# Solvability of Implicit Fractional Order Integral Equation inℓp1≤p<∞ Space via Generalized Darbo’s Fixed Point Theorem

Authors: Inzamamul Haque; Javid Ali; M. Mursaleen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations inℓp1≤p<∞ space. The fundamental tool in the presentation of our proofs is the measure of noncompactness mnc approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.


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## Body

## 1. Introduction

The integral equations have a variety of practical applications in defining specific real-world problems and situations, such as the law of physics, the theory of radioactive transmission, statistical mechanics, and cytotoxic activity (see [1–3]). Fixed point theory and mnc have several applications in solving various types of differential and integral equations [4, 5]. The existence of solutions for the system of integral equations is studied by Aghajani et al. [6]. Mursaleen and Mohiuddin [7] proved the existence theorem for the infinite system of differential equations in the space ℓp. Banas [8] studied the solution of nonlinear differential and integral equations. Arab et al. [9] proved the existence of functional integral equations using mnc. Also, Çakan [10] proved the existence of nonlinear integral equations in Banach spaces by using the technique of mnc. The notions of α-ψ and β-ψ condensing operators were recently developed in [11], and they were used to show some new fixed point results using the technique of measure of noncompactness.Fractional calculus theory and applications grew rapidly in the nineteenth and early twentieth century, and many contributors provided interpretations for fractional derivatives and integrals. Rezapour et al. [12] studied a fractional-order model for anthrax disease among animals based on the Caputo-Fabrizio derivative. The Erdélyl-Kober fractional integral is utilized in various areas of mathematics, including porous media and viscoelasticity [13–15]. The study of fractional order integral equations has become necessary due to their importance. The Erdélyl-Kober fractional operator was studied by various researchers for differential and integral equations. Darwish [16] studied the existence of a solution for Erdélyl-Kober fractional Urysohn-Volterra quadratic integral equations. Mollapouras and Ostadi [17] investigated the existence and stability of the solution of the functional integral equations of fractional order arising in physics, mechanics, and chemical reactions. Mohammadi et al. [18] and Jleli et al. [19] have been generalized Darbo’s fixed point theorem with the help of a new type of contraction operator. Motivated by these, we have generalized Darbo’s fixed point theorem by using the ψ,ϕ-type contraction [20] whose extension is also generalized and established by Jleli et al. [19] and Mohammadi et al. [18] and applied it to the existence of solutions for the infinite system of fractional order integral equations in ℓp1≤p<∞ space.


