Common Fixed Point Theorems for Generalized Geraghty ( α , ψ , φ )-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty (α, ψ, φ)-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via α−admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces.


Introduction
Fixed point theory occupies a central role in the study of solving nonlinear equations of kinds Sx = x, where the function S is characterized on abstract space X.It is outstanding that the Banach contraction principle is a standout amongst essential and principal results in the fixed point theorem.It ensures the existence of fixed points for certain self-maps in a complete metric space and provides a helpful technique to find those fixed points.Many authors studied and extended it in many generalizations of metric spaces with new contractive mappings, for example, see References [1][2][3] and the references therein.
Otherwise, Hitzler and Seda [4] introduce the notation of metric-like (dislocated) metric space as a generalization of a metric space, they introduced variants of the Banach fixed point theorem in such space.Metric like spaces were revealed by Amini-Harandi [5] who proved the existence of fixed point results.This interesting subject has been mediated by certain authors, for example, see References [6][7][8].In partial metric spaces and partially ordered metric-like spaces, the usual contractive condition is weakened and many researchers apply their results to problems of existence and uniqueness of solutions for some boundary value problems of differential and Integral equations, for example, see References [9][10][11][12][13][14][15][16][17][18][19][20][21][22] and the references therein.
By using the function β ∈ S, Geraghty [23] presented the following exceptional theorem where β ∈ S, then T has a fixed point and has to be unique.
The main results of Geraghty have engaged many of authors, see References [24][25][26] and the references therein.
Recently, Amini-Harandi and Emami [27] reconsidered Theorem 1 as the framework of partially ordered metric spaces and they presented taking into account existence theorem.
Theorem 2. Let (Y, d) be a partially ordered complete metric space.Assume S:Y → Y is a mapping such that there exists u 0 ∈ Y with u 0 Su 0 and α ∈ F such that Hence, S has a fixed point supported that either S is continuous or Y is such that if an increasing sequence {u n } → u, then u n ≤ u for all n.
In 2015, Karapinar [28] demonstrated the following specific results: Theorem 3. [28] Let (Y, σ) be a complete metric-like space.Assume that S:Y → Y is a mapping.If there exists for all u, v ∈ Y, then S has a unique fixed point u * ∈ Y with σ(u * , u * ) = 0.
The notion of quasi-contraction presented by Reference [29], is known as one of the foremost common contractive self-mappings.
A mapping S:Y → Y is expressed to be a quasi contraction if there exists 0 ≤ λ < 1 such that for any u, v ∈ Y.
In this paper, we show the generalized Geraghty (α, ψ, φ)-quasi contraction type mapping in partially ordered metric like space, then we present some fixed and common fixed point theorems for such mappings in an ordered complete metric-like space.We investigate this new contractive mapping as a generalized weakly contractive mapping in our main results, then we display an example and an application to support our obtained results.

Preliminaries
In this section, we review a few valuable definitions and assistant results that will be required within the following sections.Definition 1. [5] Let Y be a nonempty set.A function σ:Y × Y → [0, ∞) is expressed to be a metric-like space on X if for any u, v, z ∈ Y, the accompanying stipulations satisfied: The pair (Y, σ) is called a metric-like space.
Obviously, we can consider that every metric space and partial metric space could be a metric-like space.However, this assertion isn't valid.Example 1. [5] Let Y = {0, 1} and We note that σ(0, 0) ≤ σ(0, 1).So, (Y, σ) is a metric-like space and at the same time it is not a partial metric space.
Additonally, each metric-like σ on Y create a topology τ σ on Y whose use as a basis of the group of open σ-balls Let (Y, σ) be a metric-like space and f :Y → Y be a continuous mapping.Then Remark 1. [30] Let Y = {0, 1}, and σ(u, v) = 1 for each u, v ∈ Y and u n = 1 for each n ∈ N.Then, it is easy to see that u n → 0 and u n → 1 and so in metric-like spaces the limit of a convergent sequence is not necessarily unique.Lemma 1. [30] Let (Y, σ) be a metric-like space.Let {u n } be a sequence in Y such that u n → u where u ∈ Y and σ(u, u) = 0.Then, for all u, v ∈ Y, we have Then, we can consider (Y, σ) to be a metric-like space, but it does not satisfy the conditions of the partial metric space, as σ(0, 0) ≤ σ(0, 1).Samet et al. [31] displayed the definition of α-admissible mapping as followings: Definition 3. [32] Let S, T:X → X be two mappings and α:X × X → R be a function.We consider that the pair (S, T) is α-admissible if u, v ∈ X, α(u, v) ≥ 1 ⇒ α(Su, Tv) ≥ 1 and α(Tu, Sv) ≥ 1 Definition 4. [33] Let S:X → X and α:X × X → [0, ∞).Then, S is called a triangular α-admissible mapping if Definition 5. [32] Let S, T:X → X and α:X × X → [0, ∞).Then, (S, T) is called a triangular α-admissible mapping if (1) The pair (S, T) is α-admissible, (2) Let Ψ indicate the set of functions ψ:[0, ∞) → [0, ∞) that approve the following stipulations: (1) ψ is strictly continuous increasing, (2) for all t > 0 and φ(0) = 0. Definition 6. [12] Let (X, d, ) be a partially ordered metric space.Assume f , g:X → X are two mappings.Then: (1) For all x, y ∈ X are said to be comparable if x y or y x holds, (2) f is said to be nondecreasing if x y implies f x f y, (3) f , g are called weakly increasing if f x g f x and gx f gx for all x ∈ X, (4) f is called weakly increasing if f and I are weakly increasing, where I is denoted to the identity mapping on X.

