Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis

: In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are deﬁned, continuous, and bounded on the semi-inﬁnite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in BC ( R + ) . In addition, some sufﬁcient conditions are investigated to demonstrate the asymptotic stability of the solutions of that integral inclusion. Finally, some cases analyzed are in the presence and absence of the control variable, and two examples are provided in order to indicate the validity of the assumptions.


Introduction and Background
In the last few years, many researchers have focused their work on some cubic integral equations. They have extended their results for quadratic integral equations to a specific set of cubic integral equations on a bounded interval, for example, Refs. [1][2][3][4][5]. The cubic integral equations can be considered as a generalization of quadratic integral equations, which are applicable to many real-world problems. Furthermore, in [6], the authors investigated some findings for the existence of solutions to a cubic functional integral equation related to a control variable; this has broader implications than those discussed in [2][3][4][5].
The investigations in [2][3][4][5] are on bounded intervals. However, in this article, we establish our results on unbounded intervals.
Caballero et al. [1] provided the first contribution to the solvability of the cubic integral equations; they proved the existence of nondecreasing solutions to Urysohn cubic integral equations.
The importance of dealing with problems involving control variables is due to the unforeseen factors that continually upset ecosystems in the actual world, which may result in modifications to biological traits such as rates of survival. Ecology has a practical interest in the question of whether an ecosystem can withstand those unpredictable disruptive events that continue for a short period of time. In the context of control variables, the disturbance functions are what we refer to as control variables. Cichoń (1996) initiated abstract control problems for differential conclusions [7]. His results were also applied to a semilinear optimal control problem.
In [8], Chen established some averaging conditions for a nonautonomous Lotka-Volterra system that is regulated via feedback by producing a suitable Lyapunov function (Lyapunov functional).
A class of feedback-controlled nonlinear functional-integral equations exist, are asymptotically stable, and are globally attractive, as found by Nasertayoob, using the measure of noncompactness in conjunction with Darbo's fixed point theorem [9]. Moreover, under appropriate circumstances, the authors of [10] investigated whether a nonlinear neutral delay population system with feedback control has a positive periodic solution. The proof depends on the strict-set-contraction operators' fixed-point theorem [10].
A functional integral equation incorporating a control parameter function that fulfils a constraint functional equation is the subject of El-Sayed et al.'s research [11]. Additional findings of existence may be found in [12], where researchers looked at a nonlinear functional integral equation restricted by a functional equation with a parameter.
In this article, we demonstrate that solutions exist and are asymptotically stable for a class of nonlinear functional-integral inclusions with the fractal feedback control where d dr β is the fractal derivative of order β (see [13,14]) and f i (i = 1, 2, 3) and h satisfy some conditions and G is a Lipschitzian set-valued map. This is the first attempt to discuss a class of nonlinear functional-integral inclusions with fractal feedback control. the nonlinear problem affected by an external source is studied in R + . The existence of a control variable v that satisfies the fractal derivation equation is established. We study the existence and the stability of solutions for the hybrid cubic functional integral inclusion (1) in BC(R + ).
The basic tools used in our research are the fixed point theorem of the Darbo type [15] and the strategy of measure of noncompactness.
The measure of noncompactness and Darbo's fixed point theorem are beneficial methods and techniques to study the nonlinear functional-integral equations that arise in some real-world problems [16][17][18][19][20].
To be able to recall the definitions of the keywords' global attractivity, local attractivity, and asymptotic stability of the solution, see [18,27,28].
What follows in the article is arranged as follows: In Section 1, we outline some previous results to explain our motivation and the innovation of the work. Section 2 states and demonstrates the existence of a result for a single-valued problem by a direct application of Darbo's fixed point theorem [15]. Additionally, the asymptotic stability of the solution to our problem will be studied. Next, we extend our results to the multi-valued problem in Section 4. Finally, in Section 4, we will provide an illustration of our main result with two examples and with some special cases of the studied problem.

