# Analytical Solution for Differential and Nonlinear Integral Equations viaFϖe-Suzuki Contractions in Modified ϖe-Metric-Like Spaces

Authors: Hasanen A. Hammad; Manuel De la Sen; Hassen Aydi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The aim of this manuscript is to present a new space, namely, a modifiedϖe-metric-like space, and we establish some related fixed point results using extended Fϖe-Suzuki and generalized Fϖe-Suzuki contractions on the mentioned space. Here, we support our theoretical consequences in two ways: the first one consists of presenting illustrative examples and the second one consists of finding analytical solutions for some integral and differential equations in the context of the mentioned space.


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## Body

## 1. Introduction and Elementary Discussions

Solving second-order differential equations after converting to integral equations using Green’s function has become commonplace for many academic researchers because of its importance in theoretical and practical applications. One aspect of the solution of the differential and integral equations is the analytical solution, which in turn mainly supports the numerical solution used in solving dynamical systems.There are many methods for obtaining analytical solutions, including the fixed point technique. For instance, many works have focused their attention on solving the Fredholm integral equation [1] analytically and numerically by this technique [2, 3].Among the generalizations of the Banach principle [4], the notion of F-contractions was initiated by Wardkowski [5].After two years, Piri and Kuman [6] made a slight change in the principle of Wardkowski and called it an F-Suzuki contraction. It has contributed significantly to upholding the reputation of the fixed point theory in many fields and has become a great weight in the functional analysis.Definition 1.
A mappingA:ℸ→ℸ defined on the metric space ℸ,d is named as an
(i)
F-contraction if there are F∈Π and ϑ>0 so that(1)dAℓ,Aℏ>0⇒ϑ+FdAℓ,Aℏ≤Fdℓ,ℏforallℓ,ℏ∈ℸ(ii)
F-Suzuki contraction if there are F∈Π and ϑ>0 so that(2)12dℓ,Aℓ<dℓ,ℏ⇒ϑ+FdAℓ,Aℏ≤Fdℓ,ℏforallℓ,ℏ∈ℸ,where Σ is the class of functions F:0,+∞→ℝ so that
(F1)for all ℧,Ω∈ℝ+ such that ℧<Ω,F℧<FΩ
(F2)for each positive real sequence ℧p, limp→∞℧p=0ifflimn→∞F℧p=−∞
(F3)there is ω∈0,1 so that lim℧→0+℧ωF℧=0There are developments to translate fixed point theorems into nonlinear integral equations and differential equations (for related works and developments, see [7–20]).The nonlinear mapping in Banach contraction needs to be continuous. It is not applicable in the discontinuous case. In the past, Kannan [21]. was able to overcome this shortcoming by giving a fixed point result without the mapping being continuous. Variant works appeared to resolve this problem by adding conditions to the spaces (see [22–25]).Among modern spaces, ab-metric-like space was introduced by Alghmandi et al. [26] as an extension of a b-metric, which was presented by Bakhtin [27], and a metric-like, which was presented by Amini-Harandi [28]. In [26], some fixed point theorems have been provided. In recent works, many contributions on fixed point results involving different contractive conditions are given (see [29–36]).At the beginning of 2019, Parvaneh and Kadelburg [37] generalized the b-metric-like space by replacing the coefficient located in the third condition by a strictly increasing continuous function. They named it an extended b-metric-like space and studied on it some fixed point sequences for JSHR-contractive type mappings with some applications.It is noted that in this section, we did not address definitions and mathematical theorems for two reasons: first, there are large basics related to the mentioned spaces, and second, access to the main results is direct.According to the previous results, in this paper, we present fixed point consequences by usingFϖe-Suzuki contractions in the class of modified ϖe-metric-like spaces. Under the framework of the mentioned space, we apply the theoretical results to find an analytical solution for nonlinear integral equations. On the other hand, some important examples to justify our theorems are discussed.