Main Results
In this section, we present the notation of generalized Geraghty (α, ψ, φ)-quasi contraction self-mappings in partially ordered metric-like space.Then, we present some fixed and common fixed point theorems for such self-mappings.We investigate this new contractive self-mapping as a generalized weakly contractive self-mapping which is a generalization of the results of Reference [34].Results of this kind are amongst the most useful in fixed point theory and it's applications.Definition 7. Let (X, σ) be a partially ordered metric-like space and S, T:X → X be two mappings.Then, we consider that the pair (S, T) is generalized Geraghty (α, ψ, φ)-quasi contraction self-mapping if there exist holds for all elements x, y ∈ X and 0 ≤ λ < 1, where M x,y = max{σ(x, y), σ(x, Sx), σ(y, Ty), σ(Sx, y), σ(x, Ty)}.
The following two lemmas will be utilized proficiently within the verification of our fundamental result.
Proof.Since α(x 0 , Sx 0 ) ≥ 1 and S, T are α−admissible, we get By triangular α−admissibility, we get By proceeding the above process, we conclude that α(x n , x n+1 ) ≥ 1 for all n ∈ N ∪ {0}.Now, we prove that α(x n , x m ) ≥ 1, for allm, n ∈ N with n < m.Since By continuing this process, we have Lemma 4. Let (X, , σ) be a partially ordered metric-like space.Assume S, T are two self-mappings of X which the pair (S, T) is generalized (α, ψ, φ)-quasi contraction self-mappings.Fix x 1 ∈ X and define a sequence {x n } by x 2n+1 = Sx 2n and x 2n+2 = Tx 2n+1 for all n ∈ N. If lim n→∞ σ(x n , x n+1 ) = 0 and the sequence {x n } is nondecreasing, then {x n } is a Cauchy sequence.
Now, we show that the sequence {x n } is Cauchy sequence.Assume, for contradiction's sake, that {x n } isn't Cauchy sequence.Therefore, there exist > 0 and two subsequences {n k } and {m k } of the sequence {x n } such that σ(x By the above inequalities and triangle inequality property, we imply that In view of lim n→∞ σ(x n , x n+1 ) = 0 and letting k → ∞ in the above inequalities, we obtain By the triangle inequality, we have Taking the limit as k → ∞ in the above inequalities and using Equation ( 9), we get Since x n k +1 x m k and α(x n k +1 , x m k ) ≥ 1 for all k ∈ N, so by substituting x with x n k +1 and y with x m k in Equation (7), it follows that holds for all elements x, y ∈ X and 0 ≤ λ < 1, where Taking the limit as k → ∞ of the above inequality and applying Equations ( 9), (10), we get Letting k → ∞ in Equation ( 11) and using φ ∈ Φ, β ∈ S and Equation ( 12), we deduce that This is possible only if = 0. Which contradicts the positivity of .Therefore, we get the desired result.
Then, the pair (S, T) has a common fixed point z ∈ X with σ(z, z) = 0.Moreover, assume that if x 1 , x 2 ∈ X such σ(x 1 , x 1 ) = σ(x 2 , x 2 ) = 0 implies that x 1 and x 2 are comparable elements.Then the common fixed point of the pair (S, T) is unique.
