1. Introduction
Let
for
, and
. Define:
Furthermore, the product space
is given by
where:
Clearly, both and are Banach spaces.
Let
be the partial Riemann–Liouville fractional integral of order
with respect to
, with initial point zero [
1]:
for
.
Assume that
is the Lebesgue integrable and bounded on
for all
and
. In this paper, we begin to construct a unique solution in the space
using Babenko’s method and properties of the gamma function for the following generalized Abel’s integral equation of the second kind with variable coefficients for
:
where each fractional integral
carries its own weight function
, and all
satisfy a certain condition. Then, we further study the uniqueness of solutions in
for:
where
is a mapping from
to
R. Finally, the sufficient conditions are given for the uniqueness of solutions in
to the symmetric system:
where both
and
are mappings from
to
R, and all coefficient functions
are Lebesgue integrable and bounded on
. Equations (
1)–(
3) are new in the present studies, and have never been investigated before.
Clearly, Equation (
1) turns out to be:
if
and
(constant). Equation (
4) is obviously the classical Abel’s integral equation of the second kind. In 1930, Hille and Tamarkin [
2] derived its solution as
where:
is the Mittag–Leffler function.
There have been many analytic and numerical studies on Abel’s integral equation of the second kind, including its variants and generalizations in distribution [
3,
4,
5,
6,
7,
8,
9,
10,
11]. Cameron and McKee [
12] investigated the following Abel’s integral equation of the second kind, numerically based on the construction and convergence analysis of the high-order product integral:
where
is the unknown function defined on the interval
and the kernel
is Lipschitz continuous in its third variable. Pskhu [
13] constructed an explicit solution for the generalized Abel’s integral equation with constant coefficients
for
:
using the Wright function:
and convolution. Evidently, Equation (
5) is a special case of our Equation (
1) for particular values of
m,
and
. In 2019, Li and Plowman [
14] derived a convergent solution for the following Abel’s integral equation:
based on Babenko’s approach in the space
. Obviously, Equation (
6) is also a particular case of Equation (
1) with
,
,
, and
for
.
In a wide range of scientific and engineering problems, the existence of a solution to an integral equation is equivalent to the existence of a fixed point for a suitable and well-defined mapping on spaces under consideration. Fixed points are therefore essential tools in studying integral equations or systems arising from the real world. Banach’s contractive principle provides a general condition ensuring that, if it is satisfied, the iteration of the mapping produces a fixed point [
15].
Babenko’s approach [
16] is a very useful method in solving differential and integral equations, which treat differential or integral operators like variables. The method itself is similar to the Laplace transform when dealing with differential or integral equations with constant coefficients, but it also works for certain equations with distributions, such as
and
, whose Laplace transforms do not exist in the classical sense [
6,
8]. As an example, we are going to solve Equation (
4) using this technique. Clearly:
Informally:
which coincides with Hille and Tamarkin’s result provided above.
2. The Main Results
In this section, we are going to present our main outcomes with several examples for the illustration of the key theorems.
Theorem 1. Assume that , , and is Lebesgue integrable and bounded on Ω for all and . In addition, there exists such that: Then, Equation (1) has a unique solution in the space : Proof. Equation (
1) turns out to be:
Thus, by Babenko’s approach:
Obviously, there exists
such that:
for all
and
.
Then, it follows from reference [
17] that:
Since there exists
such that:
which infers that:
for all
by noting that
is an increasing function if
. Furthermore:
for
and
, since
for all
. Let:
Applying the identity:
we derive that:
We still need to show that Equation (
7) is a solution of Equation (
1). Indeed:
and:
by noting that all of the above series are uniformly and absolutely convergent in the space
due to inequality (
8).
Evidently, the uniqueness immediately follows from the fact that the homogeneous integral equation:
only has solution zero by Babenko’s method. This completes the proof of Theorem 2. □
Remark 1. Note that is not a monotone increasing function on since , and .
Example 1. Abel’s integral equation:has the following convergent solution in :where the coefficients and are given below. Proof. Clearly:
and functions
and
are Lebesgue integrable and bounded on
. By Theorem 1:
Obviously:
for
, and:
On the other hand:
for
, and:
This completes the proof of Example 1. □
Using Banach’s fixed point theorem, we are now ready to show the uniqueness of solutions in
for Equation (
2).
Theorem 2. Suppose that , and is Lebesgue integrable and bounded on Ω for and , and there exists such that: Let be defined on satisfying:and . Furthermore, assume that:where are given in Theorem 1 as Then, Equation (2) has a unique solution in . Proof. Let
. We first show that
. Indeed:
which implies that:
Define a nonlinear mapping
T on
by
Thus,
T is a mapping from
to
. We now need to show that
T is a contractive mapping. In fact:
which claims that
T is contractive since
. This completes the proof of Theorem 2. □
Example 2. Let . Then, the generalized Abel’s integral equation:has a unique solution in . Proof. Clearly,
,
,
, and:
Furthermore:
on
. Therefore,
. Obviously:
, and:
It remains to compute the value of
q:
By Theorem 2, Equation (
9) has a unique solution in
. This completes the proof of Example 2. □
Finally, we study the uniqueness of solutions of in-symmetry system (
3) in the product space
.
Theorem 3. Suppose that , , and are Lebesgue integrable and bounded on Ω for and , and there exists such that: Let and be defined on satisfying:and . Furthermore, assume that:where and are given as Then, the in-symmetry system (3) has a unique solution in . Proof. Let
. We first show that
. Indeed:
which implies that:
Similarly, .
Define a mapping
T on
as
where:
and:
and symmetrically:
Hence,
T is a mapping from
to
. It remains to show that
T is contractive. In fact:
This completes the proof of Theorem 3. □