# Incomplete Metric With The Application For Integral Equations of Volterra Type

Authors: Sumit Chandok; Ljiljana Paunović; Stojan Radenović
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In this article, we propose some weaker orthogonalτ,FT− type of contraction mappings in the setting of metric spaces endowed with an orthogonal relation, as well as certain sufficient criteria for the existence of fixed points for this class of mappings. To establish all of our results in the manuscript, we just used Wardowski’s strictly increasing property. Using the aforementioned results as an application, we demonstrate that the Volterra type integral equation has a solution and is stable in the Hyers-Ulam-Rassias-Wright sense.


---

## Body

## 1. Introduction

Gordji et al. [1] recently introduced the idea of orthogonal sets and extended the Banach contraction principle. They also demonstrated how their findings can be used to ensure the existence and uniqueness of solutions to first-order differential equations. Using Wardowski’s F-contraction notion [2–4], numerous studies showed the existence of fixed points (see [5–18]). The purpose of this study is to improve the concept of Wardowski’s contraction in metric spaces that are not complete in the sense of Gordji et al. [1]. In the context of a metric space equipped with an orthogonal relation, we describe some weaker orthogonal τ,FT− type contraction mappings. For this class of mappings, we also employ strictly increasing properties to construct some necessary requirements for the presence of fixed points. This, we feel, is a significant improvement over many previously published findings. We also enrich this paper with a nontrivial example. We use the aforementioned results to show that the Volterra type integral equation has a solution and is stable in the Hyers-Ulam-Rassias-Wright sense.Definition 1.1 (see [1]).
LetW be a nonempty set and ⊥ be a binary relation defined on W×W. W,⊥ is called an orthogonal set (O-set), if ⊥ satisfies the following condition:
there exists℘0∈W such that (for all x∈W, ℘0⊥x) or (for all x∈W, x⊥℘0).Example 1.2.
LetW=3,∞. Define ℘⊥n if and only if ℘≤n. It’s clear that 3⊥n applies to all n∈W and W,⊥ is an O-set.More fascinating examples may be found at [1].Definition 1.3 (see [1, 19]).
A sequence℘nn∈ℕ is referred to as a strongly orthogonal sequence (SO-sequence) if (for all n,l∈ℕ; ℘n⊥℘n+l) or (for all n,l∈ℕ; ℘n+l⊥℘n).
If (for alln∈ℕ,℘n⊥℘n+1) or (for all n∈ℕ,℘n+1⊥℘n), the sequence ℘nn∈ℕ is referred to as an orthogonal sequence (O-sequence).Definition 1.4 (see [1, 19]).
LetW,⊥ represent an O-set, W,δ a metric space, and W,δ,⊥ an orthogonal metric space (OMS). A mapping g:W⟶W is said to be strongly orthogonally continuous (SO-continuous) at ℘∈W, if we have g℘n⟶g℘, for each SO-sequence ℘n in W with ℘n⟶℘. In addition, g is called SO-continuous on W if g is SO-continuous for each ℘∈W.
A mappingg:W⟶W is called orthogonally continuous (⊥-continuous) at ℘∈W if we have g℘n⟶g℘, for each O-sequence ℘n in W with ℘n⟶℘. A mapping g is also called ⊥-continuous on W if g is ⊥-continuous for each ℘∈W.Every continuous mapping is clearly⊥-continuous, while the opposite is not true (see [1,19]).Definition 1.5 (see [1, 19]).
A self mappingT:W⟶W on an orthogonal set W,⊥ is called ⊥-preserving if ℘1⊥℘2, then T℘1⊥T℘2. Also, T:W⟶W is called a weakly ⊥-preserving if ℘1⊥℘2, then T℘1⊥T℘2 or T℘2⊥T℘1.Every⊥-preserving mapping is clearly weakly ⊥-preserving, but not the other way around (see [1]).Definition 1.6 (see [19]).
If every Cauchy SO-sequence is convergent, then an OMSW,δ,⊥ is called strongly orthogonally complete (SO-complete).Every complete metric space is clearly SO-complete, while the converse is not true (see [19]).


