Zeroes and Fixed Points of Different Functions via Contraction Type Conditions Muhammad Usman Ali 1 , Khanitin Muangchoo-in 2,3 , and Poom Kumam 2,3(B ) 1 Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan muh usman ali@yahoo.com 2 KMUTT-Fixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand khanitin.math@mail.kmutt.ac.th, poom.kum@kmutt.ac.th 3 KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Abstract. The purpose of this paper is to introduce some results which help us to ensure the existence of fixed points and zero points of three different functions satisfying a single contraction-type condition. We also provide an example to support our result. Keywords: φ-fixed points · (φ, ψ)-fixed points · Zero points Mathematics Subject Classification: Primary 47H10 Secondary 54H25 1 Introduction and Mathematical Preliminaries The notion of φ-fixed point of a self mapping T was introduced by Jleli et al. [1]. They said that an element x of X is φ-fixed point of T : X → X and φ : X → [0, ∞) if x ∈ F T ∩ Z φ , where F T = {x ∈ X : x = Tx} and Z φ = {x ∈ X : φ(x)=0}. They proved some theorems for the existence of φ-fixed points of a self mapping T . The contraction type conditions used in the results of Jleli et al. [1] are based on the following family of functions: A family of functions F contain the functions F : [0, ∞) × [0, ∞) × [0, ∞) → [0, ∞) such that (i) max{a, b}≤ F (a, b, c) for each a, b, c ≥ 0; (ii) F (0, 0, 0) = 0; (iii) F is continuous. Following we list the most significant result of Jleli et al. [1]. c  Springer International Publishing AG 2018 L. H. Anh et al. (eds.), Econometrics for Financial Applications, Studies in Computational Intelligence 760, https://doi.org/10.1007/978-3-319-73150-6_28