# Solution of a Fractional Integral Equation Using the Darbo Fixed Point Theorem

Authors: Bhuban Chandra Deuri; Marija V. Paunović; Anupam Das; Vahid Parvaneh
Journal: Journal of Mathematics
 (2022)

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## Abstract

The concept of measure of noncompactness in a Banach space is used in this paper to extend some tripled fixed point theorems. We prove the existence of fractional integral equation solutions using a generalized Darbo fixed point theorem. To demonstrate the validity of the main result, an example is provided.


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## Body

## 1. Introduction

Noncompactness measure has ushered in a new branch of nonlinear analysis. It covers a wide range of applications in operator theory. Noncompactness measures have a wide range of applications in FP theory and are particularly useful in differential and integral equations, as well as fractional calculus. Kuratowski [1] investigated the first definition of a noncompactness measure. In 1955, Darbo [2] ensured the existence of fixed points for some mappings using the notion of noncompactness measures, which were obtained by generalizing the Schauder FP theorem [3] and the Banach contraction principle. Many authors use the term “noncompactness measure” to make Darbo FP theorem more general.The goal of this paper is to extend Darbo’s FP theorem and to apply our findings to determine the existence of solutions of fractional integral equations.We begin with preliminaries, notations, concepts, and definitions that will be used throughout the paper.Let us have a real Banach spaceB,∥.∥, and Bg,r=v∈B:v−g≤r. Let S≠ϕ⊆B. Also, let(a)
ℝ=−∞,∞.(b)
ℝ+=0,∞.(c)
S¯= the closure of S.(d)
ConvS= the convex closure of S.(e)
GB= the set of all nonempty and bounded subsets of B.(f)
ℍB= the set of all relatively compact sets.We provide the below definition of MNC, which is referenced in [4].Definition 1.
A mappingΛ:GB⟶ℝ+ is said to be a MNC in B if it fulfills the following axioms:(i)
The family kerΛ=S∈GB:ΛS=0≠ϕ and ker Λ⊂ℍB.(ii)
S⊆P⇒ΛS≤ΛP.(iii)
ΛS¯=ΛS.(iv)
ΛConvS=ΛS.(v)
ΛLS+1−LP≤LΛS+1−LΛP for any L∈0,1.(vi)
IfSq∈GB,Sq=S¯q,Sq+1⊂Sq for q=1,2,… and limq⟶∞ΛSq=0, then S∞=∩q=1∞Sq≠ϕ.SinceΛS∞≤ΛSq for all q, ΛS∞=0, and so S∞=∩q=1∞Sq∈kerΛ.In the theory of fixed points, the Schauder FP principle and Darbo theorem are crucial.Theorem 1 (see [3]) (Schauder).
For a nonempty, bounded, closed, and convex subset (NBCCS)D of a Banach space B, if ϒ:D⟶D is a continuous and compact mapping, it must have at least one FP.Theorem 2 (see [2]) (Darbo).
For a NBCCSW of a Banach space B, if T:W⟶W is a continuous self-mapping with(1)ΛTΩ≤pΛΩ,Ω⊆W,where p∈0,1 and Λ is an arbitrary MNC on B, then T has a FP.In fractional calculus, fixed point theorems have numerous applications. Let us have a look at some of the work that has been done in this area.In [5], Sahoo et al. developed numerous new inequalities for twice differentiable convex functions that are coupled with the Hermite–Hadamard integral inequality by using an integral equality related to the k-Riemann–Liouville fractional operator. In addition, for various types of convex functions, certain fresh examples of the established conclusions are derived. This fractional integral adds the symmetric properties of Riemann–Liouville and Hermite–Hadamard inequalities. The authors in [6] explored the existence and uniqueness of solutions to two-dimensional Volterra integral equations, Riemann–Liouville integrals, and Atangana–Baleanu integral operators.Deng et al. [7] examined the existence of mild solutions for a class of impulsive neutral stochastic functional differential equations in Hilbert spaces with noncompact semigroup. The Hausdorff measure of noncompactness and the Mönch fixed point theorem are used to find sufficient conditions for the existence of mild solutions. The presence of an almost periodic solution to a fractional differential equation with impulse and fractional Brownian motion under nonlocal conditions was the subject of the essay [8].