## 2. Fixed Point Results

Here, first we introduce some weaker orthogonalτ,FT-type contraction mappings in the context of an incomplete metric, as defined in [1,19], and then construct fixed points of mappings meeting such a class of contractions.For the purpose of simplicity, we’ll suppose that an expression -∞.0 has the value −∞.We refer toF as the family of all functions F:0,+∞⟶ℝ that satisfy:For every℘1,℘2>0, ℘1>℘2 implies F℘1>F℘2.So there are bothlim℘1⟶℘2−F℘1=F℘2− and lim℘1⟶℘2+F℘1=F℘2+ for all ℘2∈0,+∞ because it is known from mathematical analysis that the following is true for all ℘2∈0,+∞ (see [20])(1)F℘2−≤F℘2≤F℘2+.Remark 2.1.
ConsiderF:0,+∞⟶ℝ to be a strictly increasing function. Then there are following two possible outcomes:(1)
F0+=lim℘⟶0+F℘=−∞(2)
F0+=lim℘⟶0+F℘=m, for some m∈ℝ (for more details see [18,20,21]).
As a result, each strictly increasing functionF:0,+∞⟶ℝ fulfils one of the two conditions either (1) or (2) (see (Aljančić [21] Proposition 1, Section 8)). Wardowski’s second and third requirements are thus unnecessary.Example 2.2.
Let functionsF1,F2,F3:0,+∞⟶ℝ defined by:(1)
F1℘=−1/℘,(2)
F2℘=ln℘,(3)
F3℘=℘+ln℘,
ThenF1,F2,F3∈ℱ, for all ℘>0.To begin, we’ll require the following result, which is an orthogonal metric space extension of [22].Lemma 2.3.
Let℘nbe a SO-sequence in an OMSW,δ,⊥such that(2)limn⟶+∞δ℘n,℘n+1=0.If a SO-sequence℘nis not a Cauchy SO-sequence inW, then there isε>0and two sequences of positive integersmlandnlsuch thatnl>ml>land the following SO-sequences tend toε+whenl⟶+∞:(3)δ℘ml,℘nl,δ℘ml,℘nl+1,δ℘ml−1,℘nl,δ℘ml−1,℘nl+1,℘℘ml+1,℘nl+1.Proof.
If℘n is not a Cauchy SO-sequence, then there exist ε>0 and two sequences of positive integers nl and ml such that(4)nl>ml>l,δ℘ml,℘nl−1<ε,δ℘ml,℘nl≥ε,
for all positive integersl. To prove (4), assume that(5)Ξl=m∈N:there existsml≥l,dxml,xm≥ε,m>ml>l.
Obviously,Ξl≠∅ and Ξl⊆ℕ. Then by the well-ordering principle, the minimum element of Ξl exists and denoted by nl, and clearly (4) holds. Then(6)ε≤δ℘ml,℘nl≤δ℘ml,℘nl−1+δ℘nl−1,℘nl<ε+δ℘nl−1,℘nl.
Using (2), we conclude that(7)liml⟶+∞δ℘ml,℘nl=ε+.
Further,(8)δ℘ml,℘nl≤δ℘ml,℘nl+1+δ℘nl+1,℘nl.
as well as(9)δ℘ml,℘nl+1≤δ℘ml,℘nl+δ℘nl,℘nl+1.
Taking the limitl⟶+∞ and using (2) and (7), we get(10)liml⟶+∞δ℘ml,℘nl+1=ε+.
Also,(11)δ℘ml+1,℘nl+1≤δ℘ml+1,℘ml+δ℘ml,℘nl+1.
(12)δ℘nl,℘ml≤δ℘nl,℘nl+1+δ℘nl+1,℘ml+1.
Now, from (11) and (12) it follows that(13)liml⟶+∞δ℘ml+1,℘nl+1=ε+.
On the same lines, we can prove that the remaining two sequences in (3) tend to ε+.In this paper the functionF:0,+∞⟶ℝ will be strict increasing and F0+=−∞.The following assumption is required in our results. LetT:W⟶W be a self mapping on a SO-complete OMS W,δ,⊥.Property.
Sthere is a SO-sequence ℘l∈W defined by ℘l+1=Tx℘l=Tl+1℘0, for an orthogonal element ℘0∈W with ℘0⊥T℘0 or T℘0⊥℘0, such that ℘l⟶℘∗∈W and ℘l⊥℘l+m or ℘l+m⊥℘l for all l,m∈ℕ, then ℘l⊥℘∗ or ℘∗⊥℘l for all l∈ℕ.Definition 2.4.
LetT:W⟶W be a self mapping on an OMS W,⊥,δ if there exist functions τ:0,+∞⟶0,+∞ and F∈F, such that for all ℘1,℘2∈W with ℘1⊥℘2 satisfying the following hypotheses:
(c1)δT℘1,T℘2>0,
(c2)liminfq⟶t+τq>0, for all t>0,
either
(c3)τδ℘1,℘2+FδT℘1,T℘2≤Fδ℘1,℘2.
or
(c4)1/2δ℘1,T℘1<δ℘1,℘2 implies τδ℘1,℘2+FδT℘1,T℘2≤Fδ℘1,℘2.
or
(c5)τδ℘1,℘2+FδT℘1,T℘2≤FM℘1,℘2, where
M℘1,℘2=maxδ℘1,℘2,δ℘1,T℘1,δ℘2,T℘2,δ℘1,T℘2+δ℘2,T℘1/2,δT2℘1,℘1+δT2℘1,T℘2/2,δT2℘1,T℘1,δT2℘1,℘2,δT2℘1,T℘2
(respectively,M℘1,℘2=maxδ℘1,℘2,δ℘1,T℘1,δ℘2,T℘2,δ℘1,T℘2+δ℘2,T℘1/2).
T is called an orthogonal τ,FT-contraction if it satisfies (c1)-(c3).
T is called an orthogonal τ,FT-Suzuki type contraction if it satisfies (c1), (c2), (c4).
