Bol. Soc. Paran. Mat. (3s.) v. 39 4 (2021): 83–95. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.41105 Multi-valued Fixed Point Theorem via F- contraction of Nadler Type and Application to Functional and Integral Equations ∗ Muhammad Shoaib, Muhammad Sarwar * and Poom Kumam abstract: In this work, using F-contraction of Nadler type, common multi- valued fixed point results in the setting of b-metric space are established. With the assistance of the determined results sufficient conditions for the existence of common solutions to the systems of functional and integral equations are studied. Key Words:b- metric space, F-contraction, Functional equations. Contents 1 Introduction and Preliminaries 83 2 Main Results 86 3 Application 90 4 Conclusion 93 1. Introduction and Preliminaries In this paper, CB(Λ) denotes the family of non-empty bounded and closed subsets of Λ. R + , N 0 and N signify the set of all non-negative real numbers, the set of non-negative integers and the set of positive integers respectively. Metric fixed point theory which is a vital class of non-linear analysis, is normally not only restricted to mathematical proposition, but also comes into action in most of the applied sides of pure sciences and technical fields. Among the top-listed significance of fixed points of contractive mappings defined for variety of spaces is the confirmation of the existence and uniqueness of solutions of differential, integral as well as functional equations. Nadler [13] elaborated and extended the Banach contraction principle [3] to set- valued mapping by using the Pompeiu-Hausdorff metric. The variability of these non-linear problems pare the way for finding out some more innovated and authen- tic tools which is currently more highlighted in the literature. Among these tools which is considered to be a novel tool is by Wardowski [18], in which the author has shown another kind of contractive mapping called F-contraction. Vetro [17] demon- strated some fixed point results for multi-valued operator using F-contraction and studied functional and integral equations. Czerwik [7] and Bakhtin [4] genralized * Corresponding author. 2010 Mathematics Subject Classification: 35B40, 35L70. Submitted December 27, 2017. Published May 27, 2018 83 Typeset by B S P M style. c  Soc. Paran. de Mat.