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Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications



Bagathi Srinuvasa Rao, Gajula Naveen Venkata Kishore, Muhammad Sarwar and Nalamalapu Konda Reddy
 
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ABSTRACT

The purpose of this paper was to prove some fixed point theorems in Sb-metric spaces by using (α, β)-admissible Geraghty type rational contractive conditions and some suitable examples have been provided with relevant to the results. Also, an application to Homotopy theory as well as integral equations were given.

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  How to cite this article:

Bagathi Srinuvasa Rao, Gajula Naveen Venkata Kishore, Muhammad Sarwar and Nalamalapu Konda Reddy, 2018. Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications. Journal of Applied Sciences, 18: 9-18.

DOI: 10.3923/jas.2018.9.18

URL: https://scialert.net/abstract/?doi=jas.2018.9.18
 
Received: December 09, 2017; Accepted: July 04, 2018; Published: October 30, 2018


Copyright: © 2018. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

INTRODUCTION

In Banch1 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 spaces2-5.

In recent time, Sedghi et al.6 described Sb-metric spaces by applying the concept of S and b-metric spaces and established some fixed point results in Sb-metric spaces. Subsequently to improve many author’s established numerous results on -metric spaces7-10.

In Geraghty11 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, Karapinar14 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, Chandok15 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 investigators17-25.

The aim of present article was to prove unique fixed point theorems for (α, β)-admissible Geraghty type contraction in ordered Sb-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: Sb: X3→[0, 1) be a mapping defined on a non-empty set X and b>1 be a given real number. Suppose that the mapping Sb satisfies the following properties:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

(Sb3) 0<Sb(l, m, n)<b (Sb(l, l, x)+Sb(m, m, x)+Sb(n, n, x)) for all l, m, n∈ X. Then, the function Sb is called a Sb-metric on X and the pair (X, Sb) is called a Sb-metric space.

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

Following example shows that a Sb-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 Sb-metric with b = 22(k-1).

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Lemma: ([6]) In a Sb-metric space, we have Sb(u, u, w)<2b Sb (u, u, v)+b2Sb(v, v, w).

Definition: ([6]) If (X, Sb) be a Sb-metric space. A sequence {xn} in X is said to be:

Sb-Cauchy sequence if, for each ∈>0, there is an integer n0 ∈ Z+ such that Sb (xn, xn, xm)<∈ for each n, m>n0
Sb-convergent to a point x∈ X if, for each ∈>0, there is an integer n0 ∈ Z+ such that Sb (xn, xn, x)<∈ or Sb (x, x, xn)<∈ for all n> n0 and denoted by limn→∞ xn = x

Definition: ([6]) A Sb-metric space (X, Sb) is called complete if every Sb-Cauchy sequence is Sb-convergent in X.

Lemma: ([6]) If (X, Sb) be a Sb-metric space with b>1 and suppose that {xn} is a Sb-convergent to x, then:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

for all y∈ X. Specifically, if x = y then limn→∞ Sb (xn, xn, 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, Sb) be a Sb-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 {xn} is a sequence in X such that xn→x∈X, α (xn, xn, xn+1)>1 and β (xn, xn, xn+1)>1 and Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications then there exist a sub sequence {} of {xn} such that:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications for all k∈ N and α (x, x, Ex)>1 and β (x, x, Ex)>1.

RESULTS AND DISCUSSIONS

Let Ω = {Ω/Ω: [0, ∞) →[0, 1)} be a family of function then {tn} be a any bounded sequence of positive reals such that (tn)→1 astn→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, Sb) be a Sb-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:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(1)

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(2)

For all x, y∈X, x is comparable to y, I = 3 or 4 and φ∈Φ, r>1. Where:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Theorem: Let (X, Sb, Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications ) be complete ordered Sb-metric space, α, β: X×X×X→R+ and E: X→X be satisfies:

E is an (α, β)-admissible mapping
E is an (α, β)-Geraghty type-I rational contractive mapping with i = 4
There exist x0 ∈X such that x0Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and ApplicationsE x0 with α (x0, x0, Ex0)>1 and β (x0, x0, Ex0)>1 for all x0 ≠ Ex0
Either E is continuous or X is (α, β)-regular

Then E has a unique fixed point in X.