## 2. An ExtendedFϖe-Suzuki Contraction

We begin this section with definitions of metric-like andb-metric-like spaces.Definition 2 (see [26]).
Letℸ be a nonempty set. A function Δ:ℸ2→ℝ+ is named as metric-like on ℸ, if for all ℧1,℧2,℧3∈ℸ:
(ℸ1)Δ℧1,℧2=0⇒℧1=℧2
(ℸ2)Δ℧1,℧2=Δ℧2,℧1
(ℸ3)Δ℧1,℧3≤Δ℧1,℧2+Δ℧2,℧3Definition 3 (see [26]).
Ab-metric-like on a nonempty set ℸ is a function ϖ:ℸ2→ℝ+ so that for all ℧1,℧2,℧3∈ℸ and a constant s≥1:
(ϖ1)ϖ℧1,℧2=0⇒℧1=℧2
(ϖ2)ϖ℧1,℧2=ϖ℧2,℧1
(ϖ3)ϖ℧1,℧3≤sϖ℧1,℧2+ϖ℧2,℧3
Here,ℸ,ϖ is named as a b-metric-like space (with constant s).For examples about metric-like andb-metric-like spaces, see [33–35].Now, we will generalize Definition3 as follows.Definition 4.
Letℸ be a nonempty set and s:ℸ×ℸ→[1,∞. A function ϖe:ℸ2→[0,∞ is called a modified ϖe-metric-like if, for all ℧1,℧2,℧3∈ℸ:
(ϖe1)ϖe℧1,℧2=0⇒℧1=℧2
(ϖe2)ϖe℧1,℧2=ϖe℧2,℧1
(ϖe3)ϖe℧1,℧3≤s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3
Here,ℸ,ϖe is named as a modified extended b-metric-like space (simply, a modified ϖe-metric-like space).Note that the class of modifiedϖe-metric-like spaces is larger than the class of b-metric-like spaces by replacing the constant s≥1 of Definition 3 by a nonconstant function s:ℸ×ℸ→[1,∞ of Definition 4.Example 5.
Letℸ=0,∞. Define ϖe:ℸ2→[0,∞ by
(3)ϖeκ,μ=0,ifκ=μ=0,μ1+μ,ifκ=0,μ≠0,κ1+κ,ifμ=0,κ≠0,κ+μ,ifκ≠0,μ≠0.
Considers:ℸ2→[1,∞ as sκ,μ=2+2κ+2μ.First,ϖe1 and ϖe2 are obvious. We need to prove ϖe3. For this, let ℧1,℧2, and ℧3 in ℸ. We state the following cases.Case 1.
℧1=℧3=0. Here, ϖe3 holds.Case 2.
℧1=0 and ℧3≠0. Then,
(4)ϖe℧1,℧3=℧31+℧3,s℧1,℧3=2+2℧3.Subcase 1.
℧2=0. We have
(5)ϖe℧1,℧3=℧31+℧3≤2+2℧30+℧31+℧3=s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3.Subcase 2.
℧2≠0. We have
(6)ϖe℧1,℧3=℧31+℧3≤2+2℧3℧21+℧2+℧2+℧3=s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3.Case 3.
℧3=0 and ℧1≠0. Proceeding similarly as in Case 2, ϖe3 holds.Case 4.
℧1≠0 and ℧3≠0. Then,
(7)ϖe℧1,℧3=℧1+℧3,s℧1,℧3=2+2℧1+2℧3.Subcase 1.
℧2=0. We have
(8)ϖe℧1,℧3=℧1+℧3≤2+2℧1+2℧3℧11+℧1+℧31+℧3=s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3.Subcase 2.
℧2≠0. We have
(9)ϖe℧1,℧3=℧1+℧3≤2+2℧1+2℧3℧1+℧2+℧2+℧3=s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3.On the other hand,ℸ,ϖe is not a b-metric-like space. We argue by contradiction by assuming that ℸ,ϖe is a b-metric-like space with a coefficient s≥1 (a constant). Then, for any real μ>0, we have
(10)ϖeμ,μ+1≤sϖeμ,0+ϖe0,μ+1.That is,(11)2μ+1≤sμ1+μ+μ+12+μ.Lettingμ→∞, we get +∞≤2s, which is a contradiction.Example 6.
Letℸ=0,1,2. Define ϖe:ℸ2→[0,∞ and s:ℸ×ℸ→[1,∞ as follows:
(12)ϖe0,0=ϖe1,1=ϖe2,2=0,ϖe0,1=ϖe1,0=12,ϖe0,2=ϖe2,0=1,ϖe1,2=ϖe2,1=3,and sκ,μ=2+κ+μ.
First, we show thatϖe is a modified ϖe-metric-like space. Trivially, the conditions ϖe1 and ϖe2 hold. For ϖe3, we get
(13)ϖe0,1=12;s0,1ϖe0,2+ϖe2,1=12.
Thus,(14)ϖe0,1≤s0,1ϖe0,2+ϖe2,1.
Again,(15)ϖe1,2=3;s1,2ϖe1,0+ϖe0,2=65,ϖe0,2=1;s0,2ϖe0,1+ϖe1,2=60.
Hence, for all℧1,℧2,℧3∈ℸ,ϖe℧1,℧3≤s℧1,℧3ϖe℧1,℧2+ϖe℧2,℧3. Then, ℸ,ϖe is a modified ϖe-metric-like space, but it is not a b-metric-like space because if we take s=2 in the inequality (13), we get
(16)ϖe0,1=12;2ϖe0,2+ϖe2,1=8.
Thus,(17)ϖe℧1,℧3≰sϖe℧1,℧2+ϖe℧2,℧3.Definition 7.
Let℧ι be a sequence in the modified ϖe-metric-like space ℸ,ϖe.
(a)
Iflimι→∞ϖe℧ι,℧=ϖe℧,℧, then ℧ι is convergent to ℧(b)
℧ι is called Cauchy if limι,j→∞ϖe℧ι,℧j exists and is finite(c)
If for each Cauchy sequence℧ι, there is ℧∈ℸ, so that limι,j→∞ϖe℧ι,℧j=ϖe℧,℧=limι→∞ϖe℧ι,℧; therefore, ℸ,ϖe is said to be completeDefinition 8.
A nonlinear self-mappingA on a modified ϖe-metric-like space ℸ,ϖe is named as an extended Fϖe-Suzuki contraction if there are Fϖe∈Πandϑ>0 so that for ℧,μ∈ℸ, the following condition holds:
(18)12ϖe℧,A℧<ϖe℧,μ⇒ϑ+FϖeϖeA℧,Aμ≤Fϖeϖe℧,μ,such that limn,m→∞s℧ι,℧j<1/η for all ℧∘∈ℸ, where 0<η<1. We consider here ℧ι=Aι℧∘, ι=1,2,⋯, where Π is the set of continuous functions Fϖe:ℝ+→ℝ so that
(I1)For all j,ℓ∈ℝ+ with j<ℓ,Fϖej<Fϖeℓ
(I2)For each positive real sequence jp, limp→∞jp=0 iff limp→∞Fϖejp=−∞
(I3)There is η∈0,1 so that limj→0+jηFϖej=0
Now, we introduce our first theorem.Theorem 9.
Letℸ,ϖe be a complete modified ϖe-metric-like space and A be an extended Fϖe-Suzuki contraction mapping, then A admits a unique fixed point.Proof.
