This paper explores the solvability of multiterm hybrid functional equations with multiple delays, addressing these equations under some nonlocal hybrid boundary conditions. By applying Schauder fixed-point theorem, we establish the existence of continuous solutions and provide sufficient requirements for the continuous dependence of the unique solution on some factors. In addition, the existence of integrable solutions is examined, broadening the theoretical applicability of these results. Finally, to demonstrate the utility of the approach, an example is provided, showcasing the effectiveness of the proposed method in handling several hybrid problems.

1. Introduction

In recent years, the study of multiterm functional equations has witnessed remarkable advancements, primarily driven by their applicability in modeling systems with memory effects, delayed responses, and nonlocal dynamics. The versatility of infinite systems and multiterm functional equations, as highlighted by Rzepka and Sadarangani [1], Banaś and Chlebowicz [2], and Banaś et al. [3], has been instrumental in capturing delayed and nonlocal dynamics across various domains where systems evolve based on both present and past states.

Moreover, the study of nonlocal boundary conditions has gained significant attention due to their relevance in real-world applications where boundary behavior depends on the system’s overall state over an interval. Unlike classical local boundary conditions, these nonlocal conditions integrate the system’s history into its boundary constraints [4, 5]. El-Sayed et al. [6] demonstrated the influence of nonlocal conditions in the context of pantograph nonlocal fractional-order problems, showcasing their impact on the existence and uniqueness of solutions under fractal–fractional feedback control.

In addition, hybrid functional differential equations have further expanded the scope of mathematical modeling by incorporating the effects of present and past states through integral and functional terms. Such equations naturally arise in diverse fields, including biology, physics, and engineering, where memory effects and delayed feedback are intrinsic to system dynamics. For instance, Karimov et al. [7] studied hybrid nonlinear generalized fractional pantograph equations, utilizing advanced analytical techniques and measures of noncompactness.

The solvability of hybrid functional equations under nonlocal boundary conditions has garnered significant attention in recent years. Fixed-point theorems, such as those of Schauder and Banach, have been instrumental in establishing the existence of solutions. Diethelm [8] demonstrated the utility of these methods in solving fractional differential equations, particularly in the context of boundary value problems involving memory-dependent terms. Similarly, Balachandran et al. [9] analyzed nonlinear fractional pantograph equations using fixed-point methods to study systems with delays and feedback mechanisms. Abdo et al. [10] extended these techniques to nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivatives, showcasing the advantages of employing nonsingular and nonlocal kernels for addressing such problems.

Recent contributions have explored hybrid functional equations in systems with fractional and nonlocal components. For example, Machado et al. [11] and Samadi et al. [12] systematically addressed the challenges of hybrid systems by proposing innovative techniques to analyze their stability and dynamics. Fundamental results by Dhage and Lakshmikantham [13] laid the groundwork for hybrid differential equations, while Mahmudov and Matar [14] investigated the existence of mild solutions for hybrid systems of arbitrary fractional orders. These findings demonstrate the increasing relevance of hybrid fractional models in various applications.

Applications of hybrid functional equations in biological systems were highlighted by Sitho et al. [15] and Ahmad et al. [16], who employed fixed-point methods to establish the existence of solutions for systems exhibiting both continuous and discrete state changes. More recently, El-Sayed et al. [6] investigated hybrid integrodifferential inclusions and multiterm fractional models, emphasizing their role in controlling some systems and analyzing asymptotic stability under various conditions.

The recent advancements in hybrid integrodifferential equations and multiterm fractional models have significantly enhanced our understanding of complex dynamic systems. For instance, El-Sayed et al. [17] explored implicit hybrid Urysohn–Stieltjes integral inclusions, addressing fractal differential constraints and highlighting their structural stability. Furthermore, El-Sayed et al. [18] developed a fractional hybrid framework with nonlocal conditions using differential feedback control, establishing robust existence and stability results.

Our manuscript builds upon these foundational and contemporary studies by addressing the solvability of multiterm hybrid functional equations under nonlocal hybrid conditions. Through a detailed examination of the interplay between fractional calculus, hybrid systems, and fixed-point theorems, this work aims to contribute to the ongoing efforts to advance the mathematical understanding and application of these equations.

