Analysis of a Fractional-Order Quadratic Functional Integro-Differential Equation with Nonlocal Fractional-Order Integro-Differential Condition

: Here, we center on the solvability of a fractional-order quadratic functional integro-differential equation with a nonlocal fractional-order integro-differential condition in the class of continuous functions. The maximal and minimal solutions will be discussed. The continuous dependence of the solutions on a few parameters will be examined. Finally, the problems of conjugate orders and integer orders, and some other problems and remarks will be discussed and presented.

Many existence and uniqueness results of differential equations and inclusions with nonlocal boundary conditions have been investigated by some researchers and using different techniques (see [4][5][6][7][8] and the references therein).
Some existence results under mixed Lipschitz and Caratheodory conditions have been discussed for functional integro-differential equations of fractional orders, and the existence of some properties of the solution has been proved in [6].
Subashini et al. [10] discussed the existence of a nonlocal fractional integro-differential problem of the Hilfer type by applying the Monch fixed-point theorem and techniques of measure of noncompactness.
An investigation described the applicability of the a priori estimate method on a nonlocal, nonlinear fractional differential equation for which the weak solution and the unique solution exist [11].

Existence of Solution
We take into account the following assumptions: (i) f 1 : I × R → R is measurable in r ∈ I ∀y ∈ R and continuous in y ∈ R ∀r ∈ I. Furthermore, ∃ a bounded measurable function a 1 : (ii) f 2 : I × R → R is measurable in r ∈ I ∀y ∈ R and continuous in y ∈ R ∀r ∈ I. Furthermore, ∃ a bounded measurable function a 2 : (iii) r 1 is a positive root of the following equation: (iv) h : I × R × R −→ R is measurable in r for all x, y ∈ R and continuous in x, y for r ∈ I, and there exists a bounded measurable function a 3 : Now, we have to prove the following auxiliary result.

Lemma 1.
The solution to the problem (1) and (2) is given in the form, if it exists, where y satisfies the equation (5) and the two systems (1) and (2), and (4) and (5) are equivalent such that δ, γ ≤ α. Proof.
Proof. We characterize a closed ball Q r 1 and an operator F 1 as and For y ∈ Q r 1 , we can obtain |y(r)| = I 1−α | f 1 r, y(r) × I β f 2 (r, I α−γ y(r)) | ≤ I 1−α |a 1 (r)| + b 1 |y(r) × I β f 2 (r, I α−γ y(r))| ) , and we deduce that So, we have This proves that F 1 : Q r 1 → Q r 1 and the family {F 1 y} is uniformly bounded on Q r 1 . Let y ∈ Q r 1 and r 1 , r 2 ∈ I, where r 2 > r 1 and | r 1 − r 2 |≤ δ; therefore, This shows that family {F 1 y} is equi-continuous on Q r 1 , and by the Arzela-Ascoli result [13], mapping F 1 is relatively compact. Let {y n } ⊂ Q r 1 be such that y n → y; then, This yields that F 1 y n (r) → F 1 y(r). Thus, mapping F 1 is continuous, and by applying the Schauder fixed-point theorem [13], y ∈ Q r 1 ⊂ C(I) is a solution of (5).
To prove that x ∈ C(I) exists and satisfies Equation (4), we prove the next result.
Proof. Let Q r 2 be the closed ball and define operator F 2 Let x ∈ Q r 2 ; then, Similarly as performed above, F 2 : Q r 2 → Q r 2 , and {F 2 x} is uniformly bounded on Q r 2 . Let x ∈ Q r 2 and r 1 , r 2 ∈ I; moreover, r 2 > r 1 and | r 1 − r 2 |≤ δ. Then, Thus, family {F 2 x} is equicontinuous on Q r 2 , and operator F 2 is relatively compact. For {x n } ⊂ Q r 2 , and x n → y; then, Using the Lebesgue dominated convergence theorem, we have This means that F 2 x n (r) → F 2 x(r). Hence, operator F 2 is continuous, and there exists x ∈ Q r 2 ⊂ C(I) that satisfies (4). Therefore, we prove the existence of a continuous solution x of Equation (4).

Absolutely Continuous Solution
By taking y(r) = dx dr (r), (1) and (2) reduce to the fractional-order quadratic integral equation y(r) = f 1 r, I 1−α y(r) × I β f 2 (r, I 1−γ y(r)) , r ∈ (0, 1] and the nonlocal integro-differential condition In the same way as in Lemma 1, we can prove the equivalence between (1) and (2), and system (9) and With the same technique as in [14], the existence of solution y ∈ L 1 (I) of (9) can be proved. So, the existence of an absolutely continuous solution of problem (1) and (2) can be proved.

Maximal and Minimal Solutions
Lemma 2. Let us assume that the conditions of Theorem 1 hold. Let x, y ∈ C(I) satisfy such that one of them is strict. Let us assume that f 1 and f 2 are monotonic non-decreasing in the second argument; thus, x(r) < y(r), r > 0.
As proved in Theorem 1, {y (r)} is equicontinuous and uniformly bounded on I; then, {y } is relatively compact, and there exists a decreasing sequence n such that n → 0, n → ∞ and lim n→∞ y n (r) exists uniformly on I. Let lim n→∞ y n (r) = q(r). Then, then, (ii) * h : I × R × R → R is measurable in r ∈ I ∀x, y ∈ R and satisfies the Lipschitz condition, with Lipschitz condition L 3 > 0.

Theorem 4.
Let us assume that (iii), (i*) are verified. If then the solution of (5) is unique.
Proof. Let us consider that assumptions (iii), (i*) are satisfied; then, Theorem 1 is verified, and the solution of Equation (5) exists. Let y 1 , y 2 be two solutions of (5), where ) . Hence, which implies the uniqueness of the solution of (5).

Corollary 1.
Let us suppose that Theorem (4) is verified. If L 3 < 1, then for every solution y ∈ C(I) of Equation (5), there exists a unique solution x of (4).

Theorem 5.
Let assumption (iv) hold; then, the unique solution of (4) depends continuously on x 0 .

Theorem 6.
Let assumption (iv) hold; then, the unique solution of the integral Equation (4) depends continuously on y.

Theorem 7.
Let assumption (iv) hold; then, the unique solution of (4) depends continuously on h.

Conclusions
Integral and differential equations contribute significantly to mathematical analysis and have numerous applications to issues in the real world. It has a wide range of uses in mechanics, population dynamics, mathematical biology, engineering, mathematical physics and other fields [1][2][3]8,[15][16][17][18].
In this study, we establish the solvability of nonlocal problem (1) and (2). The existence of solution x ∈ C(I) of problem (1) and (2) is investigated, and the existence of the maximal and minimal solutions of problem (1) and (2) is proved. Furthermore, some continuous dependency results of solution x on fractional-order derivative y(t), on parameter x 0 and on function h are also proved.
Author Contributions: These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.