## 2. Preliminaries

We introduce notations, definitions, and introductory facts in this section, which will be used throughout the paper.For bounded subsetE of a metric space Y, Kuratowski [21] measure of noncompactness is defined as
(1)αE=infε>0:E=⋃k=1nEk,diamEk≤ε,1≤k≤n<∞,where diamEk denotes the diameter of a set Ek⊂Y.The Hausdorff (or ball)mnc for E is defined as follows:
(2)χE=infε>0:Ehasafiniteε−netinY.Throughout this paper, we assumed that (X,.) is a real Banach space and ℝ+≔0,∞. Let B⊂X, we mark by B¯ and ConvB the closure and the closed convex hull of B, respectively. In addition, let MX denotes the family of nonempty bounded subset of X and NX collection of all nonempty relatively compact subsets of X.Definition 1 (see [22]).
A mappingν:MX⟶0,∞ is called the measure of noncompactness mnc in X if
(A1). The family ker ν≔B∈MX:νB=0 is nonempty and ker ν⊂NX
(A2). B1⊂B2⇒νB1≤νB2
(A3). νB¯=νB
(A4). νConvB=νB
(A5). νsB1+1−sB2≤sνB1+1−sνB2 for 0≤s≤1
(A6). If Bk∈MX, Bk=B¯k, Bk+1⊂Bk,k∈ℕ and limk⟶∞νBk=0 then B∞≔⋂k=1∞Bk is nonempty
In the Banach spaceℓp,.ℓp, the Hausdorff mncχ is given by
(3)χℓpB=limk⟶∞supy∈B∑i≥kyip1/p,B∈Mℓp.
Now, we will go over the following important theorems in fixed point theory that play a key role.Theorem 2 (see [23], Schauder).
SupposeE is a nonempty, bounded, convex, and closed subset of a Banach space X. Then, each continuous, compact map L:E⟶E has a fixed point.
The following theorem given by Darbo is a generalization of Theorem2.Theorem 3 (see [22], Darbo).
Supposeζ be a nonempty, bounded, convex, and closed subset of a Banach space X, and let L:ζ⟶ζ is a continuous operator such that there exists a constant m∈0,1 with the property χLζ≤mχζ. Then, L has a fixed point in the set ζ.
The Darbo fixed point theorem is more effective than the Schauder fixed point theorem. Because in the case of Darbo fixed point theorem, the compactness of the domain of the operator, which is essential in Schauder’s theorem, has been relaxed.
Jleli et al. obtained the following extension of the Darbo fixed point theorem.Theorem 4 (see [19]).
Supposeζ is a nonempty, bounded, closed, and convex subset of a Banach space X, and L:ζ⟶ζ be continuous operator such that for every nonempty subset Ω of ζ,
(4)νΩ.νLΩ>0⇒θνLΩ≤θνΩk
wherek∈0,1 and θ∈Δ, Δ is the class of function θ:0,∞⟶1,∞ satisfying the following condition:
For eachtn⊂0,∞, limn⟶∞θtn=1⇔limn⟶∞tn=0+.
Then,L has at least one fixed point in ζ.With a newly defined contraction operator, Mohammadi et al. have established the following generalization of Darbo’s fixed point theorem, which is also generalized the result of Wardowski’s F-contraction [24].Theorem 5 (see [18]).
Supposeζ is a nonempty, bounded, convex, and closed subset of a Banach space X, and L:ζ⟶ζ be continuous mapping such that for every nonempty subset Ω of ζ,
(5)τ+WνLΩ≤WνΩ,
whereν is the arbitrary mnc, τ is the arbitrary positive constant, W∈Γ, Γ be the set of function W:ℝ+⟶ℝ such that
(W1). W is a strictly increasing and continuous function
(W2). limk⟶∞tk=0⇔limk⟶∞Wtk=−∞, for each tk⊂ℝ+
Then,L has at least one fixed point in ζ.
We also recall the concept of a coupled fixed point for a bivariate mapping and an important theorem on the construction of a measure of noncompactness on a finite product space.Definition 6 (see [25]).
An elementw1,w2∈Z×Z is called a coupled fixed point of mapping L:Z×Z⟶Z if Lw1,w2=w1 and Lw2,w1=w2.Theorem 7 (see [26]).
Letν1,ν2,⋯νk be mnc in Banach spaces X1,X2,⋯,Xk, respectively, and the function h:ℝ+k⟶ℝ+ is a convex function and hr1,r2,⋯,rk=0 if and only if rj=0∀j=1,2,⋯,k. Then,
(6)ν~ω=hν1ω1,ν2ω2,⋯,νkωk
is amnc in X1×X2×⋯×Xk, where ωi is the natural projection of ω into Xi∀i=1,2,⋯,k.
Now, we have the following example, which was described in [6], as a result of Theorem 7.Example 1.
Consideringν be a mnc on a Banach space X, let Gw1,w2=maxw1,w2 for any w1,w2∈ℝ+2, as we can see that G is convex and Gw1,w2=0 if and only if w1=w2=0; hence, all the conditions of Theorem 7 are satisfied. Therefore, ν~E=maxνE1,νE2 defines a mnc in the space X×X where Ei,i=1,2 denote the natural projection of E. Similarly, by letting Gw1,w2=w1+w2 for any w1,w2∈ℝ+2, we have ν~E=νE1+νE2 defines a mnc in the space X×X where Ei,i=1,2 denote the natural projection of E.
In 2016, Liu et al. [20] introduced Φ, the set of function ϕ:ℝ+⟶ℝ+ such that
(ф1). ϕ is nondecreasing
(ф2). limk⟶∞ϕtk=0 if and only if limk⟶∞tk=0, for each sequence tk⊂0,∞
(ф3). ϕ is continuous
Berinde [27] introduced the class of function known as comparison function. Let Ψ be the family of all function ψ:ℝ+⟶ℝ+ satisfying the following conditions:
(ѱ1). ψ is monotone increasing
(ѱ2). limn⟶∞ψnt=0,∀t>0