Let℧∘∈ℸ and ℧ιι=1∞ defined by ℧ι+1=A℧ι=Aι+1℧∘. If there is ι∈ℕ so that ϖe℧ι,A℧ι=0. It completes the proof. Otherwise, assume that 0<ϖeAι,Γ℧ι=ϖe℧ι,℧ι+1=ϖeι; therefore, for all ι∈ℕ,(19)12ϖe℧ι,A℧ι<ϖe℧ι,A℧ι,it yields or
(20)ϑ+FϖeϖeA℧ι,A2℧ι≤Fϖeϖe℧ι,A℧ι,(21)FϖeϖeA℧ι,A2℧ι≤Fϖeϖe℧ι,A℧ι−ϑ.
By the same method, one gets(22)Fϖeϖeι=Fϖeϖe℧ι,A℧ι=FϖeϖeA℧ι−1,A2℧ι−1≤Fϖeϖe℧ι−1,A℧ι−1−ϑ≤Fϖeϖe℧ι−2,A℧ι−2−2ϑ⋮≤Fϖeϖe℧∘,A℧∘−ιϑforallι≥1.
Takingι→∞ in (22), we have
(23)limι→∞Fϖeϖeι=−∞.
So, byI2, we obtain
(24)limι→∞ϖeι=0.
ApplyingI3, there is η∈0,1 so that
(25)limι→∞ϖeιηFϖeϖeι=0.
By (22), one writes for all ι≥1,(26)ϖeιηFϖeϖeι−Fϖeϖe∘≤−ιϑϖeιη≤0.
Considering (24) and (25) and passing ι→∞ in (26), one gets
(27)limι→∞ιϖeιη=0.
By (27), there exists ι1∈ℕ so that ιϖeιη≤1 for all ι≥ι1, or
(28)ϖeι≤1ι1/ηforallι≥ι1.
Consider the integersm>ι. Applying ϖ3 and (28), one writes
(29)ϖe℧ι,℧m≤s℧ι,℧mϖe℧ι,℧ι+1+ϖe℧ι+1,℧m≤s℧ι,℧mϖe℧ι,℧ι+1+s℧ι,℧ms℧ι+1,℧mϖe℧ι+1,℧ι+2+ϖe℧ι+2,℧m≤s℧ι,℧mϖe℧ι,℧ι+1+s℧ι,℧ms℧ι+1,℧mϖe℧ι+1,℧ι+2+⋯+s℧ι,℧ms℧ι+1,℧ms℧ι+2,℧m⋯s℧m−2,℧ms℧m−1,℧mϖe℧m−1,℧m≤s℧1,℧ms℧2,℧m⋯s℧ι,℧mϖe℧ι,℧ι+1+s℧1,℧ms℧2,℧m⋯s℧ι+1,℧mϖe℧ι+1,℧ι+2+⋯+s℧1,℧ms℧2,℧m⋯s℧m−1,℧mϖe℧m−1,℧m=∑ι=1∞ϖe℧ι,℧ι+1∏j=1ιs℧j,℧m≤∑ι=1∞1ι1/η∏j=1ιs℧j,℧m≤∑ι=1∞1ι1/η1η=1η∑ι=1∞1ι1/η.
Recall that∑ι=1∞1/ι1/η converges, so ϖe℧ι,℧m→0. Therefore, ℧ι is a Cauchy sequence in the complete modified ϖe-metric-like space ℸ,ϖe; hence, there is ℧∗∈ℸ such that ℧ι→℧∗, as ι→∞. That is,
(30)limι,m→∞ϖe℧ι,℧m=limι→∞ϖe℧ι,℧∗=ϖe℧∗,℧∗=0.Next, ifFϖe is continuous, then two short cases arise.Case 1.
For eachι∈ℕ, there exists jι∈ℕ such that ℧jι=A℧∗ and jι>jι−1, where j∘=0. Therefore, one gets ℧∗=limι→∞℧jι=limι→∞A℧∗=A℧∗.Case 2.
There isι∘∈ℕ so that for all ι≥ι∘,℧ι≠A℧∗. It is clear that 1/2ϖe℧ι,A℧∗<ϖe℧ι,A℧∗ for all ι≥ι∘.By (18), we have
(31)ϑ+Fϖeϖe℧ι+1,A2℧∗=ϑ+FϖeϖeA℧ι,A2℧∗≤Fϖeϖe℧ι,A℧∗.SinceFϖe is continuous, we obtain at the limit ι→∞,or
(32)ϑ+Fϖeϖe℧∗,A2℧∗≤Fϖeϖe℧∗,A℧∗,(33)Fϖeϖe℧∗,A℧∗≤Fϖeϖe℧∗,℧∗−ϑ,which is a contradiction due to I1. Then, ϖe℧∗,A℧∗=0, which means that ℧∗=A℧∗.The two cases above lead to the existence of a fixed point ofA, i.e., ℧∗=A℧∗.Now, assume that℧1∗ and ℧2∗ are so that ℧1∗=A℧1∗≠℧2∗=A℧2∗. We have 1/2ϖe℧1∗,℧2∗<ϖe℧1∗,℧2∗, which implies by (18) that
(34)ϑ+Fϖeϖe℧1∗,℧2∗=ϑ+FϖeϖeA℧1∗,A℧2∗≤Fϖeϖe℧1∗,℧2∗.It is again a contradiction.The following examples verify all required hypotheses of Theorem9.Example 10.
Letℸ=0,∞. Define ϖe:ℸ2→ℝbyϖe℧,ℵ=℧+ℵ2 and s:ℸ×ℸ→[1,∞ by s℧,ℵ=1+℧+ℵ, for all ℧,ℵ∈ℸ. Here, ϖe is a modified extended ϖe-metric-like space. Define A:ℸ→ℸasA℧=1/3℧, for all ℧∈ℸ. It is clear that
(35)12ϖe℧,A℧=12ϖe℧,13℧=12℧+13℧2=1618℧2≤℧2≤℧+μ2=ϖe℧,μ.
Consider, for all℧,μ∈ℸ,(36)FϖeϖeA℧,Aμ=Fϖeϖe13℧,13μ=Fϖe13℧+13μ2=Fϖe19℧+μ2.
Also,(37)Fϖeϖe℧,μ=Fϖe℧+μ2.
Let the functionFϖe∈Π be defined by Fϖeℓ=lnℓ, for ℓ>0. Then,
(38)FϖeϖeA℧,Aμ−Fϖeϖe℧,μ=ln19℧+μ2−ln℧+μ2=ln1/9℧+μ2℧+μ2=ln19=−2.197≤−2.
Therefore,A is an extended Fϖe-Suzuki contraction mapping with ϑ=2. Moreover, if ℧m=1/m+1∈ℸ, we have
(39)limi,m→∞s℧m,℧i=limi,m→∞1+1m+1+1i+1=1<1η,for η∈0,1. So, all hypotheses of Theorem 9 are satisfied, and A has 0 as a unique fixed point.Example 11.
Letℸ=1/3ι−1:ι∈ℕ∪0. Suppose that ϖe:ℸ×ℸ→0,∞ and s:ℸ×ℸ→[1,∞ are functions defined by ϖe℧,μ=max℧,μ2ands℧,μ=1+μ+℧, respectively, for all μ,℧∈ℸ. Then, the pair ℸ,ϖe is a complete modified ϖe-metric-like space. Define a nonlinear mapping A:ℸ→ℸby(40)A℧=132ι,if℧∈132ι−1;ι∈ℕ,0,if℧=0.We shall prove that a mappingA is an extended Fϖe-Suzuki contraction with Fϖeℓ=lnℓ for ℓ>0 and ϑ>0, by showing the following cases.Case 1.
Let℧=1/32ι−1 and μ=1/32m−1, for m>ι≥1, one can write
(41)12ϖe℧,A℧=12ϖe132ι−1,132ι=12max132ι−1,132ι2=12132ι−12<132ι−12=max132ι−1,132m−12=ϖe℧,μ.
Consider(42)FϖeϖeA℧,Aμ=FϖeϖeA132ι−1,A132m−1=Fϖeϖe132ι,132m=Fϖemax132ι,132m2=Fϖe132ι2=ln132ι2=2ln132ι.
Also,(43)Fϖeϖe℧,μ=Fϖeϖe132ι−1,132m−1=Fϖemax132ι−1,132m−12=Fϖe132ι−12=2ln132ι−1.
By subtracting (42) and (43), we find that
(44)FϖeϖeA℧,Aμ−Fϖeϖe℧,μ=2ln132ι−ln132ι−1=2ln132ι×32ι·3−1=−2ln3<−2.Case 2.
Let℧=1/32ι−1 and μ=0. We have
(45)12ϖe℧,A℧=12max132ι−1,132ι2=12132ι−12<132n−12=max132ι−1,02=ϖe℧,μ.
Suppose that(46)FϖeϖeA℧,Aμ=Fϖeϖe132ι,0=Fϖemax132ι,02=Fϖe132ι2=ln132ι2=2ln132ι;also,
(47)Fϖeϖe℧,μ=Fϖeϖe132ι−1,0=Fϖemax132ι−1,02=Fϖe132ι−12=2ln132ι−1.
By subtracting (46) and (47), we have the same inequality (44).Case 3.
Let℧=0 and μ=1/32m−1. The proof follows immediately as Case 2. Thus, A is an extended Fϖe-Suzuki contraction mapping with ϑ=2. Here, 0 is the unique fixed point.


