Analytical Contribution to a Cubic Functional Integral Equation with Feedback Control on the Real Half Axis

: Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions (cid:66)(cid:67) ( R + ) , by applying the technique of measure of noncompactness on R + . Furthermore, we establish sufﬁcient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results.


Introduction and Background
The measure of noncompactness is essential in fixed point theory (see [1,2]).It can be utilized to demonstrate the existence results related to integral equations of numerous types; for example, we refer to [3,4].The notion of a measure of noncompactness (MNC) was introduced by Kuratowski [5] in 1930.Darbo's fixed point theorem [6], which guarantees the existence of a fixed point, is an important application of this measure because it generalizes both the Schauder fixed point and the Banach contraction principle connected with a measure of noncompactness in the Banach space BC(R + ) (of all bounded and continuous functions on R + ), has been successfully used by J. Banaś (see [7][8][9][10][11][12][13]) to obtain the existence of asymptotically stable solutions of some integral and quadratic integral equations (see [9,10]).
Nonetheless, in the more real problems, biological systems or ecosystems are always perturbed via unpredictable forces.These perturbations usually change the system parameters.In control theory, these perturbation functions can be viewed as control variables.Integrated feedback control is central to regulation, sensory adaptation, and long-term effects Robustness [14][15][16][17].
In [14], Chen obtained some averaged conditions for the permanence and global attractivity of a nonautonomous Lotka-Volterra system with feedback controls by creating an appropriate Lyapunov function (Lyapunov functional).
In [15], Nasertayoob proved the existence, asymptotic stability, and global attractivity of a class of nonlinear functional-integral equations with feedback control, using Darbo's fixed point theorem [6] associated with a measure of noncompactness.Furthermore, the existence of a positive periodic solution for a nonlinear neutral delay population system with feedback control is examined under certain conditions [16].The proof depends on the fixed-point theorem of strict-set-contraction operators [16].El-Sayed et al. [18] were concerned with a functional integral equation involving a control parameter function that satisfies a constraint functional equation.Further existence results were obtained in [19], where they studied a nonlinear functional integral equation that was constrained by a functional equation with a parameter.
Recently, cubic integral equations have gained much attention.Many authors who are concerned with studying quadratic integral equations have extended their results to some special cubic integral equations on a bounded interval, for example, [20][21][22][23].
The aim of this article is to establish existence results for solutions to the following cubic functional integral equation, which is more general than those studied in [20][21][22][23].
with a feedback control given by for r, s ∈ R + , we prove that problems (1) and ( 2) have solutions that belong to BC(R + ) × BC(R + ).Our proof depends on a suitable combination of the technique of measures of noncompactness and Darbo's fixed point principle on the real half-axis.It is the first attempt to investigate a cubic functional integral equation restricted to a control variable u on the real half-axis.Moreover, we establish some sufficient conditions for the continuous dependence of the state variable x on the control variable u.Finally, two illustrated examples and some particular cases are presented.

Theorem 1 ([6]
).Let Q be a nonempty bounded closed convex subset of the space E and let F : Q → Q be a continuous operator such that µ(FX) ≤ kµ(X) for any nonempty subset X of Q, where k ∈ [0, 1) is a constant.Then, F has a fixed point in the set Q.
Then, we can introduce the following: Additionally,

Existence of Solutions
Let B r be the ball defined by where r is a positive solution of the equation Due to consideration of the cubic functional integral Equation (1) with the feedback control (2).Based on the following hypothesis: (i) The functions φ i : R + → R + , (for i = 1, 2) are continuous, and φ i → ∞ as r → ∞.
(ii) The functions ψ : R + → R + are continuous non-decreasing and ψ(r) ≤ t. (iii) g i : R + × R, i = 1, 2 are continuous functions and there exists a positive constant l i such that |g i (r, µ) − g i (r, ν)| ≤ l i |µ − ν| for each r ∈ R + and for all µ, ν ∈ R.Moreover, the function r → g i (r, 0) ∈ BC(R + ) and we have (iv) f i : R + × R → R, i = 1, 2 are continuous functions and there exists a positive constant η i such that for each r ∈ R + and for all u, v ∈ R.Moreover, the function r → f i (r, 0) ∈ BC(R + ) and we have Theorem 2. Assume that assumptions (i)-(vi) hold.Then, the cubic functional integral Equation (1) with a feedback control (2) has a solution in the space BC(R + ).
Proof.Define an operator F by where F 1 , F 2 are given by and We shall prove that for some r > 0, FB r ⊂ B r , we have Similarly, we obtain We shall prove that for some r > 0, FB r ⊂ B r , we have Therefore, This demonstrates that the operator Let us consider the following two cases: (1 * ) Choose T > 0 such that for r ≥ T the following inequalities holds: Then, we have ) where, for > 0, we denote Taking into account the uniform continuity of the function h 1 , we deduce that ω T (h 1 , ) → 0 as → 0. Thus, in this case, by the above estimation, we obtain Finally, from the two cases (1 * )and(2 * ), we can deduce that the operator F 1 is a continuous operator by the same way we can prove that ≤ so operator F 2 is also a continuous operator.

ds
Hence, we can easily derive the following inequality: Now, considering our assumptions, we obtain the following estimate: where we denoted Clearly, given assumption (v), we know that c < 1.By a similar way as performed above, we can deduce that Now, from the estimations ( 9) and ( 12), and the definition of µ on U, we obtain Since all conditions of Theorem 1 are satisfied, then F has a fixed point (x, y) ∈ U. Consequently, the cubic functional integral Equation ( 1) has at least one solution x ∈ BC(R + ) restricted to the control variable u ∈ BC(R + ).

