1. Introduction
The existence of an integral equation solution is a true indicator of whether or not a given integral equation can be solved [1] . Over the years, numerous function space-based methods have been implemented to test the nature of an integral equation solution. Although all these various procedures have the same end goal, the function spaces and the applied fixed-point theorems are the key components that differentiate them. According to [2] , fixed-point theorems are essential in determining whether there exists a solution for a given integral equation. The most used fixed point is the Schauder fixed point [3] . Other fixed points confirm that a self-mapping on a set, which is continuous, convex, nonempty, and a compact subset of a Banach space, possesses at least a fixed point [4] . However, finding this subset of a set belonging to a certain function space, which is bounded, convex, closed, and at the same time maps itself by the operator due to an underlying integral equation, is very difficult. For example, some of the approaches that apply either classical Banach or the Schauder fixed point normally result in inaccurate results since strong hypotheses are required for the use of these fixed-point theorems. To address these inaccuracies, the techniques of measures of noncompactness and the Darbo fixed-point theorem have been successfully employed in establishing the existence of solutions, rather than relying solely on methods like the classical Banach or Schauder fixed-point theorems. Measures of noncompactness and Darbo fixed point theorem are highly valuable in functional analysis spanning areas such as metric fixed-point theory and operator theory, differential equations, functional equations, integral and integro-differential equations, optimization, and more.
With the introduction of the concept of the measure of noncompactness, there have been successful applications through the Darbo Fixed point theorem in establishing the existence of the solution of an integral equation. [5] presented an approach that depends on the measure of noncompactness and the Darbo fixed-point theorem.
[6] provided an integral-type generalization of Darbo’s theorem and applied it to establish the existence of solutions for functional integral equations. [7] presented another generalization of Darbo’s theorem along with an application. Recently, integral equations of fractional orders have been investigated in [8] and [9] using measures of noncompactness. For various types of integral equations, refer to [10] [11] [12] [13] . [14] utilized shifting distance functions to establish several new generalizations.
Although this approach has been applied to prove the existence of monotonic solutions of integral equations of various types in the space of Lebesgue integrable functions, it’s application in establishing the solvability of the convolution Volterra integral equation in the space of Lebesgue integrable functions has not been extensively studied [15] [16] [17] [18] [19] .
Therefore, in this study, considering a convolution Volterra integral equation in the form of Equation (1):
, (1)
we establish the proof of existence of solutions of the convolution integral Equation (1) using the measure of noncompactness and the Darbo fixed-point theorem in the space of Lebesgue integrable functions. The mathematical preliminaries and theoretical concepts are in section two, while the main results are presented in section three.
2. Main Concept
2.1. Basic Definitions and Preliminaries
Some of the various mathematical concepts and theorems required for the study are recalled in this section. For the purpose of this study,
represent the set of real numbers,
denotes for all and
the set of natural numbers.
Definition 1 A set
is convex if and only if for every two points in G, the line segment that connects them is entirely contained within G. That is
[20] .
Definition 2 Let X be a vector space over the field
then X is said to be a Banach space if and only if X is equipped with a norm and is also complete.
The space X is complete if for every Cauchy sequence
in X, there exist a subsequence
which converges to
.
Definition 3 Let
denote the set of real numbers and
be an interval on
. For a given nonempty, non-bounded and Lebesgue measurable subset
of
, denoted by L1, as the space of Lebesgue integrable functions on
the standard norm is given by
.
Definition 4 Suppose
and
represent a measure space. If
is said to be a measurable function, then we define
and
.
The Lebesgue space can therefore be restated in the following definition.
Definition 5 Let
be a space then this is defined as a set of
2.2. Volterra Integral Equations
Volterra Integral equation is a type of integral equation which has one of its limits to be a variable. The standard form of a Volterra integral equation is given by:
(2)
The Volterra integral equation can be of either the first or second kind, depending on the position of the unknown variable inside or outside the integral sign. When the unknown function
appears inside and outside the integral sign and
in Equation (2), the resulting integral equation is called a Volterra integral equation of the second kind and is represented by:
(3)
The convolution integral equation results from the nature of the kernel of the integral equation.
2.3. Convolution and Regularization
Theorem 1 Let
and
with
. Then for almost everywhere
the function
is integrable on
and so
in addition
and
[21] .
Theorem 2 Suppose that
is differentiable on an open interval I and
is integrable on I. Let
. If
is continuous then for
every
.
2.4. Carathéodory Conditions
The Carathéodory Conditions stipulate that in the domain of the
space, the following conditions are fulfilled:
1) The function
be defined and continuous in x for almost all t;
2) The function
be measurable in t for each x;
3)
where the function
is integrable in the Lebesgue sense on each finite interval.
