On the Boundary Value Problem of Nonlinear Fractional Integro-Differential Equations

Using Banach’s contractive principle and the Laray–Schauder fixed point theorem, we study the uniqueness and existence of solutions to a nonlinear two-term fractional integro-differential equation with the boundary condition based on Babenko’s approach and the Mittag–Leffler function. The current work also corrects major errors in the published paper dealing with a one-term differential equation. Furthermore, we provide examples to illustrate the application of our main theorems.

Fractional differential and integral equations provide powerful tools in describing and modeling many phenomenons in various fields of science and engineering, such as control theory, porous media, memory and electromagnetics [1][2][3][4]. There has been a great deal of research published on the existence and uniqueness of fractional differential and integral equations involving Riemann-Liouville or Liouville-Caputo operators with initial conditions or boundary value problems [5][6][7][8][9][10][11][12][13].
In 2022, Rezapour et al. [14] investigated the existence of solutions for a category of the multi-point boundary value problem involving a p-Laplacian differential operator with the generalized fractional derivatives depending on another function. The authors in [15] considered the existence, uniqueness and stability of a positive solution in relation to a fractional version of a variable order thermostat model equipped with nonlocal boundary values in the Caputo sense using Guo-Krasnoselskii's fixed point theorem on cones.
In 2021, Turab et al. [16] looked into the existence of solutions for a class of nonlinear boundary value problems on a hexasilinane graph with applications in chemical formulas. The authors in [17] dealt with the existence and Ulam-Hyers stability (UHs) of Caputo-type fuzzy fractional differential equations (FFDEs) with time-delays by applying Schauder's fixed point theorem and a hypothetical condition. In 2017, Sun et al. [18] studied the existence and uniqueness for the following system of FDEs with a boundary value based on Banach's contractive principle (BCP) and the Laray-Schauder fixed point theorem (L-SFTP): where ω 0 , ω 1 , · · · , ω −2 , ω q are real constants and Θ : [p, q] × R → R is a continuous function. Their work relies on Lemma 2.4 in the paper, which states the following: if and only if However, the authors consider this lemma is to be incorrect, and the term should not appear in the lemma. Indeed, plus the boundary condition only implies that To move forward, we begin by introducing several differential and integral operators, a Banach space C [p, q], the Mittag-Leffler function (the M-L function) as well as Babenko's approach (BA) in Section 2. Then, we present sufficient conditions for the existence and uniqueness of the solutions using the BCP and the L-SFPT, with illustrative examples to show the applications of the main theorems in Section 3. Finally, we summarize the entire paper in Section 4.

Preliminaries
We define the Banach space C [p, q] for ∈ N as The Riemann-Liouville fractional integral I κ p of order κ ∈ R + is defined for function Ψ(ω) as (see [1,2]) if the integral exists. In particular, from [19]. In fact, is the Dirac delta function, which is an identity in terms of convolution.
BA [20] is a useful instrument in solving differential and integral equations with initial conditions by treating bounded integral operators as normal variables. The method itself is close to the Laplace transform while dealing with differential equations with constant coefficients, but it can be applied to differential and integral equations with variable coefficients [21,22]. Evidently, it is always necessary to prove the convergence of solution series, otherwise the solution is not well-defined. To demonstrate this technique in detail, we present the following example to solve Abel's integral equation, as well as Lemma 2, which will play an important role in the subsequent section to define the nonlinear mappings.
Consider Abel's integral equation for α > 0 and a constant c where Φ is a continuous function. Clearly, Treating the factor (1 − cI α 0 ) as a normal variable, we come to The following lemma is another application of BA.
Proof. Applying the operator I κ p to both sides of the equation and using the condition Ψ (ρ) (p) = 0, for ρ = 0, 1, 2, · · · , − 2, we find Dedifferentiating the above equation − 1 times and setting ω = q, we derive that Therefore, by noting that Ψ ( −1) (q) = 0. In summary, we have Using BA (treating the factor 1 + µI κ+ζ p as a variable), we come to We note that all the above steps are reversible since BA is. It remains to be shown that all series on the right-hand side of Equation (3) are convergent in terms of the norm in C[p, q]. Clearly, This completes the proof of Lemma 2.

Remark 1.
(i) In particular, for µ = 0, the FDE with boundary value (ii) Clearly, the FDE with boundary value can be solved along the same lines. This is clearly a generalization of Equation (2).
(iii) Lemma 2 still holds for κ = using the same computation.
The following theorems will be used in Section 3 to study the existence and uniqueness.
implies that T has a fixed point.

Existence and Uniqueness of Solutions
where κ 1 , κ 2 ∈ R. Furthermore, we suppose Then, the FD system (1) has a unique solution in the space C −1 [p, q].
Proof. From Lemma 2, we define a nonlinear mapping Υ over the space from the proof of Lemma 2. Clearly, Thus Υ is a mapping from C −1 [p, q] to itself. To prove that Υ is contractive, we notice that, for Ψ, Clearly, for I 1 , we derive that Regarding I 2 , It follows from the above that Since σ < 1, Υ is contractive. By BCP, the FD system (1) has a unique solution in the space C −1 [p, q]. This completes the proof of Theorem 3. Proof. Clearly, the function Therefore σ ≤ 0.9673916 < 1. By Theorem 3, the FD system has a unique solution in C 4 [0. 6, 2.3]. This completes the proof of Example 1.
We are now ready to present the following theorem regarding existence of solutions to the FDE (1).

Theorem 4. Assume Θ : [p, q] × R → R is a continuous and bounded function and
Then, the FDE (1) with a boundary value has at least one solution in the space C −1 [p, q].
(iv) Finally, we prove that the set is bounded. For any Ψ ∈ Y, Ψ = βΥΨ. This infers that Let which claims that Hence, Y is bounded. By L-SFPT, the FDE (1) with boundary value has at least one solution in the space C −1 [p, q] ⊂ C[p, q]. This completes the proof of Theorem 4.

Example 2. The following FDE with boundary value
has at least one solution in the space C 3 [1,2].

Remark 2. Clearly, from
in Theorem 3, we imply that is a Lipschitz function due to the factor κ 2 .

Conclusions
We studied the uniqueness and existence of solutions to the nonlinear two-term fractional integro-differential equation with a boundary condition by using Babenko's approach, the Mittag-Leffler function, Banach's contractive principle and the Laray-Schauder fixed point theorem. The current work also indicated key errors in the paper (Applied Mathematics, 2017, 8, 312-323) in handling a one-term differential equation. Furthermore, we provided three examples to demonstrate the application of our main theorems using the online Mittag-Leffler calculator. Clearly, it would be interesting and challenging to study the same system with a variable coefficient µ(x).