# Existence of Solutions for Nonlinear Integral Equations in Tempered Sequence Spaces via Generalized Darbo-Type Theorem

Authors: S. A. Mohiuddine; Anupam Das; Abdullah Alotaibi
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings.


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## Body

## 1. Introduction and Preliminaries

Darbo [1] constructed the fixed point theorem, and later, researchers called this widely studied theorem by his name, that is, “Darbo fixed point theorem” wherein he enforced the technique of measure of noncompactness (shortly, MNC) while Kuratowski [2] was the first who described the idea of MNC. Many researchers are employing Darbo’s theorem to demonstrate the existence or solvability of several functional equations (linear or nonlinear) in conjunction with different kind of Banach sequence spaces or simply called Banach spaces. Recently, the infinite system of several kinds of differential equations was considered by Banas and Lecko [3], Mursaleen et al. [4, 5], and Mohiuddine et al. [6] to obtain the existence of solutions in the framework of Banach spaces, namely, the spaces c0, c, ℓp, and ℓ1 of null, convergent, absolutely p-summable, and absolutely summable sequences in conjunction with the Dorbo-type theorem. The reader can refer to the recent monographs [7, 8] on the normed/paranormed sequence spaces and related topics.The integral equations play a significant contribution in diverse branches of science and engineering as well as this theory is applicable in several real life problems such as gas kinetic theory, neutron transportation, and radiation [9]. Most recently, the researchers used different kinds of integral equations (infinite system) (see [10–12]) to demonstrate existence of solutions by means of the notion of MNC, i.e., in ℓp [13] and in Banach space [14–16].Suppose thatE is a Banach space, and suppose also that Bθ,r^=x∈E:x−θ≤r^ is a closed ball. If X≠∅⊆E, then its closure and convex closure, respectively, will write by symbols X¯ and ConvX. Further, ME will be used to denote the family of bounded (nonempty) subsets of E as well as its subfamily, NE, which consists of all relatively compact sets. The MNC is defined in [17] (see also [18]) as follows.Definition 1.
A mappingG:ME⟶ℝ+=0,∞ is called MNC in E if
(i)
J∈ME, which implies GJ=0 gives J be relatively compact(ii)
kerG=J∈ME:GJ=0≠∅. Also, kerG⊂NE(iii)
J⊆J1GJ≤GJ1(iv)
GJ¯=GJ(v)
GConvJ=GJ(vi)
GςJ+1−ςJ1≤ςGJ+1−ςGJ1,ς∈0,1(vii)
Jj∈ME,Jj=J¯j,Jj+1⊂Jj for j∈ℕ and limj⟶∞GJj=0, and then, ⋂j=1∞Jj≠∅
Note thatJ∞=⋂j=1∞Jj∈kerG. Since GJ∞≤GJj for any j, we infer that GJ∞=0.Banas and Krajewska [19] proposed the generalization of classical spaces c0, c, and ℓ∞ with the help of tempering sequence α=αii=1∞ while the tempering sequence means that αi is positive for any i∈ℕ and αi is nonincreasing, and they defined c0α, cα, and ℓ∞α which are called the tempered sequence space. Inspired by these constructions, very recently, Rebbani et al. [20] defined the tempered space ℓpα as follows:
(1)L=ρ=ρnn=1∞∈w:∑n=1∞αnpρnp<∞for1≤p<∞,where w is the space of real or complex sequences, or simply, we shall write L∶≡ℓpα. Clearly, ℓpα is a Banach space endowed with
(2)ρℓpα=∑n=1∞αnpρnp1/p.In case ofαn=1 for all n∈ℕ, the tempered space ℓpα coincides with ℓp, and, in addition, if p=1, ℓpα coincides with ℓ1. In the same paper, they gave the Hausdorff MNC χℓpα for a nonempty bounded set Bα of ℓpα (1≤p<∞) by
(3)χℓpαBα=limn⟶∞supy∈Bα∑k≥nαkpykp1/p.We will useCI,ℓpα to denote the collection of all continuous mappings from I=0,a (a>0) to ℓpα, and CI,ℓpα is a Banach space with the norm
(4)ρCI,ℓpα=sups∈Iρsℓpα,where ρs=ρnsn=1∞∈CI,ℓpα. For any nonempty bounded set Eα of CI,ℓpα and for s∈I, one defines Eαs=ρs:ρs∈Eα and hence, its MNC is given by
(5)χCI,ℓpαEα=sups∈IχℓpαEαs.Recall the theorem given in [1] as follows:Theorem 2.
Suppose thatJ is a nonempty, closed, bounded, and convex subset of E, and suppose also that S:J⟶J is a continuous mapping, and there exists κ∈0,1 satisfying
(6)GSΛ≤κGΛ,Λ⊆J.
Then,S has a fixed point.