Proof Let X0 ∈ X such that α(x0, x0, Ex0)>1 and β (x0, x0, Ex0)>1, since E is self-map, then ∃ a sequence {xn} in X such that xn+1 = E xn, n = 0, 1, 2, 3.

Case I: If xn = Exn = xn+1, then clearly xn is a fixed point of E.

Case II: Assume xn ≠ E xn, ∀n.

Since x0 Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications Ex0 = x1 and by definition of E, we have:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Since E is (α, β)-admissible mapping, α(x0, x0, Ex0) = α(x0, x0, x1)>1:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Hence by induction, we get α(xn, xn, xn+1)>1 for all n>0.

Similarly, β (xn, xn, xn+1)>1 for all n>0.

Now:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Where:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Thus:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Also:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Continuing this way we can conclude that:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

If Sb (Enx0, Enx0, En+1x0)<Sb(En+1x0, En+1x0, En+2x0), which is contradiction.

Hence Sb(En+1x0, En+1x0, En+2x0)<Sb (Enx0, Enx0, En+1x0). Thus, {Sb (Enx0, Enx0, En+1x0)} is non-increasing and must converges to a real number ξ>0. Such that limn→∞ Sb (Enx0, Enx0, En+1x0) = ξ. If ξ>0 which is contradiction. Hence ξ = 0. Thus limn→∞ Sb (Enx0, Enx0, En+1x0) = 0.

Now we prove that {Enx0} is a Cauchy sequence in (X, Sb). On contrary assume that {Enx0} is not Cauchy sequence. Then there exist ∈>0 and monotonically increasing sequence of natural numbers {mk} and {nk} such that nk>mk:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(3)

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(4)

From Eq. 3 and 4, we have:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Letting k→∞:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(5)

Where:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

But:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Now from Eq. 5:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

It is clear that:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Hence:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Which is contradiction. Hence {Enxo} is a Cauchy sequence in (X, Sb). Because of completeness of (X, Sb), there is an ν∈X with {Enxo} → ν∈(X, Sb).

Assume that E is continuous. Therefore:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

for all k∈N and α (ν, ν, E ν)>1 and β (ν, ν, E ν)>1. Since and (X, Sb) is regular, it follows is comparable to ν.

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(6)

Here:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Hence from Eq. 6:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

So, we have:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

That is:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Consequently:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

And hence Sb (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:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Where:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Which is contradiction. Unless Sb (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 Sb: X×X×X→R+ be a mapping defined as:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

where, X = [0, ∞) and Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications by pImage for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applicationsr⇔ p<r. So clearly (X, Sb, Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications) is complete ordered Sb-metric space with b = 4. Define E: X→X by:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Since (X, Sb, Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications) is complete ordered Sb-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 {pn} is a sequence in X such that α(pn, pn, pn+1)>1 and β(pn, pn, pn+1)>1 and pn →p∈ X, for all n∈N∪ {0}, then pn ⊆ [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:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Theorem: Let (X, Sb, Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications) is complete ordered Sb-metric space, E: X→X be a mapping satisfies: (I) Sb (Ex, Ex, Ey)Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and ApplicationsΩ (Sb (x, x, y)) Sb (x, x, y) for all x, y ∈ X.

(II) E is continuous or if an increasing sequence {xn}→x∈ X, then xnImage for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applicationsx ∀ n∈N. Further if x0 ∈X with x0Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and ApplicationsEx0 E. Then, E has a unique fixed point in X.

Proof: Similar proof follows from above Theorem.

