**INTRODUCTION**

In Banch^{1} introduced the notion of Banach contraction principle. It is most fundamental tool in nonlinear analysis and some results related with generalization of various type of metric spaces^{2-5}.

In recent time, Sedghi *et al*.^{6} described S_{b}-metric spaces by applying the concept of S and b-metric spaces and established some fixed point results in S_{b}-metric spaces. Subsequently to improve many author’s established numerous results on -metric spaces^{7-10}.

In Geraghty^{11} studied a generalization of Banach contraction principle. In Samet *et al*.^{12} initiated the concept of α-contractive and α-admissible mappings and proved fixed point theorems on complete metric spaces for such class of mappings. In Cho *et al*.^{13} initiated the concept of α-Geraghty contractive type mappings. On the other hand, Karapinar^{14} established the existence of a unique fixed point for a triangular α-admissible mapping which is a generalized α-ψ-Geraghty contractive type mapping. Later on, Chandok^{15} illustrated the theory of (α, β)-admissible Geraghty type contractive mappings. Also very recently, Gupta *et al*.^{16} proved some fixed point results in ordered metric spaces under the (ψ, β)-admissible Geraghty contractive type mappings.

Subsequently, this type of research has been studied by several investigators^{17-25}.

The aim of present article was to prove unique fixed point theorems for (α, β)-admissible Geraghty type contraction in ordered S_{b}-metric spaces. The obtained results generalized, unified and modified some recent theorems in the literature. Some suitable example and an applications to Homotopy theory as well as integral equations were given here to illustrate the usability of the obtained results.

Firstly, recall some definitions, lemmas and examples.

**PRELIMINARIES**

**Definition:** ([6]) Let: S_{b}: X^{3}→[0, 1) be a mapping defined on a non-empty set X and b__>__1 be a given real number. Suppose that the mapping S_{b} satisfies the following properties:

(S_{b}3) 0<S_{b}(l, m, n)__<__b (S_{b}(l, l, x)+S_{b}(m, m, x)+S_{b}(n, n, x)) for all l, m, n∈ X. Then, the function S_{b} is called a S_{b}-metric on X and the pair (X, S_{b}) is called a S_{b}-metric space.

**Remark:** ([6]) It must be noted that, the class of S_{b}-metric spaces is definitely larger than that of S-metric spaces. In fact, each S-metric space is a S_{b}-metric space whenever b = 1.

Following example shows that a S_{b}-metric space on X need not be a S-metric spaces.

**Example:** ([6]) Let (X, S) be S-metric space and S_{*}(l, m, n) = S(l, m,n)^{k} where k>1 is a real number. Note that (X, S_{*}) is not necessarily S-metric space but S_{*} is a S_{b}-metric with b = 2^{2(k-1)}.

**Definition:** ([6]) Let (X, S_{b}) be a S_{b}-metric space. Then, we define the open ball Bs_{b} (l, r) and closed ball Bs_{b} [l, r] with centre l∈ X and radius r>0 as following respectively:

and:

**Lemma:** ([6]) In a S_{b}-metric space, we have S_{b}(u, u, w)__<__2b S_{b} (u, u, v)+b^{2}S_{b}(v, v, w).

**Definition:** ([6]) If (X, S_{b}) be a S_{b}-metric space. A sequence {x_{n}} in X is said to be:

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S_{b}-Cauchy sequence if, for each ∈>0, there is an integer n_{0} ∈ Z^{+} such that S_{b} (x_{n}, x_{n}, x_{m})<∈ for each n, m__>__n_{0} |

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S_{b}-convergent to a point x∈ X if, for each ∈>0, there is an integer n_{0} ∈ Z^{+} such that S_{b} (x_{n}, x_{n}, x)<∈ or S_{b} (x, x, x_{n})<∈ for all n__>__ n_{0} and denoted by lim_{n}_{→∞} x_{n} = x |

**Definition:** ([6]) A S_{b}-metric space (X, S_{b}) is called complete if every S_{b}-Cauchy sequence is S_{b}-convergent in X.

