# Analytical Approaches on the Attractivity of Solutions for Multiterm Fractional Functional Evolution Equations

Authors: Xiangling Li; Azmat Ullah Khan Niazi; Farva Hafeez; Reny George; Azhar Hussain
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The most important objective of the current research is to establish some theoretical existence and attractivity results of solutions for a novel nonlinear fractional functional evolution equations (FFEE) of Caputo type. In this respect, we use a familiar Schauder’s fixed-point theorem (SFPT) related to the method of measure of noncompactness (MNC). Furthermore, we consider the operatorE and show that it is invariant and continuous. Moreover, we provide an application to show the capability of the achieved results.


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## Body

## 1. Introduction

During the recent years, the study of fractional evolution equations (FEE) has attracted a lot of attention. Such class pulls out the interest of such countless creators toward itself, inspired by their broad use in numerical analysis. Fractional Calculus (FC), as much as classic analytics, has discovered significant examples in the study of problem in a thermal system and mechanical system. Also, in certain spaces of sciences like control hypothesis, a fractional differential operator appears to be more reasonable to model than the old style integer order operator. Because of this, FEE has been utilized in models about organic chemistry and medication.In the last few years, the hypothesis of FEE has been scientifically explored by a major number of extremely fascinating and novel papers (see [1–3]). The existence of global attractivity solutions to the Ψ-Hilfer Cauchy fractional problem is investigated by several researchers (see [4]). Chang et al. [5] used fixed-point theorems to study the asymptotic decay of various operators, as well as the existence and uniqueness of a class of mild solutions of Sobolev fractional differential equations. In [6, 7], the theory of fractional differential equations was discussed. The Ψ-Hilfer fractional derivative was used to investigate the existence, uniqueness, and Ulam-Hyers stabilities of solutions of differential and integro-differential equations.The existence and attractivity of solutions to the following coupled system of nonlinear fractional Riemann-Liouville-Volterra-Stieltjes quadratic multidelay partial integral equations are investigated by many authors. The properties of bounded variation functions are defined by them (see [8–10]). The attractivity of solutions to the Hilfer fractional stochastic evolution equations is discussed by Yang and others. In circumstances where the semigroup associated with the infinitesimal generator is compact, they establish sufficient criteria for the global attractivity of mild solutions (see [11]). Also, mild solutions for multiterm time-fractional differential equations with nonlocal initial conditions and fractional functional equations (FFE) have been researched (see [12, 13]).A functional differential equation is a general name for a number for more specific types of DE that are used in different applications. There are delay differential equations (DDE), integro-differential equations, and so on. FC has been effectively applied in different applied zones like computational science and financial aspects. In specific circumstances, we need to solve FEE having more than one differential operator, and this kind of FEE is known as multiterm FEE. The researchers set up the existence of monotonic solution for multiterm PDE in Banach spaces, utilizing the RL-fractional derivative.The greater part of the current work is concentrated on the existence and uniqueness of the solution for FEE (see [14–16]). The goal of this study is to investigate the existence of solutions to a class of multiterm FFEE on an unbounded interval in terms of bounded and consistent capacities. We also look at several key aspects of the arrangement that are relevant to the concept of attractivity of solution.Consider IVP of the following FFEE:(1)CDβvt=Bvt+∑i=1nCDβifit,vt+f0t,vt,t>t0,1<β<2,vt=ϕt=ϕ0,v′t=ϕ′t=ϕ1,t0−ϱ≤t≤t0,where CDβ is the Caputo fractional derivative (CFD) of order β>0,ρ=constant, ϕ∈Ct0−ϱ,t0,R, and i=1,2,⋯n,CDβi is the CFD of order 0<βi<β and f:H×C−ϱ,0,R⟶R, in such a way that H=t0,∞ is a predefined function. We additionally consider for any x∈H the function vt:−ϱ,0⟶R given that vts=vt+s for every s∈−ϱ,0. We show that (1) has an attractive solution under the broad and favourable assumption using the SFPT and the concept of measure of noncompactness. We believe that by using classic SFPT and a control function, we can achieve a different result.The following is the outline for this paper. We review some essential preliminaries in Section2. In Section 3, we give a few supposition and lemmas or theorems to introduce the consequence of such section for (1) utilizing SFPT. In Section 4, we first review some assistant realities about the idea of MNC and related signs; at that point, we study the existence of solution for (1) applying a well-known Derbo-type fixed-point hypothesis along with the method of MNC. Finally, in Section 5, we discuss a useful application to represent our main result.