## 2. Main Result

We now recall some important definitions that are helpful to our work.Definition 2.
LetV be the set of all maps v:ℝ⟶ℝ satisfying(2)∑i=1∞vsn=∞,for all sn⊆ℝ.Definition 3 (see [9]).
Letℚ be the set of all functions Q:ℝ+×ℝ+⟶ℝ+ that fulfills the axioms:(1)
maxℓ1,ℓ2≤Qℓ1,ℓ2 for ℓ1,ℓ2≥0.(2)
Q is continuous.(3)
Qℓ1+ℓ2,n1+n2≤Qℓ1,n1+Qℓ2,n2.Example 1.
Qℓ1,ℓ2=ℓ1+ℓ2 is an example of the class ℚ.Using the above two classes of control functions, we prove the following results.Theorem 3.
LetW be a NBCCS of a Banach space B. Also, let Γ:W⟶W be a continuous mapping with(3)QΛΓG,ϖΛΓG≤QΛG,ϖΛG−vQΛG,ϖΛG,for all G⊆W, where Λ is an arbitrary MNC, v∈V, and Q∈ℚ. Also, let ϖ:ℝ+⟶ℝ+ be a nondecreasing continuous mapping. So, Γ has at least one FP in W.Proof.
We define the sequenceWnn as follows:(4)W0=W,Wn=ConvΓWn,n=1,2,3,….
We can easily see through induction that(5)Wn+1⊆Wn,n=0,1,2,….
IfN∈ℕ so that QΛWN,ϖΛWN=0, then ΛWN=0, that is, WN is a relatively compact set. So, by Theorem 1, Γ admits a FP in W.
Now, we may assume thatQΛWN,ϖΛWN>0 for each N∈ℕ.
On the contrary, we have(6)QΛWn+1,ϖΛWn+1=QΛΓWn,ϖΛΓWn⩽QΛWn,ϖΛWn−vQΛWn,ϖΛWn⩽QΛWn−1,ϖΛWn−1−vQΛWn−1,ϖΛWn−1,−vQΛWn,ϖΛWn⩽⋯⩽QΛW0,ϖΛW0−∑i=0nvQΛWi,ϖΛWi.
Since∑i=0nvQΛWi,ϖΛWi⟶∞, then QΛWn+1,ϖΛWn+1⟶0, as n⟶+∞.
This implies that(7)ΛWn⟶0,asn⟶+∞.
SinceWn⊇Wn+1, by Definition 1, we obtain that W∞:=∩n=1∞Wn is a nonempty, closed, and convex subset of W and W∞ is Γ invariant.
So, Theorem1 concludes that Γ has a FP in W.
Hence, we have the completed proof.The following is a crucial consequence of Theorem3.Corollary 1.
LetW be a NBCCS of a Banach space B. Also, let Γ:W⟶W be a continuous mapping with(8)ΛΓG+ϖΛΓG≤ΛG+ϖΛG−vΛG+ϖΛG,for all G⊆W, where Λ is an arbitrary MNC and v∈V. Also, let ϖ:ℝ+⟶ℝ+ be a nondecreasing continuous mapping. So, Γ has at least one FP in W.Proof.
PuttingQℓ1,ℓ2=ℓ1+ℓ2 in Theorem 3, we get the above corollary.Corollary 2.
LetW be a NBCCS of a Banach space B. Also, let Γ:W⟶W be a continuous mapping with(9)ΛΓG≤ΛG−vΛG,for all G⊆W, where Λ is an arbitrary MNC and v∈V. So, Γ has at least one FP in W.Proof.
Settingϖℓ=0 in Corollary 1, we obtain the above corollary.Corollary 3.