2.5. Superposition Operator
Suppose that a function
satisfies the Carathéodory conditions, then for the function
where
is assigned for every
which is measurable on
. The operator T is called the superposition operator generated by f. Functions of several variables are converted to a single variable by the superposition operator T [22] . The superposition operator converts the functions of several variables to a single variable function which can be observed for the L norm.
Theorem 3 (Superposition) The space L1 map continuously onto itself by the superposition operator T if and only if
(4)
and
, where
and
is a function from L1 [23] .
Next, we recall a theorem on the compactness of a measure subset X of L1.
Theorem 4 X is a compact measure if and only if X is a bounded subset of L1 comprising of function which are almost everywhere nondecreasing or nonincreasing on the interval [24] .
Also, we recall some facts on the convolution operator as indicated in [25] . Let
and
then the integral
(5)
exists for almost every
.
belongs to the space L1 where it is the linear operator which maps the space of L1 to L1. The linear operator H is also bounded and continuous since the norm
(6)
For every
. Thus,
is a convolution operator which is majored by
.
Theorem 5 Suppose
is measurable on
such that the integral operator
(7)
maps L1 into L1, then H transforms the set of nonincreasing functions from L1 into L1 if and only if for
and ,
then the assertion
is valid.
2.6. Measure of Non-Compactness
One of the most widely used techniques for proving that certain operator equation has a solution is, to reformulate the problem as a fixed-point problem and see if the latter can be solved via a fixed-point argument. Measure of non-compactness is a function defined as the family of all non-empty and bounded subset of a metric space such that it is equal to zero on the whole family of relatively compact sets [26] .
2.7. Hausdorff Measure of Non-Compactness
The Hausdorff measure of noncompactness of a nonempty and bounded subset Q of X denoted by
according to [27] is defined as the infimum of all numbers
such that Q has r-net in X.
(8)
Also, [10] defined the Hausdorff measure in space L as for
, let
(9)
and
(10)
where
denotes the Lebesgue measure of a subset D. Also given that
, then these two measures
and
are connected by the following theorem.
Theorem 6 Let Q be a nonempty, bounded and compact measure subset of L1, then
(11)
Since these measures of noncompactness are used alongside certain fixed-point theorem, the next theorem considers the fixed point which will be used in this paper.
2.8. Darbo Fixed Point Theorem
The Darbo fixed point theorem is an extension of the classical Banach contraction mapping and the Schauder fixed point theorem.
Theorem 7 (Darbo Fixed Point) Suppose Q is a nonempty, bounded, closed and convex subset of X and let
be a continuous transformation which is a contraction with respect to the measure of noncompactness μ, i.e. there exist
such that
for any nonempty subset E of Q. Then P has at least one fixed point in the set Q [28] .
2.9. Lebesgue Integration
A measurable real-valued function φ defined on a set E is said to be simple provided it takes only a finite number of real values. Suppose φ assumes distinct values
on E, then the measurability of φ, its level set
are
measurable and the canonical representation of φ on E is given by
on E [29] .
The following definitions from [29] are also recalled.
Definition 6 A bounded function f on a domain E of finite measure is said to be Lebesgue integrable over E provided its upper and lower integrals is called the Lebesgue integral and is denoted by
.
Definition 7 (Measurable function) let
be a measure space. A function
is said to be measurable if the set
is measurable for each
.
Suppose that X is a measure space, then the Lebesgue integral
can be defined for any non-negative measurable function
. Although, this will depend more on the function, the integral can be infinite but will always be well-defined as
.
3. Main Results
According to [30] , in order to establish the existence of a solution of an integral equation in the form of Equation (1), if the integral equation has a convolution kernel, then the right-hand side of Equation (1) can be defined under more general hypotheses. Therefore, the following assumptions are made for establishing the proof of existence of solution of the convolution Volterra integral equation in Equation (1):
1) Let
be such that f is continuous and bounded on
.
2)
3)
satisfies the Carathéodory condition.
4)
is increasing and absolutely continuous such that there exist u such that
.
5)
Theorem 8 There exists at least one solution for Equation (1) that is
which is almost everywhere nondecreasing on
if and only if the assumptions (1) - (5) are satisfied.