## 3. An Extended GeneralizedFϖe-Suzuki Contraction

Definition 12.
A self-mappingA on a modified extended b-metric-like space ℸ,ϖe is called an extended generalized Fϖe-Suzuki contraction if there are Fϖe∈Π and ϑ>0 such that, if for all ℧,ℵ∈ℸ, the following hypothesis is satisfied
(48)12ϖe℧,A℧<ϖe℧,ℵ⇒ϑ+FϖeϖeA℧,Aℵ≤Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ.Remark 13.
(i)
Every extendedFϖe-Suzuki contraction is an extended generalized Fϖe-Suzuki contraction(ii)
Suppose thatA is an extended generalized Fϖe-Suzuki contraction, by Definition 12, for all ℧,ℵ∈ℸ, we get A℧≠Aℵ and 1/2ϖe℧,A℧<ϖe℧,ℵ. Thus,(49)FϖeϖeA℧,Aℵ<ϑ+FϖeϖeA℧,Aℵ≤Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ
By conditionI1, for all ℧,ℵ∈ℸ with A℧≠Aℵ, we have
(50)ϖeA℧,Aℵ≤maxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ.Note that the inverse of the above remark is generally incorrect.Example 14.
Letℸ=0,∞. Define ϖe:ℸ2→ℝbyϖe℧,ℵ=℧+ℵ2 and s:ℸ×ℸ→[1,∞bys℧,ℵ=1+℧+ℵ, for all ℧,ℵ∈ℸ. Here, ϖe is a modified extended ϖe-metric-like space. Define A:ℸ→ℸ as
(51)A℧=0,if0≤℧<1,12,if℧≥1.
Note thatA is not an extended Fϖe-Suzuki contraction. Indeed, for 0≤℧<1 and ℵ=1, we can write 1/2ϖe℧,A℧=1/2ϖe℧,0=1/2℧2<℧+12=ϖe℧,ℵ and
(52)maxϖe℧,1,ϖe℧,A℧1+ϖe℧,A℧,ϖe1,A11+ϖe1,A1,ϖe℧,A1+ϖe1,A℧42+℧=maxϖe℧,1,ϖe℧,01+ϖe℧,0,ϖe1,A11+ϖe1,A1,ϖe℧,A1+ϖe1,042+℧≥ϖe1,A1=ϖe1,12=94>14=ϖeA℧,Aℵ.
Let the functionFϖe∈Π be defined by Fϖeℓ=lnℓ, for ℓ>0. Then,
(53)FϖeϖeA℧,Aℵ−Fϖeϖe℧,ℵ≤FϖeϖeA℧,A1−Fϖeϖe1,A1=Fϖe14−Fϖe94=ln14×49=−ln9<−2.
Therefore,A is an extended generalized Fϖe-Suzuki contraction forϑ=2.The following theorem is the main consequence of this part.Theorem 15.
Letℸ,ϖe be an extended b-metric-like space and A be an extended generalized Fϖe-Suzuki contraction self-mapping, then A has a unique fixed point, provided that limi,ms℧i,℧m≤1/η,for0<η<1.Proof.
By the first lines of proof of Theorem9, we build a sequence ℧ιι=1∞ as ℧ι+1=A℧ι=Aι+1℧∘. Here, we consider ι∈ℕ∪0, 0<ϖe℧ι,A℧ι=ϖe℧ι,℧ι+1, so
(54)12ϖe℧ι,A℧ι<ϖe℧ι,A℧ι.
Applying conditions (48) and ϖe3, we get
(55)ϑ+Fϖeϖe℧ι+1,℧ι+2=ϑ+FϖeϖeA℧ι,AA℧ι≤Fϖemaxϖe℧ι,A℧ι,ϖe℧ι,A℧ι1+ϖe℧ι,A℧ι,ϖeA℧ι,A2℧ι1+ϖeA℧ι,A2℧ι,ϖe℧ι,A2℧ι+ϖeA℧ι,A℧ι4s℧ι,A℧ι=Fϖemaxϖe℧ι,℧ι+1,ϖe℧ι,℧ι+11+ϖe℧ι,℧ι+1,ϖe℧ι+1,℧ι+21+ϖe℧ι+1,℧ι+2,ϖe℧ι,℧ι+2+ϖe℧ι+1,℧ι+14s℧ι,℧ι+1≤Fϖemaxϖe℧ι,℧ι+1,ϖe℧ι,℧ι+11+ϖe℧ι,℧ι+1,ϖe℧ι+1,℧ι+21+ϖe℧ι+1,℧ι+2,s℧ι,℧ι+1ϖe℧ι,℧ι+1+ϖe℧ι+1,℧ι+2+2s℧ι,℧ι+2ϖe℧ι,℧ι+14s℧ι,℧ι+2=Fϖemaxϖe℧ι,℧ι+1,ϖe℧ι,℧ι+11+ϖe℧ι,℧ι+1,ϖe℧ι+1,℧ι+21+ϖe℧ι+1,℧ι+2,3ϖe℧ι,℧ι+1+ϖe℧ι+1,℧ι+24≤Fϖemaxϖe℧ι,℧ι+1,ϖe℧ι+1,℧ι+2.
Now, ifϖe℧ι,℧ι+1<ϖe℧ι+1,℧ι+2, then
(56)Fϖeϖe℧ι+1,℧ι+2≤Fϖeϖe℧ι+1,℧ι+2−ϑ,which is a contradiction due to I1, so we should write
(57)Fϖeϖe℧ι+1,℧ι+2≤Fϖeϖe℧ι,℧ι+1−ϑ,∀ι∈ℕ∪0.
By the same manner,(58)Fϖeϖe℧ι,℧ι+1≤Fϖeϖe℧ι−1,℧ι−ϑ,∀ι∈ℕ∪0.
From (57) and (58), one can write
(59)Fϖeϖe℧ι+1,℧ι+2≤Fϖeϖe℧ι−1,℧ι−2ϑ,∀ι∈ℕ∪0.
Repeating the same scenario, we have(60)Fϖeϖe℧ι,℧ι+1≤Fϖeϖe℧∘,℧1−ιϑ,∀ι∈ℕ∪0.
The proof of Theorem9, namely, relations (22)–(29), yields that ℧ι is Cauchy sequence in ℸ,ϖe, which is complete; hence, there is ℧∗∈ℸ so that ℧ι→℧∗ as ι→∞. That is,
(61)limι,m→∞ϖe℧ι,℧m=limι→∞ϖe℧ι,℧∗=ϖe℧∗,℧∗=0.
Now, ifA is continuous, by (24) we get
(62)ϖeA℧∗,℧∗=limι→∞ϖeA℧ι,℧ι=limι→∞ϖe℧ι+1,℧ι=0.
Thus,A℧∗=℧∗; that is, ℧∗ is a fixed point of A.
Next, in the case thatFϖe is continuous, we claim that
(63)ϖe℧m,℧∗≤ϖe℧∗,A℧m,∀m∈ℕ∪0.
By the fact1/2ϖ℧m,A℧m<ϖ℧m,A℧m and using (48), we obtain that