## 2. Preliminaries

In this section, we discuss some known definitions. Likewise, we define a few ideas identified with (1) along with SFPT.Definition 1 (see [17]).
For a functionf, the fractional integral of order β with t0∈R is defined as
(2)Iβft=1Γβ∫t0tfst−s1−βds,t>t0,β>0,given that the R.H.S is pointwise characterized on t0,∞ where Γ· is the usual gamma function.Definition 2 (see [17]).
The RL-derivative of orderm−1<β<m with t0∈R for a function f∈Cmx0,∞,R can be composed as
(3)Dβft=1Γm−βdmdtm∫t0tfst−sβ+1−mds,t>t0,m∈ℕ.Definition 3 (see [17]).
Caputo derivative of orderm−1<β<m for a function f∈Cm+1t0,∞,R can be composed as
(4)CDβft=Dβfs−∑h=1m−1fht0Γh−β+1s−t0h−βt,t>t0,m∈ℕ.Definition 4 (see [18]).
The solutionvx of IVP (1) is supposed to be attractive if ∃ a constant term c0t0>0 in such a way that
(5)ϕs≤c0s∈t0−ϱ,t0.This means that vt⟶0 as like t⟶∞.Definition 5 (see [19]).
The solutionvt of IVP (1) is supposed to be attractive, if
(6)limt⟶∞vt−wt=0,for some arrangement w=wt of IVP (1).Theorem 6 (SFP theorem [20].
IfV is nonempty, closed, bounded convex subset of Banach space Y and K:V⟶V is totally continuous, at that point K has a fixed point in V.


