1 Introduction

The objective of the axes of this study first is a direct generalization of the research [1] which refers to the investigation of the EUS to the initial value problem (IVP) involving neutral functional differential equations (FDEs) with sequential fractional orders and the -Caputo operator (1) (2) (3) where denote -Caputo fractional derivatives, 0 < α₁, α₂ < 1, are functions, and .

Let be defined on : Then for r ∈ J, we denote by Differential functional equations play a crucial role across various domains, including control theory, neural networks, and epidemiology [2]. Specifically, delay differential equations prove invaluable in understanding the dynamics of real populations, handling both finite and infinite delay terms. The presence of delays in derivatives poses challenges, prompting an alternative approach through the exploration of neutral functional differential equations. Additionally, fractional derivatives become instrumental in capturing hereditary as well as memory effects in diverse materials and processes. Thus, making the investigation of functional neutral differential equations that involve fractional derivatives a vital area of research. Hybrid problems, which involve the combination of different mathematical techniques or models to solve complex physical phenomena, have applications in various fields. Some examples of physical applications of hybrid problems include Fluid-structure interaction, electromagnetic-structural interaction, thermo-mechanical analysis, coupled heat and mass transfer, multiphase flow with chemical reactions, biomechanics. Comprehensive information on this topic can be found in referenced texts [3–7].

Significant progress in fractional calculus, especially concerning initial value problems (IVPs) of fractional differential equations (FDEs), has been made in recent years. Contributions from various authors, such as Kilbas et al. [8], Lakshmikantham et al. [9], Miller and Ross [10], Podlubny [11], Samko et al. [12], Diethelm [13], along with related papers [14–23], have enriched this field. Riemann-Liouville and Caputo type fractional derivatives dominate this research, while the Hadamard derivative introduces a unique perspective with its logarithmic function containing an arbitrary exponent. See [24–26] and references therein for a comprehensive descriptions of the Hadamard fractional integral and derivative.

In a previous study [27–30], researchers focused on an IVP involving fractional functional and neutral functional differential equations with infinite delay of the Riemann-Liouville type. Subsequent investigations [31] extended to IVPs concerning functional and neutral functional differential equations, including fractional order of Hadamard type. Another study [32] addressed an initial value problem involving retarded functional Caputo-type fractional impulsive differential equations with variable moments.

Fixed-point theory, a mathematical branch, explores the existence and properties of fixed points in mappings or functions. With applications across mathematics and various fields like economics, computer science, physics, engineering, and social sciences, fixed-point theory provides a framework for studying solutions to equations and mappings. The current research focuses on a novel class of sequential fractional neutral functional differential equations, particularly of the -Caputo type. Utilizing fixed point theorems by Banach and Krasnoselskii [33] together with the nonlinear version of Leray-Schauder type introduced by [34], this study advances our understanding of these equations.

The paper’s remaining sections are organized as follows: Section 2 reviews crucial preliminaries for subsequent analysis. Section 3 examines the existence and uniqueness of solutions for the presented problem in Eqs (1) and (2). Section 4 presents the existence results for the problem. Section 5 introduces a generalization incorporating an initial value integral condition, along with illustrative examples showcasing the study’s findings. Finally, Section 6 constructs additional illustrative examples to reinforce the obtained results.

2 Essentiel preliminaries

In the following section, we will introduce the notation, definitions, and preliminary facts that are essential for the subsequent analysis. We denote by the Banach space consisting of all continuous functions from the interval J to the real numbers . The norm associated with this space is given by Also C_τ is endowed with norm

Let be increasing and ∀r.

Definition 1.1 ([8, 35]). The -Riemann-Liouville fractional integral (-RLFI) of order α > 0 for a CF is referred to as

Definition 1.2 ([8, 35]). The -Caputo fractional derivative (-CFD) of order α > 0 for a CF is the aim of where

Lemma 1.3 ([8, 35]). Let q, ℓ > 0, and . Then, , and by assuming

we have

1. ,
2. ,
3. ,
4. ,
5. for k ∈ {0, …, n − 1}, n ∈ N, n − 1 < q ≤ n.

Lemma 1.4 ([8, 35]). Let α ∈ (n − 1, n), β > 0, , , . So has the unique solution and with ℓ = 0, 1, …, n − 1.

Besides, and

Lemma 1.5. The function is a solution of the problem (4) iff (5) Proof. The solution of -Caputo differential Eq in (4) can be written as (6) upon considering the condition , with being an arbitrary constant, we determine that σ₁ can be expressed as . Subsequently, we derive the following result: The equation above allows us to determine that and the validity of (5) is demonstrated. The reverse can be deduced through straightforward calculations.

3 Existence and uniqueness result

Here, we prove the EUS for IVP defined by Eqs (1)–(3).

Definition 2.1. A function , it is referred to as a solution spanning from (1)–(3) if meets the equation on J, by on and .

The following theorem provides a uniqueness outcome based on the given assumptions.

1. (Ω1) there exists ℓ > 0 such that
2. (Ω2) there exists a nonnegative constant k such that

Theorem 2.2. Provided that conditions (Ω1) and (Ω2) are satisfied. If (7) in such a case, a unique solution for IVP defined by Eqs (1)–(3) exists over the interval .

Proof. Let’s examine the operator. characterized by (8) In order to demonstrate that the operator is a contraction, consider . Then, we can observe the following: Consequently we obtain Considering (7), this implies the contraction of the operator . Consequently, by Banach’s contraction principle, possesses a unique fixed point. Thus, demonstrating the EUS to the problem (1)–(3) over the interval .