**Lemma:** ([6]) If (X, S_{b}) be a S_{b}-metric space with b__>__1 and suppose that {x_{n}} is a S_{b}-convergent to x, then:

and:

for all y∈ X. Specifically, if x = y then lim_{n}_{→∞} S_{b} (x_{n}, x_{n}, y) = 0.

**Definition:** Let E: X→X be a self-mapping and α, β: X×X×X→R^{+} defined on non-empty set X. Then, the mapping E is said to be (α, β)-admissible mapping, if α(x, x, y)__>__1 and β(x, x, y)__>__1 implies α (Ex, Ex, Ey)__>__1 and β (Ex, Ex, Ey)__>__1 for all x, y∈X.

**Definition:** Let (X, S_{b}) be a S_{b}-metric space, α, β: X×X×X→[0, ∞] be a mappings defined on non-empty set X. We say that X is a (α, β)-regular if {x_{n}} is a sequence in X such that x_{n}→x∈X, α (x_{n}, x_{n}, x_{n+1})__>__1 and β (x_{n}, x_{n}, x_{n+1})__>__1 and then there exist a sub sequence {} of {x_{n}} such that:

and:

and for all k∈ N and α (x, x, Ex)__>__1 and β (x, x, Ex)__>__1.

**RESULTS AND DISCUSSION**S

Let Ω = {Ω/Ω: [0, ∞) →[0, 1)} be a family of function then {t_{n}} be a any bounded sequence of positive reals such that (t_{n})→1 ast_{n}→0. Let Φ = {Φ: Φ: [0, ∞)→[0, ∞)} be a family of functions such that Φ is continuous, strictly increasing and Φ(0) = 0.

**Definition:** Let (X, S_{b}) be a S_{b}-metric space, α, β: X×X×X→R^{+} and E: X→X is said to be (α, β)-Geraghty type-I and type-II rational contractive mapping if, there exists Ω∈Ω such that for all x, y∈X, satisfies the following conditions:

For all x, y∈X, x is comparable to y, I = 3 or 4 and φ∈Φ, r__>__1. Where:

**Theorem: **Let (X, S_{b}, ) be complete ordered S_{b}-metric space, α, β: X×X×X→R^{+} and E: X→X be satisfies:

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E is an (α, β)-admissible mapping |

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E is an (α, β)-Geraghty type-I rational contractive mapping with i = 4 |

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There exist x_{0} ∈X such that x_{0}E x_{0} with α (x_{0}, x_{0}, Ex_{0})__>__1 and β (x_{0}, x_{0}, Ex_{0})__>__1 for all x_{0} ≠ Ex_{0} |

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Either E is continuous or X is (α, β)-regular |

Then E has a unique fixed point in X.

Proof Let X_{0} ∈ X such that α(x_{0}, x_{0}, Ex_{0})__>__1 and β (x_{0}, x_{0}, Ex_{0})__>__1, since E is self-map, then ∃ a sequence {x_{n}} in X such that x_{n+1} = E x_{n}, n = 0, 1, 2, 3.

**Case I:** If x_{n} = Ex_{n} = x_{n+1}, then clearly x_{n} is a fixed point of E.

**Case II:** Assume x_{n} ≠ E x_{n}, ∀n.

Since x_{0} Ex_{0} = x_{1} and by definition of E, we have:

Since E is (α, β)-admissible mapping, α(x_{0}, x_{0}, Ex_{0}) = α(x_{0}, x_{0}, x_{1})__>__1:

Hence by induction, we get α(x_{n}, x_{n}, x_{n+1})__>__1 for all n__>__0.

Similarly, β (x_{n}, x_{n}, x_{n+1})__>__1 for all n__>__0.

Now:

Where:

Thus:

Also:

Continuing this way we can conclude that:

If S_{b} (E^{n}x_{0}, E^{n}x_{0}, E^{n+1}x_{0})__<__S_{b}(E^{n+1}x_{0}, E^{n+1}x_{0}, E^{n+2}x_{0}), which is contradiction.