## 3. Attractivity of Solutions with Schauder’s Fixed-Point Principle

The Schauder fixed-point theorem states that any compact convex nonempty subset of a normed space has the fixed-point property, which is one of the most well-known conclusions in fixed-point theory. It is also true in spaces that are locally convex. The Schauder fixed-point theorem has recently been extended to semilinear spaces. The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension.This section contains the following information: we examine (1) utilizing the SFPT under the following suppositions:(H1) The functionfit,vt is Lebesgue measurable in terms of t for every i=1,2,⋯n, on t0,∞, and fit,ϕ is continuous in terms of ϕ on C−ρ,0,R.(H2) There is a function that is strictly nonincreasingJ:R⟶R which disappears at infinity in such a way that
(7)ϕmt0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds≤Jt−t0,∀t∈H=t0,∞,m=0,1.(H3)∃ a constant α in such a way that for every i=1,2,3⋯n, we have fi∈L1/αH,C−ϱ,0,R with
(8)α∈0,min0≤i≤nβ−βi.By condition (H1), IVP (1) is equal to the following condition:
(9)vx=ϕmt0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds,t>t0,m=0,1,ϕmt,t∈t0−ϱ,t0,where β0=0 and 0<βi<β for i=1,2,3⋯n. We define the operator E as
(10)Evx=ϕmt0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds,t>t0,m=0,1,ϕmt,t∈t0−σ,t0,for each v∈Ct0−ρ,∞,R.Consider the IVP of the following FFEE:(11)CDβvt=Bvt+∑i=1nCDβifit,vt+f0t,vt,t>t0,1<β<2,vt=ϕt=ϕ0,v′t=ϕ′t=ϕ1,t0−ϱ≤t≤t0.The above system is equal to the following integral:(12)vt=ϕ0+ϕ1t+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds,t∈0,∞,provided that the integral (12) exists.Theorem 7.
If (12) holds, then
(13)vt=Cβtϕ0+Kβtϕ1+∫t0tt−sβ−1Pβt−sfis,vsds,t∈0,∞,where
(14)Cβt=∫0∞MβθCtβθdθ,Kβt=∫0tCβsds,Pβt=∫0∞βθMβθStβθdθ.Proof.
Letλ>0, then
(15)νλ=∫0∞e−λsvsds,μλ=∫0∞e−λsfisds.
Applying the Laplace transform to (12), we get
(16)υλ=λβ−1λβ−B−1ϕ0+λβ−2λβ−B−1ϕ1+λβ−B−1μλ,for t≥0.
(17)νλ=λβ/2−1∫0∞e−λβ/2tCtϕ0dt+λ−1λβ/2−1∫0∞e−λβ/2tCtϕ1dt+∫0∞e−λβ/2tStμtdt.
Let(18)ϕβθ=βθβ+1Mβθ−β,θ∈0,∞,and its Laplace transform is given by
(19)∫0∞e−λθϕβθdθ=e−λβ,β∈12,1.Using (19), we have
(20)λβ−1∫0∞e−λβtCtϕ0dt=∫0∞βλtβ−1e−λtβCtβϕ0dt=∫0∞−1λddt∫0∞e−λtθϕβθdθCtβϕ0dt=∫0∞∫0∞−λθ−λe−λtθϕβθCtβϕ0dt=∫0∞∫0∞θϕβθe−λtθCtβϕ0dtdθ=∫0∞∫0∞ϕβθe−λtCtβθβϕ0dθdt=∫0∞e−λt∫0∞ϕβθCtβθβϕ0dθdt=L∫0∞MβθCtβθϕ0dθλ=LCβtϕ0λ.
SinceLg1tλ=λ−1, according to the Laplace convolution theorem, we have
(21)λ−1λβ−1∫0∞e−λβtCtϕ1dt=Lg1tλ∗LCβtϕ1λ,(22)λ−1λβ−1∫0∞e−λβtCtϕ1dt=Lg1∗Cβtϕ1λ.
Similarly,(23)∫0∞e−λβtStμλdt=∫0∞βtβ−1e−λtβStβμλdt=∫0∞∫0∞βtβ−1ϕβθe−λtθStβμλdtdθ=∫0∞∫0∞βtβ−1θβϕβθe−λtθStβθβμλdtdθ=∫0∞e−λt∫0∞βtβ−1θβϕβθStβθβμλdθdt=L∫0∞βtβ−1MβθStβθdθλ·Lfitλ=L∫t0tt−sβ−1Pβt−sfisdsλ.
Combining equations (20), (22), and (23), we have
(24)vt=Cβtϕ0+∫t0tCβsϕ1ds+∫t0tt−sβ−1Pβt−sfis,vsds.The above system can also be written as
(25)vt=Cβtϕ0+∫t0tCβsϕ1ds+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds,t>t0.
Thus, the proof is complete.Lemma 8.
Assume thatfit,vt fulfills conditions (H1)-(H3). At that point, (1) has minimum one solution in Ct0−ρ,∞,R.Proof.
Define a setP⊂Ct0−ϱ,∞,R by