## 2. Dorbo-Type Fixed Point Theorems

In order to discuss our Dorbo-type theorems, we first recall the set of functions which has been recently used in [13] as follows: Consider the function M:ℝ+×ℝ+⟶ℝ+ such that
(1)
maxϑ1,ϑ2≤Mϑ1,ϑ2 for ϑ1,ϑ2≥0(2)
M is continuous and nondecreasing(3)
Mϑ1+ϑ1,ν1+ν2≤Mϑ1,ν1+Mϑ2,ν2hold. We will denote the collection of such functions by M. The example of aforesaid kind of function is Mϑ1,ϑ2=ϑ1+ϑ2.Theorem 3.
Consider a Banach spaceE, a nonempty, closed, bounded convex set D⊆E, and an arbitrary MNC G. Also, consider a continuous mapping T:D⟶D satisfying the inequality
(7)αMGTX,γGTX≤αMGX,γGX−βMGX,γGX,for any X≠∅⊆D, where M∈M and α,β,γ:ℝ+⟶ℝ+ are functions such that α,γ are continuous on ℝ+ and β is lower semicontinuous which satisfies the relations
(8)β0=0andβx>0x>0.
Then,(9)ThasatleastonefixedpointinD.Proof.
Consider a sequenceDnn=1∞ such that D1=D and Dn+1=ConvTDn for n∈ℕ. One can find that TD1=TD⊆D=D1,D2=ConvTD1⊆D=D1. We obtain in a similar way that D1⊇D2⊇D3⊇⋯⊇Dn⊇Dn+1⊇⋯. If there exists n0∈ℕ satisfying GDn0=0, then Dn0 is a compact set. With a view of Schauder theorem [21], T has a fixed point in D⊆E.
Further, assume thatGDn>0 for n∈ℕ. Clearly, GDnn=1∞ is nonnegative, decreasing, and bounded below sequence. Therefore, GDnn=1∞ is convergent and
(10)limn⟶∞GDn=r≥0,say.
Inequality (7) gives
(11)αMGDn+1,γGDn+1=αMGConvTDn,γGConvTDn=αMGTDn,γGTDn≤αMGDn,γGDn−βMGDn,γGDn.
If possible, assumer>0. Letting limsupn⟶∞ in the last inequality, one obtains
(12)limsupn⟶∞αMGDn+1,γGDn+1≤limsupαMGDn,γGDnn⟶∞−limsupn⟶∞βMGDn,γGDn,which yields
(13)αMr,γr≤αMr,γr−βMr,γr.
It follows from the inequality (13) that
(14)βMr,γr≤0.
Consequently, we getβMr,γr=0. So, γr=r=0. Therefore, we have
(15)limn⟶∞GDn=0.
Using the factDn⊇Dn+1 and Definition 1, we fairly have
(16)D∞=⋂j=1∞Dn⊆D,which is nonempty, convex, closed subset of D and D∞ is T invariant. By taking into account Schauder theorem [21], we conclude that (9) holds.Theorem 4.
Consider a Banach spaceE, a nonempty, closed, bounded convex set D⊆E, and an arbitrary MNC G. Also, consider a continuous mapping T:D⟶D satisfying the inequality
