Research Article A New Approach for the Approximations of Solutions to a Common Fixed Point Problem in Metric Fixed Point Theory Ishak Altun, 1,2 Nassir Al Arifi, 3 Mohamed Jleli, 4 Aref Lashin, 5,6 and Bessem Samet 4 1 College of Science, King Saud University, Riyadh, Saudi Arabia 2 Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey 3 College of Science, Geology and Geophysics Department, King Saud University, Riyadh 11451, Saudi Arabia 4 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 5 College of Engineering, Petroleum and Natural Gas Engineering Department, King Saud University, Riyadh 11421, Saudi Arabia 6 Faculty of Science, Geology Department, Benha University, Benha 13518, Egypt Correspondence should be addressed to Bessem Samet; bsamet@ksu.edu.sa Received 3 July 2016; Accepted 18 September 2016 Academic Editor: Filomena Cianciaruso Copyright © 2016 Ishak Altun et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We provide sufcient conditions for the existence of a unique common fxed point for a pair of mappings ,:→, where  is a nonempty set endowed with a certain metric. Moreover, a numerical algorithm is presented in order to approximate such solution. Our approach is diferent to the usual used methods in the literature. 1. Introduction and Problem Formulation Let (,) be a complete metric space and ,:→ be two given operators. In this paper, we are interested on the problem: Find ∈ such that =, =. (1) We provide sufcient conditions for the existence of one and only one solution to (1). Moreover, we present a numeri- cal algorithm in order to approximate such solution. Our approach is diferent to the existing methods in the literature. System (1) arises in the study of diferent problems from nonlinear analysis. For example, when we deal with the solvability of a system of integral equations, such problem can be formulated as a common fxed point problem for a pair of self-mappings ,:→, where  and  are two operators that depend on the considered problem. For some examples in this direction, we refer to [1–5] and references therein. Te most used techniques for the solvability of problem (1) are based on a compatibility condition introduced by Jungck [6]. Such techniques are interesting and can be useful for the solvability of certain problems (see [6–9] and references therein). However, two major difculties arise in the use of such approach. At frst, the compatibility condition is not always satisfed, and in some cases it is not easy to check such condition. Moreover, the numerical approximation of the common fxed point is constructed via the axiom of choice using certain inclusions, which makes its numerical implementation difcult. In this paper, problem (1) is investigated under the following assumptions. Assumption (A1). We suppose that  is equipped with a partial order ⪯. Recall that ⪯ is a partial order on  if it satisfes the following conditions: (i) ⪯, for every ∈. (ii) ⪯ and ⪯ imply that ⪯, for every (,,)∈ ××. (iii) ⪯ and ⪯ imply that =, for every (,)∈ ×. Hindawi Publishing Corporation Journal of Function Spaces Volume 2016, Article ID 6759320, 5 pages http://dx.doi.org/10.1155/2016/6759320