(64)ϑ+FϖeϖA℧m,A2℧m≤Fϖemaxϖe℧m,A℧m,ϖe℧m,A℧m1+ϖe℧m,A℧m,ϖeA℧m,A2℧m1+ϖeA℧m,A2℧m,ϖe℧m,A2℧m+ϖeA℧m,A℧m4s℧m,A℧m≤Fmaxϖe℧m,A℧m,ϖe℧m,A℧m1+ϖe℧m,A℧m,ϖeA℧m,A2℧m1+ϖeA℧m,A2℧m,s℧m,A℧mϖe℧m,A℧m+ϖeA℧m,A2℧m+2s℧m,A℧mϖe℧m,A℧m4s℧m,A℧m≤Fmaxϖe℧m,A℧m,ϖe℧m,A℧m1+ϖe℧m,A℧m,ϖeA℧m,A2℧m1+ϖeA℧m,A2℧m,3ϖe℧m,A℧m+ϖeA℧m,A2℧m4≤Fmaxϖe℧m,A℧m,ϖeA℧m,A2℧m.
Ifϖe℧m,A℧m<ϖeA℧m,A2℧m, then we have
(65)FϖeϖA℧m,A2℧m≤FϖeϖA℧m,A2℧m−ϑ,a contradiction due to I1. So, we should write
(66)FϖeϖA℧m,A2℧m≤Fϖeϖe℧m,A℧m−ϑ.
SinceFϖe is continuous and strictly increasing, it follows that
(67)ϖeA℧m,A2℧m<ϖe℧m,A℧m.Now, to ensure the existence of a fixed point, two cases arise as follows.Case 1.
For eachι∈ℕ, there is jι∈ℕ so that ℧jι=A℧∗ and jι>jι−1, where j∘=1. Then, we have ℧∗=limι→∞℧jι=limι→∞A℧∗=A℧∗, i.e., ℧∗ is a fixed point of A.Case 2.
There isι∘∈ℕ such that for all ι≥ι∘,℧ι+1≠A℧∗. It is clear that 1/2ϖe℧ι,A℧∗<ϖe℧ι,A℧∗ for all ι≥ι∘.By (48) and (63), we get
(68)ϑ+FϖeϖeA℧ι,A2℧∗≤Fϖemaxϖe℧ι,A℧∗,ϖe℧ι,A℧ι1+ϖe℧ι,A℧ι,ϖeA℧∗,A2℧∗1+ϖeA℧∗,A2℧∗,ϖe℧ι,A2℧∗+ϖeA℧∗,A℧ι4s℧ι,A℧∗≤Fϖemaxϖe℧ι,A℧∗,ϖe℧ι,A℧ι1+ϖe℧ι,A℧ι,ϖeA℧∗,A2℧∗1+ϖeA℧∗,A2℧∗,2ϖe℧ι,A℧∗+ϖeA℧∗,A2℧∗+ϖe℧ι,A℧ι4≤Fϖemaxϖe℧ι,A℧∗,ϖe℧ι,A℧ι,ϖeA℧∗,A2℧∗<Fϖemaxϖe℧ι,A℧∗,ϖe℧ι,A℧ι,ϖe℧∗,A℧∗.SinceFϖe is continuous, we find at the limit ι→∞, or
(69)ϑ+Fϖeϖe℧∗,A2℧∗≤Fϖeϖe℧∗,A℧∗,(70)Fϖeϖe℧∗,A℧∗≤Fϖeϖe℧∗,℧∗−ϑ,which is a contradiction. So, ϖe℧∗,A℧∗=0, which leads to ℧∗=A℧∗.The two cases above ensure the existence of a fixed point ofA.To ensure the uniqueness, suppose that℧∗,υ∗ are distinct fixed points of A. Hence, 1/2ϖe℧∗,υ∗<ϖe℧∗,υ∗, which implies that
(71)ϑ+Fϖeϖe℧∗,υ∗=ϑ+FϖeϖeA℧∗,Aυ∗≤Fϖemaxϖe℧∗,υ∗,ϖe℧∗,A℧∗1+ϖe℧∗,A℧∗,ϖeυ∗,Aυ∗1+ϖeυ∗,Aυ∗,ϖe℧∗,Aυ∗+ϖeυ∗,A℧∗4s℧∗,υ∗≤Fϖeϖe℧∗,υ∗,which is a contradiction again. Hence, the fixed point is unique.In the following, we justify all required hypotheses of Theorem15.Example 16.
Suppose thatℸ=0,∞. Define functions ϖe:ℸ2→ℝ and s:ℸ×ℸ→[1,∞byϖe℧,ℵ=℧+ℵ2 and s℧,ℵ=1+℧+ℵ, respectively. Then, ℸ,ϖe is a complete modified ϖe-metric-like space. Define A:ℸ→ℸ by
(72)A℧=0,if℧∈0,14,14ι,if℧∈14,∞,ι∈ℸ.
Define the functionFϖe∈Π by Fϖeℓ=lnℓ for ℓ>0 and ϑ>0. We state the following.Case 1.
Let℧=1/4ι−1 and ℵ=1/4m−1, for m>ι≥2. Now, for ι=1 and m=2, we have ℧=1 and ℵ=1/4. Therefore,
(73)12ϖe℧,A℧=12ϖe1,A1=121+142=2532<2516=1+142=ϖe℧,ℵ.
Let(74)FϖeϖeA℧,Aℵ=FϖeϖeA1,A14=Fϖeϖe14,116=Fϖe5162=2ln516=−2.326,as well as,
(75)Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ=Fϖemaxϖe1,14,ϖe1,A11+ϖe1,A1,ϖe1/4,A1/41+ϖe1/4,A1/4,ϖe1,A1/4+ϖe1/4,A14s1,1/4=Fϖemax2516,2541,25281,3532304=Fϖe2516=ln2516=0.0446.
So, we get(76)FϖeϖeA℧,Aℵ−Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ=−2.326−0.044=−2.37<−2.Case 2.
Let℧=1/4 and ℵ=0. So, we get
(77)12ϖe℧,A℧=12ϖe14,116=25512<116=14+02=ϖe℧,ℵ.
Consider(78)FϖeϖeA℧,Aℵ=FϖeϖeA14,A0=Fϖeϖe116,0=Fϖe1162=2ln116=−5.545.
Additionally,(79)Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ=Fϖemaxϖe14,0,ϖe1/4,A1/41+ϖe1/4,A1/4,ϖe0,A01+ϖe0,A0,ϖe1/4,A0+ϖe0,A1/44s1/4,0=Fϖemax116,25281,0,171280=Fϖe25281=ln25281=−2.419.
Subtracting the two relations, we have(80)FϖeϖeA℧,Aℵ−Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ=−5.545+2.419=−3.126<−2.
From the above, we deduce that the mappingA is an extended generalized Fϖe-Suzuki contraction with ϑ=2. Moreover, if ℧m=1/4m∈ℸ, we have
(81)limi,m→∞s℧m,℧i=limi,m→∞1+14m+14i=1<1η,for η∈0,1. Hence, the requirements of Theorem 15 hold; therefore, A has a unique fixed point. Here, it is 0.