LetW be a NBCCS of a Banach space B. Also, let Γ:W⟶W be a continuous mapping with(10)ΛΓG≤σΛG,for all G⊆W, where Λ is an arbitrary MNC and σ=k/k+1∈0,1. So, Γ has at least one FP in W.Proof.
Settingvs=1/k+1s in Corollary 2, we obtain the above corollary.Definition 4.
(see [10]). A mapping J:W×W×W⟶W is called to have a tripled fixed point a,x,ℏ∈W3 if Ja,x,ℏ=a, Ja,x,ℏ=x and Ja,x,ℏ=ℏ.Theorem 4 (see [4]).
LetΛ1,Λ2,…,Λn be an MNC in B1,B2,…,Bn, respectively. Additionally, suppose that the mapping P:ℝ+ρ⟶ℝ+ is convex with Py1,y2,…,yρ=0⇔yσ=0 for σ=1,2,3,…,ρ. Then, ΛˇΩ=PΛ1Ω1,Λ2Ω2,…,ΛnΩn will be an MNC in B1×B2×⋯×Bn.Example 2 (see [11]).
LetPa,x,ℏ=a+x+ℏ, for a,x,ℏ∈ℝ+3. Now, Pa,x,ℏ=a+x+ℏ=0⇔a=x=ℏ=0. As P is convex which fulfills all conditions of Theorem 4, ΛˇΩ=PΛ1Ω1,Λ2Ω2,Λ3Ω3 is an MNC on B1×B2,×B3, where Ωσ is the natural projection of Ω into Bσ for σ=1,2,3.Example 3 (see [12]).
LetPa,x,ℏ=maxa,x,ℏ, for a,x,ℏ∈ℝ+3. Now, Pa,x,ℏ=maxa,x,ℏ=0⇔a=x=ℏ=0. As P is convex which fulfills all conditions of Theorem 4, ΛˇΩ=PΛ1Ω1,Λ2Ω2,Λ3Ω3 is an MNC on B1×B2,×B3, where Ωσ is the natural projection of Ω into Bσ for σ=1,2,3.Theorem 5.
LetW be a NBCCS of a Banach space B. Also, let J:W×W×W⟶W be a continuous mapping with(11)QΛJϖ1×ϖ2×ϖ3,ϖΛJϖ1×ϖ2×ϖ3≤μ3Λϖ1×ϖ2×ϖ3+ϖΛϖ1×ϖ2×ϖ3,for all ϖ1×ϖ2×ϖ3⊆W, where Λ is an arbitrary MNC and ϖ and Q are as in Theorem 1. Also, let μa+x+ℏ≤μa+μx+μℏ;a,x,ℏ≥0 and ϖa+x+ℏ≤ϖa+ϖx+ϖℏ; a,x,ℏ≥0. So, J has at least a tripled fixed point in W.Proof.
We consider a functionJˇ:W3⟶W3 by(12)Jˇϖ1,ϖ2,ϖ3=Jϖ1,ϖ2,ϖ3,Jϖ2,ϖ3,ϖ1,Jϖ3,ϖ1,ϖ2,for all ϖ1,ϖ2,ϖ3∈W. It is trivial that Jˇ is continuous.
SinceJ is continuous, assume that ϖ⊂W3 is nonempty. We have(13)Λˇϖ=Λϖ1+Λϖ2+Λϖ3,where ϖ1,ϖ2,ϖ3 represent W’s natural projections.
Now, we get(14)QΛˇJˇϖ,ϖΛˇJˇϖ⩽QΛˇJϖ1×ϖ2×ϖ3×Jϖ2×ϖ3×ϖ1×Jϖ3×ϖ1×ϖ2,ϖΛˇJϖ1×ϖ2×ϖ3×Jϖ2×ϖ3×ϖ1×Jϖ3×ϖ1×ϖ2=QΛJϖ1×ϖ2×ϖ3+ΛJϖ2×ϖ3×ϖ1+ΛJϖ3×ϖ1×ϖ2,ϖΛJϖ1×ϖ2×ϖ3+ΛJϖ2×ϖ3×ϖ1+ΛJϖ3×ϖ1×ϖ2≤QΛJϖ1×ϖ2×ϖ3+ΛJϖ2×ϖ3×ϖ1+ΛJϖ3×ϖ1×ϖ2,ϖΛJϖ1×ϖ2×ϖ3+ΛJϖ2×ϖ3×ϖ1+ΛJϖ3×ϖ1×ϖ2≤QΛJϖ1×ϖ2×ϖ3,ϖΛJϖ1×ϖ2×ϖ3+QΛJϖ2×ϖ3×ϖ1,ϖΛJϖ2×ϖ3×ϖ1+QΛJϖ3×ϖ1×ϖ2,ϖΛJϖ3×ϖ1×ϖ2≤μΛϖ1+Λϖ2+Λϖ3+ϖΛϖ1+Λϖ2+Λϖ3=μΛˇϖ+ϖΛˇϖ=μQΛˇϖ,ϖΛˇϖ.
We can conclude from Theorem1 that Jˇ has a minimum of one FP in W3.
Now, from Theorem1, J admits a tripled fixed point.


