Malaya Journal of Matematik, Vol. 8, No. 4, 1735-1742, 2020 https://doi.org/10.26637/MJM0804/0069 Common fixed point theorem for four self maps of S-metric spaces by employing compatibility of type(R) V. Naga Raju 1 * and G. Upender Reddy 2 Abstract In this paper, we define an almost generalized weakly contraction condition and prove a common fixed point theorem for two pairs of compatible mappings of type(R) in an S-metric space. As an application, we deduce a common fixed point result for four finite families of self mappings. Keywords S-metric space, common fixed point, associated sequence, compatibility mappings of type(R). AMS Subject Classification 54H25, 47H10. 1 Department of Mathematics, University College of Science, Osmania University, Hyderabad, India. 2 Department of Mathematics, Nizam College, Osmania University, Hyderabad, India. *Corresponding author: 1 viswanag2007@gmail.com Article History: Received 24 April 2020; Accepted 09 September 2020 c 2020 MJM. Contents 1 Introduction ...................................... 1735 2 Preliminaries ..................................... 1735 3 Main Results ..................................... 1737 4 Conclusion ....................................... 1742 References ....................................... 1742 1. Introduction In an attempt to generalize the notion of metric space, several mathematicians proposed different generalizations and studied fixed and common fixed point results under different contractive conditions. In this direction, Mustafa and Sims [1] introduced G-metric spaces as a generalization of metric spaces in 2006 and proved existence of some fixed points. In 2012, Sedghi, Shobe and Aliouche [2] introduced S-metric spaces by generalizing G-metric spaces and investigated some of its properties. But, in 2014 Dung, Hieu and Radojevic [3] showed by an example that the class of S-metric spaces and the class of G-metric spaces are not the same. Thereafter, in 2014, S.Sedghi, N.V.Dung [4] generalized some results in [2] and in [5], J.K. Kim, S. Sedghi, N. Shobkolaei established a common fixed point theorem for R-weakly commuting maps in S-metric spaces. For more fixed point results on S-metric spaces, we refer to ([6]-[10]). On the other hand, in 2004, Rohan et al. [11] combined the definitions of compatible, compatible of type(P) and introduced compatible of type(R) in Banach spaces and studied some fixed point results for such mappings. The aim of this paper is to define an almost weakly generalized contraction and compatibility of type(R) in S-metric spaces and to prove a common fixed point theorem using associated sequence[12] with respect to four self maps. 2. Preliminaries In this section, we recall some definitions and results which will be used in the results. Definition 2.1 ([2]). Let X be a non-empty set. Then we say that a function S : X 3 → [0, ∞) is an S -metric on X iff it satisfies the following for all α , β , γ and θ in X , (P1) S(α , β , γ )= 0 iff α = β = γ (P2) S(α , β , γ ) ≤ S(α , θ , θ )+ S(β , θ , θ )+ S(γ , θ , θ ). Here (X , S) is called a S -metric space. Example 2.2 ([2]). Let (X , d ) be a metric space. Define S : X 3 → [0, ∞) by S(α , β , γ )= d (α , β )+ d (β , γ )+ d (γ , α )