(17)αGTX+γGTX≤αGX+γGX−βGX+γGX,for X≠∅⊆D, where α,β,γ:ℝ+⟶ℝ+ are functions such that α,γ are continuous on ℝ+ and β is lower semicontinuous satisfies relation (8). Then, (9) holds.Proof.
This result can be obtained by considering the functionMν1,ν2=ν1+ν2 in Theorem 3.Theorem 5.
Consider a Banach spaceE, a nonempty, closed, bounded convex set D⊆E, and an arbitrary MNC G. Also, consider a continuous mapping T:D⟶D satisfies the inequality
(18)MGTX,γGTX≤ηMGX,γGXX≠∅⊆D,M∈M,where γ,η:ℝ+⟶ℝ+ are two functions such that γ is continuous and η is nondecreasing satisfying
(19)limn⟶∞ηnx=0x≥0.
Then, (9) holds.Proof.
ConsiderDnn=1∞ such that
(20)D1=DandDn+1=ConvTDnn∈ℕ.
Then, we see that(21)TD1=TD⊆D=D1andD2=ConvTD1⊆D=D1.
Continuing in this way, we obtain(22)D1⊇D2⊇D3⊇⋯⊇Dn⊇Dn+1⊇⋯.
If there existsn0∈ℕ satisfying the condition GDn0=0, then the set Dn0 is compact. By taking into account Schauder theorem [21], we conclude that (9) holds.We now assumeGDn>0 (n∈ℕ). Consequently, a sequence GDnn=1∞ is decreasing and bounded below. Thus, GDnn=1∞ is convergent and so
(23)limn⟶∞GDn=r≥0,say.With a view of (18), one writes
(24)MGDn+1,γGDn+1=MGConvTDn,γGConvTDn=MGTDn,γGTDn≤ηMGDn,γGDn≤η2MGDn−1,γGDn−1⋯⋯≤ηnMGD,γGD.Suppose thatr>0 (if possible). We obtain by letting n⟶∞ together with (19) and (23) in the inequality (24) that
(25)ηnMGD,γGD⟶0,which yields
(26)Mr,γr=0.We therefore haveγr=r=0, so limn⟶∞GDn=0. With the help of (22), we obtain nonempty, convex, closed set D∞⊆D which is T invariant. Hence, by Schauder theorem [21], we reach to the desired result.Theorem 6.
Consider a Banach spaceE, a nonempty, closed, bounded convex set D⊆E, and an arbitrary MNC G. Also, consider a continuous mapping T:D⟶D satisfies the inequality
(27)MGTX,γGTX≤kMGX,γGX0≤k<1,X≠∅⊆D,M∈M,where a function γ:ℝ+⟶ℝ+ is continuous. Then, (9) holds.Proof.
This can be easily obtained by considering(28)ητ=kτ0≤k<1,∀τ≥0.in Theorem 5, above.Theorem 7.
Consider a Banach spaceE, a nonempty, closed, bounded convex set D⊆E, and an arbitrary MNC G. Also, consider a continuous mapping T:D⟶D having the property
(29)GTX+γGTX≤kGX+γGX0≤k<1,X≠∅⊆D,where γ is a continuous function. Then, (9) holds.Proof.
By using the functionMx,y=x+y, the proof is obtained as an immediate consequence of Theorem 6.