## 3. Measure of Noncompactness onC0,T

LetB=Cℐ be the space of real continuous functions on ℐ, where ℐ=0,T, which is equipped with(15)μ=supμt:t∈ℐ,μ∈B.LetJ≠ϕ⊆B be bounded. For μ∈J and δ>0, denote by Λμ,δ the modulus of the continuity of μ, i.e.,(16)Λμ,δ=supμh1−μh2:h1,h2∈ℐ,h1−h2≤δ.Moreover, we set(17)ΛJ,δ=supΛμ,δ:μ∈J;Λ0J=limδ⟶0ΛJ,δ.It is generally known that the mappingΛ0 is a MNC in B, and ΓJ=1/2Λ0J will be the Hausdorff MNC (see [4]).


## 4. Solvability of Fractional Integral Equations

In this part, we show how our conclusions concerning the existence of a solution to a fractional integral equation in a Banach space can be applied.Consider the following fractional integral equation [13]:(18)ψh=ψ0+fh,ψh+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ,where 0≤χ<1,ψ0=ψ0≥0,h∈ℐ=0,T.Let(19)Qr0=ψ∈B:ψ≤r0.Assume that(A)
f:ℐ×ℝ⟶ℝ is a continuous function and there exists a constant β1≥0 satisfying(20)fh,ψ−fh,ψ1≤β1ψ−ψ1,h∈ℐ;ψ,ψ1∈ℝ.Also,(21)F^=supfh,0:h∈ℐ.(B)
σ:ℐ×ℝ+⟶ℝ+ is a continuous function and there exists a nondecreasing function ϖ:ℝ+⟶ℝ+ satisfying(22)σh,ψ≤ϖψ;h,ψ∈ℐ×ℝ.(C)
There exists a positive solutionr0 for the following inequality:(23)ψ0+β1r0+F^+ϖr0Γχ+1.Tχ≤r0.Theorem 6.
If constraints (A)–(C) hold, equation (18) has at least one solution in B.Proof.
Consider the following operatorP:B⟶B such that(24)Pψh=ψ0+fh,ψh+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ.Step 1.
We show thatP maps Qr0 into Qr0. Let ψ∈Qr0, and we now have(25)Pψh≤ψ0+fh,ψh+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ≤ψ0+fh,ψh−fh,0+fh,0+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ≤ψ0+β1ψh+F^+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ.
Also,(26)1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ≤1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ≤ϖψΓχ∫0hh−ℓχ−1dℓ≤ϖψΓχ+1Tχ.
Hence,∥ψ∥<r0 gives(27)Pψ≤ψ0+β1r0+F^+ϖr0Γχ+1.Tχ≤r0.
Due to assumption (C),P maps Qr0 into Qr0.Step 2.
