Common Fixed Points for Mappings under Contractive Conditions of ( α , β , ψ )-Admissibility Type

In this paper, we introduce the notion of (α, β, ψ)-contraction for a pair of mappings (S, T) defined on a set X. We use our new notion to create and prove a common fixed point theorem for two mappings defined on a metric space (X, d) under a set of conditions. Furthermore, we employ our main result to get another new result. Our results are modifications of many existing results in the literature. An example is included in order to show the authenticity of our main result.


Introduction and Preliminaries
The importance of fixed point theories lies in finding and proving the uniqueness of solutions for many questions of Applied Sciences such as Physics, Chemistry, Economics, and Engineering.The pioneer mathematician in the area of fixed point theory was Banach, who established and proved the first fixed point theorem named the "Banach contraction theorem" [1].After that, many authors formulated and established many contractive conditions to modify the Banach contraction theorem in many different directions.Khan [2] introduced the altering distance mapping to formulate a new contractive condition in fixed point theory in order to extend the Banach fixed point theorem to new forms.For some extension to the Banach contraction theorem, we ask the readers to see References [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].Recently, Abodyeh et al. [21] introduced a new notion, named almost perfect function, to formulate new contractive conditions to modify and extend some fixed point theorems known in the literature.Now, we mention the notions of altering distance function and almost perfect function:

2.
ψ is a nondecreasing and continuous function.

If for all sequence (s
One of the most important notions in fixed point theory to derive new contractive conditions is α-admissibility, which were introduced by Samet et al. [22].Then, E. Karapıner et al. [23] generated the concept of triangular α-admissibility.In meantime, Abdeljawad [24] expanded the notion of α-admissibility to a pair of functions.For some fixed point theorems on α-admissibility, we direct readers to read References [25][26][27][28][29][30][31]. The notions of α−admissibility mapping and α-admissibility for a pair of mappings are introduced as follows: Definition 3 ([22]).Let S be a self-mapping on X and α: X × X → R + ∪ {0} be a function.Then, S is called α-admissible if for all v, w ∈ X with α(v, w) ≥ 1 it holds α(Sv, Sw) ≥ 1.
In our work we need the following definitions: Definition 6 ([30]).Let d be a metric on a set X and α, β: X × X → R + ∪ {0} be functions.Then, X is called α, β-complete if and only if {x n } is a Cauchy sequence in X and α(x n , x n+1 ) ≥ β(x n , x n+1 ) for all n ∈ N imply (x n ) converges to some x ∈ X. Definition 7 ([30]).Let d be a metric on a set X and α, η: X × X → R + ∪ {0} be functions.A self-mapping T on X is called α, β-continuous if {x n } is a sequence in X, x n → x as n → ∞ and α(x n , x n+1 ) ≥ β(x n , x n+1 ) for all n ∈ N imply Tx n → Tx as n → ∞.
In this paper, we introduce a new contractive condition of type (α, β, ψ)-admissibility for a pair of mappings (S, T) defined on a set X. We utilize our new contractive condition to formulate and prove a common fixed point theorem for two self-mappings defined on a metric space (X, d) under a set of conditions.Then, we utilize our main result to obtain some fixed point results.
This paper is divided into three sections.In the first section, we collect all necessary definitions and preliminaries that cover the subject of our paper.In Section 2, we give our new definitions and our main result.In addition, we give an example to validate our main result.In Section 3, we write our conclusion.
Then, e v+w ≥ e v .So v + w ≥ v and hence w is a nonnegative real number.Therefore α(Tv, Sw) ≥ β(Tv, Sw).

Definition 9.
Let ψ be a nondecreasing function on R + ∪ {0}.We call ψ a perfect function if the following conditions hold: Then, ψ is a perfect function.
The main result of this paper is: On the set X, let α, β : X × X → R + ∪ {0} be two functions and S, T: X → X be two mappings.Assume there exists a metric d on X such that the following hypotheses hold: 1. (X, d) is an α, β-complete metric space.
Then, both mappings S and T have a common fixed point.
For n ∈ N ∪ {0}, we get on ψ implies that x 2n+1 = x 2n+2 , a contradiction.Therefore, max{kψ(d(x 2n , x 2n+1 )), kψ(d(x 2n+1 , x 2n+2 ))} = kψ(d(x 2n , x 2n+1 )). Hence, Using arguments similar to the above, we may show that Combining Equations ( 3) and ( 4) together, we reach By recurring Equation ( 5) n-times, deduce On allowing n → +∞ in Equation ( 6), we get Condition (2) on the function ψ implies that We intend to prove that (x n ) is a Cauchy sequence in X, take n, m ∈ N with m > n.We divide the proof into four cases: Case 1: n is an odd integer and m is an even integer.
Therefore, there exist s ∈ N and an odd integer h such that n = 2s + 1 and m = n By permitting n, m → +∞ in above inequalities and considering Equation ( 7), we have The properties of ψ imply that Case 2: n and m are both even integers.
Applying the triangular inequality of the metric d, we have Letting n → +∞ and in view of Equations ( 8) and ( 9), we get lim n,m→+∞ d(x n , x m ) = 0.
Case 3: n is an even integer and m is an odd integer.
Applying the triangular inequality of the metric d, we have On permitting n → +∞ and considering Equations ( 8) and ( 9), we get lim n,m→+∞ d(x n , x m ) = 0.
Case 4: n and m are both odd integers.Applying the triangular inequality of the metric d, we have On permitting n → +∞ and in view of Equations ( 8) and ( 9), we get lim n,m→+∞ d(x n , x m ) = 0. Combining all cases with each other, we conclude that lim n,m→+∞ d(x n , x m ) = 0.
Thus, we conclude that (x n ) is a Cauchy sequence in X.The α, β-completeness of the metric space (X, d) ensures that there is x ∈ X such that x n → x.Using the α, β-continuity of the mappings S and T, we deduce that x 2n+1 = Sx 2n → Sx and x 2n+2 = Tx 2n+1 → Tx.By uniqueness of limit, we obtain Sx = Tx = x.Thus, x is a fixed point of S.
Then S and T have a common fixed point.
To prove (6), let z, w ∈ X be such that α(z, w) ≥ β(z, w).Hence, S and T satisfy Definition 2.3 for k = 4 5 .Therefore, S and T satisfy all the conditions of Theorem 1.Therefore, S and T have a common fixed point.

1.
By taking S = T in Theorem 1 and Corollary 1, we can formulate and get some fixed point results.

Conclusions
New notions of (α, β)-admissibility and (α, β)-contraction for a pair of self-mappings on a set X are given.According to these notions, we introduced and proved our main result.Additionally, we gave an example to validate our main result.