## 3. Existence of Solutions for Integral Equation

We are studying the existence of solutions for an infinite system of the nonlinear integral equation which is considered as follows:(30)Ωnξ=Fnξ,Ωξ,∫0ξGnξ,s,Ωsdsn∈ℕ,where Ωξ=Ωnξn=1∞, ξ∈I=0,a,a>0.To discuss the result of this section, our assumptions are as below:(1)
Forn∈ℕ, the functions Fn:I×CI,ℓpα×ℝ⟶ℝ are continuous with
(31)∑n≥1αnpFnξ,Ω0,0p⟶0∀ξ∈I,where
(32)Ω0=Ωn0ξn=1∞∈CI,ℓpαandΩn0ξ=0∀n∈ℕ,t∈I.Moreover, these exist continuous functionsAn,Bn:I⟶ℝ+ such that the inequality
(33)Fnξ,Ωξ,p−Fnξ,Ω¯ξ,p¯p≤AnξΩnξ−Ω¯nξp+Bnξp−p¯pholds, where Ω¯ξ=Ω¯nξn=1∞∈CI,ℓpα.(2)
Forn∈ℕ, the functions Gn:I×I×CI,ℓpα⟶ℝ are continuous. Also, there exists Lk satisfying
(34)Lk=sup∑n≥kαnpBnξ∫0ξGnξ,s,Ωsdsp:ξ∈I.Further,(35)supn∈ℕLn=Landlimn⟶∞Lk=0(3)
Define an operatorH on I×CI,ℓpα to CI,ℓpα as follows
(36)ξ,Ωξ⟶HΩξ=Fnξ,Ωξ,∫0ξGnξ,s,Ωsdsn=1∞.(4)
Let(37)supAnξ:ξ∈I,n∈ℕ=A,^such that 0<2A∧1/p<1 and
(38)B^=sup∑n≥1αnpBnξ:ξ∈I.Theorem 8.
Under assumptions (1)–(4), the system
(39)Ωnξ=Fnξ,Ωξ,∫0ξGnξ,s,Ωsdshas at least one solution in CI,ℓpα, where
(40)Ωξ=Ωnξn=1∞,ξ∈I=0,a,a>0.Proof.
For arbitrary fixedξ∈I,
(41)Ωξℓpαp=∑n≥1αnpFnξ,Ωξ,∫0ξGnξ,s,Ωsdsp≤2p∑n≥1αnpFnξ,Ωξ,∫0ξGnξ,s,Ωsds−Fnξ,Ω0,0p+2p∑n≥1Fnξ,Ω0,0p≤2p∑n≥1αnpAnξΩnξp+Bnξ∫0ξGnξ,s,Ωsdsp≤2pA^Ωξℓpαp+2pL,which yields
(42)Ωξℓpαp≤2pL1−2pA^=rp,say.
It follows from (42) that
(43)Ωξℓpα≤r,and hence,
(44)ΩCI,ℓpα≤r.
Let us define nonempty set(45)B=Ωξ∈CI,ℓpα:ΩCI,ℓpα≤r,ξ∈I,which is closed, bounded, and convex subset of CI,ℓpα. By assumption (3) and for arbitrary fixed ξ∈I, we write
(46)HΩξ=HnΩξn=1∞=Fnξ,Ωξ,∫0ξGnξ,s,Ωsdsn=1∞Ωξ∈B.
Also,(47)∑n=1∞αnpHnΩξp=∑n≥1αnpFnξ,Ωξ,∫0ξGnξ,s,Ωsdsp<∞.
Hence,HΩξ∈ℓpα.
Since(48)HΩξℓpα≤r⇒HΩCI,ℓpα≤r,we have that H maps B into B. We are now claiming that H is continuous on B. For this, suppose ϵ>0 and Ωξ=Ωnξn=1∞,Ω¯ξ=Ω¯nξn=1∞∈B satisfying
(49)Ω−Ω¯CI,ℓpα<ϵ2A∧1/p=δ.
For arbitrary fixedξ∈I,
(50)HnΩξ−HnΩ¯ξp=Fnξ,Ωξ,∫0ξGnξ,s,Ωsds−Fnξ,Ω¯ξ,∫0ξGnξ,s,Ω¯sdsp≤A^Ωnξ−Ω¯nξp+Bnξ∫0ξGnξ,s,Ωsds−∫0ξGnξ,s,Ω¯sdsp≤A^Ωnξ−Ω¯nξp+Bnξ∫0ξGnξ,s,Ωs−Gnξ,s,Ω¯sdsp.
Considering the factΩ−Ω¯CI,ℓpα<δ and Gn is continuous, we get
(51)Gnξ,s,Ωs−Gnξ,s,Ω¯s<ε21/pB∧+11/pa,and so,
(52)∫0ξGnξ,s,Ωs−Gnξ,s,Ω¯sds<ϵ21/pB∧+11/pa∫0ξds<ϵa21/pB∧+11/pa=ϵ21/pB∧+11/p.
It follows from (50) and (52) that
(53)∑n=1∞αnpHnΩξ−HnΩ¯ξp≤A^∑n=1∞αnpΩnξ−Ω¯nξp+ϵp2B^+1∑n=1∞αnpBnξ<A^ϵp2A^+ϵpB^2B^+1<ϵp,which gives
(54)HΩξ−HΩ¯ξℓpαp<ϵp.
Therefore,(55)HΩ−HΩ¯CI,ℓpα<ϵwheneverΩ−Ω¯CI,ℓpα<δ.
Hence,H is continuous on B.
Now, for arbitrary fixedξ∈I and BΩ=Ωξ:Ωξ∈B, we write
(56)χℓpαHBξ=limn⟶∞supΩξ∈Bξ∑k=n∞αkpΩkξp1/p=limn⟶∞supΩξ∈Bξ∑k=n∞αkpFnξ,Ωξ,∫0ξGnξ,s,Ωsdsp1/p≤limn⟶∞supΩξ∈Bξ2p∑k=n∞αkpA∧Ωkξp+Lk1/p,or