Hence S_{b}(E^{n+1}x_{0}, E^{n+1}x_{0}, E^{n+2}x_{0})__<__S_{b} (E^{n}x_{0}, E^{n}x_{0}, E^{n+1}x_{0}). Thus, {S_{b} (E^{n}x_{0}, E^{n}x_{0}, E^{n+1}x_{0})} is non-increasing and must converges to a real number ξ__>__0. Such that lim_{n}_{→∞} S_{b} (E^{n}x_{0}, E^{n}x_{0}, E^{n+1}x_{0}) = ξ. If ξ>0 which is contradiction. Hence ξ = 0. Thus lim_{n}_{→∞} S_{b} (E^{n}x_{0}, E^{n}x_{0}, E^{n+1}x_{0}) = 0.

Now we prove that {E^{n}x_{0}} is a Cauchy sequence in (X, S_{b}). On contrary assume that {E^{n}x_{0}} is not Cauchy sequence. Then there exist ∈>0 and monotonically increasing sequence of natural numbers {m_{k}} and {n_{k}} such that n_{k}>m_{k}:

and:

From Eq. 3 and 4, we have:

Letting k→∞:

Where:

But:

Now from Eq. 5:

It is clear that:

Hence:

Which is contradiction. Hence {E^{n}x_{o}} is a Cauchy sequence in (X, S_{b}). Because of completeness of (X, S_{b}), there is an ν∈X with {E^{n}x_{o}} → ν∈(X, S_{b}).

Assume that E is continuous. Therefore:

Now, assume that X is (α- β)regular. Therefore, there exists a sub sequence {x_{nk}} of {x_{n}} such that:

for all k∈N and α (ν, ν, E ν)__>__1 and β (ν, ν, E ν)__>__1. Since and (X, S_{b}) is regular, it follows is comparable to ν.

Suppose E ν≠ν. From (1) and by the definition of φ, by known Lemma:

Here:

Hence from Eq. 6:

So, we have:

That is:

Consequently:

And hence S_{b} (ν, ν, Eν) = 0 that is ν = Eν. Therefore, ν is fixed point of E.

Further to prove the uniqueness, suppose that ν* is also anther fixed point of E such that ν ≠ ν* and α (ν, ν, E ν)__>__1, α (ν*, ν*, E ν*)__>__1 and β (ν, ν, E ν)__>__1, β (ν*, ν*, E ν*)__>__1.

Consider:

Where:

Which is contradiction. Unless S_{b} (ν, ν, ν*) = 0, that is ν = ν*. Hence E has a unique fixed point.

**Corollary:** In the hypothesis of above Theorem, replace i = 3 in place of i = 4. Then, E has a unique fixed point.

**Example:** Let S_{b}: X×X×X→R^{+} be a mapping defined as:

where, X = [0, ∞) and by pr⇔ p__<__r. So clearly (X, S_{b}, ) is complete ordered S_{b}-metric space with b = 4. Define E: X→X by:

Also define α, β: X×X×X→R^{+} and Ω: [0, ∞]→[0, 1), φ: [0, ∞)→[0, ∞) as:

and:

Since (X, S_{b}, ) is complete ordered S_{b}-metric space. We show that E is an (α, β)-admissible mapping. Let p, q∈ X, if α(p, p, q)__>__1 and β(p, p, q)__>__1 then p, q∈ [0,1]. On the other hand, for all p ∈ [0,1] then E(p)__<__1. It follows α(Ep, Ep, Eq)__>__1 and β(Ep, Ep, Eq)__>__1. Therefore, the predication holds. In support of the above argument α(0, 0, E0)__>__1 and β(0, 0, E0)__>__1. Now, if {p_{n}} is a sequence in X such that α(p_{n}, p_{n}, p_{n+1})__>__1 and β(p_{n}, p_{n}, p_{n+1})__>__1 and p_{n} →p∈ X, for all n∈N∪ {0}, then p_{n} ⊆ [0, 1] and hence p∈ [0, 1]. This implies α(p, p, Ep)__>__1 and β(p, p, Ep)__>__1 . Let p, q∈ [0, 1]. Then:

Hence, the given inequality is satisfied. Otherwise α(p, p, Ep) β(q, q, Eq) = 0. Then:

Therefore, all the conditions are satisfied of above Theorem and 0 is unique fixed point of E.