Example: Let Sb: X×X×X → R+ be a mapping defined as:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

where, X= [0, ∞) and Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications by pImage for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applicationsr⇔ p<r. So clearly (X, Sb, Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications) is complete ordered Sb-metric space with b = 4. Define E: X→X by:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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, Sb) be complete Sb-metric space, U and be an open and closed subset of X such that Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications Assume that α, β: X×X×X→ℝ+, Hb: ×[0,1] → X be an (α, β)-admissible operator satisfying the following conditions:

u≠,Hb (u, κ) for each u∈ ∂U and κ ∈ [0,1] (Here ∂U is boundary of U in X)
α (u, u, Hb (u, κ)) β(v, v, Hb (v, κ))φ(4b5 Sb(Hb(u, κ), Hb(u, κ), Hb(u, κ))
<Ω (φ (Sb(u, u, v)) φ(Sb(u ,u, v))

For all u, v∈Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications and κ ∈ [0, 1], where Ω∈Ω and φ∈Φ:

There exist Mb>0 such that Sb (Hb (u, κ), Hb (u, κ), Hb (u, ζ))<Mb|κ-ζ|. For every u ∈ and κ, ζ ∈ [0, 1]. Then, Hb (.,0) has a fixed point ⇔ Hb (.,1) has a fixed point

Proof: Let the set B = {κ ∈ [0, 1]: u = Hb (u, κ) for some u ∈ U}. Since Hb (.,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 Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applicationswith κn→κ ∈ [0, 1] as n →∞.

Now, κ ∈ B must be shown. Since κn ∈ B for n = 0, 1, 2, 3,….. there exists un ∈ U with un+1 = Hb (un, κn). Since Hb is (α, β)-admissible operator:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Hence by induction α(un, un, un+1)>1 for all n>0. Similarly, β(un, un, un+1)>1 for all n>0. Consider:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Letting n→∞:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

By the hypothesis, we have:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Therefore:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

In the above Inequality:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Since Ω∈Ω, it follows:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Which yields:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(7)

Now, {un} is a Sb-Cauchy sequence in (X, Sb) is to be shown. On contrary assume that {un} is not a Sb-Cauchy sequence.

There is an ε>0 and monotone increasing sequence of natural numbers {mk} and {nk} such that nk>mk:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(8)

and:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(9)

Therefore, from Eq. 8 and 9:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Letting k→∞:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(10)

But:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

From Eq. 10:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

That is:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Which implies:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Consequently, we obtain:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and hence:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

It is a contradiction.

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(11)

Now:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

So:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

That is:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

implies:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Consequently, we get:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

and hence Sb (Hb (η, κ), Hb (η, κ), η) = 0. It follows that Hb (η, κ) = η.

Thus κ∈ B. Clearly B is closed in [0, 1]. Let κ0∈ B. Then there exists u0∈ U such that u0 = Hb (u0, κ0). Since U is open, then there exists r>0 such that Choose κ∈ (κ0-∈, κ0+∈) such that:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Then, for:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Therefore:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Thus for each fixed κ∈ (κ0-∈, κ0+∈), Hb (, ;κ):Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications(u0, r)→Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications (u0, r). Then, all the conditions of Theorem (4.1) holds. Thus, we conclude that Hb (. ;κ) has a fixed point in Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications. But this must be in U. Therefore, κ∈B for κ∈ (κ0-∈, κ0+∈). Hence(κ0-∈, κ0+∈)⊆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:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(12)

where, K:I×R→R is a continuous function and x0∈ R. Let Ω: [0, ∞)→ [0, 1), φ: [0, ∞)→[0, ∞) be a two functions defined as:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

And consider the following conditions:

If there exist a function θ: R3→R such that there is an x1 ∈ C(I), for all t∈I, we have:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

For all t∈I and for all x, y ∈ C(I ), θ(x(t), x (t), y(t))>0⇒:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

θ (xn, xn, xn+1)>0, limn→∞ inf θ (xn, xn, x)>0

Then, (12) has a unique solution.

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications
(13)

Now:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

Thus:

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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

Image for - Fixed Point Theorems in Ordered Sb-Metric Spaces by Using (α, β)-Admissible Geraghty Contraction and Applications

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 Sb-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 Sb-metric spaces. This study will help researchers to generalized different contractions in Sb-metric spaces with applications to integral equations as well as Homotopy theory. Thus, a new framework on Sb-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.

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