T is called a generalized orthogonal τ,FT-contraction (respectively, weak orthogonal τ,FT-contraction) if it satisfies (c1), (c2), (c5).We are now ready to provide our first outcome.Theorem 2.5.
LetT:W⟶W be a ⊥-preserving, an orthogonal τ,FT-contraction and satisfy Property S on a SO-complete OMS W,δ,⊥ (not necessarily a complete metric space). Then T has a unique fixed point.Proof.
Let℘0∈W be such that ℘0⊥T℘0 or T℘0⊥℘0. Take ℘1:=T℘0, ℘2:=T℘1=T2℘0. In this manner, we define a SO-sequence ℘n in W by ℘n+1=T℘n=Tn+1℘0 for all n∈ℕ∪0.
If℘l=℘l+1 for some l∈ℕ∪0, then ℘l=℘l+1=T℘l. So, we take ℘n≠℘n+1, and γn=δ℘n+1,℘n=δT℘n,T℘n−1, for all n∈ℕ. As T is an orthogonal τ,FT-contraction, for every n∈ℕ, we obtain(14)τδ℘n,℘n−1+FδT℘n,T℘n−1≤Fδ℘n,℘n−1.
Therefore,τγn−1+Fγn≤Fγn−1, for every n≥1.
From (c2),c>0 and n0∈ℕ exist, with τγn>c for all n≥n0. As a result, for all n≥n0 we have(15)c+Fγn+1≤τγn+Fγn+1≤Fγn.
AsF∈F, sequence γn is strictly decreasing and converging to some Λ≥0, for all n≥n0.
Takingn⟶+∞ in (15), we get(16)c+FΛ+≤FΛ+.
which is a contradiction and henceγn⟶0.
Now we must demonstrate that℘n is a Cauchy SO-sequence. Assume, on the other hand, that ℘n is not a Cauchy SO-sequence. Putting ℘1=℘ml, ℘2=℘nl in (c3), we have(17)τδ℘ml,℘nl+FδT℘ml,T℘nl≤Fδ℘ml,℘nl.
It implies that(18)τδ℘ml,℘nl+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl.
Since the SO-sequence℘n is not a Cauchy SO-sequence, by Lemma 2.3, we have δ℘ml,℘nl and δ℘ml+1,℘nl+1 tend to ε+, as l⟶+∞.
Now, using (18) and (c2), there exist c>0 and l1∈ℕ such that we get(19)c+Fδ℘ml+1,℘nl+1≤τδ℘ml,℘nl+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl.
wheneverl≥l1. That is,(20)c+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl,
forl≥l1. Taking l⟶+∞, in the last relation, we get(21)c+Fε+≤Fε+.
which contradicts to our assumption. This establishes that the SO-sequence℘n is a Cauchy SO-sequence. Since W is SO-complete, there exists ℘∗∈W such that limn⟶+∞℘n=℘∗.
By using our other assumption,℘n=T℘n−1=Tn℘0⊥℘∗ or ℘∗⊥℘n=T℘n−1=Tn℘0. Using (c3), we get(22)τδ℘n,℘∗+FδT℘n,T℘∗≤Fδ℘n,℘∗orτδ℘n,℘∗+Fδ℘n+1,T℘∗≤Fδ℘n,℘∗.
Using (c2) andF∈F, we obtain δ℘n+1,T℘∗<δ℘n,℘∗. Taking n⟶+∞, we get ℘∗=T℘∗, that is ℘∗ is a fixed point of T.
Now we’ll show that℘∗ is the only unique fixed point of T. Suppose that y∗ is another fixed point of T with y∗≠℘∗. By our choice of ℘0, ℘0⊥y∗ or y∗⊥℘0. Since T is ⊥-preserving, we have T℘0⊥T℘∗ and T℘0⊥Ty∗ or T℘∗⊥T℘0 and Ty∗⊥T℘0. Therefore,(23)τδ℘∗,y∗+Fδ℘∗,y∗=τδ℘∗,y∗+FδT℘∗,Ty∗≤Fδ℘∗,y∗.
which is a contradiction asτδ℘∗,y∗>0. Hence ℘∗=y∗ and T has a unique fixed point.
Now, we give the following result onτ,FT-Suzuki type contraction which is related to the generalization and the improvement of Theorem 2.5.Theorem 2.6.
LetT:W⟶Wbe a⊥-preserving, an orthogonalτ,FT-Suzuki type contraction and satisfy PropertySon a SO-complete OMSW,δ,⊥. ThenThas a unique fixed point.Proof.
On the similar lines of Theorem2.5, we may assume that γn=δ℘n+1,℘n=δT℘n,T℘n−1>0, for all n∈ℕ.
SinceT is an orthogonal τ,FT-Suzuki type contraction, for every n∈ℕ, we have 1/2δ℘n,℘n+1=1/2δ℘n,T℘n<δ℘n,℘n+1. So from (c4), we get(24)τδ℘n,℘n−1+FδT℘n,T℘n−1≤Fδ℘n,℘n−1.
Therefore, we haveτγn−1+Fγn≤Fγn−1, for all n∈ℕ. Now, using the similar comments in Theorem 2.5, we get γn⟶0.
Now we must demonstrate that℘n is a Cauchy SO-sequence. Assume, on the other hand, that ℘n Â is not a Cauchy SO-sequence.
So by Lemma2.3, we have δ℘ml,℘nl and δ℘ml+1,℘nl+1 tend to ε+, as l⟶+∞.
Therefore, it follows that there is somel1∈ℕ such that 1/2δ℘ml,℘nl+1<δ℘ml,℘nl, for all l∈ℕ with l≥l1. Then, by substituting ℘1=℘ml, ℘2=℘nl in (c4) for l≥l1, we have(25)τδ℘ml,℘nl+FδT℘ml,T℘nl≤Fδ℘ml,℘nl.