**Theorem: **Let (X, S_{b}, ) is complete ordered S_{b}-metric space, E: X→X be a mapping satisfies: (I) S_{b} (Ex, Ex, Ey)Ω (S_{b} (x, x, y)) S_{b} (x, x, y) for all x, y ∈ X.

(II) E is continuous or if an increasing sequence {x_{n}}→x∈ X, then x_{n}x ∀ n∈N. Further if x_{0} ∈X with x_{0}Ex_{0} E. Then, E has a unique fixed point in X.

**Proof:** Similar proof follows from above Theorem.

**Example:** Let S_{b}: X×X×X → R^{+} be a mapping defined as:

where, X= [0, ∞) and by pr⇔ p__<__r. So clearly (X, S_{b}, ) is complete ordered S_{b}-metric space with b = 4. Define E: X→X by:

for all p∈ X, also define Ω: [0, ∞)→[0, 1), by:

Then, by above Theorem, 0 is unique fixed point of E.

**Theorem:** In the hypotheses of above Theorem, replace (2) in place of (1). Then, E has a unique fixed point.

**Corollary:** In the hypotheses of above Theorem, replace i = 3 in place of i = 4. Then, E has a unique fixed point.

**APPLICATIONS**

**Application to homotopy**

**Theorem:** Let (X, S_{b}) be complete S_{b}-metric space, U and be an open and closed subset of X such that Assume that α, β: X×X×X→ℝ^{+}, H_{b}: ×[0,1] → X be an (α, β)-admissible operator satisfying the following conditions:

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u≠,Hb (u, κ) for each u∈ ∂U and κ ∈ [0,1] (Here ∂U is boundary of U in X) |

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α (u, u, H_{b} (u, κ)) β(v, v, H_{b} (v, κ))φ(4b^{5} S_{b}(H_{b}(u, κ), H_{b}(u, κ), H_{b}(u, κ)) |

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__<__Ω (φ (S_{b}(u, u, v)) φ(S_{b}(u ,u, v)) |

For all u, v∈ and κ ∈ [0, 1], where Ω∈Ω and φ∈Φ:

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There exist M_{b}__>__0 such that S_{b} (H_{b} (u, κ), H_{b} (u, κ), H_{b} (u, ζ))__<__M_{b}|κ-ζ|. For every u ∈ and κ, ζ ∈ [0, 1]. Then, H_{b} (.,0) has a fixed point ⇔ H_{b} (.,1) has a fixed point |

**Proof:** Let the set B = {κ ∈ [0, 1]: u = H_{b} (u, κ) for some u ∈ U}. Since H_{b} (.,0) has a fixed point in U, so 0 ∈ B.

Now, prove B is closed as well as open in [0, 1] and hence by the connectedness B = [0, 1]. Let with κ_{n}→κ ∈ [0, 1] as n →∞.

Now, κ ∈ B must be shown. Since κ_{n} ∈ B for n = 0, 1, 2, 3,….. there exists u_{n} ∈ U with u_{n+1} = H_{b} (u_{n}, κ_{n}). Since H_{b} is (α, β)-admissible operator:

and:

Hence by induction α(u_{n}, u_{n}, u_{n+1})__>__1 for all n__>__0. Similarly, β(u_{n}, u_{n}, u_{n+1})__>__1 for all n__>__0. Consider:

Letting n→∞:

By the hypothesis, we have:

Therefore:

In the above Inequality:

Since Ω∈Ω, it follows:

Which yields:

Now, {u_{n}} is a S_{b}-Cauchy sequence in (X, S_{b}) is to be shown. On contrary assume that {u_{n}} is not a S_{b}-Cauchy sequence.

