Research Article Fixed Point Theorems for Manageable Contractions with Application to Integral Equations N. Hussain, 1 I. Iqbal, 2 Badriah A. S. Alamri, 1 and M. A. Kutbi 1 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, University of Sargodha, Sargodha, Pakistan Correspondence should be addressed to N. Hussain; nhusain@kau.edu.sa Received 2 September 2016; Accepted 14 November 2016; Published 16 February 2017 Academic Editor: Adrian Petrusel Copyright © 2017 N. Hussain et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper we utilize the concept of manageable functions to defne multivalued  ∗ − ∗ manageable contractions and prove fxed point theorems for such contractions. As applications we deduce certain fxed point theorems which generalize and improve Nadler’s fxed point theorem, Mizoguchi-Takahashi’s fxed point theorem, and some other well-known results in the literature. Also, we give an illustrating example showing that our results are a proper generalization of Nadler’s theorem and provide an application to integral equations. 1. Introduction and Preliminaries Te Banach contraction principle [1] is an elementary result in metric fxed point theory. Tis golden principle has been broadened in several directions by diferent authors (see [1– 18]). An interesting generalization is the elongation of the Banach contraction principle to multivalued maps, known as Nadler’s fxed point theorem [19] and Mizoguchi-Takahashi’s fxed point theorem [20]. In 2012, Samet et al. [18] defned --contractive and -admissible mappings and then Salimi et al. [17] generalized this idea by introducing function  and established fxed point theorems. Further Hasanzade Asl et al. [13] extended these notions to multivalued functions by introducing the concepts of  ∗ --contractive and  ∗ - admissible for multivalued mappings and proved some fxed point results. Hussain et al. [14] modifed the notions of  ∗ -admissible as follows. Defnition 1 (see [14]). Let T : X →2 X be a multifunction on a metric space (X,) and , : X × X → R + be two functions, where  is bounded; then T is an  ∗ -admissible mapping with respect to  if (,)≥(,) implies that  ∗ (T, T) ≥  ∗ (T, T), ,∈ X, (1) where  ∗ (A, B)= inf ∈A,∈B (,),  ∗ (A, B)= sup ∈A,∈B (,). (2) Further, Ali et al. [3] generalized the results of Hussain et al. and introduced the following defnition. Defnition 2 (see [3]). Let :→2  be a closed valued mapping on a metric space (,) and ,  :  ×  → R + Hindawi Journal of Function Spaces Volume 2017, Article ID 7943896, 10 pages https://doi.org/10.1155/2017/7943896