It implies that(26)τδ℘ml,℘nl+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl.
Now, using (c2) and (26), there exist c>0 and l2∈ℕ such that we get(27)c+Fδ℘ml+1,℘nl+1≤τδ℘ml,℘nl+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl.
wheneverl≥l2. That is,(28)c+Fδ℘ml+1,℘nl+1≤Fδ℘ml,℘nl.for l≥l2. Taking l⟶+∞ in the obtained last relation, we get(29)c+Fε+≤Fε+.
which contradicts to our assumption. This establishes that the SO-sequence℘n is a Cauchy SO-sequence. Since W is an SO-complete, there exists ℘∗∈W such that limn⟶+∞℘n=℘∗.
Now, we claim that for alln∈ℕ,(30)12δ℘n,T℘n<δ℘n,℘∗or12δT℘n,T2℘n<δT℘n,℘∗.
Now, again we supposee that there is somem∈ℕ such that(31)12δ℘m,T℘m≥δ℘m,℘∗or12δT℘m,T2℘m≥δT℘m,℘∗.
Therefore,2δ℘m,℘∗≤δ℘m,T℘m≤δ℘m,℘∗+δ℘∗,T℘m. It implies(32)δ℘m,℘∗≤δ℘∗,T℘m.
It follows from (31) and (32) that(33)δ℘m,℘∗≤δ℘∗,T℘m≤12δT℘m,T2℘m.
Since1/2δ℘m,T℘m<δxm,Txm, τδ℘m,T℘m+FδT℘m,T2℘m≤Fδ℘m,T℘m. Using (c2), we get FδT℘m,T2℘m<FT℘m,T℘m. Hence by using property of F, we get(34)δT℘m,T2℘m<δ℘m,T℘m.
It follows from (31), (33) and (34),(35)δT℘m,T2℘m<δ℘m,T℘m≤δ℘m,℘∗+δ℘∗,T℘m≤12δT℘m,T2℘m+12δT℘m,T2℘m=δT℘m,T2℘m,
which is a contradiction. Hence (30) holds.
By our assumption,℘n=T℘n−1=Tn℘0⊥℘∗ or ℘∗⊥℘n=T℘n−1=Tn℘0. So from (30) and (c4), for every n∈ℕ, either τδ℘n,℘∗+FδT℘n,T℘∗≤Fδ℘n,℘∗, or τδ℘n+1,℘∗+FδT2℘n,T℘∗≤Fδ℘n+1,℘∗ holds. Also, we can rewrite it as(36)τδ℘n,℘∗+Fδ℘n+1,T℘∗≤Fδ℘n,℘∗orτδ℘n+1,℘∗+Fδ℘n+2,T℘∗≤Fδ℘n+1,℘∗.
Further, using (c2) andF∈F, we get(37)δ℘n+1,T℘∗<δ℘n,℘∗orδ℘n+2,T℘∗<d℘n+1,℘∗.
Takingn⟶+∞, we get ℘∗=T℘∗ in both the cases.
It is easy to see the uniqueness of a fixed point ofT.Theorem 2.7.
LetT:W⟶Wbe a⊥-preserving, generalized orthogonalτ,FT-contraction and satisfy PropertySon a SO-complete OMSW,δ,⊥. ThenThas a unique fixed point.Proof.
On the similar lines of Theorem2.5, we may assume that γn=δ℘n+1,℘n=δT℘n,T℘n−1>0, for all n∈ℕ.
SinceT is a generalized orthogonal τ,FT-contraction, for every n∈ℕ, we get(38)τδ℘n,℘n−1+FδT℘n,T℘n−1≤FM℘n,℘n−1.where(39)M℘n,℘n−1=maxδ℘n−1,℘n,δ℘n−1,℘n,δ℘n,℘n+1,δ℘n−1,℘n+1+02,δ℘n+1,℘n−1+02,δ℘n+1,℘n,δ℘n+1,℘n,0=maxδ℘n−1,℘n,δ℘n,℘n+1,δ℘n−1,℘n+12≤maxδ℘n−1,℘n,δ℘n,℘n+1.
It is clear thatmaxδ℘n−1,℘n,δ℘n,℘n+1=δ℘n−1,℘n, otherwise we get a contradiction.
Therefore,τγn−1+Fγn≤Fγn−1, for all n∈ℕ. Now, using the similar comments in Theorem 2.5, we get γn⟶0.
Now we must demonstrate that℘n is a Cauchy SO-sequence. Assume, on the other hand, that ℘n Â is not a Cauchy SO-sequence. So by Lemma 2.3, we have δ℘ml,℘nl and δ℘ml+1,℘nl+1 tend to ε+, as l⟶+∞.
Putting℘1=℘nl, ℘2=℘ml in (c5), we have(40)τδ℘nl,℘ml+FδT℘nl,T℘ml≤FM℘nl,℘ml,where(41)M℘nl,℘ml=maxδ℘nl,℘ml,δ℘nl,℘nl+1,δ℘ml,℘ml+1,δ℘nl,℘ml+1+δ℘ml,℘nl+12,δ℘nl+2,℘nl+δ℘nl+2,℘ml+12,δ℘nl+2,℘ml+1,δ℘nl+2,℘ml,δ℘nl+2,℘ml+1.
Using Lemma2.3, we have liml⟶+∞M℘nl,℘ml=ε+>0.
Now, takingl⟶+∞, using (40) and (c2), there exist c>0 and l1∈ℕ such that we get(42)c+Fε+≤Fε+,
which contradicts to our assumption. This establishes that the SO-sequence℘n is a Cauchy SO-sequence. Since W is-complete, there exists ℘∗∈W such that limn⟶+∞℘n=℘∗.
By using our assumption,℘n=T℘n−1=Tn℘0⊥℘∗ or ℘∗⊥℘n=T℘n−1=Tn℘0. Using (c5), we get(43)τδ℘n,℘∗+FδT℘n,T℘∗≤FM℘n,℘∗orτδ℘n,℘∗+Fδ℘n+1,T℘∗≤FM℘n,℘∗.where M℘n,℘∗=maxδ℘n,℘∗,δ℘n,℘n+1,δ℘∗,T℘∗,δ℘n,T℘∗+δ℘∗,℘n+1/2,δ℘n+2,℘n+δ℘n+2,T℘∗/2,δ℘n+2,℘n+1,δ℘n+2,℘∗,δ℘n+2,T℘∗.
Now takingn⟶+∞, using (c2) and F∈F, we get ℘∗=T℘∗, that is ℘∗ is a fixed point of T. It is easy to see that ℘∗ is a unique fixed point of T.Remark 2.8.
Every weak orthogonalτ,FT-contraction is a generalized orthogonal τ,FT-contraction. So Theorem 2.7 is also true if we take weak orthogonal τ,FT-contraction.


