Abstract
This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval and, secondly, the existence of nonincreasing solutions in unbounded interval . Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution on some parameters. Finally, some examples are provided to verify our investigation.
Keywords:
constrained problem; Hyers–Ulam stability; measure of noncompactness; implicit hybrid functional integral equation; asymptotic stability and dependency MSC:
93B05; 47H10; 26A33; 93C27; 47H08; 34K401
1. Introduction
The study of integral equations serves as a significant mathematical tool in both pure and applied analysis. This is particularly notable in fields such as mechanical vibrations, engineering, and mathematical physics. Previous research efforts dedicated to addressing these types of problems are available in the literature (see [,,,]), especially Volterra equations with linear functionals and a small parameter, which are considered in [].
The principle tools applied in our study are Darbo’s fixed-point theorem [] and the strategy of MNC.
The MNC and Darbo’s fixed-point theorem are useful techniques to discuss the nonlinear functional integral equations that appear in many real-world problems [,,,,].
The application of the MNC technique within the Banach space has proven to be highly effective in establishing the existence and stability of solutions for a wide class of functional equations. Prominently, J. Banaś has successfully utilized this technique in his research (e.g., [,]). Moreover, some literature explores the implementation of this approach in studying various functional equations (see [,]).
Constrained problems play a crucial role in the mathematical representation of life problems. By converting these problems into mathematical models [,]. The importance of dealing with problems involving constraints or control variables is due to the unexpected factors ceaselessly disturbing biological systems within the genuine world; this may result in alterations to biological characteristics, such as rates of survival. Ecology has a practical interest in the question of whether an ecosystem can withstand those unpredictable, disruptive events that proceed for a brief period of time. In the context of control variables, the disturbance functions are what we refer to as control variables.
In [], Chen built up a few averaging conditions for a nonautonomous Lotka–Volterra system that is controlled through criticism by creating an appropriate Lyapunov function (Lyapunov functional).
A family of feedback-controlled nonlinear functional integral equations exist, is asymptotically stable, and is globally attractive as demonstrated by Nasertayoob, utilizing the MNC in conjunction with Darbo’s fixed-point theorem []. Additionally, under suitable circumstances, it was investigated in [] whether a nonlinear neutral delay population system with a feedback control has a positive periodic solution. The existence of a positive periodic solution for a nonlinear neutral delay population system with a feedback control is considered in [].
El-Sayed et al. [] conducted a research study on a constraint functional equation. Further investigations on the existence of solutions can be found in [], where researchers examined a nonlinear functional integral equation under the constraint of a parameter functional equation.
The authors in [] extensively investigated its solvability, asymptotic stability, and continuous dependence of the solution on some parameters. They utilized the technique of (MNC) within the space .
In this study, our focus is on examining the constrained problem of the delay functional equation,
subject to the quadratic integral constraint
Our aim here is to investigate the existence of nondecreasing solutions, , and nonincreasing solutions, , by the De Blasi (MNC) and Darbo’s fixed-point theorem []. Sufficient conditions for the uniqueness of the solution and the continuous dependence of the unique solution on the parameter and functions and of problems (1) and (2) will be studied. Next, we explore the equivalence between problem (1) and (2) and the implicit hybrid functional integral equation (IHFIE).
is established. The Hyers–Ulam stability of problem (1) and (2) and of the IHFIE (3) will be studied. The asymptotic stability and the continuous dependence of the solution on the parameter and functions and will be proved. Finally, some examples are given to illustrate our results.
The importance of examining hybrid functional integral and quadratic functional integral problems is found in the reality that this category includes distinctive energetic frameworks in specific cases. This class of hybrid differential equations includes the perturbations of original differential equations in several ways.
Theorem 1
([]). Let U be a nonempty, bounded, closed, and convex subset of a Banach space ∁ and let be a continuous mapping. Assume that there exists a constant such that for any nonempty subset ∧ of U, where η is an MNC defined in ∁. Then, ℵ has at least one fixed point in U.
2. Solvability in Bounded Interval
Let be the class of Lebesgue integrable functions on , , with the standard norm
2.1. Quadratic Functional Integral Constraint
Let us examine Equation (2) within the framework of the following assumptions:
- (i)
- , and , are Carathéodory functions [], and there exist the integrable functions and nonnegative constants , and such that
- (ii)
- f and , are nondecreasing for every nondecreasing argument such that and for all , , implies
- (iii)
- There exists a positive root of the algebraic equation
Theorem 2.
