Rend. Circ. Mat. Palermo DOI 10.1007/s12215-015-0211-0 Weakly ( s, r )-contractive multi-valued operators Tayyab Kamran 1,2 · Sharafat Hussain 1 Received: 2 February 2015 / Accepted: 8 June 2015 © Springer-Verlag Italia 2015 Abstract In this paper we introduce the notion of weakly (s, r )-contractive multi-valued operator and establish some fixed point theorems for this operator. Our results generalize the results of Popescu. Keywords Multi-valued maps · (s, r )-contraction · Weakly Picard operator Mathematics Subject Classification 47H10 · 54H25 1 Introduction and preliminaries The Banach fixed point theorem [1] says that every contraction on a complete metric space has a unique fixed point. To get an analog result for multi-valued mappings, one has to equip the powerset of a set with some suitable metric. One such metric is a Hausdorff metric. The study of fixed points for multi-valued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin [5]. Following the Banach contraction principle Nadler [6] introduced the concept of multi-valued contraction and established that a multi- valued contraction possesses a fixed point in a complete metric space. Reich [8, 9] generalized the contractive condition given by Nadler [6]. Rus [10] introduced the notion of a multi- valued weakly Picard operator. Jleli et al. [2] extended multi-valued weakly Picard operator in the setting of partial Hausdorff metric spaces. Popescu [7] introduced the notion of (s, r )- contractive multi-valued operators and showed that they are weakly Picard operators. He B Tayyab Kamran tayyabkamran@gmail.com Sharafat Hussain sharafat185@gmail.com 1 Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan 2 Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad, Pakistan 123