## 3. Consequences of Fixed Point Results

In this section, we discuss some of the ramifications of the preceding section’s findings.First, we’ll illustrate how our findings allow us to formulate coupled fixed point theorems in O-complete orthogonal metric spaces using our results. The following definition emerges first.LetG:W×W⟶W be a given mapping. We say that ℘1,℘2∈W×W is a coupled fixed point of G if G℘1,℘2=℘1 and G℘2,℘1=℘2.Our result is based on the following simple lemma which tells a coupled fixed point is a fixed point (see Samet et al. [23]).Lemma 3.1.
LetG:W×W⟶Wbe a given mapping. Define the mappingT:Y=W×W⟶Y=W×WbyT℘1,℘2=G℘1,℘2,G℘2,℘1, for all℘1,℘2∈W×W. Then,℘1,℘2is a coupled fixed point ofGif and only if℘1,℘2is a fixed point ofT.Theorem 3.2.
LetG:W×W⟶W be a self mapping on a SO-complete OMS W,δ,⊥. Assume the following assumptions are true:(i)
there existsτ:0,+∞⟶0,+∞, such that for allx,y,u,v∈Wwithx⊥y,u⊥v,liminfq⟶t+τq>0, for allt>0,δGx,y,Gu,v>0,(44)τδx,y,u,v+FδGx,y,Gu,v≤Fδx,y,u,v,where F∈F,(ii)
Gis⊥-preserving,(iii)
If there exist SO-sequencesxn,yn∈Wdefined byxn+1=Gxn,yn=Gn+1x0,y0,yn+1=Gyn,xn=Gn+1y0,x0for orthogonal elementsx0,y0∈Xwithx0,y0⊥Gx0,y0orGx0,y0⊥x0,y0, such thatyn⟶y∗∈W,xn⟶x∗∈Wandyn⊥yn+loryn+l⊥yn,xn⊥xn+lorxn+l⊥xn, for alln,l∈ℕ, thenxn⊥x∗orx∗⊥xn,yn⊥y∗ory∗⊥yn, for alln∈ℕ.
ThenG has a coupled fixed point.Proof.
Here takeY=W×W,δ is SO -complete OMS. Define the mapping T:Y⟶Y by T℘1,℘2=G℘1,℘2,G℘2,℘1, for all ℘1,℘2∈W×W. From (44), we have(45)τδξ,η+FδTξ,η≤Fδξ,η,for all ξ=ξ1,ξ2,η=η1,η2∈Y. So using Theorem 2.5, we get the result.Remark 3.3.
On the same lines of Theorem3.2, we can prove other coupled fixed point results.We get the following result by takingτq=β>0 in Theorems 2.5 and 2.6.Corollary 3.4.
LetT:W⟶Wbe a self mapping on a SO-complete OMSW,δ,⊥. Assume the following assumptions hold:(i)
there exists someβ>0, such that for all℘1,℘2∈Wwith℘1⊥℘2,δT℘1,T℘2>0,(46)β+FδT℘1,T℘2≤Fδ℘1,℘2,
or(47)12δ℘1,T℘1<δ℘1,℘2impliesβ+FδT℘1,T℘2≤Fδ℘1,℘2.where F∈F,(ii)
Tis⊥-preserving,(iii)
PropertyS.
ThenT has a unique fixed point.The following outcome is a direct result of Corollary3.4.Corollary 3.5.
LetT:W⟶W be a self mapping on a SO-complete OMS W,δ,⊥. Assume the following assumptions hold:(i)
Tis⊥-preserving,(ii)
PropertyS,(iii)
there exists someβi>0,i=1,2,3,4,5such that for all℘1,℘2∈Wwith℘1⊥℘2,δT℘1,T℘2>0,1/2δx,Tx<δ℘1,℘2any of the following contracting conditions are true:(48)β1+δT℘1,T℘2≤δ℘1,℘2;β2−1δT℘1,T℘2≤−1δ℘1,℘2;β3−1δT℘1,T℘2+δT℘1,T℘2≤δT℘1,T℘2−1δ℘1,℘2;β4+11−eδT℘1,T℘2≤−11−eδ℘1,℘2;β5+1e−δT℘1,T℘2−eδT℘1,T℘2≤−1e−δ℘1,℘2−eδ℘1,℘2.In each of these circumstances,T has a unique fixed point.Proof.
The proof follows directly from Corollary3.4, as each functions F1γ=γ, F2γ=−1/γ, F3γ=−1/γ+γ, F4γ=1/1−eγ and F5γ=1/e−γ−eγ, where γ=dx,y>0 is strictly increasing on 0,+∞.Forτs=β>0 in Theorem 2.7, we have the following result.Corollary 3.6.
LetT:W⟶W be a self mapping on a SO-complete OMS W,δ,⊥. Assume the following assumptions hold:(i)
there exists someβ>0, such that for all℘1,℘2∈Wwith℘1⊥℘2,δT℘1,T℘2>0,(49)β+FδT℘1,T℘2≤FM℘1,℘2,
whereM℘1,℘2=maxδ℘1,℘2,δ℘1,T℘1,δ℘2,T℘2,δ℘1,T℘2+δ℘2,T℘1/2,δT2℘1,℘1+δT2℘1,T℘2/2,δT2℘1,T℘1,δT2℘1,℘2,δT2℘1,T℘2.andF∈F,(ii)
Tis⊥-preserving,(iii)
PropertyS.
ThenT has a unique fixed point.Corollary 3.7.
LetT:W⟶W be a self mapping on a SO-complete OMS W,δ,⊥. Assume the following assumptions hold:(i)
Tis⊥-preserving,(ii)
PropertyS,(iv)
there exists someβi>0,i=1,2,3such that for all℘1,℘2∈Wwith℘1⊥℘2,δT℘1,T℘2>0, the following contractive conditions hold(50)β1+δT℘1,T℘2≤M℘1,℘2;β2−1δT℘1,T℘2+δT℘1,T℘2≤MT℘1,T℘2+1M℘1,℘2;β3−e1/δT℘1,T℘2+eδT℘1,T℘2≤−e1/M℘1,℘2+eM℘1,℘2,where M℘1,℘2=maxδ℘1,℘2,δ℘1,T℘1,δ℘2,T℘2,δ℘1,T℘2+δ℘2,T℘1/2,δT2℘1,℘1+δT2℘1,T℘2/2,δT2℘1,T℘1,δT2℘1,℘2,δT2℘1,T℘2.Then in each of these casesT has a unique fixed point.Proof.
The proof immediately follows from Corollary3.6, as each functions F1γ=γ, F2γ=−1/γ+γ, and F3γ=−e1/γ+eγ, where γ=dx,y>0 is strictly increasing on 0,+∞.TakingFγ=lnγ, γ>0 as a result of Corollary 3.5, we get the following result.Corollary 3.8.
LetT:W⟶W be a self mapping on a SO-complete OMS W,δ,⊥. Assume the following assumptions hold:(i)
there exists someβ>0, such that for all℘1,℘2∈Wwith℘1⊥℘2,δT℘1,T℘2>0,(51)δT℘1,T℘2≤e−βδ℘1,℘2,where F∈F,(ii)
Tis⊥-preserving,(iii)
PropertyS.Then the mappingT has a unique fixed point.Corollary 3.9.
LetT:W⟶Wbe a self mapping on a complete metric spaceW,δ. Assume that there exists someβ>0, such that for all℘1,℘2∈W,TsatisfiesδT℘1,T℘2>0,(52)β+FδT℘1,T℘2≤Fδ℘1,℘2,where F∈F. Then the mapping T has a unique fixed point.Proof.
Define a binary relation onW by ℘1⊥℘2 if and only if δT℘1,T℘2>0impliesβ+FδT℘1,T℘2≤Fδ℘1,℘2.
SinceT satisfies (52), we have ℘0⊥℘2, for any fixed ℘0∈W and for all ℘2∈W. Thus W,⊥ is an O-set and it is easy to see the O-completeness of W. Furthermore, T is ⊥-continuous, ⊥-preserving and T satisfies (46). Hence using Corollary 3.4, we get the result.Example 3.10.
LetW=0,12 with usual metric δ. Define the binary relation ⊥ on W by ℘1⊥℘2 if xy≤℘1∨℘2 where ℘1∨℘2=℘1or℘2. Then W,δ,⊥ is O-complete OMS. Define the mapping T:W⟶W by(53)T℘1=℘13,0≤℘1≤30,℘1>3.
Let℘1⊥℘2. We may assume that ℘1℘2≤℘1, without loss of generality. Then the following cases are satisfied:Case I.
If℘1=0 and 0≤℘2≤3, then T℘1=0 and T℘2=℘2/3.Case II.
If℘1=0 and ℘2>3, then T℘1=0=T℘2.Case III.
If℘2≤2 and ℘1≤3 then T℘2=℘2/3 and T℘1=℘1/3.Case IV.
If℘2≤2 and ℘1>3 then ℘1−℘2>℘2, T℘2=℘2/3 and T℘1=0.
From all these cases, we obtainT℘1−T℘2≤1/3℘1−℘2 for all ℘1,℘2∈W with ℘1⊥℘2.
It is easy to see thatT is ⊥-preserving and ⊥-continuous. Also 0 is a fixed point of the mapping T.Remark 3.11.
On the lines of Corollary3.9, we can easily say that our results extend the corresponding results of [2–4, 12, 15, 18].Our results are more general than the results of many researchers (see [2–4, 12, 14, 18] and references cited therein) as we use only strictly increasing condition of Wardowski’s function. Our theorems are, therefore, legitimate generalisations of Wardowski’s fixed point theorem.