## 4. Supportive Applications

This part is considered as the strength of the paper, where we use the results presented in Theorems9 and 15 to get the analytical solutions both of the Fredholm integral equation and the second-order differential equation, respectively. For this purpose, we will divide this section into two parts as follows.


### 4.1. Analytical Solution of Fredholm Integral Equation

Let the Fredholm integral equation given by(82)℧η=∫uvΦη,ζ,℧ζdζ,for all η,ζ∈[u,v, where Fϖe:u,v→ℝ and Φ:u,v×u,v×ℝ→ℝ are continuous functions.Letℸ=Cu,v,ℝ be the set of all real continuous functions defined on u,v, endowed with
(83)ϖe℧,ℵ=℧+ℵ∞2forall℧,ℵ∈ℸ,where ℧∞=supη∈[u,v℧ηe−ηϑ with s℧,ℵ=1+℧+ℵ, where s:ℸ×ℸ→[1,∞. Note that ℸ,ϖe is a complete modified ϖe-metric-like space.Now, the following is the main result of this part.Theorem 17.
LetA be self-mapping on the complete modified ϖe-metric-like space ℸ,ϖe. Assume that
(i)
for eachη,ζ∈[u,v and ℧,ℵ∈ℸ,(84)12℧η+∫uvΦη,ζ,℧ζdζ∞2≤℧η∞+ℵη∞2(ii)
for allη,ζ∈[u,v, there is a constant ϑ∈ℝ+ such that(85)Φη,ζ,℧ζ+Φη,ζ,ℵζ≤e−ϑ/2v−u℧ζ+ℵζ
Then, there exists a solution of the problem (82).Proof.
Consider the nonlinear self-mappingA:ℸ→ℸ given as
(86)A℧η=∫uvΦη,ζ,℧ζdζ.
Clearly, if℧∗=A℧∗, then it is a solution of the problem (82).
Let℧,ℵ∈ℸ, so, by condition (i), we deduce that 1/2ϖe℧η,A℧η<ϖe℧η,ℵη. After applying the condition (ii), for any ℧η,ℵη∈ℸ, we can write
(87)A℧η+Aℵη2≤∫uvΦη,ζ,℧ζ+Φη,ζ,ℵζdζ2∫uve−ϑ/2v−u℧ζ+ℵζdζ2≤∫uve−ϑ/2v−u×e−2ϑη×e2ϑη×℧ζ+ℵζdζ2≤e−ϑv−u2×ϖe℧,ℵ×e2ϑη∫uvdζ2≤e−ϑϖe℧,ℵ×e2ϑη,so we have
(88)A℧η+Aℵη×e−ϑη2≤e−ϑϖe℧,ℵ,which leads to
(89)A℧η+Aℵη∞2≤e−ϑϖe℧,ℵ.
It yields that(90)ϖeA℧η,Aℵη≤e−ϑϖe℧,ℵ.
TakingFϖeℓ=lnℓ for ℓ>0, one gets or
(91)lnϖeA℧η,Aℵη≤lne−ϑϖe℧,ℵ,(92)ϑ+lnϖeA℧η,Aℵη≤lnϖe℧,ℵ.
Equivalently,(93)ϑ+FϖeϖeA℧η,Aℵη≤Fϖeϖe℧,ℵ.
By Theorem9, A admits a fixed point, which is a solution of the problem (82).


