Asymptotic Stability and Dependency of a Class of Hybrid Functional Integral Equations

: Here, we discuss the solvability of a class of hybrid functional integral equations by applying Darbo’s ﬁxed point theorem and the technique of the measure of noncompactness (MNC). This study has been located in space BC ( R + ) . Furthermore, we prove the asymptotic stability of the solution of our problem on R + . We introduce the idea of asymptotic dependency of the solutions on some parameters for that class. Moreover, general discussion, examples, and remarks are demonstrated.


Introduction
The study of delay functional integral equations has received much consideration over the last few decades.Further studies and results for such kinds of problems may be found in [1][2][3] and the references therein.
The technique of MNC [4] in the Banach space BC(R + ) had been effectively utilized by J. Banaś (see [5,6]) for demonstrating that asymptotic stable solutions for various functional equations have been established (see [7,8]).
Quadratic integral equations continuously emerge in numerous problems, such as the theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory.
Although the existence results in each of these monographs are included [1,[11][12][13], their primary goal was to show a unique method or strategy as well as results pertaining to different existence for particular quadratic integral equations.
The significance of the investigations of hybrid functional integral and quadratic functional integral problems locates within the reality that this type involves different dynamic systems in particular cases.This class of hybrid differential equations involves the perturbations of original differential equations in several ways.A sharp classification of distinctive sorts of perturbations of differential equations shows up in Dhage [14], which can be treated with hybrid fixed point theory.
Here, consider the class of hybrid functional integral equation Our aim here is to establish the solvability and discuss some asymptotic stability facts of the solution ϑ ∈ BC(R + ) of ( 1).The main tool in our study is applying Darbo's fixed point [4] and MNC technique.
Furthermore, the asymptotic dependency of ϑ ∈ BC(R + ) on the parameter λ ≥ 0 and on the functions g 1 , g 2 , β 2 , and β 4 has been studied.Some special cases and examples have been discussed. Let and, easily, we obtain as a solution of (1), which implies that ν is a solution of We arrange our article just like that: we conclude the solvability of (3) in BC(R + ), and then the asymptotic stability of the solution ν ∈ BC(R + ) of ( 3) is discussed in Section 2. The main theorems for the existence of the solutions ϑ ∈ BC(R + ) and the asymptotic stability and dependency of the solution ϑ ∈ BC(R + ) on the parameter λ ≥ 0, and on the functions g 1 , g 2 , β 2 and β 4 have been established.Finally, some general remarks and comments will be provided.
The class BC(R + ) of all bounded and continuous functions in R + , with an internal composition law is noted by (.) : X × X −→ X, (x, y) −→ x.y, which is associative and bilinear.
A normed algebra is an algebra endowed with a norm satisfying the following property: For all x, y ∈ X, we have x.y ≤ x .y .
A complete normed algebra is called a Banach algebra.Now, let ϑ ∈ Y ⊆ BC(R + ) and ε ≥ 0 be provided, defined as ω T (ϑ, ε), T ≥ 0, the modulus of continuity of the function ϑ on [0, T] and The MNC on BC(R + ) has the form [22,23] Theorem 1 ([4]).Let C be a nonempty, bounded, closed, and convex subset of a Banach space ε and let Y : C −→ C be a continuous mapping.Assume that there exists a constant K ∈ [0, 1) such that µ(Y∧) ≤ Kµ(∧) for any nonempty subset ∧ of C, where µ is an MNC defined in ε.Then, Y has at least one fixed point in C.

Existence of Solutions
To achieve our goals, assume that ∀(t, ξ, υ, w), (t, (v) For a positive constant r satisfying the equation From Equation (4), we receive In the same manner, from Equation (5), we receive Theorem 2. Suppose that (i) − (iv) hold.Then, we have a solution ν ∈ BC(R + ) for (3).

Proof.
Let Associate the operator For ν ∈ Q r , then Thus, the mapping F draws the set Q r into Q r .
Then, we have Now, from ( 11) and ( 12), we obtain Consequently, we deduce the next result.

Asymptotic Dependency
Now, replace the assumption (iv) by (iv) * as follows: where lim t→∞ β j (t) Theorem 6. Suppose that Theorem 4 is verified and then the asymptotic dependency of the solution of (2) on the function g 1 occurred; that is, and then Then, By the same manner, we can prove the asymptotic dependency on the function g 2 .Example 1.An example of g 1 can be g 1 (r, u) = γre −r + b 2 u, and then and then and then Analogously, the asymptotic dependency on the function β 4 can be proved.= .

Comments and Remarks
The problem (1) contains several key problems that appear in classical analysis.
Here, we provide a few special problems.
Example 3. Taking into account the next problem the functions g 1 , g 2 , β 2 and β 4 .Furthermore, we can discuss other asymptotic dependency results on the other parameters of (3).Finally, we discussed the exceptional cases, and examples are provided to illustrate our results.