## 4. Applications

The application of the acquired results is demonstrated in this section.


### 4.1. Solution of Volterra type Integral equation

Here, we show how to apply the existence of a fixed point forτ,FT-contractions can be applied to the following Volterra type equation:(54)℘t=∫0tKt,s,℘sds+bt,t∈I,where I=0,T, T>0, K:I×I×ℝ⟶ℝ, b:I⟶ℝ.The following assumptions must be made in order to obtain our claims:(A1)b, K are SO-continuous functions.(A2) there is a strictly increasing SO-sequenceαn satisfying α0=0, αn≥1, αn−αn−1≤1, αn⟶+∞ such that for all s,t∈I and ℘1,℘2∈ℝ such that ℘1−℘2<αneT with ℘1⊥℘2 defined by ℘1℘2≥℘1 or ℘1℘2≥℘2 and n∈ℕ, we have(55)Kt,s,℘1−Kt,s,℘2≤αn1+αnαn−αn−1e−tαn℘1−℘2.LetW=CI be a complete normed linear space, which contains all continuous functions ℘:I⟶ℝ that have Bielecki’s norm: ℘=supt∈Ie−t℘t.We’re now in a position to state our initial conclusion on existence.Theorem 4.1.
If (A1) and (A2) hold, the nonlinear integral problem (4.1) has a unique solution inW.Proof.
Define the operatorA:W⟶W as(56)A℘t=∫0tKt,s,℘sds+bt,℘∈W.A solution of the (54) will be a fixed point of the operator A.
Define the orthogonality relation⊥ on W by ℘1⊥℘2⟺℘1t℘2t≥℘1t or ℘1t℘2t≥℘2t for all ℘1,℘2∈X, t∈I, in order to satisfy all of the requirements of Theorem 2.5.
Considerτ:0,+∞⟶0,+∞ of the form(57)τt=−t+α1,0<t<α1,−t+αn,αn−1≤t<αn,n≥2.
HereA is ⊥-preserving. For each ℘1,℘2∈X with ℘1⊥℘2 and t∈I, we have(58)A℘1t=∫0tKt,s,℘1sds+bt≥1.It follows that A℘1tA℘2t≥A℘2t, so A℘1t⊥A℘2t.
Next we claim thatA is orthogonal τ,FT-contraction. Take a function Ft=−1/t, t>0. Fix n≥2 and take any ℘1,℘2∈X with ℘1⊥℘2 such that αn−1≤℘1−℘2<eTαn. Take note that for each s∈I, we get(59)℘1s−℘2s≤essups∈Ie−s℘1s−℘2s<esαn≤eTαn.Therefore, we get(60)A℘1t−A℘2t≤∫0tKt,s,℘1s−Kt,s,℘2sds≤αn1+αnαn−αn−1e−tαn∫0t℘1s−℘2sds,t∈I.Next, we see that 1+℘1−℘2αn−℘1−℘2<1+αnαn−αn−1, and since αn+1>1, −sαn+1≤−s for all s∈I. Hence, we have(61)A℘1t−A℘2t≤αn1+℘1−℘2αn−℘1−℘2e−tαn∫0t℘1s−℘2sds=αn1+℘1−℘2αn−℘1−℘2e−tαn∫0t℘1s−℘2se−sαn+1esαn+1ds≤αn℘1−℘21+℘1−℘2αn−℘1−℘2e−tαn∫0tesαn+1ds<αn℘1−℘21+‖℘1−℘2‖αn−℘1−℘2e−tαn1αn+1etαn+1=αn℘1−℘21+℘1−℘2αn−℘1−℘2etαn+1−αn1αn+1.
Using the properties of sequenceαn, we get(62)e−tA℘1t−A℘2t≤℘1−℘21+℘1−℘2αn−℘1−℘2,t∈I.By considering the supremum with respect to t in the aforementioned inequality, we get orthogonal τ,FT-contraction. For n=1, the calculations are the same. The proof comes to a finish with the Theorem 2.5.