Assume that (i)–(iii) are met. If , then (2) has at least one nondecreasing solution .
Proof.
Let be a closed ball of all the nondecreasing functions
and the operator
Then, we deduce that transforms the nondecreasing functions into functions of the same type.
Let , then
and
Hence,
and .
Now, let and , then
Applying the Lebesgue Theorem [], we obtain
This means that . Hence, the operator is continuous.
Now, let be a nonempty subset of . Fix and take a measurable set such that . Then, for any , we have
then
and
This implies
where is the Hausdorff MNC [,,,].
Since , it follows from Darbo’s theorem [] that is a contraction and that it has at least one fixed point in . Then, there exists at least one nondecreasing solution to (2). □
Corollary 1.
Let the assumptions of Theorem 2 hold; then, the solution to (2) satisfies .
Proof.
The results follow from the monotonicity of the solution ⋎ on I. □
2.2. The Delay Functional Equation
Now, consider (1) under these assumptions:
- (iv)
- , , are increasing and absolutely continuous, and there exist two constants and such that a.e on I.
- (v)
- and are Carathéodory functions [], and there exist two integrable functions and two constants i = 1, 2 such that
- (vi)
Theorem 3.
Let the assumptions of Theorem 2 be satisfied. Assume assumptions (iv)–(vi) are met; then, there exists at least one nondecreasing solution to (1).
Proof.
Let be a closed ball of all nondecreasing functions
and
Then, we deduce that A transforms the nondecreasing functions into functions of the same type.
Let . Then,
and
Then,
Hence, .
Now, let and . Then,
and
Apply the Lebesgue Theorem []. Then,
This means that . Hence, the operator A is continuous.
Now, let ℧ be nonempty subset of . Fix and take a measurable set such that . Then, for any , we obtain
Then,
and
This implies
where is the Hausdorff MNC [,,,].
Since , it follows Darbo’s theorem; A is a contraction and has at least one fixed point in . Then, there exists at least one nondecreasing solution to (1). □
2.3. Uniqueness of the Solution
Now, consider the following assumptions:
- , and are measurable in and satisfy the Lipschitz condition such that
- and are measurable in and satisfy the Lipschitz condition such that
So, we have proved the following Lemma.
Lemma 1.
Assumptions and imply assumptions and , respectively.
Theorem 4.
2.4. Hyers–Ulam Stability
Now, replace the assumption by as follows:
- is measurable in for any and continuous in for all , and is measurable in for any and continuous in for all . Moreover, there exist a bounded and measurable such that , and they satisfy the Lipschitz condition such thatMoreover, is nondecreasing for every nondecreasing argument.
Definition 1.
Theorem 5.
Proof.
□
2.5. Continuous Dependence on Constraint
Theorem 6.
where is a solution to
Let the assumptions of Theorem 4 be satisfied for and , i = 1, 2. Then the unique solution depends continuously on , and in the sense that
Proof.
Then,
and
Hence,
□
2.6. Dependence of on ⋎
Theorem 7.
Let the assumptions of Theorem 5 be satisfied. Then, the unique solution depends continuously on ⋎ in the sense that
where is the solution to
and is the solution to
Proof.
□
Corollary 3.
Proof.
From Theorem 6, the results follow. □
3. Hybrid Functional Integral Equation
Let be the solution to (3).
Then, we have proved the following.
Corollary 5.
Let the assumptions of Theorems 2–6 be satisfied. Then,
Hyers–Ulam Stability
Definition 2.
Theorem 8.
Let the assumptions of Theorem 5 be satisfied; then, the IHFIE (3) is Hyers–Ulam stable.
4. Solvability in Unbounded Interval )
Here, we examine the existence of solutions, , to (2). Take into account the following assumptions:
- (vii)
- and , i = 1, 2 are Carathéodory functions [], and there exist bounded and integrable functions where , , , and such that, and, for all , , implies
- (viii)
- , i = 1,2 are Carathéodory functions [], and there exist bounded and integrable functions and such that
- (ix)
- There exists a positive root of the algebraic equation
Theorem 9.
Proof.
Let be a closed ball of all nonincreasing functions
Associate the operator
Then, we deduce that K transforms nonincreasing functions into other nonincreasing functions, and is nonincreasing on .
Let . Then,
Then,
Thus, the operator K maps the ball into itself. Moreover, our assumptions imply that the operator K is continuous on .
