On the De Blasi Measure of Noncompactness and Solvability of a Delay Quadratic Functional Integro-Differential Equation

: Quadratic integro-differential equations have been discussed in many works, for instance. Some analytic results on the existence and the uniqueness of problem solutions to quadratic integro-differential equations have been investigated in different classes. Various techniques have been applied such as measure of noncompactness, Schauder’s ﬁxed point theorem and Banach contraction mapping. Here, we shall investigate quadratic functional integro-differential equations with delay. To prove the existence of solutions of the quadratic integro-differential equations, we use the technique of De Blasi measure of noncompactness. Moreover, we study some uniqueness results and continuous dependence of the solution on the initial condition and on the delay function. Some examples are presented to verify our results.

Each of these monographs contains some existence results, and the main objective is to present a technique to obtain some results concerning various quadratic integral equations.
In this paper, we study the quadratic integro-differential equations by considering the initial value problem of the implicit quadratic integro-differential equation with delay.
g(s, x(s))ds , a.e. t ∈ (0, 1] satisfying an initial condition We present a new quadratic integro-differential, where the derivative of the function x is multiplied by an integral term involving the function x. Let dx dt = y(t) then we can deduce that and (1) can be written as y(t) = f t, y(t).
The existence of non-decreasing solutions y ∈ L 1 [0, 1] of (4) will be studied by the De Blasi measure of non-compactness. Additionally, we shall prove the continuous dependence of the solution of the problems (1) and (2) on the delay function and on the functions g. Consequently, the existence of a solution x ∈ AC[0, 1] of the problems (1) and (2) will be studied.
(ii) f , g : I × R → R + are Carathéodory functions, which are measurable in t ∈ I ∀ x ∈ R and continuous in x ∈ R ∀ t ∈ I, and there exist m i : Moreover, f is non-decreasing for every non-decreasing x, i.e., for almost all t 1 , t 2 ∈ I satisfying t 1 ≤ t 2 and for all ).
(iii) Let r α be a positive root of the following equation Now, the following lemma can be proved. (1) and (2) is equivalent to the coupled system of integral Equations (3) and (4).

Lemma 1. The problems
Proof. It is clear that the solution of the problems (1) and (2) is given by the coupled system of integral Equations (3) and (4). Conversely, let the solution y ∈ L 1 (I) of (4) exist, then from (3) and d dt x(t) , the solution x ∈ AC[0, 1] of the problems (1) and (2) will exist. Now, we have the following existences theorem.
then the Equation (4) has a solution y ∈ L 1 (I), which is non-decreasing.
Proof. Let Q r be a closed ball containing all non-decreasing functions on I by and define the supper position operator F Fy(t) = f t, y(t). φ(t) 0 g s, x 0 + s 0 y(θ)dθ ds , y ∈ Q r . Now, let y ∈ Q r , then Now, let {y n } ⊂ Q r , and y n → y, then Applying Lebesgue dominated convergence Theorem [22], then from our assumptions we get lim n→∞ Fy n (t) = f t, lim n→∞ y n (t).
i.e., Fy n (t) → Fy(t) implies the operator F is continuous.
Taking Ω be a non empty subset of Q r . Let > 0 be fixed number and take a measurable set D ⊂ I such that measure D ≤ . Then, for any y ∈ Ω, then applying the De Blasi measure of noncompactness [23][24][25][26].
we obtain Then implies where χ is the Hausdorff measure of noncompactness [23][24][25][26], which is defined by and has a fixed point y ∈ Q r . Then, there exists a solution y ∈ L 1 (I) for (4). Hence, ∃ a solution x ∈ AC(I) of the problems (1) and (2).

Uniqueness of the Solution
Now, assume that: The assumption (ii) * implies the assumptions (ii).
Proof. From Remark 1, the assumptions of Theorem 1 are satisfied and the solution of (4) exists. Let y 1 , y 2 be two solutions in Q r of the integral Equation (4), then Hence, then y 1 = y 2 and the solution of (4) is unique. As a result, the uniqueness of the solution of (1) and (2) is proved. Proof. Given δ > 0 and |x 0 − x * 0 | ≤ δ and let x * be the solution of (1) and (2) corresponding to initial value x * 0 , then

Continuous Dependence
However, Hence, Theorem 4. Suppose that the conditions of Theorem 2 hold, then the unique solution of the problems (1) and (2) depends continuously on g.

Proof.
Given δ > 0 and |g(t, x(t)) − g * (t, x(t))| ≤ δ and let x * be the solution of (1) and (2) corresponding to g * (t, x(t)), then However, Hence, Theorem 5. Suppose that the conditions of Theorem 2 hold, then the unique solution of the problems (1) and (2) depends continuously on the delay function φ.

Examples
Example 1. Let the following differential equation satisfying the initial data Then Easily we can verify all conditions of Theorem 1. Then, the initial value problems (7) and (8) with initial data Then, x(t)) = t 3 + 1 2 x(t), φ(t) = t β t ∈ I, β ≥ 1.
Obviously we can verify all conditions of Theorem 1. Then, the initial value problems (9) and (10) has a solution.

Conclusions
In this paper, we have studied a delay quadratic functional integro-differential Equation (1). We have investigated the solvability of the problem (1) and (2)