### 4.2. Hyers-Ulam-Rassias-Wright Stability

In fixed point theory, generalization of Ulam stability [16,24] has piqued the interest of various scholars (see [25–27]). In this section, we will look at the Hyers-Ulam-Rassias-Wright stability of the integral (54).The following series representation defines the Wright function (see [28]):(63)ϕσ,κ;z=∑l=0∞zll!Γσl+κ,for σ>−1,κ>0,z∈ℝ. It is an entire function of order 1/1+σ.If (54) meets the following criteria, it is called Hyers-Ulam-Rassias-Wright stable:for eachε>0 and for every solution ℘∈W, there is a constant δ>0 satisfying(64)℘t−∫0tKt,s,℘sds−bt≤εϕσ,κ;z.there exists somev∈W satisfying vt⊥℘t and(65)vt=∫0tKt,s,vsds+bt,such that(66)v−℘≤δεϕσ,κ;z.Theorem 4.2.
The fixed point problemA℘=℘, whereA℘t=∫0tKt,s,℘sds+bt,℘∈W, is Hyers-Ulam-Rassias-Wright stable, under the hypothesis of Theorem 4.1.Proof.
On the account of Theorem4.1, we guarantee a unique ℘2∗∈W such that ℘2∗=T℘2∗=∫0tKt,s,℘2∗sds+bt, that is, ℘2∗∈W forms a solution of ℘t=A℘t=∫0tKt,s,℘sds+bt. Let ε>0 and ℘1∗∈W be an ε-solution, that is,(67)℘1∗−A℘1∗=℘1∗t−∫0tKt,s,℘1∗sds+bt≤εϕσ,κ;z.Using Theorem 4.1, we have(68)℘2∗−℘1∗=A℘2∗−℘1∗≤A℘2∗−A℘1∗+A℘1∗−℘1∗≤e−τλ℘2∗−℘1∗+εϕσ,κ;z,for someλ>0.Therefore, ℘2∗−℘1∗≤1/1−λe−τεϕσ,κ;z=δεϕσ,κ;z, where δ=1/1−λe−τ>0. As a result, the (54) is Hyers-Ulam-Rassias-Wright stable.


### 4.3. Differential equations

We’ll now show that the differential equation below has a solution:(69)2yy+1y′=2y+122t+Gt,y,where y is evaluated at each t, G:−α,α×−α,α⟶ℝ+ is SO-continuous, α>0 has a positive solution in C+=ξ∈C:ξ≥0, where C is a subset of the Banach space W of continuous functions ξ:−β,β⟶ℝ, 0<β<α, with the supremum norm C:=ξ∈X:ξ0=0,ξ≤α.Define the orthogonality relation⊥ on C+ by x⊥y if and only if xtyt≥xt or xtyt≥yt for all t∈−α,α.The (69) can be simplified in the following form(70)2yy+1y′2y+12=2t+Gt,y.Further, we obtain(71)t2+∫0tGs,ysds=∫0t2ysys+12ys+12y′sds=∫0yt2vv+12v+12dv=y2t2yt+1.To satisfy the hypotheses of Theorem2.5, we demonstrate that the operator Ay:=y−y2/2y+1=y2+y/2y+1 is an orthogonal τ,FT-contraction on C+ for τs=1/s+1 and Fs=−1/s, s>0.For everyy,x∈C+ with x⊥y or y⊥x and t∈−α,α, we have(72)Ax−Ay=1+2xy+y+xx−y1+2y+2x+4xy.Here, we found x−y≤y+x+2xy, which, when combined with the fact that a function t↦1−t/1+2t,t≥0, is decreasing provides the following(73)Ax−Ay≤x−y1−x−y1+2x−y.Further, using the increasing function t↦t1+t/1+2t, t≥0, we get(74)Ax−Ay≤supt∈0,αxt−yt1−supt∈0,αxt−yt1+2supt∈0,αxt−yt=1+x−yx−y1+2x−y.Now, if Ax≠Ay, we get operator A is an orthogonal τ,FT-contraction for τs=1/s+1 and Fs−1/s, s>0.