(26)P=v:v∈Ct0−ϱ,∞,R,vt≤Jt−t0∀t≥t0.P is clearly a nonempty, convex, closed, and bounded subset of Ct0−ϱ,∞,R. To show that (1) has a solution, it just necessities to prove that in P, the operator E has a fixed point. To begin with, we prove that P is E-invariant. This is without any problem acquired by condition (H2). Now, we should explain that E is continuous. For this, suppose that vmm∈ℕ is a sequence of a function to such an extent that vm∈P∀m∈ℕ and vm⟶v as m⟶∞. Clearly, by the continuity fit,vt, we get
(27)limm⟶∞fit,vtm=fit,vt∀t>t0,i=1,2,⋯n.Assume thatε>0 is given. After all, J is strongly decreasing. At that point for some T>t0, we have
(28)Jt−t0<ε2,∀t>T.Fort0<t≤T, we get
(29)Evmt−Evt≤∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsm−fs,vsds≤∑i=1n1Γβ−βi∫t0tt−sβ−βi−1/1−α1−α∫t0tfis,vsm−fs,vs1/αdsα≤∑i=1n1−α1−αΓβ−βiβ−βi−α1−αT−t0β−βi−α∫t0tfis,vsm−fs,vs1/αdsα≤∑i=1n1−α1−αΓβ−βiβ−βi−α1−αT−t0β−βi,supt0≤t≤Tfis,vsm−fs,vs,which disappear when m⟶∞. Then again, since P in E-invariant, at that point, (28) yields
(30)Evmt−Evt≤2Jt−t0<ε,∀t>T.Thus, fort>t0, this implies that
(31)Evmt−Evt⟶0asm⟶∞.Ifx∈t0−ρ,t0, we clearly have Evmt−Evt=0. Therefore, the continuity E has been proven. Then, we prove that EP is equicontinuous. Assume that ε>0 is given, t1,t2∈t0,T where T>t0 is picked with the end of goal that (28) holds. Applying (H3), we get
(32)Evt2−Evt1≤∑i=1n1Γβ−βi∫t0t2t2−sβ−βi−1fis,vsds−∫t0t1t1−sβ−βi−1fis,vsds≤∑i=1n1Γβ−βi∫t0t1t2−sβ−βi−1−t1−sβ−βi−1fis,vsds+∫t1t2t2−sβ−βi−1fis,vsds≤∑i=1n1Γβ−βi∫t0t1t1−sβ−βi−1−t2−sβ−βi−11/1−αds1−α∫t0t1fis,vs1/αdsα+∫t1t2t2−sβ−βi−1/1−αds1−α∫t1t2fis,vs1/αdsα≤∑i=1n1Γβ−βi∫t0t1t1−sβ−βi−1/1−α−t2−sβ−βi−1/1−αdu1−α∫t0t1fis,vs1/αdsα+∫t1t2t2−sβ−βi−1/1−αds1−α∫t1t2fis,vs1/αdsα≤∑i=1n1−α1−αΓβ−βiβ−βi−α1−αt2−t1β−βi−α/1−α+t1−t0β−βi−α/1−α−t2−t0β−βi−α/1−α1−α×∫t0Tfis,vs1/αdsα+t2−t1β−βi−α∫t0Tfis,vs1/αdsα⟶0ast1⟶t2.Ift1,t2>T, at that point, since P is E-invariant and using(28), we get
(33)Evt2−Evt1≤∑i=1n1Γβ−βi∫t0t2t2−sβ−βi−1fis,vsds−∫t0t1t1−sβ−βi−1fis,vsds≤Jt1−t0+Jt2−t0<ε.Ift0<t1<T<t2, it can be seen that t1⟶t2 which implies that t1⟶T∧t2⟶T; then, according to the above discussion, we have got
(34)Evt2−Evt1≤Evt2−EvT+EvT−Evt1⟶0ast2⟶t1.Thus, we resolve thatEP is equicontinuous on t0,T∀T>0. Since EP⊂P and from the set P, it is clear that
(35)limT⟶∞supv∈EPsupvt:t>T=0.Hence,EP is a moderately smaller set in Ct0−ρ,∞,R and all requirements of SFPT are satisfied. In this set, the operator E maps on P and has a fixed point. This reality indicates that (1) has at least one solution in P.Theorem 9.
Assume that conditions (H1)-(H2) are fulfilled; at that point, IVP (1) accepts at the minimum one attractive solution by Definition 4.Proof.
The previous lemma states that there is at least one solution of (1) that belongs to P in (Lemma 8). Then, use the property of function J, to show attractivity. As a result, at ∞, all of the functions in P vanish, and therefore, the result of (1) is ⟶0 as x⟶∞.
So, the proof is complete.Remark 10.
The conclusion of Theorem9 does not imply that solutions are globally attractive in the sense of Definition 5.