## 3. Main Result

Theorem 8.
SupposeE is a nonempty, bounded, convex, and closed subset of a Banach space X, and S:E⟶E be continuous mapping such that
(7)νSQ>0⇒ϕνSQ≤ψϕνQ
for all subsetQ of E, where ϕ∈Φ and ψ∈Ψ and ν is an arbitrary mnc. Then, S has at least one fixed point.Proof.
By induction, we define a sequenceEk by letting E0=E and Ek=ConvSEk−1,k≥1. We have SE0=SE⊆E=E0, E1=ConvSE0⊆E=E0; therefore, by continuing this process, we obtain
(8)E0⊇E1⊇E2⊇⋯⊇Ek⊇Ek+1.
Let there exists an integerK≥0 such that νEK=0. So, EK is relatively compact and SEK⊆ConvSEK=EK+1; hence, Theorem 2 yields that S has a fixed point. So we suppose that νEk>0 for k∈ℕ. By Equation (7), we have
(9)ϕνEk+1=ϕνConvSEk=ϕνSEk≤ψϕνEk≤ψϕνConvSEk−1=ψϕνSEk−1≤ψ2ϕνEk−1⋯≤ψk+1ϕνE0.
Thus, we have(10)ϕνEk+1≤ψk+1ϕνE0.
Lettingk⟶∞ in (10) and applying Ψ2, we have
(11)ϕνEk+1=0ask⟶∞.
By usingΦ2, we obtained
(12)limk⟶∞νEk+1=0.
SinceEk+1⊆Ek and SEk⊆Ek for all k≥1, then from A6, E∞=⋂k=1∞Ek is nonempty convex closed set, invariant under S and belong to ker ν. Now, from Theorem 2, S has a fixed point.Remark 9.
We can get the Darbo’s fixed point Theorem3, if we take ψs=λs,λ∈0,1,s>0 and ϕs=s:0,∞⟶0,∞ in above theorem.Remark 10.
We get Theorem4 if we take ψs=λs,λ∈0,1,s>0ϕs=lnθs:0,∞⟶0,∞ in Theorem 8.Remark 11.
We get Theorem5 if we take ψs=e−τs,s,τ>0 and ϕs=eWs:0,∞⟶0,∞ in Theorem 8.Theorem 12.
LetE be a nonempty, closed, and convex subset of a Banach space X, and S:E×E⟶E be continuous operator such that
(13)ϕνSω1×ω2≤12ψϕνω1+νω2,
for all subsetω1,ω2 of E, where ν arbitrary mnc and ϕ and ψ are as in Theorem 8. Then, S has at least a coupled fixed point.Proof.
From Example1, we have ν~ω=νω1+νω2 defines a mnc in the space X×X where ωi,i=1,2 denote the natural projection of ω. Now, consider S~:E×E⟶E×E defined by
(14)S~w1,w2=Sw1,w2,Sw2,w1,
which is clearly continuous. We show thatS~ satisfies all the conditions of Theorem 8. Let ω⊂E×E be nonempty subset. From Equation (13) we have
(15)ϕν~S~ω≤ϕν~Sω1×ω2×Sω2×ω1≤ϕνSω1×ω2+νSω2×ω1≤12ψϕνω1+νω2+12ψϕνω2+νω1≤ψϕνω1+νω2=ψϕν~ω.
Therefore, all the conditions of Theorem8 are satisfied. Hence, S~ has a fixed point which implies that S has a coupled fixed point.


