A New Approach for the Approximations of Solutions to a Common Fixed Point Problem in Metric Fixed Point Theory

1College of Science, King Saud University, Riyadh, Saudi Arabia 2Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450 Yahsihan, Kirikkale, Turkey 3College of Science, Geology and Geophysics Department, King Saud University, Riyadh 11451, Saudi Arabia 4Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia 5College of Engineering, Petroleum and Natural Gas Engineering Department, King Saud University, Riyadh 11421, Saudi Arabia 6Faculty of Science, Geology Department, Benha University, Benha 13518, Egypt


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 sufficient conditions for the existence of one and only one solution to (1).Moreover, we present a numerical algorithm in order to approximate such solution.Our approach is different to the existing methods in the literature.
System (1) arises in the study of different 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 fixed 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][2][3][4][5] and references therein.
The 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][7][8][9] and references therein).However, two major difficulties arise in the use of such approach.At first, the compatibility condition is not always satisfied, and in some cases it is not easy to check such condition.Moreover, the numerical approximation of the common fixed point is constructed via the axiom of choice using certain inclusions, which makes its numerical implementation difficult.
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 satisfies the following conditions: (i)  ⪯ , for every  ∈ .
Assumption (A2).The operator  :  →  is level closed from the left; that is, the set is nonempty and closed.
In order to make the lecture for the reader easy, let us give an example.
We endow  with the partial order ⪯ given by Next, define the operator  :  →  by Clearly,  :  →  is well-defined.Now, consider the set that is, Let {  } ⊂ lev  ⪯ be a sequence that converges to some  ∈  (with respect to ); that is, Since the uniform convergence implies the point-wise convergence, for all  ∈ [0, 1], we have lim Moreover, for all  ∈ [0, 1], Therefore, which proves that  :  →  is level closed from the left.
Remark 2. Note that the fact that  :  →  is level closed from the left does not imply that lev  ⪰ = { ∈  :  ⪰ } (12) is closed.Several counterexamples can be obtained.We invite the reader to check this fact by himself.
Assumption (A3).For every  ∈ , we have In order to fix our next assumption, we need to introduce the following class of mappings.We denote by Ψ the set of functions  : [0, ∞) → [0, ∞) satisfying the conditions: Here,   is the th iterate of .Any function  ∈ Ψ is said to be a (c)-comparison function.
We have the following properties of (c)-comparison functions.

Assumption (A4).
There exists a function  ∈ Ψ such that, for every (, ) ∈  × , we have Now, we are ready to state and prove our main result.

A Common Fixed Point Result and Approximations
Our main result is given by the following theorem.
The proof is complete.
Observe that Theorem 4 holds true if we replace Assumption (A2) by the following.Taking  =   (the identity operator), we obtain immediately from Theorem 4 (or from Theorem 5) the following fixed point result.