## 4. Uniform Local Attractivity of Solutions with Measure of Noncompactness

The purpose of this section is to look at the solution of (1) in the Banach space (BS), BCRt0−ρ consisting of every single real functions characterized, continuous as well as bounded on Rt0−ρ=t0−ρ,∞ by means of the strategy of MNC. It is concentrated on an alternate method to develop some adequate conditions solvability of (1). We assemble a few definitions and assistant realities which will be required further on.LetF be a BS and ConvY and Y¯ represent the convex closure and closure of Y as a subset of F. Further, mF represents the group of all bounded subsets of E, and the nF represents its subfamily which contains all relatively compact sets. Also, assume that the closed ball is By,r where center=y, radius=r, and Br represents the ball Bξ,r with the end of goal that ξ is the zero component of the BS of F.Definition 11.
ν:mF⟶R+ is supposed to be MNC in F if it fulfills the following criteria:
(i)
The familykerν=Y∈mF:νY=0 is nonempty and kerν∈nF(ii)
Y⊆Z⇒νY≥νZ.(iii)
νY=νY¯(iv)
νConvY=νY(v)
∀λ∈0,1(36)νλY+1−λZ≤λνY+1−λνZ.(vi)
IfYmm∈ℕ is a closed sequence set from mF in such a way that(37)Ym+1⊂Ym∀m=1,2,⋯,limm⟶∞νYm=0,then
(38)Y∞=⋂m=1∞Ynisnon‐empty.
As a result, thekerν family is referred to as the kernel of MNC of ν.Definition 12.
InF, let ν be an MNC. So the mapping S:C⊆F⟶F is supposed to be a νF-contraction if ∃ a constant term 0<b<1 in such way
(39)νSD≤bνD.
D⊆C is a bounded closed subset.Remark 13.
As pointed out in [21], global attractivity of solutions implies local attractivity, while the converse is not true.Theorem 14 (see [22]).
Suppose thatC is a nonempty, bounded, convex, and closed subset of BS of F, and assume that S:C⟶C is a continuous function which fulfills
(40)νSD≤ϕνD,for every D⊆C, where ν represents an arbitrary MNC and ϕ:R+⟶R+ represents a monotone nondecreasing function with limm⟶∞ϕmt=0∀t∈R+. At that point, S has minimum one fixed point in C.We will work in BS,BCRt0−ρ where t0 and ρ are given in (1). The functional space is furnished with the standard norm which is v=supvt:t≥t0−ρ. For this reason, we present a MNC in the space BCRt0−ρ which is built like the one in the space BCR+. Suppose that B is a bounded subset in BS of BCRt0−ρ and T>t0−ρ is given. For v∈B and ε>0, we denote by ωt0−ρTv,ε the modulus of continuity of the function v on t0−ρ,T, i.e.,
(41)ωt0−ρTv,ε=supvt−vs:t,s∈t0−ρ,T,t−s≤ε.Now, suppose that we take(42)ωt0−ρTB,ε=supωt0−ρTv,ε:v∈B,ωt0−ρTB=limε⟶0ωt0−ρTB,ωt0−ρB=limT⟶∞ωt0−ρTB.Ift>t0−ρ is a fixed number, we use the term
(43)Bt=vt:v∈B,as well as
(44)diamBt=supvt−wt:v,w∈B.Considerν defined on the family mBCRt0−ρ by formula
(45)νB=ωt0−ρB+limt⟶∞supdiamBt.(H4) For alli=1,2,⋯n,fi:Rt0−ρ×Ct0−ρ,t0,R⟶R is continuous and there is a continuous function ei:Rt0−ρ⟶R in such a way that
(46)fit,v−fit,w≤eitJv−w,where J:R+⟶R+ is a function that is superadditive, i.e, Jb+Jc≤Jb+c,∀b,c≥0.(H5) Assume that∀i=1,2⋯n such that the following constant exists:
(47)Bi=supt∈H∫t0tt−sβ−βi−1eisds<∞,Ci=supt∈H∫t0tt−sβ−βi−1fis,0ds<∞,limm⟶∞λBmJmt=0∀t>0,where
(48)λB=∑i=1nBiΓβ−βi<1.(H6)∃ a nonnegative result r0 of the following inequality
(49)supt∈t0−ρ,t0ϕt+λBJr+λC≤r,where
(50)λB=∑i=1nCiΓβ−βi.Theorem 15.
Under the supposition (H4)-(H6), equation (1) has minimum one solution in BCRt0−ρ. In addition, solution of (1) is uniformly locally attractive.Proof.