## 4. Application

The fractional integral of a functionh∈L1a,b (the space of Lebesgue measurable functions) of order γ with respect to monotone, continuous derivative function w is defined by [28]
(16)Ia,wγhz=1Γγ∫azw′shswz−ws1−γds,γ>0,−∞≤a<b≤∞.Nieto and Samet [29] discussed the existence of solutions of the generalized implicit fractional integral equation
(17)ys=Gs,ys,g∫asw′rfs,r,yrws−wr1−γdr,s∈a,ρ,whereρ>0,a≥0,γ∈0,1,G:a,ρ×ℝ×ℝ⟶ℝ,g:ℝ⟶ℝ,w:a,ρ⟶ℝ and f:a,ρ×a,ρ×ℝ⟶ℝ.Also, the existence of a solution for an infinite system of implicit fractional integral equations in the Banach spacec0 and ℓ1 was studied by Das et al. [30].Motivated by these, we discuss the existence of a solution for the infinite system of implicit fractional integral equations in the Banach spaceℓp,p≥1.Consider the following infinite system of fractional integral equations(18)yks=Gks,ys,∫asw′rfks,r,yrws−wr1−γdr,k∈ℕ,whereγ∈0,1,s∈a,ρ,ρ>0,a≥0,Gk:a,ρ×ℝ×ℝ⟶ℝ,w:a,ρ⟶ℝ and fk:a,ρ×a,ρ×ℝ⟶ℝ,ys=yksk=1∞∈Ca,ρ,ℓp.


### 4.1. Existence of Solution onCa,ρ,ℓp

Assume thatM.1. The functionsGk:a,ρ×Ca,ρ,ℓp×ℝ⟶ℝk∈ℕ are continuous and satisfy
(19)Gks,ys,u−Gks,y¯s,vp≤αksyks−y¯ksp+βksu−vp,forys=yksk=1∞,y¯s=y¯ksk=1∞∈Ca,ρ,ℓp and αk,βk:a,ρ⟶ℝ+k∈ℕ are continuous functions. Also,
(20)∑k=1∞Gs,y0,0pconverge to zero for alls∈a,ρ, where y0=yk0sk=1∞∈Ca,ρ,ℓp such that yk0s=0 for all s∈a,ρ,k∈ℕM.2. The functionsfk:a,ρ×a,ρ×Ca,ρ,ℓp⟶ℝk∈ℕ are continuous and there exists
(21)Rk=supfks,r,yr:s,r∈a,ρ:yr∈Ca,ρ,ℓpwherek∈ℕ. Also, supk∈ℕR=R^M.3. The functionw:a,ρ⟶ℝ+ is in C1 and nondecreasingM.4. Define an operatorP from a,ρ×Ca,ρ,ℓp×ℝ to Ca,ρ,ℓp as follows:
(22)s,ys↦Pyswhere(23)Pys=Gks,ys,∫asw′rfks,r,yrws−wr1−γdrk=1∞M.5. Let(24)sups∈a,ραks=α^k,supk∈ℕαk=α^,0<α^<2−pAlso,∀s∈a,ρ, ∑k=1∞βks is convergent and
(25)∑k=1∞βks≤β^.AssumeBp=s∈Ca,ρ,ℓp:∥s∥Ca,ρ,ℓp≤κ0.Theorem 13.
Under the assumption (M.1)-(M.5), Equation (18) has at least one solution in Ca,ρ,ℓp.Proof.
For anys∈a,ρ,
(26)∥ys∥ℓpp=∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdrp=∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y0s,0+Gks,y0s,0p≤2p∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y0s,0p+∑k=1∞Gks,y0s,0p≤2p∑k=1∞αkyksp+βks∫asw′rfks,r,yrws−wr1−γdrp≤2pα^k∥ys∥ℓp+2p∑k=1∞βks∫asw′rfks,r,yrws−wr1−γdrp≤2pα^k∥ys∥ℓp+2p∑k=1∞βks∫asw′rR∧ws−wr1−γdrp≤2pα^k∥ys∥ℓp+2pβ^R∧p∫asw′rws−wr1−γdrp≤2pα^k∥ys∥ℓp+2pβ^R∧pwρ−waγpγp.
Therefore,(27)1−2pα^k∥ys∥ℓp≤2β∧1/pR^wρ−waγγimplies
(28)∥ys∥ℓp≤2β∧1/pR^wρ−waγγ1−2pα^k=κsay.
Hence,∥ys∥Ca,ρ,ℓp≤κ0.
Let us take an operatorP:a,ρ×Ca,ρ,ℓp⟶Ca,ρ,ℓp defined by
(29)Pys=Gks,ys,∫asw′rfks,r,yrws−wr1−γdrk=1∞=Pkysk=1∞,
whereys∈Bp,s∈a,ρ.
Since for eachs∈a,ρ, we have by assumption (M.4) that
(30)∑k=1∞Pkysp<∞.
Hence,Pys∈Ca,ρ,ℓp.
Lety¯s=y¯ksk=1∞∈Bp and ε>0 with ∥y−y¯∥Ca,ρ,ℓp<ε/2α^=δ.
Fors∈a,ρ, and by using Hölder inequality with 1/p+1/q=1, we have
(31)Pkys−Pky¯sp=Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y¯s,∫asw′rfks,r,y¯rws−wr1−γdrp≤αksyks−y¯ksp+βks∫asw′rfks,r,yr−fks,r,y¯rws−wr1−γdrp≤αksyks−y¯ksp+s−ap/qβks∫asw′rws−wr1−γdrpfks,r,yr−fks,r,y¯rp.
Sincefk are continuous for all k∈ℕ, hence for ∥y−y¯∥Ca,ρ,ℓp<ε/2α^=δ, we have
(32)fks,r,yr−fks,r,y¯rp<εγp2β^s−ap/qwρ−waγp.
Therefore,(33)∑k=1∞Pkys−Pky¯sp≤α^∑k=1∞yks−y¯ksp+εγp2β^s−ap/qwρ−waγp×β^s−ap/qwρ−waγpγp≤α^∥y−y¯∥Ca,ρ,ℓp+ε2<ε2+ε2=ε.
Therefore,∥Py−Py^∥Ca,ρ,ℓp<ε when ∥y−y¯∥Ca,ρ,ℓp<δ; hence, P is continuous on Bp.
Now,(34)χℓpPBpp=limk⟶∞supy∈Bp∑i≥k∞Gis,ys,∫asw′rfis,r,yrws−wr1−γdrp≤limk⟶∞supy∈Bp2pα^∑i≥k∞yisp+2p∑i≥k∞βis∫asw′rfks,r,yrws−wr1−γdrp≤limk⟶∞supy∈Bp2pα^∑i≥k∞yisp+2pR∧pwρ−waγpγp∑i≥k∞βis=2pα^χℓpBpp.
Hence,(35)χℓpPBp≤2α∧1/pχℓpBp.
Letm=2α∧1/p, so m∈0,1.
Therefore,(36)χℓpPBp≤mχℓpBp,m∈0,1.
Thus, from assumption (M.5) and Remark9, operator P has at least one fixed point in Bp⊆Ca,ρ,ℓp. Hence, Equation (18) has a solution in Ca,ρ,ℓp. This completes the proof.


