Commun. Korean Math. Soc. 35 (2020), No. 4, pp. 1107–1121 https://doi.org/10.4134/CKMS.c190388 pISSN: 1225-1763 / eISSN: 2234-3024 COMMON FIXED POINT FOR GENERALIZED MULTIVALUED MAPPINGS VIA SIMULATION FUNCTION IN METRIC SPACES Swati Antal and U. C. Gairola Abstract. The purpose of this paper is to introduce the notion of gen- eralized multivalued Z-contraction and generalized multivalued Suzuki type Z-contraction for pair of mappings and establish common fixed point theorems for such mappings in complete metric spaces. Results obtained in this paper extend and generalize some well known fixed point results of the literature. We deduce some corollaries from our main result and provide examples in support of our results. 1. Introduction One of the fundamental and most useful results in fixed point theory is Banach Contraction Principle [8]. This result has been extended in many di- rections for single and multivalued cases on a metric space. In 1969, Nadler [19] introduced the notion of multivalued contraction mapping and show that such mapping has a fixed point on complete metric space. Then many fixed point theorems have been proved by various authors as a generalization of the Nadler’s theorem (see [4, 6, 9–11, 15, 17, 18]). Recently, F. Khojasteha et al. [14] introduced the notion of a simulation func- tion with a view to consider a new class of contraction called Z -contraction. They studied the existence and uniqueness of fixed point for Z -contraction type operators. Using the idea of a simulation function, different contractive condi- tions can be expressed in a simple and unified way. This class of Z -contraction includes a large type of non-linear contraction existing in the literature (see [1–3, 7, 13, 16, 20, 22, 23]). In this paper we introduce the notion of the generalized multivalued Z - contraction and generalized multivalued Suzuki type Z -contraction for pair of Received November 10, 2019; Revised May 24, 2020; Accepted June 4, 2020. 2010 Mathematics Subject Classification. Primary 47H10, 54H25. Key words and phrases. Complete metric space, multivalued mapping, generalized Z- contraction for pair of mappings, generalized Suzuki type Z-contraction for pair of mappings, simulation function, common fixed point. c 2020 Korean Mathematical Society 1107