To begin, we will look at the operatorE, which was defined by the formula in the previous section:
(51)Evt=ϕmt0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds,t>t0,m=0,1,ϕmt,t∈t0−σ,t0.∀v∈BCRt0−ρ. From the condition (H4)-(H6), the function Ev is continuous on Rt0−ρ. We note that BCRt0−ρ in E-invariant. For any v∈BCRt0−ρ and t>t0, we have
(52)Evt≤ϕ0t0+ϕ1t0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vsds≤ϕ0t0+ϕ1t0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vs−fis,0+fis,0ds≤ϕ0t0+ϕ1t0+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1eisJvsds+∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,0ds≤ϕ0t0+ϕ1t0+λBJv+λC.The above expression shows that Ev is bounded on the interval t0,∞, and connecting with the fact that ϕ∈Ct0−ρ,t0,R, we infer that Ev∈BCRt0−ρ; thus, E changes BCRt0−ρ into itself. Then again, utilize condition (H6)∃ a number r0>0 which appreciates in (49). For such digit, the operator E changes the ball Br0 of BCRt0−ρ into itself. Consider a nonempty subset Y of the ball Br0 as well as fix x,y∈Y in whatever way you want. At that point, for fixed t>t0, we get
(53)Evt−Ewt≤∑i=1n1Γβ−βi∫t0tt−sβ−βi−1fis,vs−fis,wsds≤∑i=1n1Γβ−βi∫t0tt−sβ−βi−1eisJvs−wsds,Evt−Ewt≤λBJv−w,which implies that
(54)limt⟶∞diamEYt≤λBJlimt⟶∞diamYt.Additionally, let us takeT>t0 as a fixed and ε>0. Assume that v∈Y is chosen and suppose that t1,t2∈t0,T to such an extent that t1−t2≤ε. Without loss of consensus, we may suppose that t1<t2. At that point, thinking about our hypothesis, we obtain
(55)Evt2−Ewt1≤∑i=1n1Γβ−βi∫t0t2t2−sβ−βi−1fis,vsds−∫t0t1t1−sβ−βi−1fis,vsds≤∑i=1n1Γβ−βi∫t0t1t1−sβ−βi−1−t2−sβ−βi−1fis,vsds+∫t1t2t2−sβ−βi−1fis,vsds≤∑i=1n1Γβ−βi∫t0t1t1−sβ−βi−1−t2−sβ−βi−1fis,vsds+∫t1t2t2−sβ−βi−1eisJvs+fis,0ds≤∑i=1nω1Tfi,βi,ε+ω2Tfi,βi,εΓβ−βi+λBJωt0−ρTv,ε,for v∈Y⊆Br0, where the symbols used in above term are given by
(56)ω2Tfi,βi,ε=sup∫t1t2t2−sβ−βi−1eisJr0+fis,0ds:t1,t2∈t0−ρ,T,t1−t2≤ε.Now, consider that the functionfis,vs is uniformly continuous on t0−ρ,T×Br0∀i=1,2,⋯n; we simply get the following expression:
(57)ωt0−ρTEY≤λBJωt0−ρTY,which along with (54) and superadditive of J implies that
(58)νEY=ωt0−ρEY+limt⟶∞supdiamEYt≤λBJωt0Y+λBJlimt⟶∞supdiamYt≤λBJνY.Now,ν as given by (45) defines a MNC on BCRt0−ρ; at that point, the inequality along with Theorem 14 shows that (1) has a solution in BS.To show that all solutions of (1) are consistently locally attractive, let us put Br01=ConvEBr0 and Br02=ConvEBr01 and so on, where Br0=ball and r0=radius and center=0 in the space BCRt0−ρ. We basically see that Br01⊂Br0 and Br0m+1⊂Br0m for m=1,2,⋯, and furthermore, the set of this sequence is convex, closed, and nonempty. Moreover, in the light of the current inequality, we get
(59)νBr0m+1≤λBmJmνBr0forn=1,2,⋯.Combining all the facts thatνBr0≥0 as well as condition (H5) with the recent above inequality, we have
(60)limm⟶∞νBr0m=0.We can derive from the definition of MNC thatB=⋂m=1∞Br0m is convex, nonempty, closed, and bounded. B is an E-invariant set, and the operator E is continuous on it. In addition, remembering the reality that B∈kerν and the set belongs to kerν, we infer that all solutions of (1) are consistently locally attractive.Remark 16.
Note that (1) has at least one attractive solution in the sense of Definition 4.