Inspired by these studies, we demonstrate the existence of solutions for a class of multiterm hybrid functional equations

()

restricted to the nonlocal hybrid condition

()

For τ ∈ J = [0, T], where D^δ is the Caputo fractional derivative of order δ ∈ {α, β} and 0 < α, β ≤ 1, with η₁ ∈ C(J × R, R) and η₂ ∈ C(J × R, R\{0}) are given continuous functions. Furthermore, the functions Ϝ ∈ C(J × R^k) and g_i ∈ C(J² × R, R), for i = 1, 2, …, k, are Carathéodory function, measurable in τ, ς ∈ J for all ν ∈ R and continuous in ν ∈ R for all τ, ς ∈ J.

In our research, we utilize a technique centered on the method of Schauder fixed-point theorem to investigate the existence of continuous solutions for a multiterm hybrid functional equation with multiple delays under nonlocal hybrid boundary conditions. The structure of the multiterm equation (1), together with the nonlocal boundary condition (2), presents some existence and uniqueness results.

The rest of this work is structured as follows: Section 2 demonstrates an existence theorem for multidimensional functional equations involving multiple delays and investigates the asymptotic stability of the corresponding solution. Subsequently, a particular case and an illustrative example are discussed in Section 4. In conclusion, Section 4 highlights the primary result through an example and explores various special cases pertinent to the problem under consideration.

2. Multiterm Functional Equation With Multidelays

Let

()

Then, we can deduce that any solution of equation (1) is given by

()

where ν is a solution of the following multiterm functional equation:

()

with a nonlocal condition

()

Analyze the multiterm problems (5) and (6) based on criteria outlined in the following.

i.
The functions ϕ_i, ψ_i : [0, ∞)⟶[0, ∞), for i = 1, 2, …, k, are continuous and satisfy
()
ii.
Ϝ : J × R^k⟶R is a Carathéodory function, measurable in ρ ∈ J for all ν_i ∈ R^k and continuous in ν_i ∈ R^k for all ρ ∈ J. There exist positive constants l and Ϝ^∗ such that
()
iii.
The functions g_i : J² × R⟶R, for i = 1, 2, …, k, are Carathéodory function, measurable in ρ, ζ ∈ J for all ν ∈ R and continuous in ν ∈ R for all ρ, ζ ∈ J. There exist measurable, bounded functions a_i, b_i : R₊ × R₊⟶R₊ such that
()
and
()
where a = max{a_i} and b = max{b_i}.
iv.
The function φ : J × R²⟶R is measurable in ρ for every ν, w ∈ R and continuous in ν, w for ρ ∈ J. Moreover, there exist a bounded, measurable function n : J⟶R and a positive constant m > 0 such that
()
where
()
v.
There exists a positive solution r of the quadratic equation such that
()

This leads us to the next lemma.

Lemma 1. If the solution of the problems (5) and (6) exists, then it can be expressed by the fractional-order integral equation

()

where ω represents the solution of the equation

()

and the two nonlocal problems (5), (6), (14), and (15) are equivalent.

Proof 1. Let ν be a solution of problems (5) and (6). By performing the operation I^1−α on each side of equation (5), we get

()

Letting D^αν(ρ) = ω(ρ), we conclude that

()

Now,

()

Hence, substituting equation (6) into equation (17), we have

()

Now, putting ω(ρ) = D^αν(ρ) in equation (14), equations (6) and (15) are reduced to

()

Operating by I^α on the above equation, then by differentiation, we obtain equation (5). This establishes the equivalence of the two systems represented by equations (5) and (6), as well as equations (14) and (15).

Theorem 1. Assume that conditions (i) − (v) are met, then the problem equations (5) and (6) have at least one continuous solution ω ∈ C(J, R).

Proof 2. Let Q_r denote the closed ball

()

where r satisfy condition (v) and defined the operator A₁ as

()

Since by (ii) and (iii), functions Ϝ(ρ, ω₁, ω₂, …, ω_k) and g_i(ρ, ζ, ω) for i = 1, 2, …, k are continuous, this implies that A₁ω ∈ C(J, R).

We shall prove that for some r > 0, A₁Q_r ⊂ Q_r. For

()

Taking into account the assumption (v), the estimate above indicates that the operator A₁ maps the ball Q_r into itself, as there exists a positive solution r that satisfies the inequality.