### 4.2. Particular Cases

Consider the following infinite system of fractional integral equations involving the Riemann-Liouville fractional integral(37)yks=Gks,ys,∫asfks,r,yrs−r1−γdr,k∈ℕ,whereγ∈0,1,s∈a,ρ,ρ>0,a≥0,Gk:a,ρ×ℝ×ℝ⟶ℝ,w:a,ρ⟶ℝ and fk:a,ρ×a,ρ×ℝ⟶ℝ,ys=yksk=1∞∈Ca,ρ,ℓp.Takingws=s in Theorem 13, we obtain the following existence result for the integral Equation (37).Corollary 14.
Under the assumption (M.1)-(M.5), Equation (37) has at least one solution in Ca,ρ,ℓp.Consider the following infinite system of fractional integral equation involving the Hadamard fractional integral(38)yks=Gks,ys,∫aslnrsfks,r,yrsdr,k∈ℕ,whereγ∈0,1,s∈a,ρ,ρ>0,a≥0,Gk:a,ρ×ℝ×ℝ⟶ℝ,w:a,ρ⟶ℝ and fk:a,ρ×a,ρ×ℝ⟶ℝ,ys=yksk=1∞∈Ca,ρ,ℓp,Takingws=lns in Theorem 13, we obtain the following existence result for the integral Equation (38).Corollary 15.
Under the assumption (M.1)-(M.5), Equation (38) has at least one solution in Ca,ρ,ℓp.