## 5. Example

Example 17.
Consider the FFEE:(61)CD3/2vt=CD4/3s+1−1/2e−sinvs−1t+sinvt−13vt−1+t+1−4/5,t>0,vt=te−t,v′t=t2e−t,t∈−1,0.We can clearly see that condition (H1) holds. To show that condition (H2) is fulfilled, sincev0=0 and v′0=0, we have the following expression ∀t>0:(62)1Γ3/2∫0tt−s−1/2sinvs−13vs−1+s+1−4/5ds+1Γ3/6∫0tt−s−5/6s+1−1/2e−sinvs−1ds≤1Γ3/2∫0tt−s−1/2s−4/5ds+1Γ3/6∫0tt−s−5/6s−1/2ds=t−3/10Γ3/2∫01s−4/51−s−1/2ds+t−1/3Γ3/6∫01s−1/21−s−5/6ds.For anyβ,γ∈R, the following identity is obtained:
(63)∫01sβ−11−sγ−1ds=ΓβΓγΓβ+γ.So (62) implies that
(64)1Γ3/2∫0tt−s−1/2sinvs−13vs−1+s+1−4/5ds+1Γ3/6∫0tt−s−5/6s+1−1/2e−sinvs−1ds≤Jt,where
(65)Jt=Γ1/5Γ7/10t−3/10+πΓ2/3t−1/3,t>0.Clearly, this is a nonincreasing function on ℝ and demonstrates that condition (H2) is true. Finally, it is necessary to show that condition (H3) is valid. Assume for a moment that γ=1/10∈0,min1/2,1/6; at that point, we have
(66)∫0∞sinvs−13vs−1+s+1−4/51/γds≤∫0∞s+1−8ds=17,which proves that f0s,vs=sinvs−1/3vs−1+s+1−4/5∈L1/γH,C−1,0,R. Likewise, for f1s,vs=s+1−1/2e−sinvs−1, we deduce that
(67)∫0∞s+1−1/2e−sinvs−11/γds≤∫0∞s+1−5ds=14.This shows thatf1s,vs∈L1/γH,C−1,0,R. Hence, all conditions are satisfied, so the solution of (1) is existent and also attractive.


## 6. Conclusion

The main conclusion of this study is that the multiterm fractional functional evolution equation belongs to a specific class of attractivity. The goal of this study is to investigate the existence of solutions to a class of multiterm FFEE on an unbounded interval in terms of bounded and consistent capacities. We look at some key aspects of the arrangement that are connected to the concept of solution attractivity. We use a familiar Schauder fixed-point theorem (SFPT) related to the method of measure of noncompactness (MNC). We go over some of the auxiliary realities surrounding the concept of MNC and related signs. Using a well-known Derbo-type fixed-point hypothesis and the MNC technique, we investigate the existence of a solution for (1).



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