Now, let be nonempty subset of . Fix and take a measurable set such that . Then, for any , we obtain
Then,
and
Next, fixing , we obtain
However,
Then,
Since , it follows from Darbo’s theorem that K is a contraction and that it has at least one fixed point in . Then, there exists at least one nonincreasing solution to (2). □
Corollary 6.
Let the assumptions of Theorem 9 hold. Then, from (12), we can deduce that
Now, consider the functional Equation (1) under these assumptions:
- (x)
- , , are increasing and absolutely continuous, and there exist two constants , such that a.e on .
- (xi)
- and are Carathéodory functions [], and there exists a bounded and integrable function and a bounded and measurable , , and , such that
- (xii)
Theorem 10.
Let the assumptions of Theorem 9 be satisfied. Assume that (x)–(xii) are met. Then, there exists at least one nonincreasing solution to (1).
Proof.
Let be a closed ball of all of the nonincreasing functions
Associate the operator
then we deduce that H transforms nonincreasing functions into other nonincreasing functions, and is nonincreasing on .
Let . Then,
Then,
and
Hence, the operator H maps the ball into itself. Moreoover, our assumptions imply that the operator H is continuous on .
Now, let be a nonempty subset of . Fix and take a measurable set such that . Then, for any , we obtain
Then,
and
Next, fixing , we obtain
However,
and, since , then
According to (15) and (16), we obtain
Finally, we obtain
where is the Hausdorff MNC [,,,].
Since , it follows from Darbo’s theorem that H is a contraction and that it has at least one fixed point in . Then, there exists at least one nonincreasing solution to (1). □
Corollary 7.
4.1. Asymptotic Stability
Now, replace assumptions , , and with , , and , as follows:
- and , are measurable in and satisfy the Lipschitz condition,
- and are measurable in and satisfies Lipschitz condition,
- .
Lemma 2.
Assumptions and imply assumptions and , respectively.
Theorem 11.
4.2. Continuous Dependence on Some Results
Now, replace assumption with , as follows:
- , are Carathéodory functions [] and satisfy the Lipschitz condition
Theorem 12.
where is the solution to
and is the solution to
Let (vii)–(viii) and (xi)–(xii) occur. Then, the solution to problems (1) and (2) exhibits asymptotic dependency on the parameter ⋋ and on the functions and such that
Proof.
Firstly,
Then,
and
Hence,
Now, let and be two solutions to problems (1) and (2) and . Then,
Then,
and
From (19), we have
Then,
□
Now, from Theorems 9–12, we have the following corollaries.
Corollary 8.
The IHFIE (3) has at least one nonincreasing solution .
Corollary 9.
The IHFIE (3) is asymptotically stable.
Corollary 10.
The IHFIE (3) is asymptotically dependent on the parameter ⋋ and on the functions and .
5. Examples
Example 1.
Now, we find an implicit hybrid functional integral equation that is equivalent to problems (20) and (21).
Example 2.
Consider the following an implicit hybrid functional integral equation
Set
Then, (22) has at least one solution .
Example 3.
Example 4.
Consider the following an implicit hybrid functional integral equation
Set
Then, (25) has at least one solution .
6. Conclusions
In this study, we have conducted a thorough examination of the constrained problem involving the nonlinear delay functional Equation (1) subject to the quadratic functional integral constraint (2). To address this problem, we employed the technique associated with MNC.
In this investigation, we discussed two cases: In the first case, we studied the existence of nondecreasing solutions, ⋎, on a bounded domain for constraint (2). Then, we studied the existence of nondecreasing solutions, , for (1). Moreover, we established some sufficient conditions to guarantee the uniqueness of the solution and its dependence on the parameter ⋋ as well as the functions and . We also studied the equivalence between problems (1) and (2) and the IHFIE (3). Furthermore, we thoroughly investigated the Hyers–Ulam stability of problems (1) and (2) and IHFIE (8). In the second case, we established the solvability and asymptotic stability and dependency of the nonincreasing solution to problems (1) and (2) on the parameter and the functions and . Finally, we provided some illustrative examples to demonstrate the practical application and validity of our obtained results.
Author Contributions
Conceptualization, A.M.A.E.-S.; Methodology, M.M.S.B.-A. and E.M.A.H.; Validation, M.M.S.B.-A. and E.M.A.H.; Formal analysis, A.M.A.E.-S.; Investigation, A.M.A.E.-S.; Writing—original draft, M.M.S.B.-A.; Writing—review & editing, E.M.A.H. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
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