Qualitative Study for a Delay Quadratic Functional Integro-Differential Equation of Arbitrary (Fractional) Orders

: Symmetry analysis has been applied to solve many differential equations, although deter-mining the symmetries can be computationally intensive compared to other solution methods. In this work, we study some operators which keep the set of solutions invariant. We discuss the existence of solutions for two initial value problems of a delay quadratic functional integro-differential equation of arbitrary (fractional) orders and its corresponding integer orders equation. The existence of the maximal and the minimal solutions is proved. The sufﬁcient condition for the uniqueness of the solutions is given. The continuous dependence of the unique solution on some data is studied. The continuation of the arbitrary (fractional) orders problem to the integer order problem is investigated.

Quadratic integral equations have achieved high attention because of their useful application and problems concerning the real world. These types of equations have been studied by many authors and in different classes, see [8][9][10][11][12][13][14][15][16][17][18][19][20]. Each of these monographs contains existence results, but their main objectives were to present special methods or techniques and results concerning various existences for certain quadratic integral equations.
In [21], an infinite system of singular integral equations was discussed. In [22], some integro-differential equations of fractional orders involving Carathéodory nonlinearities were studied. In [18], the existence of at least a positive nondecreasing solution for an initial value problem of a quadratic integro-differential equation by applying the technique of measure of noncompactness was established.
Recently, the existence results for fractional order quadratic functional integro-differential equation were studied and some attractivity results were obtained [23].
Consider the two initial value problems of the delay quadratic functional integrodifferential equation of arbitrary (fractional) orders dx dt = f t, D α x(t).
and its corresponding integer orders equation g(s, x(s))ds , t ∈ (0, 1] (iii) g : I × R → R is measurable in t ∈ I for any x ∈ R and continuous in x ∈ R for all t ∈ I. Moreover, there exists a bounded measurable function m : I → R and a positive constant b 2 such that |g(t, x)| ≤ |m(t)| + b 2 |x| ≤ a + b 2 |x|, a = sup t∈I |m(t)|.
(iv) There exists a positive root r α of the algebraic equation Lemma 1. Problem (1) with (3) is equivalent to the integral equation where y is the solution of the integral equation Proof. Let x be a solution of (1) with (3). Operating by I 1−α on both sides of the Equation (1); we can obtain Let D α x(t) = y(t); we obtain and Let y ∈ C(I) be a solution of (6); then Now, we have the following existences theorem.
Proof. Let Q r α be the closed ball and the operator F Now, let y ∈ Q r α ; then This proves that F : Q r α → Q r α and the class of functions {Fy} is uniformly bounded on Q r α . Now, let y ∈ Q r α and t 1 , t 2 ∈ I, such that t 2 > t 1 and | t 1 − t 2 |≤ δ; then )ds.
This means that the class of functions {Fy} is equicontinuous on Q r α and by the Arzela-Ascoli Theorem [13], the operator F is relatively compact. Now, let {y n } ⊂ Q r α , and y n → y; then Applying the Lebesgue dominated convergence theorem [13], then from our assumptions we get This means that Fy n (t) → Fy(t). Hence the operator F is continuous. Now, by the Schauder fixed point theorem [13] there exists at least one fixed point y ∈ Q r α ⊂ C(I) of the integral Equation (6). Consequently there exists at least one solution x ∈ C(I) of the problem (1) with (3).

Maximal and Minimal Solutions
Lemma 2. Let the assumptions of Theorem 1 be satisfied. Assume that x, y are two continuous functions on I satisfying where one of them is strict. Let the functions f and g be monotonically nondecreasing; then Proof. Let the conclusion (7) be not true; then there exists t 1 such that From the monotonicity of f and g, we get . Proof. Firstly, we prove the existence of the maximal solution of Equation (6).
Let > 0; then It is easy to show that Equation (8) has a solution y ∈ C(I). Now, let 1 , 2 > 0 such that 0 < 2 < 1 < ; then and from Lemma 2, we obtain Now, the family {y (t)} is uniformly bounded as follows: Also, the family {y (t)} is equicontinuous as follows: )ds.
Then {y (t)} is equicontinuous and uniformly bounded on I; then {y } is relatively compact by the Arzela-Ascoli theorem [13]; then there exists a decreasing sequence n such that n → 0, n → ∞ and lim n→∞ y n (t) exists uniformly on I; let lim n→∞ y n (t) = q(t). Now, form the continuity of f , g and the Lebesgue dominated convergence theorem [13]; we have which implies that q(t) is a solution of Equation (6). Finally, let us prove that q(t) is the maximal solution of Equation (6). To do this, let y(t) be any solution of Equation (6); then Applying Lemma 2, we get From the uniqueness of the maximal solution, it is clear that y (t) → q(t) uniformly on I as → 0; thus q is the maximal solution of Equation (6).
By a similar way we can prove the existence of the minimal solution. Consequently there exist maximal and minimal solutions of problem (1) with (3).

