +966 2 6952000. E-mail addresses: alatif@kau.edu.sa, Latifmath@yahoo.com (A. Latif), aabdou@kau.edu.sa (A.A.N. Abdou). 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.10.017