### 4.3. Numerical Example

The following example illustrates the result of Theorem13Example 2.
Let us take the following infinite system of fractional integral equations:(39)yks=coss2yks4k2+s+2k3es∫0srsinykrs2−r23/4k3+r2dr,
wheres∈a,ρ=0,1,k∈ℕ.
Here,Gks,ys,u=coss2yks/4k2+s+u/k3es,fks,r,yr=sinykr/k3+r2,ws=s2,γ=1/4,a=0 and τ=1 also take p=2.
It is easy to see thatGk is continuous for all k∈ℕ and
(40)Gks,ys,u−Gks,y¯s,v2=coss2yks4k2+s−coss2y¯ks4k2+s+uk3es−vk3es2≤2coss24k2+s2yks−y¯ks2+2k3es2u−v2.
Here,(41)αks=2coss24k2+s2,α^k=14k4,α^=14,βks=2k3es2,∑k=1∞βks=2es2∑k=1∞1k6≤2e2∑k=1∞1k6~0.541π6945=β^,∑k=1∞Gks,y0,02=0.
The functionws=s2 is in C1 and nondecreasing and fk are continuous ∀k∈ℕ and fks,r,yr=sinykr/k3+r2≤1/k3. So, R=1/k3 and R^=1.
Ify∈Ca,ρ,ℓ2, then
(42)∑k=1∞coss2yks4k2+s+2k3es∫0srsinykrs2−r23/4k3+r2dr2≤22∑k=1∞coss2yks4k2+s2+22∑k=1∞2k3es∫0srsinykrs2−r23/4k3+r2dr2≤44+s2∑k=1∞yks2+16e2∑k=1∞∫0srsinykrs2−r23/4k3+r2dr2≤∑k=1∞yks2+16e2∑k=1∞∫0srykrs2−r23/4k3+r2dr2≤∑k=1∞yks2+16e2∫0srs2−r23/4k3+r2dr2∑k=1∞yks2≤yCa,ρ,ℓ2+256e2yCa,ρ,ℓ2=1+256e2yCa,ρ,ℓ2<∞.
Hence, (M.4) is satisfied. Thus, all conditions of Theorem13 hold; therefore the system (39) has a solution ys=yks in Ca,ρ,ℓ2.