Uniqueness of the Solution
Now, consider the following assumptions: (ii) * f , g : I × R → R are measurable in t ∈ I ∀x ∈ R and satisfy From the assumption (ii) * we have So, we can prove the following Lemma.
Proof. From Lemma 3 the assumptions of Theorem 1 are satisfied and the solution of integral Equation (6) exists. Let y 1 , y 2 be two solutions of integral Equation (6); then ) . Hence, Then the solution of Equation (6) is unique. Consequently, the solution of problem (1) with (3) is unique. Proof. Let δ > 0 be given such that |x 0 − x * 0 | ≤ δ and let x * be the solution of (1) with (3), corresponding to initial value x * 0 ; then ) . Hence, Theorem 5. Let the assumptions of Theorem 3 be satisfied; then the unique solution of problem (1) with (3) depends continuously on the function g.

Continuous Dependence on the Delay Function φ
). Hence, Example 1. Consider the following initial value problem with initial data Then It is clear that all assumptions of Theorem 1 are verified, for t = 1 then From (4) we can deduce that r α satisfies the quadratic equation then r α = 0.04 and r α = 0.74. Then the initial value problems (11) and (12) have at least one solution.

Integer-Orders Problem
Consider now the initial value problems (2) and (3) under the assumptions (i), (iii) and the following assumption: (ii) * * f : I × R → R is continuous and there exists an integrable function v : I → R and a positive constant b 1 such that (iv) * There exists a positive root r 1 of the algebraic equation Lemma 4. Let the assumptions (i), (ii) * * and (iii) be satisfied; then the continuation of Equation (1) as α → 1 is Equation (2).
Proof. From Theorem 1 the solution y of integral Equation (6) exists and is continuous and from Lemma 1 d dt x(t) exists and is continuous. Then from the properties of the fractional derivative [7] we have D α x(t) → d dt x(t) as α → 1. Then Equation (1)→(2) as α → 1. Now, the following lemma can be proved. (2) and (3) are equivalent to the integral equation

Lemma 5. Problems
where Now, we have the following existences theorem.
Proof. Let Q r 1 be the closed ball and define the operator F by Now, let y ∈ Q r 1 ; then and Now, let y ∈ Q r 1 and define θ 1 (δ) = sup [0, r 1 ]}; then from the uniform continuity of the function f : I × Q r 1 → R, and our assumptions, we deduce that θ 1 (δ), θ 2 (δ) → 0 as δ → 0 independently of y ∈ Q r 1 . Then we have This means that the class of functions {Fy} is equicontinuous on Q r 1 and by the Arzela-Ascoli theorem [13], the operator F is relatively compact. Now, let {y n } ⊂ Q r 1 , and y n → y; then lim n→∞ Fy n (t) = lim n→∞ f t, y n (t).
Applying the Lebesgue dominated convergence theorem [13], from our assumptions we get This means that Fy n (t) → Fy(t). Hence, the operator F is continuous. Then by the Schauder fixed point theorem [13] there exists at least one fixed point y ∈ C(I) of Equation (15). Consequently, there exists at least one solution x ∈ C(I) of problems (2) and (3).

Maximal and Minimal Solutions
By the same way as Lemma 2 and Theorem 2, we can prove Lemma 6 and Theorem 8.

Lemma 6.
Let the assumptions of Theorem 7 be satisfied. Assume that x, y are two continuous functions on I satisfying

Uniqueness of the Solution
Theorem 9. Let assumptions (i), (ii) * and (iv) * be satisfied. If then the solution of problems (2) and (3) is unique.
Proof. Let y 1 , y 2 be two solutions of functional integral Equation (15); then Hence, Then the solution of = functional integral Equation (15) is unique. Consequently, the solution of problems (2) and (3) is unique.

Continuous Dependence
Let α → 1. By the same way as Theorems 4-6, we can prove that the unique solution of problems (2) and (3) depends continuously on the parameter x 0 and on the functions g, φ.

Example 2.
Consider the following initial value problem of the delay quadratic integro-differential equation with initial data Here, x(s))ds t ∈ I, β ≥ 1, It is clear that our assumptions of Theorem (7) are satisfied for t = 1; then f * = 1 96 , a = 1 4 and b 1 = b 2 = 1 2 and r 1 satisfies then r 1 = 0.02. Therefore, by applying this to Theorem 7, the given initial value problem has a unique solution.
Conclusions Quadratic integro-differential equations have been discussed in many literature studies, for instance [18,21,22,[24][25][26]. Many real problems have been modelled by Integrodifferential equations and have been studied in different classes. Various techniques have been applied such as measure of noncompactness, Schauder's fixed point theorem and Banach contraction mapping.
In this paper, we have investigated the existences of the solutions of the initial value problem of the delay quadratic functional integro-differential equation of fractional of arbitrary (fractional) orders (1) with (3) and we have proved the existence of the maximal and minimal solutions. Moreover, we have discussed the uniqueness and the continuous dependence of the solution on x 0 , the function g and on the delay function φ.
For the continuation of problem (1) with (3) to problems (2) and (3) as α → 1, we have shown that the function f should satisfy the Lipschitz condition (9).
Finally, problem (1) with (3) can be studied for all values of α ∈ (0, 1] when the function f satisfies the Lipschitz condition (9). Moreover, some examples have been demonstrated to verify the results.
We can also extend the results presented in this paper to more generalized fractional differential equations.

Conflicts of Interest:
The authors declare no conflict of interest.