## 4.1. Solution of Volterra type Integral equation

Here, we show how to apply the existence of a fixed point forτ,FT-contractions can be applied to the following Volterra type equation:(54)℘t=∫0tKt,s,℘sds+bt,t∈I,where I=0,T, T>0, K:I×I×ℝ⟶ℝ, b:I⟶ℝ.The following assumptions must be made in order to obtain our claims:(A1)b, K are SO-continuous functions.(A2) there is a strictly increasing SO-sequenceαn satisfying α0=0, αn≥1, αn−αn−1≤1, αn⟶+∞ such that for all s,t∈I and ℘1,℘2∈ℝ such that ℘1−℘2<αneT with ℘1⊥℘2 defined by ℘1℘2≥℘1 or ℘1℘2≥℘2 and n∈ℕ, we have(55)Kt,s,℘1−Kt,s,℘2≤αn1+αnαn−αn−1e−tαn℘1−℘2.LetW=CI be a complete normed linear space, which contains all continuous functions ℘:I⟶ℝ that have Bielecki’s norm: ℘=supt∈Ie−t℘t.We’re now in a position to state our initial conclusion on existence.Theorem 4.1.
If (A1) and (A2) hold, the nonlinear integral problem (4.1) has a unique solution inW.Proof.
Define the operatorA:W⟶W as(56)A℘t=∫0tKt,s,℘sds+bt,℘∈W.A solution of the (54) will be a fixed point of the operator A.
Define the orthogonality relation⊥ on W by ℘1⊥℘2⟺℘1t℘2t≥℘1t or ℘1t℘2t≥℘2t for all ℘1,℘2∈X, t∈I, in order to satisfy all of the requirements of Theorem 2.5.
Considerτ:0,+∞⟶0,+∞ of the form(57)τt=−t+α1,0<t<α1,−t+αn,αn−1≤t<αn,n≥2.
HereA is ⊥-preserving. For each ℘1,℘2∈X with ℘1⊥℘2 and t∈I, we have(58)A℘1t=∫0tKt,s,℘1sds+bt≥1.It follows that A℘1tA℘2t≥A℘2t, so A℘1t⊥A℘2t.
Next we claim thatA is orthogonal τ,FT-contraction. Take a function Ft=−1/t, t>0. Fix n≥2 and take any ℘1,℘2∈X with ℘1⊥℘2 such that αn−1≤℘1−℘2<eTαn. Take note that for each s∈I, we get(59)℘1s−℘2s≤essups∈Ie−s℘1s−℘2s<esαn≤eTαn.Therefore, we get(60)A℘1t−A℘2t≤∫0tKt,s,℘1s−Kt,s,℘2sds≤αn1+αnαn−αn−1e−tαn∫0t℘1s−℘2sds,t∈I.Next, we see that 1+℘1−℘2αn−℘1−℘2<1+αnαn−αn−1, and since αn+1>1, −sαn+1≤−s for all s∈I. Hence, we have(61)A℘1t−A℘2t≤αn1+℘1−℘2αn−℘1−℘2e−tαn∫0t℘1s−℘2sds=αn1+℘1−℘2αn−℘1−℘2e−tαn∫0t℘1s−℘2se−sαn+1esαn+1ds≤αn℘1−℘21+℘1−℘2αn−℘1−℘2e−tαn∫0tesαn+1ds<αn℘1−℘21+‖℘1−℘2‖αn−℘1−℘2e−tαn1αn+1etαn+1=αn℘1−℘21+℘1−℘2αn−℘1−℘2etαn+1−αn1αn+1.
Using the properties of sequenceαn, we get(62)e−tA℘1t−A℘2t≤℘1−℘21+℘1−℘2αn−℘1−℘2,t∈I.By considering the supremum with respect to t in the aforementioned inequality, we get orthogonal τ,FT-contraction. For n=1, the calculations are the same. The proof comes to a finish with the Theorem 2.5.


## 4.2. Hyers-Ulam-Rassias-Wright Stability

In fixed point theory, generalization of Ulam stability [16,24] has piqued the interest of various scholars (see [25–27]). In this section, we will look at the Hyers-Ulam-Rassias-Wright stability of the integral (54).The following series representation defines the Wright function (see [28]):(63)ϕσ,κ;z=∑l=0∞zll!Γσl+κ,for σ>−1,κ>0,z∈ℝ. It is an entire function of order 1/1+σ.If (54) meets the following criteria, it is called Hyers-Ulam-Rassias-Wright stable:for eachε>0 and for every solution ℘∈W, there is a constant δ>0 satisfying(64)℘t−∫0tKt,s,℘sds−bt≤εϕσ,κ;z.there exists somev∈W satisfying vt⊥℘t and(65)vt=∫0tKt,s,vsds+bt,such that(66)v−℘≤δεϕσ,κ;z.Theorem 4.2.
The fixed point problemA℘=℘, whereA℘t=∫0tKt,s,℘sds+bt,℘∈W, is Hyers-Ulam-Rassias-Wright stable, under the hypothesis of Theorem 4.1.Proof.
On the account of Theorem4.1, we guarantee a unique ℘2∗∈W such that ℘2∗=T℘2∗=∫0tKt,s,℘2∗sds+bt, that is, ℘2∗∈W forms a solution of ℘t=A℘t=∫0tKt,s,℘sds+bt. Let ε>0 and ℘1∗∈W be an ε-solution, that is,(67)℘1∗−A℘1∗=℘1∗t−∫0tKt,s,℘1∗sds+bt≤εϕσ,κ;z.Using Theorem 4.1, we have(68)℘2∗−℘1∗=A℘2∗−℘1∗≤A℘2∗−A℘1∗+A℘1∗−℘1∗≤e−τλ℘2∗−℘1∗+εϕσ,κ;z,for someλ>0.Therefore, ℘2∗−℘1∗≤1/1−λe−τεϕσ,κ;z=δεϕσ,κ;z, where δ=1/1−λe−τ>0. As a result, the (54) is Hyers-Ulam-Rassias-Wright stable.


## 4.3. Differential equations

We’ll now show that the differential equation below has a solution:(69)2yy+1y′=2y+122t+Gt,y,where y is evaluated at each t, G:−α,α×−α,α⟶ℝ+ is SO-continuous, α>0 has a positive solution in C+=ξ∈C:ξ≥0, where C is a subset of the Banach space W of continuous functions ξ:−β,β⟶ℝ, 0<β<α, with the supremum norm C:=ξ∈X:ξ0=0,ξ≤α.Define the orthogonality relation⊥ on C+ by x⊥y if and only if xtyt≥xt or xtyt≥yt for all t∈−α,α.The (69) can be simplified in the following form(70)2yy+1y′2y+12=2t+Gt,y.Further, we obtain(71)t2+∫0tGs,ysds=∫0t2ysys+12ys+12y′sds=∫0yt2vv+12v+12dv=y2t2yt+1.To satisfy the hypotheses of Theorem2.5, we demonstrate that the operator Ay:=y−y2/2y+1=y2+y/2y+1 is an orthogonal τ,FT-contraction on C+ for τs=1/s+1 and Fs=−1/s, s>0.For everyy,x∈C+ with x⊥y or y⊥x and t∈−α,α, we have(72)Ax−Ay=1+2xy+y+xx−y1+2y+2x+4xy.Here, we found x−y≤y+x+2xy, which, when combined with the fact that a function t↦1−t/1+2t,t≥0, is decreasing provides the following(73)Ax−Ay≤x−y1−x−y1+2x−y.Further, using the increasing function t↦t1+t/1+2t, t≥0, we get(74)Ax−Ay≤supt∈0,αxt−yt1−supt∈0,αxt−yt1+2supt∈0,αxt−yt=1+x−yx−y1+2x−y.Now, if Ax≠Ay, we get operator A is an orthogonal τ,FT-contraction for τs=1/s+1 and Fs−1/s, s>0.



---

*Source: 5959903-2022-05-27.xml*