Asymptotically Stable Solutions of a Generalized Fractional Quadratic Functional-Integral Equation of Erdélyi-Kober Type

We study a generalized fractional quadratic functional-integral equation of Erdelyi-Kober type in the Banach space . We show that this equation has at least one asymptotically stable solution.

If  = 1 and (, ) = , we obtain a quadratic Urysohn-Volterra integral equation of fractional order studied by Banas' and O'Regan in [23] while in the case where  = 1, (, ) = , and (, , ) = V(, ), we get a fractional quadratic integral equation of Hammerstein-Volterra type studied by Darwish in [22].Moreover, in the case where  = 1, we obtain the quadratic functional-integral equation of fractional order studied by Darwish and Sadarangani in [28].
The aim of this paper is to prove the existence of solutions of (1) in the space of real functions, defined, continuous, and bounded on an unbounded interval.Moreover, we will obtain some asymptotic characterization of solutions of (1).Our proof depends on suitable combination of the technique of measures of noncompactness and the Schauder fixed point principle.

Notation and Auxiliary Facts
This section is devoted to collecting some definitions and results which will be needed further on.First, we recall from [33][34][35] that the Erdélyi-Kober fractional integral of a continuous function  is defined as 2

Journal of Function Spaces
When  = 1, we obtain Riemann-Liouville fractional integral; that is, Now, let (, ‖ ⋅ ‖) be an infinite dimensional Banach space with zero element .Let (, ) denote the closed ball centered at  with radius .The symbol   stands for the ball (, ).
If  is a subset of , then  and Conv denote the closure and convex closure of , respectively.Moreover, we denote by M  the family of all nonempty and bounded subsets of  and by N  its subfamily consisting of all relatively compact subsets.
Next we give the definition of the concept of a measure of noncompactness [36].
) is said to be a measure of noncompactness in  if it satisfies the following conditions.
In what follows we will work in the Banach space (R + ) consisting of all real functions defined, bounded, and continuous on R + .This space is equipped with the standard norm Next, we give the construction of the measure of noncompactness in (R + ) which will be used as main tool of the proof of our main result; see [37,38] and references therein.
Let us fix a nonempty and bounded subset  of (R + ) and numbers  > 0 and  > 0. For arbitrary function  ∈  let us denote by   (, ) the modulus of continuity of the function  on the interval [0, ]; that is, Further, let us put Let us mention that the kernel ker  ∞ 0 consists of all nonempty and bounded sets  such that functions belonging to  are locally equicontinuous on R + .On the other hand, the kernel ker  is the family containing all nonempty and bounded sets  in the space (R + ) such that the thickness of the bundle formed by the graphs of functions belonging to  tends to zero at infinity.
Finally, with the help of the above quantities we can define a measure of noncompactness as The function  is a measure of noncompactness in the space (R + ) [36,37].
In the end of this section, we recall the definition of the asymptotic stability solutions which will be used in the proof of our main result.To this end we assume that Ω is a nonempty subset of the space (R + ).Let  : Ω → (R + ) be a given operator.We consider the following operator equation: Definition 2. One says that solutions of (9) are asymptotically stable if there exists a ball ( 0 , ) such that Ω ∩ ( 0 , ) ̸ = 0 and such that for each  > 0 there exists  > 0 such that for arbitrary solutions  = (),  = () of this equation belonging to Ω ∩ ( 0 , ) the inequality |() − ()| ≤  is satisfied for any  ≥ .

The Existence and Asymptotic Stability of Solutions
In this section we will study (1) assuming that the following hypotheses are satisfied.
for all ,  ∈ R + such that  ≥  and for all  ∈ R. and Now, we are in a position to state and prove our main result.
Theorem 3. Let the hypotheses (ℎ 1 )−(ℎ 6 ) be satisfied.Then (1) has at least one solution  ∈ (R + ) and all solutions of this equation belonging to the ball   0 are asymptotically stable.
Proof.Denote by F the operator associated with the righthand side of (1).Then, (1) takes the form where Here,  and  are the superposition operators, generated by the functions  = (, ) and  = (, ) involved in (1), defined by respectively, where  = () is an arbitrary function defined on R + (see [39]).
Solving ( 1) is equivalent to finding a fixed point of the operator F defined on the space (R + ).
For convenience, we divide the proof into several steps.
Step 1 (F is continuous on R + ).To prove the continuity of the function F on R + it suffices to show that if  ∈ (R + ), then U is continuous function on R + , thanks to (ℎ 1 ), (ℎ 2 ), and (ℎ 3 ).For this purpose, take an arbitrary  ∈ (R + ) and fix  > 0 and  > 0.
Let us denote then we obtain Thus where In view of the uniform continuity of the function  on [0, ]×[0, ]×[−‖‖, ‖‖] we have that   ‖‖ (, ) → 0 as  → 0. From the above inequality we infer that the function U is continuous on the interval [0, ] for any  > 0. This yields the continuity of U on R + and, consequently, the function F is continuous on R + .
Step 2 (F is bounded on R + ).In view of our hypotheses for arbitrary  ∈ (R + ) and for a fixed  ∈ R + we have Hence, F is bounded on R + , thanks to hypothesis (ℎ 5 ).
Step 3 (F maps the ball   0 into itself).Steps 2 and 3 allow us to conclude that the operator F transforms (R + ) into itself.Moreover, from the last estimate we have From the last estimate with hypothesis (ℎ 6 ) we deduce that there exists  0 > 0 such that the operator F maps   0 into itself.
Step 5 (an estimate of F with respect to the modulus of continuity  ∞ 0 ).Take arbitrary numbers  > 0 and  > 0. Choose a function  ∈  and take  1 ,  2 ∈ [0, ] such that | 2 −  1 | ≤ .Without loss of generality we can assume that  2 >  1 .Then, taking into account our hypotheses and ( 21), we have It is easy to see that  < ∞ because (, , ) is bounded on and ) ≤   /Γ( + 1).Therefore, from the last estimate we derive the following one: Hence we have Step 6 (F is contraction with respect to the measure of noncompactness ).From ( 27) and (34) and the definition of the measure of noncompactness  given by formula (8), we obtain Step 7. We construct a nonempty, bounded, closed, and convex set  on which we will apply a fixed point theorem.
8 (F is continuous on the set ).Let us fix a number  > 0 and take arbitrary functions ,  ∈  such that ‖ − ‖ ≤ .Using the fact that  ∈ ker  and keeping in mind the structure of sets belonging to ker  we can find a number  > 0 such that for each  ∈  and  ≥  we have that |()| ≤ .Since F maps  into itself, we have that F, F ∈ .Thus, for  ≥  we get     (F) () − (F) ()     ≤ |(F) ()| +     (F) ()     ≤ 2.