Asymptotic Stability
The following corollary can now be deduced from the proof of Theorem 2.
Corollary 1.The solution x ∈ BC(R + ) of the cubic functional integral Equation (1) with the feedback control (2) is asymptotically stable; that is to say, ∀ > 0, there exists T( ) > 0 and r > 0, such that, if any two solutions of the cubic functional integral Equation (1) x, x 1 , which are restricted to the two control variables u, u 1 , respectively, then (x, u), (x 1 , u 1 ) ∈ U satisfy Proof.Under the assumptions of Theorem 2 and similarly to how relations ( 5) and ( 6) are evaluated, we have for r > T( ).

Particular Cases and Example
(I) In case g i (r, x) = 1, i = 1, 2, then our problem ( 1) and ( 2) has been reduced to the system of functional equations (II) Let f i (r, x) = 1, i = 1, 2, then the functional integral equation For x and g 2 (r, u) = u, , then we have the quadratic functional integral equation of Urysohn-type = 1 and g 2 (r, u) = u, then we obtain the cubic functional integral equation of Urysohn-type ]ds), r ≥ 0 (13) with a feedback control given by ]ds), r ≥ 0 ( 14) So, we investigate the solvability of the cubic functional integral Equation ( 13) with a feedback control ( 14) on the space BC(R + ).Considering these functional equations is a particular case of the cubic functional integral Equation ( 1) with a feedback control (2), where , .
. To confirm the assumption (iv), notice that Moreover, we have Easily, we determine r = 0.272031 Consequently, all the conditions of Theorem 2 are satisfied.Hence, we conclude that the cubic functional integral Equation ( 13) with a feedback control ( 14) has at least one asymptotically stable solution in the space BC(R + ).

Uniqueness of the Solution
Let us state the following assumption: for each r ∈ R + and for all u, v ∈ R.Moreover, the function r → h i (r, s, 0) ∈ BC(R + ) and we have Theorem 3. Assume that (i)-(iv) and (v * ) hold.If the following is true: then there exists a unique solution for the cubic functional integral Equation (1) with a feedback control (2) in the space BC(R + ).
Proof.For v 1 = (x 1 , u 1 ), v 2 = (x 2 , u 2 ), two solutions for the problem (1) and (2), we have Taking the supremum r ∈ R + , we obtain Similarly, Taking the supremum r ∈ R + , we obtain Therefore, Then, This proves the existence of a unique solution for the cubic functional integral Equation (1) with a feedback control (2) in the space BC(R + ).
On the other hand, we have where l = 1 2 and M = π 2 , Further, notice that the function h(r, s, x) satisfies assumption (v * ), where Putting γ = 0.025, H = 0.05.We easily determine r = 0.213469, where r is a positive solution of the Equation (3) and Thus, in view of Theorem 3, we conclude that the problem ( 16) and ( 17) has a unique solution.

Continuous Dependency of the State Variable on the Control Variable
We now demonstrate that the solution of the problem (1) and ( 2) is continuously dependent on the control variable u.

Definition 2. The solution of the cubic functional integral Equation (1) depends continuously on the control variable
where x * and u * satisfy the following x * (r) = f 1 (r, x * (φ 1 (r))).g 1 r, u * (φ 1 (r)) Theorem 4. Let the assumptions of Theorem 3 be satisfied, if condition (3) is satisfied.Then, the state variable x is continuously dependent on the control variable u.

Discussions and Conclusions
The present paper aims to propose a feedback control u(t) to a cubic functional integral Equation (1), and it seems natural to choose the control variable u in the form (2). We have discussed the solvability of a cubic functional integral Equation (1) with feedback control (2) on the real half-axis by applying the procedure linked to measures of noncompactness in BC(R + ) by a specific modulus of continuity.Our study has lied in the space of bounded continuous functions BC(R + ).This investigation for the problem (1) and ( 2) is equivalent to that of a coupled system of two functional integral Equations ( 1) and ( 2).The authors have initiated applying the technique associated with measures of noncompactness on a coupled system of integral equations in BC(R + ) in [24].This article is a continuation of applying the technique of MNC on a cubic functional integral equation, but in our case, Equation ( 1) is restricted to a feedback control (2).Furthermore, some particular cases and examples are illustrated.Finally, the continuous dependence of the state variable x on the control variable u is studied.
2 are a Carathéodory function, which are measurable in r, s ∈ R + × R + , ∀u, ∈ R and continuous in u ∈ R, ∀ r, s ∈ R + , and there exists measurable and bounded functions a i , b i : R+ × R + → R such that |h i (r, s, u)| ≤ a i (r, s) + b i (r, s)|u|, r ∈ J 2 are continuous functions and there exists a positive constant γ i , with γ = max{γ 1 , γ 2 }, such that