Single Valued Problem
To achieve our aims, we initially study the single-valued problem corresponding to a class of nonlinear functional-integral inclusions (1) using a fractal feedback control .
depending on the following conditions: (i) Given continuous functions φ i : (ii) The functions ψ : R + → R + are continuous, non-decreasing, and ψ(r) ≤ r.

Remark 1.
For any function x belonging to BC(R + ), the solution to a fractal feedback control indicated by v x (r), and it can be stated as follows: Chen [8] prove that with the positive initial condition v x (0) > 0, the solution v x (r) is globally attractive and bounded above by positive constants. Then Theorem 1. Assume that assumptions (i)-(vii) hold. Then, the Equation (2) has at least one solution x = x(t) that belongs to the space in the space BC(R + ). Additionally, the solutions to Equation (2) are locally attractive.
Proof. Let B r be the ball described by By defining the operator F on the space BC(R + ) by Fx(r) corresponding to our assumptions, note that for any function x ∈ BC(R + ), the function Fx is continuous on the interval [0, ∞). We will demonstrate that for some r > 0, FB r ⊂ B r , we obtain Taking assumption (iv) into consideration, from the above estimate, we conclude that the operator F transforms the ball B r into itself. There exists a positive solution r = r 0 to the equation Now, we show that F is continuous on the ball B r . In order to do this, let us fix > 0 and select x, y ∈ B r such that x − y ≤ . Then, for r ∈ I, we obtain Take into account the next two cases (i) Select T > 0 so that for r ≥ T, the given inequalities are true: Then, we have Considering that the function h has uniform continuity, we conclude that Therefore, using the above estimate in this case, we have Finally, from the two cases (i) and (ii), and considering the previous information, we come to the conclusion that the operator F continuously maps the ball B r into itself. Now, let us take a nonempty subset X of B r . Let T > 0 and > 0 be given, and choose a function x ∈ X and r 1 , where we denote We therefore, arrive at the following estimate: subsequently, depending on the function g, f i : [0, ∞) × B r → R, (i = 1, 2) are uniform continuity, conditions (iii) and (iv), we have to deduce θ v x (δ), θ g (δ) and θ f i (δ) → 0, as . As a result, when we combine the facts with the estimate (3), we obtain Consequently, we obtain In the following, we take a nonempty set X ⊂ B r . Then for any x, y ∈ X, and fixed r ≥ 0, we obtain Hence, it is simple to arrive at the following inequality: Now, considering our conditions, we discover this estimate: Then lim where we denote Clearly, given assumption (vi), we know that c < 1. Finally, linking (4) and (5), using the formula that describes the measure of noncompactness [29,30], we arrive at the following inequality.
Now, taking into account the condition that c = M + lV + l r|[k 1 + k 2 r] + l k [N + η r][η 3 + k 1 + k 2 r]) < 1, and Darbo's fixed point theorem [15], we deduce that the ball B r has a fixed point x for the operator F. Clearly, the functional integral Equation (2) has a solution x. Additionally, consider that the ball B r contains the image of the space BC(R + ) under the operator F. We conclude that B r contains the set FixF of all the fixed points of F. It is evident that all solutions to Equation (2) are included in the set Fix F. On the other hand, we determine that the family kerµ includes the set Fix F [30]. Now, taking into account the description of sets belonging to kerµ (the kernel kerµ of this measure includes the nonempty and bounded subsets X of BC(R + ) such that the thickness of the bundle generated by functions from X decreases to zero at infinity, and functions from X are locally equicontinuous on R + [26]), we conclude that all solutions to Equation (2) are globally asymptotically stable.

Multi-Valued Problem
The multi-valued problems have attracted significant attention during the last few decades. The literature on this topic is now much enriched and contains a variety of results ranging from existence theory to the methods of solution for such problems (see [31][32][33]). Now, consider the nonlinear functional-integral inclusions (1) with feedback control under the following assumption: (v * ) Let G(r, x) : R + × R + × R + → 2 R + satisfy the following assumptions: (a) The set G(r, x, y) is a nonempty, closed, and convex subset for all (r, x) ∈ R + × R + × R + . (b) The set-valued map G : [0, ∞) × R + × R + → 2 R + is continuous and Lipschitzian set-valued map with a nonempty compact convex subset of 2 R + , with a Lipschitz constant l > 0, such that H d (G(r, x 1 , y 1 ), G(r, x 2 , y 2 )) ≤ l(|x 1 − y 1 | + |x 2 − y 2 |), x i , y i ∈ R + , i = 1, 2.