(57)χℓpαHBξ≤2A∧1/pχℓpαBξ.
Operatingsupξ∈I on both sides of (57), we obtain
(58)supξ∈IχℓpαHBξ≤2A∧1/psupξ∈IχℓpαBξ.
We thus have(59)χCI,ℓpαHB≤2A∧1/pχCI,ℓpαB.
As0<2A∧1/p<1, so applying Theorem 7 for γ≡0 gives that H has at least one fixed point on B⊆CI,ℓpα, i.e., the considered system admits a solution in CI,ℓpα.Example 1.
In order to demonstrate Theorem8, we consider an infinite system of integral equation as follows:
(60)Ωnξ=ξ3Ωnξ61+ξn2+1n3∫0ξcosΩns5+sin∑j=1nΩjsds,for ξ∈0,1=I and n∈ℕ. For this demonstration, write
(61)Fnξ,Ωξ,pnΩξ=ξ3Ωnξ61+ξn2+pnΩξn3,pnΩξ=∫0ξcosΩns5+sin∑j=1nΩjsds,Gnξ,s,Ωξ=cosΩns5+sin∑j=1nΩjs.
Further, takea=1 and let αn=1/n, n∈ℕ. If Ωξ∈ℓpα for some fixed ξ∈I, then
(62)∑n≥1αnpFnξ,Ωξ,pnΩξp=∑n≥11npξ3Ωnξ61+ξn2+1n3∫0ξcosΩns5+sin∑j=1nΩjsdsp≤2p∑n≥116pn3pΩnξp+2p∑n≥11n4p=13p∑n≥11n3pΩnξp+2p∑n≥11n4p≤13p∑n≥11npΩnξp+2p∑n≥11n4p=13pΩξℓpαp+2p∑n≥11n4p<∞,as the series
(63)∑n≥11n4pconvergesforp≥1.
Therefore, for arbitrary fixedξ∈I, one has
(64)Fnξ,Ωξ,pnΩξn=1∞∈ℓpα,and hence,
(65)Fnξ,Ωξ,pnΩξn=1∞∈CI,ℓpα.
LetΩ¯ξ=Ω¯nξn=1∞∈CI,ℓpα. Then,
(66)Fnξ,Ωξ,pnΩξ−Fnξ,Ω¯ξ,pnΩ¯ξp≤13pΩnξ−Ω¯nξp+2ppnΩξ−pnΩ¯ξp.
Here(67)Anξ=13pandBnξ=2p.
Also,(68)A^=13p⇒2A∧1/p=23<1,∑n≥1αnpFnξ,Ω0,0p⟶0∀ξ∈I.
Again,(69)∑n≥kαnpBnξPnΩξp≤2p∑n≥k1np,Lk≤sup2p∑n≥k1np:ξ∈I.
We can find that(70)Lk⟶0k⟶∞andL=2p∑n≥11np.
Moreover, we have(71)∑n≥1αnpBnξ=2p∑n≥11np,,B^=sup2p∑n≥11np:ξ∈I=2p∑n≥11np<∞.
The functionsFn and Gn are continuous for all n∈ℕ as well as the conditions (1)–(4) are fulfilled so with a view of Theorem 8, we reach to our conclusion that the considered system (60) admits a solution in CI,ℓpα.


## 4. Concluding Remarks

In this work, we linked three different disciplines such as the concept of measure of noncompactness (MNC), the theory of existence of solutions for functional equations, and the Banach space theory, particularly, in tempered sequence spaces. We first discussed some generalized Dorbo-type fixed point theorems by considering the arbitrary MNC and then discussed the existence of solutions for nonlinear integral equation (infinite system) by taking aforesaid newly investigated Dorbo-type theorem in tempered sequence spaces. Finally, we constructed an illustrative example by taking an integral equation to validate our result.It is worth noting to the reader that one can obtain the results of Section2 by taking into account another suitable function instead of M:ℝ+×ℝ+⟶ℝ+ and consider two dimensional integral (or fraction integral) equation to extend the results of Section 3.



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