()

Let us now consider a nonempty subset X of Q_r. We will fix ϵ > 0 and select ν ∈ X along with ρ₁, ρ₂ ∈ J such that |ρ₂ − ρ₁|≤ϵ. Assume, without loss of generality, that ρ₁ ≤ ρ₂. Then,

()

This implies that the family of functions {A₁ω} is equicontinuous on . By the Arzelà-Ascoli Theorem [19], it follows that the operator A₁ is relatively compact.

Now, let {ω_n} ⊂ Q_r be with ω_n⟶ω, so

()

then

()

We find that A₁ω_n(ρ)⟶A₁ω(ρ).

Thus, the operator A₁ is continuous. Applying the Schauder fixed-point theorem [20], we conclude that there is at least one fixed point ω ∈ Q_r ⊂ C(J, R) for equation (15). Hence, there exists at least one solution ω ∈ C(J, R) of equation (15).

To confirm the existence of solutions ν ∈ C(J, R) satisfying equation (14), this leads to the following existence theorem.

Theorem 2. Let the assumption (iv) hold, then there is at least one continuous solution, ν ∈ C(J, R) of equation (14).

Proof 3. Let Q_r2 be the closed ball

()

and the operator A₂

()

Let , then by similar calculations as done in proof of Theorem 2.

()

and

()

This proves that . Moreover, the family of functions {A₂ν} remains uniformly bounded over .

Suppose and ρ₁, ρ₂ ∈ I with ρ₂ > ρ₁ and |ρ₁ − ρ₂|≤β, then

()

This observation shows that {A₂ν} forms an equicontinuous class on . Therefore, by the Arzelá-Ascoli Theorem [19], A₂ is a relatively compact operator.

For , ν_n⟶ν, then

()

Utilizing the Lebesgue Dominated Convergence Theorem, we deduce

()

As a result, we obtain that A₂ν_n(ρ)⟶A₂ν(ρ). Therefore, the operator A₂ is continuous, and by applying the Schauder fixed-point theorem [20], we conclude that equation (14) has at least one solution ν ∈ C(J, R).

3. Uniqueness of the Solution

We now proceed by outlining the following assumptions:

vi.
Ϝ : J × R^k⟶R is a continuous function. There exists a positive constant l such that
()
for each ρ ∈ J and all ν_i, w_i ∈ R. Based on assumption (i)^∗, it follows that
()
where Ϝ^∗ = sup_ρ∈J|{Ϝ(ρ, 0, 0, …, 0)}|.
vii.
g_i : J × R × R⟶R are measurable in ρ ∈ J, for all ν ∈ R, and satisfy the Lipschitz condition
()
Based on assumption (ii)^∗, it follows that
()
where a_i = sup_ρ∈J|a_i(ρ, ζ)|.
viii.
φ : J × R²⟶R is measurable in ρ ∈ J, for all ν ∈ R, and satisfies the Lipschitz condition with Lipschitz constant m > 0,
()
From assumption (iii)^∗, we have
()
where n = sup_ρ∈J|n(ρ)|.

Theorem 3. Assume that the conditions of Theorem 2 are fulfilled and replace conditions (ii) − (iv) by (vi) − (viii). If

()

Then, the solution of equation (15) is unique.

Proof 4. Assume that the conditions of Theorem 2 hold and let ω₁ and ω₂ represent two solutions of equation (15). Then,

()

Since T^1−αlk(a + br) + T^1−αrkb < Γ(2 − α), then the solution of the functional integral equation (15) is unique.

Corollary 1. For each solution ω ∈ C(J, R) of equation (15), there exists a unique solution ν ∈ C(J, R) for equation (14), provided that m < 1.

Proof 5. Let ω ∈ C(J, R) be a solution of equation (15), and suppose ν₁ and ν₂ are two solutions of equation (14). Then, we have

()

then

()

This implies that equation (14) has a unique solution.

4. Continuous Dependence

Definition 1. The unique solution ν ∈ C(J, R) of equation (14) depends continuously on ν₀, if ϵ > 0, ∃ δ > 0 such that

()

where ν^∗ is a solution of

()

Theorem 4. If assumption (vi) holds, the unique solution of equation (14) depends continuously on the parameter ν₀.

Proof 6. Consider ν(τ) as a solution of the integral equation (14). Then,

()

Hence,

()

and

()

Definition 2. The unique solution ν ∈ C(J, R) of equation (14) depends continuously on ω, if ϵ > 0, ∃ δ > 0 such that

()

where ν^∗ is the unique solution of the integral equation

()

Theorem 5. Let assumption (vi) be satisfied, then the unique solution of the integral equation (14) depends continuously on ω.