We show thatP is continuous on Qr0. Let δ>0 and ψ,ψ1∈Qr0 such that ψ−ψ1<δ. For all h∈ℐ, we have(28)Pψh−Pψ1h≤ψ0−ψ0+fh,ψh−fh,ψ1h+1Γχ∫0hh−ℓχ−1σℓ,ψℓdℓ−1Γχ∫0hh−ℓχ−1σℓ,ψ1ℓdℓ≤β1ψh−ψ1h+1Γχ∫0hh−ℓχ−1σℓ,ψℓ−σℓ,ψ1ℓdℓ≤β1ψ−ψ1+1Γχ∫0hh−ℓχ−1σℓ,ψℓ−σℓ,ψ1ℓdℓ<β1ψ−ψ1+1ΓχΛr0δ∫0hh−ℓχ−1dℓ<β1ψ−ψ1+1Γχ+1Λr0δTχ,where(29)Λr0δ=supσℓ,ψℓ−σℓ,ψ1ℓ:ψ−ψ1≤δ;ℓ∈I;ψ,ψ1∈−r0,r0.
Hence,ψ−ψ1<δ gives(30)Pψh−Pψ1h<β1δ+1Γχ+1Λr0δTχ.
Asδ⟶0, we get Pψh−Pψ1h⟶0.
This clearly proves thatP is continuous on Qr0.Step 3.
An estimation ofP with respect to Λ0: now, assume that Δ≠ϕ⊆Qr0. Let δ>0 be arbitrary. Also, choose ψ∈Δ with h1,h2∈ℐ such that h2−h1≤δ and h2≥h1.
Now,(31)Pψh2−Pψh1=ψ0+fh2,ψh2+1Γχ∫0h2h2−ℓχ−1σℓ,ψℓdℓ−ψ0−fh1,ψh1−1Γχ∫0h1h1−ℓχ−1σℓ,ψℓdℓ≤fh2,ψh2−fh1,ψh1+1Γχ∫0h2h2−ℓχ−1σℓ,ψℓdℓ−∫0h1h1−ℓχ−1σℓ,ψℓdℓ≤fh2,ψh2−fh1,ψh1+1Γχ∫h1h2h2−ℓχ−1σℓ,ψℓdℓ+∫0h1h2−ℓχ−1−h1−ℓχ−1σℓ,ψℓdℓ≤fh2,ψh2−fh2,ψh1+fh2,ψh1−fh1,ψh1+ϖψΓχ∫h1h2h2−ℓχ−1dℓ+∫0h1h2−ℓχ−1−h1−ℓχ−1dℓ≤β1ψh2−ψh1+fh2,ψh1−fh1,ψh1+ϖr0Γχ+1h2χ−h1χ≤β1Λψ,δ+Λfr0,δ+ϖr0Γχ+1h2χ−h1χ,where(32)Λfr0,δ=supfh2,ψh1−fh1,ψh1:h2−h1≤δ;h1,h2∈ℐ;ψ≤r0,Λψ,δ=supψh2−ψh1≤δ:h2−h1≤δ;h1,h2∈ℐ.
Asδ⟶0, h2⟶h1, so we get(33)limδ⟶0ϖr0Γχ+1h2χ−h1χ⟶0.
Hence,(34)Pψh2−Pψh1≤β1Λψ,δ+Λfr0,δ,i.e.,(35)ΛPψ,δ≤β1Λψ,ε+Λfr0,δ.
By the uniform continuity off on ℐ×−r0,r0, we now obtain limδ⟶0Λfr0,δ⟶0, as δ⟶0.
Takingsupψ∈Δ and δ⟶0, we get(36)Λ0PΔ≤β1Λ0Δ.
Hence, by Corollary3, P has a FP in Δ⊆Qr0.
That is, equation (18) has a solution in B.Example 4.
Consider the following fractional integral equation:(37)ψh=ψ2+ψ10+h4+1Γ1/2∫0hh−ℓ1/2sin−1ψ2ℓ1−ℓ2dℓ,for h∈0,2=ℐ, which is a particular case of equation (18).
Here,(38)ψ0=ψ2,fh,ϕh=ψ10+h4,χ=12,σℓ,ψℓ=sin−1ψ2ℓ1−ℓ2.
Also, it is trivial thatf is continuous and satisfies(39)fh,ψh−fh,ψ1h≤ψ−ψ110.
Therefore,β1=1/10.
Ifψ≤r0, then(40)ψ0=r02,F^=r010,σℓ,ψ≤ψ2.
So,(41)ϖr0=r02.
Putting these values in the inequality of assumption (C), we get(42)r02+110r0+r010+r02Γ3/221/2≤r0⇒r02Γ3/221/2≤310r0⇒r0≤3Γ3/21021/2.
However, assumption (C) is also fulfilled forr0=3Γ3/2/1021/2.
We can see that all of Theorem5’s assumptions are achieved, from (A) to (C). Equation (37), according to Theorem 5, has a solution in B=Cℐ.



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