There is an ε>0 and monotone increasing sequence of natural numbers {m_{k}} and {n_{k}} such that n_{k}>m_{k}:

and:

Therefore, from Eq. 8 and 9:

Letting k→∞:

But:

From Eq. 10:

That is:

Which implies:

Consequently, we obtain:

and hence:

It is a contradiction.

Hence {u_{n}} is a S_{b}-Cauchy sequence in (X, S_{b}). By completeness there exists η∈U such that:

Now:

So:

That is:

implies:

Consequently, we get:

and hence S_{b} (H_{b} (η, κ), H_{b} (η, κ), η) = 0. It follows that H_{b} (η, κ) = η.

Thus κ∈ B. Clearly B is closed in [0, 1]. Let κ_{0}∈ B. Then there exists u_{0}∈ U such that u_{0} = H_{b} (u_{0}, κ_{0}). Since U is open, then there exists r>0 such that Choose κ∈ (κ_{0}-∈, κ_{0}+∈) such that:

Then, for:

Letting n→∞ and applying φ on both sides, then:

Therefore:

Thus for each fixed κ∈ (κ_{0}-∈, κ_{0}+∈), H_{b} (, ;κ):(u_{0}, r)→ (u_{0}, r). Then, all the conditions of Theorem (4.1) holds. Thus, we conclude that H_{b} (. ;κ) has a fixed point in . But this must be in U. Therefore, κ∈B for κ∈ (κ_{0}-∈, κ_{0}+∈). Hence(κ_{0}-∈, κ_{0}+∈)⊆B. Clearly B is open in [0, 1].

Similar process can be used to prove the converse.

**Applications to integral equations**

**Theorem:** Consider the I.V.P:

where, K:I×R→R is a continuous function and x_{0}∈ R. Let Ω: [0, ∞)→ [0, 1), φ: [0, ∞)→[0, ∞) be a two functions defined as:

And consider the following conditions:

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If there exist a function θ: R^{3}→R such that there is an x_{1} ∈ C(I), for all t∈I, we have: |

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For all t∈I and for all x, y ∈ C(I ), θ(x(t), x (t), y(t))__>__0⇒: |

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For any point x of a sequence {x_{n}} of points in C(I) with: |

θ (x_{n}, x_{n}, x_{n+1})__>__0, lim_{n→∞} inf θ (x_{n}, x_{n}, x)__>__0

Then, (12) has a unique solution.

**Proof:** The integral equation of I.V.P (12) is:

Let X = C (I) be the space of all continuous functions defined on I and let S_{b} (x, y, z) = (|y+z-2x|+|y-z|)^{2} for x, y, z ∈X. Then (X, S_{b}) is a complete S_{b}-metric space, also define E: X→X by:

Now:

Thus:

With θ(x(t), y(t))__>__0 for all t∈I. Define α, β: X×X×X→[0, ∞) by:

Then, obviously E is an (α, β)-admissible, for all, y∈X, then:

It follows from Eq. 2, E has a unique fixed point in X.

**CONCLUSION**

This study presents some fixed point results by using (α, β)-admissible Geraghty type rational contractive conditions defined on ordered S_{b}-metric spaces and suitable examples that supports the main results. Also, applications to Homotopy theory as well as integral equations are provided.

**SIGNIFICANCE STATEMENT**

This study proposed a framework to established fixed point results by using (α, β)-admissible Geraghty type rational contractions in ordered S_{b}-metric spaces. This study will help researchers to generalized different contractions in S_{b}-metric spaces with applications to integral equations as well as Homotopy theory. Thus, a new framework on S_{b}-metric spaces may be arrived at.

**ACKNOWLEDGMENT**

The authors are very thankful to the reviewers and editors for their valuable comments, remarks and suggestions which improved the paper in good form.