Proof 7. Let ν(ρ) be a solution of the integral equation (14), then

()

Hence,

()

Definition 3. A unique solution ν ∈ C(J, R) of equation (14) is said to depend continuously on the function φ if for any ϵ > 0, there exists a δ > 0 such that

()

where ν^∗ represents the unique solution of the integral equation given by

()

Theorem 6. Assuming that condition (vii) holds, the unique solution of equation (14) is continuously dependent on the function φ.

Proof 8. Let ν(ρ) denote the unique solution of equation (14). So,

()

Hence,

()

5. Discussion and Insights

In this section, we investigate the existence of an integrable solution for equation (4) and present specific illustrations to demonstrate key concepts.

5.1. Integrable Solutions of Equation (4)

Firstly, we examined the existence of integrable solutions of equation (4) under the following assumption:

•
The functions η₁ : J × R⟶R\{0} and η₂ : J × R⟶R satisfy Carathéodory conditions, and there exist constants l₁, l₂, N₁, and N₂ such that
()
and
()

Theorem 7. Assume that the function ν ∈ C(J, R) and that the assumptions of Theorem 2 hold under (iii^∗) instead of (i^∗). Then, equation (4) has at least one solution μ ∈ L₁(J, R).

Proof 9. We define the operator A^∗ as

()

and the set

()

where M = sup_ρ∈I|ν(ρ)|. Clearly, the set

is nonempty, bounded, closed, and convex.

Let . Using assumption (iii^∗), we find

()

Hence,

()

This shows that and that A^∗μ is uniformly bounded on . In addition, by assumption (iii^∗), we observe that A^∗ is continuous on the set .

Next, we prove that A^∗ is compact. Let . Then, A^∗(ψ) is bounded in L₁(J, R). For , we have

()

and

()

Now, for ν ∈ C(J, R), M = sup_ρ∈I |ν(ρ)| and η₁, η₂ ∈ L₁(J, R), then

()

Hence,

()

Then, by Kolmogorov’s compactness criterion [21], we deduce that A^∗(Ω) is relatively compact, that is, A^∗ is a compact operator.

According to Schauder’s fixed-point theorem [20], there exists at least one fixed point μ of Hence, there exists at least one integrable solution of problem (4).

Theorem 8. For the function ν ∈ C(J, R). Let the assumptions of Theorems 2 and 7 be satisfied. Then, the multiterm functional problems (1) and (2) has at least one solution μ ∈ L₁(J, R) which is the solution of problem (4), where ν is the solution of equation (14).

5.2. Uniqueness of the Solution of Equation (4)

In this subsection, we demonstrate that the solution of the functional equation (4) is unique under the following assumptions:

•
The functions η₁ : R₊ × R⟶R and η₂ : R₊ × R⟶R\{0} are continuous and satisfy the following Lipschitz conditions:
()

Theorem 9. The solution to equation (4) is unique for any ν ∈ C(J, R), provided that the conditions of Corollary 1 and assumption (vi) hold, with the additional constraint that l₁ + M l₂ < 1.

Proof 10. From assumption (vi) and Corollary 1, we know that the solution μ(ρ) of equation (4) is continuous.

Let μ₁ and μ₂ be two solutions of equation (4). For ρ ∈ I, we have the following:

()

Setting M = sup_ρ∈I|ν(ρ)|, we obtain

()

Therefore,

()

This proves the uniqueness of the solution for problem (4). Since l₁ + M l₂ < 1, the operator is a contraction, ensuring the uniqueness of the solution in L₁(J, R).

Thus, equation (4) has a unique solution for all μ ∈ L₁(J, R).

With this, we can deduce a unique result for the solutions of multiterm functional problems (1) and (2).

Theorem 10. Let the assumptions of Corollary 1 and Theorem 9 be satisfied. Then, the solution of multiterm functional problems (1) and (2) is unique.

5.3. Continuous Dependency of the Solution Equation (4)

Definition 4. The solution of equation (4) continuously depends on the function ν if ∀ ϵ > 0, there exists a δ > 0 such that

()

Theorem 11. Suppose that the conditions of Theorem 9 are met. The solution of equation (4) exhibits continuous dependence on the function ν.