4 Existence results

This section is devoted to the presentation of the findings regarding the existence of solutions for the initial value problem outlined in Eqs (1)–(3). The next result is based on the nonlinear version of the Leray-Schauder theorem.

Lemma 3.1 ([34]). Consider E as a Banach space, C as a closed and convex subset of C, and U as an open subset of C with the inclusion of 0 ∈ U. Assuming that R is a continuous and compact mapping (meaning is a subset of C that is relatively compact), then either:

1. (i) R has a fixed point in , or
2. (ii) there is a (the boundary of U in C) and λ ∈ (0, 1) with .

The following assumptions are required for the upcoming theorem:

1. (Ω3) There are continuous functions ;
2. (Ω4) There is a continuous nondecreasing function H : [0, ∞)→(0, ∞) and a function satisfying
3. (Ω5) There exist constants and d₂ ≥ 0 such that
4. (Ω6) There exists a constant such that where

Theorem 3.2. With the suppositions (Ω3)–(Ω6) hold, IVP (1)–(3) possesses a solution within .

Proof. We will demonstrate that is continuous and completely continuous.

Step 1: is continuously defined. Consider a sequence such that in . So Because both and are continuous functions, we obtain as n → ∞.

Step 2: The operator transforms bounded sets into bounded sets within the function space . Sufficiently to prove for any θ > 0 there exists such that for each , we have . By (Ω4) and (Ω5), for each r ∈ J, we get Thus

Step 3: Now we show that the image of a bounded set by is an equicontinuous subset of . Indeed; suppose that r₁, r₂ ∈ J, r₁ < r₂, is a bounded subset of , and take . We have

As r₁ approaches r₂, the right-hand side of the mentioned inequality approaches zero. Equicontinuity in cases where r₁ is less than r₂ and when r₁ is less than or equal to zero while r₂ is greater than or equal to zero is readily apparent.

As a result of Steps 1 to 3, applying the theorem of Arzelá-Ascoli leads to the fact that , is continuous as well as completely continuous.

Step 4: We demonstrate the existence of an open set with for λ ∈ (0, 1) and . Let and for some 0 < λ < 1. Then, for each r ∈ J, we have Based on the assumptions we have made, for every r ∈ J, we achieve we can also write it as follows

In view of (Ω6), there exists such that . Let us set It’s worth noting that the operator , exhibits both continuity and complete continuity. Given the selection of U, there exists no on the boundary ∂U such that for any λ in (0, 1). Consequently, employing the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we conclude that possesses a fixed point , which serves as a solution to the problem described in (1)–(3). This successfully concludes the proof.

Lemma 3.3 (KFPT [33]). Consider S as a closed, bounded, convex, and non-empty subset of a Banach space X. Operators and are defined as follows:

• whenever ;
• is continuous and compact;
• is a contraction.

Then there exists z ∈ S so that .

Theorem 3.4. Assume that the conditions (Ω2) and (Ω3) are satisfied and that

(Ω7) , , for all , and .

Then we have at least one solution of problem (1)–(3) on , if (9) Proof. We define the operators Ξ₁ and Ξ₂ by (10) (11) Setting and choosing (12) we consider . For any , we have This shows that . Using (9) one can deduce that Ξ₁ is a contraction.

Since is continuous, the operator Ξ₂ is continuous. Additionally, Ξ₂ is uniformly bounded on by Next, we will establish the compactness of the operator Ξ₂. To do this, we introduce the following definitions and consequently, for r₁ < r₂, we have which is independent of and tends to zero as r₂ − r₁ → 0. Thus, Ξ₂ is equicontinuous. So Ξ₂ is relatively compact on . Hence, by the Arzelá-Ascoli theorem, Ξ₂ is compact on . Thus all the assumptions of Lemma 3.3 are satisfied. So the conclusion of Lemma 3.3 implies that the problem (1)–(3) has at least one solution on

5 Initial value integral

The findings presented in this work can be expanded to encompass scenarios involving an initial value integral condition structured as follows: (13) In the context where , the variable η within Eq (8) will be substituted with . As a consequence, we can express the existence and uniqueness statement for the problem encompassing Eqs (1) and (2)–(13) in the following manner.

Theorem 4.1. Given the fulfillment of conditions (Ω1) and (Ω2), we also make the additional assumption that

(Ω8) there exists a nonnegative constant m such that

Hence the problem (1) and (2)–(13) has a unique solution on if

The proof of the aforementioned theorem closely resembles that of Theorem 2.2.

The analogous form of the existence results, as seen in Theorems 3.2 and 3.4, can be developed for the problem described by Eqs (1) and (2)–(13) in a comparable fashion.

6 Example

In this section, we provide an example to demonstrate how our primary findings can be applied effectively. We will examine a fractional functional differential equation to illustrate this concept, let (14) (15) (16) Let For and r ∈ J, we have and Therefore, conditions (Ω1) and (Ω2) are satisfied with and respectively. Since , hence as asserted by Theorem 2.2, the problem encompassing Eqs (14)–(16) possesses a sole, distinct solution within the interval [1 − τ, e].

Also , and . The conditions stated in Theorem 3.4 are evidently met. As a direct consequence of the theorem’s conclusion, a solution to the problem described in Eqs (14)–(16) is guaranteed to exist within the interval [1 − τ, e].

7 Conclusion

In our research, we touched on the theory of existence and uniqueness of a kind of complex equations included neutral functional differential equations by attracting the generalized fractional derivation -Caputo derivative.

Acknowledgments

This research has been funded by Scientific Research Deanship at University of Ha’il—Saudi Arabia through project number RG-23 036.