### 4.2. Analytical Solution of Second-Order Differential Equation

Consider the second-order differential equation given as follows:(94)℧′′η=−Φη,℧η,η∈0,γ,℧0=℧η=0,where Φ:0,γ×ℝ→ℝ is a continuous function.Resolving the problem (94) is equivalent to resolving the following integral equation:
(95)℧η=∫0ηϕη,ζΦζ,℧ζdζ,∀η∈0,γ,where ϕ is Green’s function defined by
(96)ϕη,ζ=η1−ζ,if0≤η≤ζ≤γ,ζ1−η,if0≤ζ≤η≤γ,and Φ is a function as in Theorem 17. Hence, if ℧∈C0,γ, then ℧ is a solution of the problem (94) if and only if ℧ is a solution of the problem (95).Letℸ=C0,γ,ℝ be the set of all continuous functions defined on 0,γ, and define a norm ℧ϑ=maxη∈0,γ℧ηe−1/2ηϑ, for arbitrary η≥1. Obviously, ·ϑ is equivalent to the maximum norm · on ℸ, and ℸ is endowed with the extended generalized ϖeϑ-metric-like as
(97)ϖeϑ℧,ℵ=℧+ℵϑ2=maxη∈0,γ℧η+ℵη2e−ηϑforall℧,ℵ∈ℸandeηϑ≥1.Then,ℸ,ϖe is a complete modified ϖe-metric-like space with s℧,ℵ=1+℧+ℵ. Our main theorem is as follows.Theorem 18.
Suppose thatℸ,ϖe is a complete modified b-metric-like space and A is a nonlinear self-mapping on ℸ, then (95) possesses a unique solution ℧∈C0,γ,ℝ, if
(a1) for eachη,ζ∈[0,γ and ℧,ℵ∈ℸ,(98)12℧η+∫0ηϕη,ζΦζ,℧ζdζϑ≤℧+ℵϑ
(a2)Φ∈C0,γ×ℝ and ϕ∈C0,γ×[0,γ
(a3)Φ satisfies
(99)Φζ,℧ζ+Φζ,ℵζ2≤ϑe−ϑψ℧,ℵ,
for allζ∈[0,γ and ℧,ℵ∈ℝ, where
(100)ψ℧,ℵ=max℧+ℵ2,℧+A℧21+℧+A℧2,ℵ+Aℵ21+ℵ+Aℵ2,℧+Aℵ2+ℵ+A℧241+℧+ℵ
(a4)max∫0ηϕη,ζdζ≤1, for all η∈[0,γProof.
Consider on the setℸ, the mapping A as
(101)A℧η=∫0ηϕη,ζΦζ,℧ζdζ,for all η∈[0,γ and ℧∈ℸ. The solution of (95) is also a fixed point of A on ℸ. By condition (a1) and the definition of A, we can write 1/2ϖeϑ℧η,A℧η<ϖeϑ℧η,ℵη.
Let℧,ℵ∈ℸ. By the hypotheses (a2)-(a4), we have
(102)A℧η+Aℵη2=∫0ηϕη,ζΦζ,℧ζ+Φζ,ℵζdζ2≤∫0ηϕη,ζ2Φζ,℧ζ+Φζ,ℵζ2dζ≤∫0ηΦζ,℧ζ+Φζ,ℵζ2dζ≤∫0ηϑe−ϑψ℧ζ,ℵζdζ≤ϑe−ϑ∫0ηeζϑmax℧+ℵ2e−ζϑ,℧+A℧2e−2ζϑ1+℧+A℧2e−ζϑ,ℵ+Aℵ2e−2ζϑ1+ℵ+Aℵ2e−ζϑ,℧+Aℵ2+ℵ+A℧241+℧+ℵe−ζϑdζ≤ϑe−ϑ∫0ηeζϑmaxϖeϑ℧,ℵ,ϖeϑ℧,A℧1+ϖeϑ℧,A℧,ϖeϑℵ,Aℵ1+ϖeϑℵ,Aℵ,ϖeϑ℧,Aℵ+ϖeϑℵ,A℧4s℧,ℵdζ=ϑe−ϑψ℧,ℵ∫0ηeζϑdζ=ϑe−ϑψ℧,ℵ×eηϑϑ−1≤ϑe−ϑψ℧,ℵ×eηϑϑ≤e−ϑψ℧,ℵeηϑ.
Hence, for all℧,ℵ∈ℸ,(103)A℧η+Aℵη2×e−ηϑ≤e−ϑψ℧,ℵ,which yields
(104)ϖeϑA℧,Aℵ=maxη∈0,γA℧η+Aℵη2×e−ηϑ≤e−ϑψ℧,ℵ.
That is,(105)ϑ+lnϖeϑA℧,Aℵ≤lnψ℧,ℵ.
Defining the functionFϖeα=lnα,α>0 in (105), such that Fϖe∈Π, we have
(106)ϑ+FϖeϖeA℧,Aℵ≤Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ.
Hence, all requirements of Theorem15 hold and A is an extended generalized F-Suzuki contraction; hence, Γ possesses a fixed point ℧∈ℸ, which is a solution of the problem (95).


## 4.1. Analytical Solution of Fredholm Integral Equation

Let the Fredholm integral equation given by(82)℧η=∫uvΦη,ζ,℧ζdζ,for all η,ζ∈[u,v, where Fϖe:u,v→ℝ and Φ:u,v×u,v×ℝ→ℝ are continuous functions.Letℸ=Cu,v,ℝ be the set of all real continuous functions defined on u,v, endowed with
(83)ϖe℧,ℵ=℧+ℵ∞2forall℧,ℵ∈ℸ,where ℧∞=supη∈[u,v℧ηe−ηϑ with s℧,ℵ=1+℧+ℵ, where s:ℸ×ℸ→[1,∞. Note that ℸ,ϖe is a complete modified ϖe-metric-like space.Now, the following is the main result of this part.Theorem 17.
LetA be self-mapping on the complete modified ϖe-metric-like space ℸ,ϖe. Assume that
(i)
for eachη,ζ∈[u,v and ℧,ℵ∈ℸ,(84)12℧η+∫uvΦη,ζ,℧ζdζ∞2≤℧η∞+ℵη∞2(ii)
for allη,ζ∈[u,v, there is a constant ϑ∈ℝ+ such that(85)Φη,ζ,℧ζ+Φη,ζ,ℵζ≤e−ϑ/2v−u℧ζ+ℵζ
Then, there exists a solution of the problem (82).Proof.
Consider the nonlinear self-mappingA:ℸ→ℸ given as
(86)A℧η=∫uvΦη,ζ,℧ζdζ.
Clearly, if℧∗=A℧∗, then it is a solution of the problem (82).
Let℧,ℵ∈ℸ, so, by condition (i), we deduce that 1/2ϖe℧η,A℧η<ϖe℧η,ℵη. After applying the condition (ii), for any ℧η,ℵη∈ℸ, we can write
(87)A℧η+Aℵη2≤∫uvΦη,ζ,℧ζ+Φη,ζ,ℵζdζ2∫uve−ϑ/2v−u℧ζ+ℵζdζ2≤∫uve−ϑ/2v−u×e−2ϑη×e2ϑη×℧ζ+ℵζdζ2≤e−ϑv−u2×ϖe℧,ℵ×e2ϑη∫uvdζ2≤e−ϑϖe℧,ℵ×e2ϑη,so we have
(88)A℧η+Aℵη×e−ϑη2≤e−ϑϖe℧,ℵ,which leads to
(89)A℧η+Aℵη∞2≤e−ϑϖe℧,ℵ.
It yields that(90)ϖeA℧η,Aℵη≤e−ϑϖe℧,ℵ.
TakingFϖeℓ=lnℓ for ℓ>0, one gets or
(91)lnϖeA℧η,Aℵη≤lne−ϑϖe℧,ℵ,(92)ϑ+lnϖeA℧η,Aℵη≤lnϖe℧,ℵ.
Equivalently,(93)ϑ+FϖeϖeA℧η,Aℵη≤Fϖeϖe℧,ℵ.
By Theorem9, A admits a fixed point, which is a solution of the problem (82).