Proof 11. Let μ and μ^∗ represent the solutions of equation (4) and for δ > 0 with |ν−ν^∗|≤δ, we get

()

Now, for ρ ∈ J, let H = sup_ρ∈J|μ(ρ)| and M = sup_ρ∈J|ν(ρ)|. Then, we have

()

Thus,

()

Hence, the solution of equation (4) depends continuously on the function ν.

5.4. Continuous Dependency for Multiterm Functional Problems (1) and (2)

From Corollary 1 and Theorem 11, we derive the following corollaries for the continuous dependency of solutions of multiterm functional problems (1) and (2) on the constant ν₀ and function θ

Corollary 2. Let the assumptions of Theorems 4 and 11 be satisfied. Then, the solution of multiterm functional problems (1) and (2) depends continuously on constant ν₀.

Corollary 3. Let the assumptions of Theorems 6 and 11 be satisfied. Then, the solution of multiterm functional problems (1) and (2) depends continuously on constant φ.

5.4.1. Example

Consider the following multiterm functional equation:

()

with the nonlocal integrodifferential condition

()

Now, we investigate the solvability of the integral equation (76) under the condition (77). Take into account that this multiterm functional equation is a particular case of equation (5) with

()

Obviously, the function Ϝ is mutually continuous. Currently, for any ν₁, …, ν_k, w₁, …, w_k ∈ R and ρ ∈ [0, 1]. Condition (ii) is satisfied. Furthermore,

()

where l = (1/2) and Ϝ(ρ, 0, …, 0) = (ρ/1 + ρ³)sin(ρ).

()

This indicates that we can insert a_i(ρ, ζ) = (ρ(2ρ − ζ)/2(1 + ρ⁴)) and b_i(ρ, ζ) = (ρ/2π(ρ² + 1)(ζ + 1)).

Moreover, we have a = 0.14246919 and b = 0.0906987.

Finally, let us pay attention to the equation

()

which has the following roots:

()

and it is easily seen that root r = 1.3241 of previous equation satisfies the inequality

()

Thus, by the application of Theorem 3, the multiterm functional equation (76) has a unique solution.

6. Conclusion

This study has explored the solvability of multiterm hybrid functional equations featuring multiple delays and nonlocal hybrid boundary conditions. Leveraging the Schauder fixed-point theorem, we established the existence of continuous solutions and identified conditions ensuring their continuous dependence on key parameters. An illustrative example is presented to demonstrate our findings. These findings provide a robust theoretical framework for analyzing dynamic systems characterized by delayed feedback and memory effects.

The theoretical insights and practical examples can illustrate the potential of hybrid functional models to effectively represent complex real-world systems with memory effects and nonlocal interactions. Research on hybrid and fractional functional systems is rapidly growing because these systems are very good at modeling real-world complexity, especially where memory, delay, nonlinearity, and discrete-continuous dynamics are involved [22–25].

Future work will extend these findings to other classes of hybrid and fractional functional systems with applications in various scientific and engineering domains [26–28].

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

All authors contributed equally and significantly to writing this article. All authors read and approved the final manuscript.

Funding

The authors received no financial support for this manuscript.

Open Research

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Innovative Aspects of the Solvability of Multiterm Hybrid Functional Equations via Nonlocal Hybrid Conditions

Abstract

1. Introduction

2. Multiterm Functional Equation With Multidelays

3. Uniqueness of the Solution

4. Continuous Dependence

5. Discussion and Insights

5.1. Integrable Solutions of Equation (4)

5.2. Uniqueness of the Solution of Equation (4)

5.3. Continuous Dependency of the Solution Equation (4)

5.4. Continuous Dependency for Multiterm Functional Problems (1) and (2)

5.4.1. Example

6. Conclusion

Conflicts of Interest

Author Contributions

Funding

Open Research

Data Availability Statement

References

References

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Innovative Aspects of the Solvability of Multiterm Hybrid Functional Equations via Nonlocal Hybrid Conditions

Abstract

1. Introduction

2. Multiterm Functional Equation With Multidelays

3. Uniqueness of the Solution

4. Continuous Dependence

5. Discussion and Insights

5.1. Integrable Solutions of Equation (4)

5.2. Uniqueness of the Solution of Equation (4)

5.3. Continuous Dependency of the Solution Equation (4)

5.4. Continuous Dependency for Multiterm Functional Problems (1) and (2)

5.4.1. Example

6. Conclusion

Conflicts of Interest

Author Contributions

Funding

Open Research

Data Availability Statement

References

References

Related

Information