Proof Let the right-hand side of Equation (1) be represented by operator P, therefore,
(12)
which implies that:
(13)
Let the nonlinear Volterra integral operator as a result of Equation (1) be presented by Equation (14):
(14)
According to [21] the nonlinear Volterra integral operator can be expressed in terms of the convolution operator as a result of Equation (1) which is given by:
(15)
and F, the superposition operator due to Equation (1) is also given by
(16)
Therefore, Equation (1) can be written in the form:
(17)
Next, in order to show that the operator Px will transform any ball of radius r, (Br) into itself, it is established that for
, the function Px belongs to L when assumptions (1) - (5) are satisfied and will also imply that if there exist a ball Br, Px transforms the ball into itself. Therefore,
(18)
(19)
(20)
In order to apply the superposition theorem, Equation (20) is expanded to separate the norms of the convolution and the superposition operators.
(21)
Applying the superposition theorem on the superposition operator in Equation (21) results in:
(22)
(23)
from assumption (4), Equation (23) is rewritten as:
(24)
The theorem for Lebesgue integration by substitution (Theorem 2) is applied in order to convert the function of several variables under the integral sign in Equation (24) to a function of single variable as indicated in Equation (25)
(25)
Expressing
in terms of the norm on the space of Lebesgue integrable functions, results in Equation (26)
(26)
Since the operator Px maps the space of Lebesgue integrable functions into itself by the superposition operator, then for any
, if
, then the operator P assumes the radius of x that is
. This is due to the fact that P is an operator and cannot have a norm so it assumes the norm defined on the space of Lebesgue integrable functions.
Therefore, the exact value of the radius, r of the ball Br is deduced from Equation (26) by assuming that
and also
.
Hence from Equation (26):
(27)
(28)
(29)
Hence
(30)
Therefore, let the radius r of a ball be defined by:
(31)
Then, given that
then it can be concluded that
. This means that P maps the ball Br into itself since substituting the value of the radius r in Equation (31) in place of x in Equation (26) results in the inequality,
(32)
(33)
(34)
(35)
Therefore,
(36)
Next, in order to apply the Hausdorff measure of non-compactness and the Darbo fixed point theorem, Lemma 1 is established since the Darbo fixed point theorem is applied on sets which are closed, bounded, convex and a compact measure.
Lemma 1 Let
consisting of all functions which are almost everywhere positive and nondecreasing on
then M is closed, bounded, convex subset of
and a compact measure.
Proof Suppose for
there exist
then x is bounded for all functions of M with respect to time if and only if
.
Then, for M to be closed, there exist a sequence
such that
and the sequence
converges to a point in
as
.
Furthermore, to show also that M contains functions which are nondecreasing, let
such that
and
for
.
Thus, for every
(37)
(38)
(39)
Since
almost everywhere on
and also
then
(40)
Therefore,
. This implies that x is nondecreasing on
and as such M is closed. Next, for M to be convex, Let
for
then
for all
Let
for all
and
hence
(41)
(42)
Thus, the convexity of M is established.
Again, the subset M is a compact measure as a result of Theorem 4, since it is bounded and contains functions which are nondecreasing almost everywhere on
. Therefore,
implies that
is nondecreasing and positive almost everywhere on
. Hence Px is also nondecreasing and positive on
. Also, since
and P is nondecreasing and positive on
, it can be concluded also that
.
In order to apply the Hausdorff measure of noncompactness, let
, which is nonempty and
. Then for
and for a set
if meas
then from Equation (17)
(43)
(44)
Applying the superposition operator on Equation (44), Equation (45) is obtained.
(45)
To convert the function of several variables to a function of single variable under the integral sign in Equation (45), assumption (4) is applied on Equation (45) which generates into Equation (46) as follows:
(46)
(47)
Applying the Hausdorff measure of noncompactness in Equation (9) to Equation (47)
(48)
Therefore, the measure for the last inequality becomes:
(49)
Furthermore, fixing
so that the lower limit of the integral equation could be any value apart from zero, Equation (47) becomes:
(50)
Applying the measure of noncompactness in Equation (10) to Equation (50) result in Equation (51):
(51)
Therefore,
(52)
Combining Equations (49) and (52), the measure of noncompactness is given by
(53)
By assumption (4), applying the Darbo fixed point theorem in Theorem 7 implies that, there exist at least one fixed point for the operator P in M. This also implies that, there exist a solution for the integral Equation (1) since the condition for the Darbo fixed point theorem is satisfied.
4. Conclusion
The study proved the existence of solution of the convolution Volterra integral equation in Equation (1). For a set
which is a compact measure, bounded and convex, the condition for the Darbo fixed point theorem is satisfied. This indicates the presence of a fixed point for the convolution Volterra integral equation after the Hausdorff measure of noncompactness was applied to obtain the measure of the set
. The presence of the fixed point is an indication that there exists at least one solution to the convolution Volterra integral equation.