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

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 ﬁ xed-point theorem (SFPT) related to the method of measure of noncompactness (MNC). Furthermore, we consider the operator E and show that it is invariant and continuous. Moreover, we provide an application to show the capability of the achieved results.


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][2][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][9][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 timefractional 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][15][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: where C D β is the Caputo fractional derivative (CFD) of order β > 0, ρ = constant, ϕ ∈ Cð½t 0 − ϱ, t 0 Þ, RÞ, and i = 1, 2, ⋯n, C D β i is the CFD of order 0 < β i < β and f : H × Cð½−ϱ, 0, RÞ ⟶ R, in such a way that H = ðt 0 ,∞Þ is a predefined function. We additionally consider for any x ∈ H the function v t : ½−ϱ, 0 ⟶ R given that v t ðsÞ = vðt + sÞ for every s ∈ ½−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 Section 2. 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 Derbotype fixed-point hypothesis along with the method of MNC. Finally, in Section 5, we discuss a useful application to represent our main result.

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 function f , the fractional integral of order β with t 0 ∈ R is defined as given that the R.H.S is pointwise characterized on ½t 0 , ∞Þ where Γð·Þ is the usual gamma function.
Theorem 6 (SFP theorem [20]. If V 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.

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 fixedpoint 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 fixedpoint 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 function f i ðt, v t Þ is Lebesgue measurable in terms of t for every i = 1, 2, ⋯n, on ½t 0 , ∞Þ, and f i ðt, ϕÞ is continuous in terms of ϕ on Cð½−ρ, 0, RÞ.

2
Journal of Function Spaces (H2) There is a function that is strictly nonincreasing J : R ⟶ R which disappears at infinity in such a way that (H3) ∃ a constant α in such a way that for every i = 1, 2, 3 ⋯ n, we have By condition (H1), IVP (1) is equal to the following condition: where for each v ∈ Cð½t 0 − ρ,∞Þ, RÞ.
Consider the IVP of the following FFEE: The above system is equal to the following integral: provided that the integral (12) exists. where Proof. Let λ > 0, then Applying the Laplace transform to (12), we get for t ≥ 0. Let and its Laplace transform is given by Using (19), we have Since L½g 1 ðtÞðλÞ = λ −1 , according to the Laplace convolution theorem, we have 3 Journal of Function Spaces Similarly, Combining equations (20), (22), and (23), we have The above system can also be written as Thus, the proof is complete. Proof. Define a set P ⊂ Cð½t 0 − ϱ,∞Þ, RÞ by P is clearly a nonempty, convex, closed, and bounded subset of Cð½t 0 − ϱ,∞Þ, RÞ. 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 ðv m Þ m∈ℕ is a sequence of a function to such an extent that v m ∈ P∀m ∈ ℕ and v m ⟶ v as m ⟶ ∞. Clearly, by the continuity f i ðt, v t Þ, we get Assume that ε > 0 is given. After all, J is strongly decreasing. At that point for some T > t 0 , we have which disappear when m ⟶ ∞. Then again, since P in E -invariant, at that point, (28) yields Thus, for t > t 0 , this implies that If x ∈ ½t 0 − ρ, t 0 , we clearly have j½Ev m ðtÞ − ½EvðtÞj = 0. Therefore, the continuity E has been proven. Then, we prove that EðPÞ is equicontinuous. Assume that ε > 0 is given, t 1 , t 2 ∈ ðt 0 , T where T > t 0 is picked with the end of goal that (28) holds. Applying (H3), we get 4 Journal of Function Spaces If t 1 , t 2 > T, at that point, since P is E-invariant and using(28), we get If t 0 < t 1 < T < t 2 , it can be seen that t 1 ⟶ t 2 which implies that ðt 1 ⟶ TÞ∧ðt 2 ⟶ TÞ; then, according to the above discussion, we have got Thus, we resolve that EðPÞ is equicontinuous on ½t 0 , T ∀T > 0. Since EðPÞ ⊂ P and from the set P, it is clear that Hence, EðPÞ is a moderately smaller set in Cð½t 0 − ρ,∞Þ , RÞ 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. 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 Theorem 9 does not imply that solutions are globally attractive in the sense of Definition 5.

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), BCðR t 0 −ρ Þ consisting of every single real functions characterized, continuous as well as bounded on R t 0 −ρ = ½t 0 − ρ,∞Þ 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. Let F be a BS and ConvY and Y represent the convex closure and closure of Y as a subset of F. Further, m F represents the group of all bounded subsets of E, and the n F represents its subfamily which contains all relatively compact sets. Also, assume that the closed ball is Bðy, rÞ where center = y, radius = r, and B r represents the ball Bðξ, rÞ with the end of goal that ξ is the zero component of the BS of F. Definition 11. ν : m F ⟶ R + is supposed to be MNC in F if it fulfills the following criteria: (i) The family kerν = fY ∈ m F : νðYÞ = 0g is nonempty and kerν ∈ n F (ii) Y ⊆ Z ⇒ νðYÞ ≥ νðZÞ.
(iii) νðYÞ = νð YÞ (iv) νðConvYÞ = νðYÞ As a result, the kerðνÞ family is referred to as the kernel of MNC of ν.
Definition 12. In F, let ν be an MNC. So the mapping S : C ⊆ F ⟶ F is supposed to be a ν F -contraction if ∃ a constant term 0 < b < 1 in such way 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 that C is a nonempty, bounded, convex, and closed subset of BS of F, and assume that S : C ⟶ C is a continuous function which fulfills for every D ⊆ C, where ν represents an arbitrary MNC and ϕ : R + ⟶ R + represents a monotone nondecreasing function with lim m⟶∞ ϕ m ðtÞ = 0∀t ∈ R + . At that point, S has minimum one fixed point in C.
We will work in BS, BCðR t 0 −ρ Þ where t 0 and ρ are given in (1). The functional space is furnished with the standard norm which is kvk = sup fvðtÞ: t ≥ t 0 − ρg. For this reason, we present a MNC in the space BCðR t 0 −ρ Þ which is built like the one in the space BCðR + Þ. Suppose that B is a bounded subset in BS of BCðR t 0 −ρ Þ and T > t 0 − ρ is given. For v ∈ B and ε > 0, we denote by ω T t 0 −ρ ðv, εÞ the modulus of continuity of the function v on ½t 0 − ρ, T, i.e.,