For the rest, for each n assume that (σ n = 0).16), we find that ψ(σ n ) < λψ(σ n ) which is a contradiction with respect to 0 ≤ λ < 1.We deduce max{σ n−1 , σ n , Thus, It is clear that γ < 1.Therefore, the sequence {σ(x n , x n+1 )} is a decreasing sequence.Thus, there exists r ≥ 0 such that lim Now, we show that r = 0. Presume to the contrary, that is r > 0. Since β ∈ S and by using the condition of Theorem 4 and taking the limit as k → ∞ in Equation ( 18), we conclude which could be a contradiction.So r = 0.Then, Lemma 4 implies that {x n } is a Cauchy sequence and from the completeness of (X, σ), then there exists a x * ∈ X in order that Whereas, S and T are continuous, we conclude By Lemma 1 and Equation ( 19), we obtain that and lim By merging Equations ( 20) and ( 22), we deduce that σ(x * , Tx * ) = σ(Sx * , x * ).In addition, by Equations ( 21) and ( 23), we deduce that σ(Sx * , x * ) = σ(Sx * , Tx * ).So Presently, we display that σ(x * , Tx * ) = 0. Assume the opposite, that is, σ(x * , Tx * ) > 0, we get where Therefore, from Equation ( 25), we get Since ψ ∈ Ψ, we have σ(x * , Tx * ) < λσ(x * , Tx * ) which is a discrepancy.Thus, we have σ(x * , Tx * ) = 0. Hence, Tx * = x * .From Equation ( 24), we deduce that σ(x * , Sx * ) = 0. Therefore, Sx * = x * .Hence, x * is a common fixed point of S and T. To demonstrate the uniqueness of the common fixed point, we suppose that x is another fixed point of S and T. Directly, we prove that σ( x, x) = 0. Assume the antithesis, that is, σ( x, x) > 0. Since x x, we get which is a discrepancy.Thus, σ( x, x) = 0. Therefore, by the further conditions on X, we deduce that x * and x are comparable.Presently, suppose that σ(x * , x) = 0. Then which is a discrepancy with the condition of Theorem 4. Therefore, σ(x * , x) = 0. Hence, x * = x.Thus, S and T have a unique common fixed point.
It is additionally conceivable to expel the continuity of S and T by exchanging a weaker condition.(C) If {x n } is a nondecreasing sequence in X such that α(x n , x n+1 ) ≥ 1 for all n ∈ N ∪ {0} and x n → u ∈ X as n → ∞, then there exists a subsequence {x n l } of {x n } such that x n l u for all l.Theorem 5. Let (X, σ) be a partially ordered metric-like space.Assume that S, T:X → X are two self-mappings fulfilling the following conditions: (1) the pair (S, T) is triangular α-admissible, (2) there exists an x 0 ∈ X such that α(x 0 , Sx 0 ) ≥ 1, (3) the pair (S, T) is a generalized Geraghty (α, ψ, φ)-quasi contraction non-self mapping, (4) the pair (S, T) is weakly increasing, (5) (C) holds.
Then, the pair (S, T) has a common fixed point v ∈ X with σ(v, v) = 0.Moreover, suppose that if x 1 , x 2 ∈ X such σ(x 1 , x 1 ) = σ(x 2 , x 2 ) = 0 implies that x 1 and x 2 are comparable.Then, the common fixed point of the pair (S, T) is unique.
Proof.Here, we define {x n } as in the proof of Theorem 4. Clearly {x n } is a Cauchy sequence in X, then there exists v ∈ X in order that lim As a result of the condition of Equation ( 5), there exists a subsequence {x n l } of {x n } in order that x n l v for all l.Therefore, x n l and v are comparable.In addition, from Equation ( 13) on taking limit as n → ∞ and using Equation ( 27), we get From the definition of α yields that α(x n l , v) ≥ 1 for all l.Now by applying Equation ( 6), we have where Letting l → +∞ and using Equations ( 27) and ( 28), we have Case I: Assume that lim l→∞ M x 2n l ,v = σ(v, Tv).
Regarding the concept of ψ, we deduce that σ(v, Tv) < λσ(v, Tv) which is a discrepancy.Hence, we get that σ(v, Tv) = 0.As a result of (σ 1 ), we have v = Tv.
Case II: Assume that lim l→∞ M x 2n l ,v = σ(v, Sv).Then, arguing like above, we get v = Sv.Thus, v = Sv = Tv.Uniqueness of the fixed point is follows from the Theorem 4. This completes the proof.
Hence, the hypotheses of Corollary 1 hold with ψ(t) = t, λ = 1 2 and φ(t) = t 2 .Thus, S has a unique fixed point, that is, the integral Equation (40) has a unique solution in X.

Conclusions
We have introduced some common fixed point results for generalized (α, ψ, φ)-quasi contraction self-mapping in partially ordered metric-like spaces.We have generalized weakly contractive mapping as we used quasi contraction self-mapping, α-admissible mapping, triangular α-admissible mapping and ψ, φ as strictly increasing and continuous functions.We have provided an example and application to show the superiority of our results over corresponding (common) fixed point results.Alternatively, we suggest finding new results by replacing the single-valued mapping with multi-valued mapping.Furthermore, we suggest generalizing more results in other spaces like b-metric space, metric-like space, and others.Otherwise, we suggest using our main results for non-self-mapping to establish the existence of an optimal approximate solution.