## 4.2. Analytical Solution of Second-Order Differential Equation

Consider the second-order differential equation given as follows:(94)℧′′η=−Φη,℧η,η∈0,γ,℧0=℧η=0,where Φ:0,γ×ℝ→ℝ is a continuous function.Resolving the problem (94) is equivalent to resolving the following integral equation:
(95)℧η=∫0ηϕη,ζΦζ,℧ζdζ,∀η∈0,γ,where ϕ is Green’s function defined by
(96)ϕη,ζ=η1−ζ,if0≤η≤ζ≤γ,ζ1−η,if0≤ζ≤η≤γ,and Φ is a function as in Theorem 17. Hence, if ℧∈C0,γ, then ℧ is a solution of the problem (94) if and only if ℧ is a solution of the problem (95).Letℸ=C0,γ,ℝ be the set of all continuous functions defined on 0,γ, and define a norm ℧ϑ=maxη∈0,γ℧ηe−1/2ηϑ, for arbitrary η≥1. Obviously, ·ϑ is equivalent to the maximum norm · on ℸ, and ℸ is endowed with the extended generalized ϖeϑ-metric-like as
(97)ϖeϑ℧,ℵ=℧+ℵϑ2=maxη∈0,γ℧η+ℵη2e−ηϑforall℧,ℵ∈ℸandeηϑ≥1.Then,ℸ,ϖe is a complete modified ϖe-metric-like space with s℧,ℵ=1+℧+ℵ. Our main theorem is as follows.Theorem 18.
Suppose thatℸ,ϖe is a complete modified b-metric-like space and A is a nonlinear self-mapping on ℸ, then (95) possesses a unique solution ℧∈C0,γ,ℝ, if
(a1) for eachη,ζ∈[0,γ and ℧,ℵ∈ℸ,(98)12℧η+∫0ηϕη,ζΦζ,℧ζdζϑ≤℧+ℵϑ
(a2)Φ∈C0,γ×ℝ and ϕ∈C0,γ×[0,γ
(a3)Φ satisfies
(99)Φζ,℧ζ+Φζ,ℵζ2≤ϑe−ϑψ℧,ℵ,
for allζ∈[0,γ and ℧,ℵ∈ℝ, where
(100)ψ℧,ℵ=max℧+ℵ2,℧+A℧21+℧+A℧2,ℵ+Aℵ21+ℵ+Aℵ2,℧+Aℵ2+ℵ+A℧241+℧+ℵ
(a4)max∫0ηϕη,ζdζ≤1, for all η∈[0,γProof.
Consider on the setℸ, the mapping A as
(101)A℧η=∫0ηϕη,ζΦζ,℧ζdζ,for all η∈[0,γ and ℧∈ℸ. The solution of (95) is also a fixed point of A on ℸ. By condition (a1) and the definition of A, we can write 1/2ϖeϑ℧η,A℧η<ϖeϑ℧η,ℵη.
Let℧,ℵ∈ℸ. By the hypotheses (a2)-(a4), we have
(102)A℧η+Aℵη2=∫0ηϕη,ζΦζ,℧ζ+Φζ,ℵζdζ2≤∫0ηϕη,ζ2Φζ,℧ζ+Φζ,ℵζ2dζ≤∫0ηΦζ,℧ζ+Φζ,ℵζ2dζ≤∫0ηϑe−ϑψ℧ζ,ℵζdζ≤ϑe−ϑ∫0ηeζϑmax℧+ℵ2e−ζϑ,℧+A℧2e−2ζϑ1+℧+A℧2e−ζϑ,ℵ+Aℵ2e−2ζϑ1+ℵ+Aℵ2e−ζϑ,℧+Aℵ2+ℵ+A℧241+℧+ℵe−ζϑdζ≤ϑe−ϑ∫0ηeζϑmaxϖeϑ℧,ℵ,ϖeϑ℧,A℧1+ϖeϑ℧,A℧,ϖeϑℵ,Aℵ1+ϖeϑℵ,Aℵ,ϖeϑ℧,Aℵ+ϖeϑℵ,A℧4s℧,ℵdζ=ϑe−ϑψ℧,ℵ∫0ηeζϑdζ=ϑe−ϑψ℧,ℵ×eηϑϑ−1≤ϑe−ϑψ℧,ℵ×eηϑϑ≤e−ϑψ℧,ℵeηϑ.
Hence, for all℧,ℵ∈ℸ,(103)A℧η+Aℵη2×e−ηϑ≤e−ϑψ℧,ℵ,which yields
(104)ϖeϑA℧,Aℵ=maxη∈0,γA℧η+Aℵη2×e−ηϑ≤e−ϑψ℧,ℵ.
That is,(105)ϑ+lnϖeϑA℧,Aℵ≤lnψ℧,ℵ.
Defining the functionFϖeα=lnα,α>0 in (105), such that Fϖe∈Π, we have
(106)ϑ+FϖeϖeA℧,Aℵ≤Fϖemaxϖe℧,ℵ,ϖe℧,A℧1+ϖe℧,A℧,ϖeℵ,Aℵ1+ϖeℵ,Aℵ,ϖe℧,Aℵ+ϖeℵ,A℧4s℧,ℵ.
Hence, all requirements of Theorem15 hold and A is an extended generalized F-Suzuki contraction; hence, Γ possesses a fixed point ℧∈ℸ, which is a solution of the problem (95).


## 5. Conclusion

A modifiedϖe-metric-like space is presented, and related fixed point results via it are discussed. Nontrivial examples are conducted for supporting the mentioned space and theorems. Thereafter, by using a fixed point technique, a simple and efficient solution for the integral and differential equations is found in the setting of a modified ϖe-metric-like space. A lot of authors connected fixed point techniques and classical integral equations in various abstract spaces such as metric spaces, b-metric spaces, and partial metric spaces. We also follow the same method in the new space. In the literature, our obtained applications are an extension and/or a generalization of many existing classical integral and differential equations. The observed results of this paper open new framework research avenues for
(i)
fixed point techniques for solving Volterra-Fredholm integral equation in a modifiedϖe-metric-like space(ii)
collocation-type methods for Volterra-Hammerstein integral equations in modifiedϖe-metric-like spaces



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