Existence Theorem
Now, from the main results obtained in Section 2, we deduce the following results for feedback control functional-integral inclusions (1).

General Discussion and Examples
In this section, we present some cases in the absence and presence of feedback control. Moreover, two illustrative examples are presented.

Remark 2. Replace assumption (i) by
In this case, we have that φ i (r) for i = 1, 2, 3, 4 is continuous and φ i (r) ≤ r, and under assumptions (i), (ii * ), and (iii)-(iv) the functional Equation (2) has at least one solution x = x(r), which belongs to the space in the space BC(R + ). •

(2*) Retarded functional integral inclusion with feedback control
Let φ i (r) = r − r i , r ≥ r i > 0, i = 1, 2, 3, 4 and φ i (r) = 0, r < r i , i = 1, 2, 3, 4. Then, we have the functional retarded integral inclusion with feedback control (3*) For α = 0, then we obtain the functional retarded integral inclusion with feedback control which gives (4*) Let f 1 (r, x) = 1, f 2 (r, x)x, g(r, v, x) = x + v, and φ i (r) = r; then, the cubic integral inclusion (1) takes the form with dv(r) • In the case of the absence of control variable v(r) = 0, we obtain some particular cases which that useful for the theory of qualitative analysis of some functional integral equations and important for some models and real problems.
(2) Let f 1 (r, x) = 1, f 2 (r, x) = g(r, x) = x, and φ i (r) = r; then, the cubic integral inclusion (1) takes the form and under the assumptions of Theorem 2, then the cubic integral inclusion (9) has at least one asymptotically stable solution x ∈ BC(R + ).
Moreover, we have k 1 = 0.14246919... and k 2 = 0.0906987. Finally, let us pay attention to the cubic equation of Theorem 2, which has the form η 2 l k 2 r 3 + (k 1 η 2 + k 2 N 2 + η 2 V 2 )l r 2 (12) and has the following root r 1 = −21.3032, r 2 = 0.3.59388, and r 3 = 11.8394, and it is easily seen that the root r 0 = 0.3.59388 of the previous equation satisfies the inequality such that As a result, all the prerequisites of Theorem 2 are met. Hence, we conclude that inclusion (11) has at least one solution in the space BC(R + ). Moreover, the solutions are locally attractive.
When f 2 (t, x) = 1, we obtain Chandrasekhar integral equation. In the radiative transfer, some problems are reduced by S. Chandrasekhar [34] to the well-known integral equation of Chandrasekhar type (see [35,36]).

Conclusions
It is known that in the more reasonable position, unanticipated factors may continuously disrupt a physical system. The alteration of the system's parameters typically causes these disruptions (perturbations). These perturbation functions can be thought of as control variables in the terminology of control theory. In particular, control variables may be taken into account while using integral equations.
The feedback control mechanism may be used in conjunction with harvesting, culling, or other biological control techniques. We recommend the reader to [8][9][10] for literature on feedback control system stability.
Inspired by these applications and by the existence results of differential and integral equations involving control variables that are obtained in [8,11,12], we have established the existence and the asymptotic stability of the solutions for a nonlinear cubic functional integral inclusion with a feedback control on the real half-line, using the technique associated with a measure of noncompactness [37,38].
Furthermore, by selecting an appropriate noncompactness measure, we have proved that such solutions are asymptotically stable in the Banach space BC(R + ).
Our discussion is located in the class of bounded continuous functions BC(R + ). Our work can be considered a formal generalization of the results of the theory of differential equations and inclusions Refs. [7,[39][40][41][42][43].  Acknowledgments: The authors are thankful to Mieczysław Cichoń, for interesting discussions on the subject and help. Special thanks are going to the referee for his/her valuable suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.