## 4.1. Existence of Solution onCa,ρ,ℓp

Assume thatM.1. The functionsGk:a,ρ×Ca,ρ,ℓp×ℝ⟶ℝk∈ℕ are continuous and satisfy
(19)Gks,ys,u−Gks,y¯s,vp≤αksyks−y¯ksp+βksu−vp,forys=yksk=1∞,y¯s=y¯ksk=1∞∈Ca,ρ,ℓp and αk,βk:a,ρ⟶ℝ+k∈ℕ are continuous functions. Also,
(20)∑k=1∞Gs,y0,0pconverge to zero for alls∈a,ρ, where y0=yk0sk=1∞∈Ca,ρ,ℓp such that yk0s=0 for all s∈a,ρ,k∈ℕM.2. The functionsfk:a,ρ×a,ρ×Ca,ρ,ℓp⟶ℝk∈ℕ are continuous and there exists
(21)Rk=supfks,r,yr:s,r∈a,ρ:yr∈Ca,ρ,ℓpwherek∈ℕ. Also, supk∈ℕR=R^M.3. The functionw:a,ρ⟶ℝ+ is in C1 and nondecreasingM.4. Define an operatorP from a,ρ×Ca,ρ,ℓp×ℝ to Ca,ρ,ℓp as follows:
(22)s,ys↦Pyswhere(23)Pys=Gks,ys,∫asw′rfks,r,yrws−wr1−γdrk=1∞M.5. Let(24)sups∈a,ραks=α^k,supk∈ℕαk=α^,0<α^<2−pAlso,∀s∈a,ρ, ∑k=1∞βks is convergent and
(25)∑k=1∞βks≤β^.AssumeBp=s∈Ca,ρ,ℓp:∥s∥Ca,ρ,ℓp≤κ0.Theorem 13.
Under the assumption (M.1)-(M.5), Equation (18) has at least one solution in Ca,ρ,ℓp.Proof.
For anys∈a,ρ,
(26)∥ys∥ℓpp=∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdrp=∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y0s,0+Gks,y0s,0p≤2p∑k=1∞Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y0s,0p+∑k=1∞Gks,y0s,0p≤2p∑k=1∞αkyksp+βks∫asw′rfks,r,yrws−wr1−γdrp≤2pα^k∥ys∥ℓp+2p∑k=1∞βks∫asw′rfks,r,yrws−wr1−γdrp≤2pα^k∥ys∥ℓp+2p∑k=1∞βks∫asw′rR∧ws−wr1−γdrp≤2pα^k∥ys∥ℓp+2pβ^R∧p∫asw′rws−wr1−γdrp≤2pα^k∥ys∥ℓp+2pβ^R∧pwρ−waγpγp.
Therefore,(27)1−2pα^k∥ys∥ℓp≤2β∧1/pR^wρ−waγγimplies
(28)∥ys∥ℓp≤2β∧1/pR^wρ−waγγ1−2pα^k=κsay.
Hence,∥ys∥Ca,ρ,ℓp≤κ0.
Let us take an operatorP:a,ρ×Ca,ρ,ℓp⟶Ca,ρ,ℓp defined by
(29)Pys=Gks,ys,∫asw′rfks,r,yrws−wr1−γdrk=1∞=Pkysk=1∞,
whereys∈Bp,s∈a,ρ.
Since for eachs∈a,ρ, we have by assumption (M.4) that
(30)∑k=1∞Pkysp<∞.
Hence,Pys∈Ca,ρ,ℓp.
Lety¯s=y¯ksk=1∞∈Bp and ε>0 with ∥y−y¯∥Ca,ρ,ℓp<ε/2α^=δ.
Fors∈a,ρ, and by using Hölder inequality with 1/p+1/q=1, we have
(31)Pkys−Pky¯sp=Gks,ys,∫asw′rfks,r,yrws−wr1−γdr−Gks,y¯s,∫asw′rfks,r,y¯rws−wr1−γdrp≤αksyks−y¯ksp+βks∫asw′rfks,r,yr−fks,r,y¯rws−wr1−γdrp≤αksyks−y¯ksp+s−ap/qβks∫asw′rws−wr1−γdrpfks,r,yr−fks,r,y¯rp.
Sincefk are continuous for all k∈ℕ, hence for ∥y−y¯∥Ca,ρ,ℓp<ε/2α^=δ, we have
(32)fks,r,yr−fks,r,y¯rp<εγp2β^s−ap/qwρ−waγp.
Therefore,(33)∑k=1∞Pkys−Pky¯sp≤α^∑k=1∞yks−y¯ksp+εγp2β^s−ap/qwρ−waγp×β^s−ap/qwρ−waγpγp≤α^∥y−y¯∥Ca,ρ,ℓp+ε2<ε2+ε2=ε.
Therefore,∥Py−Py^∥Ca,ρ,ℓp<ε when ∥y−y¯∥Ca,ρ,ℓp<δ; hence, P is continuous on Bp.
Now,(34)χℓpPBpp=limk⟶∞supy∈Bp∑i≥k∞Gis,ys,∫asw′rfis,r,yrws−wr1−γdrp≤limk⟶∞supy∈Bp2pα^∑i≥k∞yisp+2p∑i≥k∞βis∫asw′rfks,r,yrws−wr1−γdrp≤limk⟶∞supy∈Bp2pα^∑i≥k∞yisp+2pR∧pwρ−waγpγp∑i≥k∞βis=2pα^χℓpBpp.
Hence,(35)χℓpPBp≤2α∧1/pχℓpBp.
Letm=2α∧1/p, so m∈0,1.
Therefore,(36)χℓpPBp≤mχℓpBp,m∈0,1.
Thus, from assumption (M.5) and Remark9, operator P has at least one fixed point in Bp⊆Ca,ρ,ℓp. Hence, Equation (18) has a solution in Ca,ρ,ℓp. This completes the proof.


## 4.2. Particular Cases

Consider the following infinite system of fractional integral equations involving the Riemann-Liouville fractional integral(37)yks=Gks,ys,∫asfks,r,yrs−r1−γdr,k∈ℕ,whereγ∈0,1,s∈a,ρ,ρ>0,a≥0,Gk:a,ρ×ℝ×ℝ⟶ℝ,w:a,ρ⟶ℝ and fk:a,ρ×a,ρ×ℝ⟶ℝ,ys=yksk=1∞∈Ca,ρ,ℓp.Takingws=s in Theorem 13, we obtain the following existence result for the integral Equation (37).Corollary 14.
Under the assumption (M.1)-(M.5), Equation (37) has at least one solution in Ca,ρ,ℓp.Consider the following infinite system of fractional integral equation involving the Hadamard fractional integral(38)yks=Gks,ys,∫aslnrsfks,r,yrsdr,k∈ℕ,whereγ∈0,1,s∈a,ρ,ρ>0,a≥0,Gk:a,ρ×ℝ×ℝ⟶ℝ,w:a,ρ⟶ℝ and fk:a,ρ×a,ρ×ℝ⟶ℝ,ys=yksk=1∞∈Ca,ρ,ℓp,Takingws=lns in Theorem 13, we obtain the following existence result for the integral Equation (38).Corollary 15.
Under the assumption (M.1)-(M.5), Equation (38) has at least one solution in Ca,ρ,ℓp.


## 4.3. Numerical Example

The following example illustrates the result of Theorem13Example 2.
Let us take the following infinite system of fractional integral equations:(39)yks=coss2yks4k2+s+2k3es∫0srsinykrs2−r23/4k3+r2dr,
wheres∈a,ρ=0,1,k∈ℕ.
Here,Gks,ys,u=coss2yks/4k2+s+u/k3es,fks,r,yr=sinykr/k3+r2,ws=s2,γ=1/4,a=0 and τ=1 also take p=2.
It is easy to see thatGk is continuous for all k∈ℕ and
(40)Gks,ys,u−Gks,y¯s,v2=coss2yks4k2+s−coss2y¯ks4k2+s+uk3es−vk3es2≤2coss24k2+s2yks−y¯ks2+2k3es2u−v2.
Here,(41)αks=2coss24k2+s2,α^k=14k4,α^=14,βks=2k3es2,∑k=1∞βks=2es2∑k=1∞1k6≤2e2∑k=1∞1k6~0.541π6945=β^,∑k=1∞Gks,y0,02=0.
The functionws=s2 is in C1 and nondecreasing and fk are continuous ∀k∈ℕ and fks,r,yr=sinykr/k3+r2≤1/k3. So, R=1/k3 and R^=1.
Ify∈Ca,ρ,ℓ2, then
(42)∑k=1∞coss2yks4k2+s+2k3es∫0srsinykrs2−r23/4k3+r2dr2≤22∑k=1∞coss2yks4k2+s2+22∑k=1∞2k3es∫0srsinykrs2−r23/4k3+r2dr2≤44+s2∑k=1∞yks2+16e2∑k=1∞∫0srsinykrs2−r23/4k3+r2dr2≤∑k=1∞yks2+16e2∑k=1∞∫0srykrs2−r23/4k3+r2dr2≤∑k=1∞yks2+16e2∫0srs2−r23/4k3+r2dr2∑k=1∞yks2≤yCa,ρ,ℓ2+256e2yCa,ρ,ℓ2=1+256e2yCa,ρ,ℓ2<∞.
Hence, (M.4) is satisfied. Thus, all conditions of Theorem13 hold; therefore the system (39) has a solution ys=yks in Ca,ρ,ℓ2.


## 5. Conclusion

In this article, we established a generalization of Darbo’s fixed point theorem and applied it to the infinite system of fractional order integral equation by another function which is the generalization of many other fractional integrals.The main contribution of this paper is that it extends the results of Jlei et al. and Mohammadi et al. to established a generalization of Darbo’s fixed point and applied it to investigate the existence of the solution of an infinite system of fractional integral equations inℓp,1≤p<∞, which is a more general space than ℓ1.



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