#### 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 operator and show that it is invariant and continuous. Moreover, we provide an application to show the capability of the achieved results.

#### 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 -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 -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: where is the Caputo fractional derivative (CFD) of order , , and is the CFD of order and , in such a way that is a predefined function. We additionally consider for any the function given that for every . 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 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 function , the fractional integral of order with is defined as
given that the R.H.S is pointwise characterized on where is the usual gamma function.

*Definition 2 (see [17]). *The RL-derivative of order with for a function can be composed as

*Definition 3 (see [17]). *Caputo derivative of order for a function can be composed as

*Definition 4 (see [18]). *The solution of IVP (1) is supposed to be attractive if a constant term in such a way that
This means that as like .

*Definition 5 (see [19]). *The solution of IVP (1) is supposed to be attractive, if
for some arrangement of IVP (1).

Theorem 6 (SFP theorem [20]. *If is nonempty, closed, bounded convex subset of Banach space and is totally continuous, at that point has a fixed point in .*

#### 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 function is Lebesgue measurable in terms of for every , on , and is continuous in terms of on .

(H2) There is a function that is strictly nonincreasing which disappears at infinity in such a way that

(H3) a constant in such a way that for every , we have with

By condition (H1), IVP (1) is equal to the following condition: where and for . We define the operator as for each .

Consider the IVP of the following FFEE:

The above system is equal to the following integral: provided that the integral (12) exists.

Theorem 7. *If (12) holds, then
where
*

*Proof. *Let , then
Applying the Laplace transform to (12), we get
for .
Let
and its Laplace transform is given by
Using (19), we have
Since , according to the Laplace convolution theorem, we have
Similarly,
Combining equations (20), (22), and (23), we have
The above system can also be written as
Thus, the proof is complete.

Lemma 8. *Assume that fulfills conditions (H1)-(H3). At that point, (1) has minimum one solution in .*

*Proof. *Define a set by

is clearly a nonempty, convex, closed, and bounded subset of . To show that (1) has a solution, it just necessities to prove that in , the operator has a fixed point. To begin with, we prove that is -invariant. This is without any problem acquired by condition (H2). Now, we should explain that is continuous. For this, suppose that is a sequence of a function to such an extent that and as . Clearly, by the continuity , we get

Assume that is given. After all, is strongly decreasing. At that point for some , we have

For , we get which disappear when . Then again, since in -invariant, at that point, (28) yields

Thus, for , this implies that

If , we clearly have . Therefore, the continuity has been proven. Then, we prove that is equicontinuous. Assume that is given, where is picked with the end of goal that (28) holds. Applying (H3), we get

If , at that point, since is -invariant and using(28), we get

If , it can be seen that which implies that ; then, according to the above discussion, we have got

Thus, we resolve that is equicontinuous on . Since and from the set , it is clear that

Hence, is a moderately smaller set in and all requirements of SFPT are satisfied. In this set, the operator maps on and has a fixed point. This reality indicates that (1) has at least one solution in .

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 in (Lemma 8). Then, use the property of function , to show attractivity. As a result, at , all of the functions in vanish, and therefore, the result of (1) is as .

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.

#### 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), consisting of every single real functions characterized, continuous as well as bounded on 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 be a BS and and represent the convex closure and closure of as a subset of . Further, represents the group of all bounded subsets of , and the represents its subfamily which contains all relatively compact sets. Also, assume that the closed ball is where , , and represents the ball with the end of goal that is the zero component of the BS of .

*Definition 11. * is supposed to be MNC in if it fulfills the following criteria:
(i)The family is nonempty and (ii).(iii)(iv)(v)(vi)If is a closed sequence set from in such a way thatthen
As a result, the family is referred to as the of MNC of .

*Definition 12. *In , let be an MNC. So the mapping is supposed to be a -contraction if a constant term in such way
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 is a nonempty, bounded, convex, and closed subset of BS of , and assume that is a continuous function which fulfills
for every , where represents an arbitrary MNC and represents a monotone nondecreasing function with . At that point, has minimum one fixed point in .*

We will work in BS, where and are given in (1). The functional space is furnished with the standard norm which is . For this reason, we present a MNC in the space which is built like the one in the space . Suppose that is a bounded subset in BS of and is given. For and , we denote by the modulus of continuity of the function on , i.e.,

Now, suppose that we take

If is a fixed number, we use the term as well as

Consider defined on the family by formula

(H4) For all is continuous and there is a continuous function in such a way that where is a function that is superadditive, i.e,

(H5) Assume that such that the following constant exists: where

(H6) a nonnegative result of the following inequality where

Theorem 15. *Under the supposition (H4)-(H6), equation (1) has minimum one solution in . In addition, solution of (1) is uniformly locally attractive.*

*Proof. *To begin, we will look at the operator , which was defined by the formula in the previous section:

. From the condition (H4)-(H6), the function is continuous on . We note that in -invariant. For any and , we have The above expression shows that is bounded on the interval , and connecting with the fact that , we infer that ; thus, changes into itself. Then again, utilize condition (H6) a number which appreciates in (49). For such digit, the operator changes the ball of into itself. Consider a nonempty subset of the ball as well as fix in whatever way you want. At that point, for fixed , we get which implies that

Additionally, let us take as a fixed and . Assume that is chosen and suppose that to such an extent that . Without loss of consensus, we may suppose that . At that point, thinking about our hypothesis, we obtain for , where the symbols used in above term are given by

Now, consider that the function is uniformly continuous on ; we simply get the following expression: which along with (54) and superadditive of implies that

Now, as given by (45) defines a MNC on ; 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 and and so on, where and and in the space . We basically see that and for , and furthermore, the set of this sequence is convex, closed, and nonempty. Moreover, in the light of the current inequality, we get

Combining all the facts that as well as condition (H5) with the recent above inequality, we have

We can derive from the definition of MNC that is convex, nonempty, closed, and bounded. is an -invariant set, and the operator is continuous on it. In addition, remembering the reality that and the set belongs to , 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:

We can clearly see that condition (H1) holds. To show that condition (H2) is fulfilled, since and , we have the following expression

For any , the following identity is obtained: