1. Introduction
In science and technology, the solution of integral and integro-differential equations plays a significant role. A differential equation, an integral equation, or an integro-differential equation is obtained whenever a physical system is modeled in the differential context. On the other hand, Hilger [
1] first introduced a time scale (or a measure chain) in 1988. Several researchers have built on different perspectives of the theory as Hilger developed the concepts of a derivative and integral on a time scale [
2,
3,
4,
5,
6]. It has been demonstrated that time scales can be applied to any area that can be represented using discrete or continuous models.
Fixed point theory provides a basis in solving existence and uniqueness problems involving all types of differential and integral equations. Many researchers have examined the question of the existence and uniqueness of integrodifferential equations (see [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16]). Recently, the authors in [
17] discussed a particular kind of integro-dynamic equation.
Fixed point theory, on the other hand, is a significant idea with multiple applications in diverse fields of mathematics. The existence of fixed points has broad implications in a variety of disciplines of analysis and topology. It has its own implications and has progressed immensely over the last one and half centuries.
Banach’s contraction principle, however, in the case of a metric space setting, is the basis of metric fixed point theory. Banach’s contraction principle is a very useful tool in nonlinear analysis since it is an easy and flexible tool for defining existence and uniqueness theorems for operator equations. This fact prompted researchers to seek to expand and extrapolate it in order to broaden its scope of application as far as possible.
In recent times, we have seen a number of generalized metric spaces, such as extended
b-metric spaces [
18], controlled metric type spaces [
19], double-controlled metric type spaces [
20], controlled
b-Branciari metric type spaces [
21], triple controlled metric type spaces [
22], and extended hexagonal
b-metric spaces [
23]; several authors concentrated their attention in order to acquire fixed point theorems in these kinds of spaces.
Although only a few investigations have been carried out on the existence and uniqueness of solutions of integro-dynamic equations, the main purpose of this research is to discuss the existence and uniqueness of Volterra–Fredholm integro-dynamic equations on time scales. We analyze the problem in the framework of complete triple controlled metric type spaces and apply a fixed point theorem with a contractive condition involving the controlled functions.
2. Preliminaries
We begin with a brief overview of the fundamental concepts of time scales.
Definition 1 ([
4,
7]).
1. A time scale is an arbitrary non-empty closed subset of the real numbers. A time scale is usually denoted by the symbol 2. For , the forward jump operator is defined as 3. For , the backward jump operator is defined as Remark 1 ([
17]).
It is easy to see that, for any , we have and Definition 2 ([
4,
7]).
We define the set as Definition 3 ([
4,
7]).
Let be a function and let . We define to be the number, provided that it exists, as follows: for any , there is a neighborhood U of t, for some , such that is called the delta or Hilger derivative of g at t.
g is the delta or Hilger differentiable or, in brielf, differentiable in if exists for all
The function is said to be the delta derivative or Hilger derivative or, in brief, the derivative, of g in
Remark 2 ([
17]).
If , then the delta derivative coincides with the classical derivative. Note that the delta derivative is well-defined. It is worth noting that the nabla derivative, which is given in [
7], is another form of derivative defined on time scales. We suggest some recent research upon these two types of fractional-order derivatives in [
4,
7,
24].
Definition 4 ([
4,
7]).
A function is called a delta antiderivative of provided that holds for all . Then, the delta integral of g is defined by Eventually, we define the monomials on time scales and consider a few of its properties.
Definition 5 ([
4,
5]).
The following is a recursive definition of monomials on time scales.for
Theorem 1 ([
5]).
For each , the inequalityholds for each On the other hand, M. Frechet [
25] developed the well-known conception of metric space as an outgrowth of conventional distance. In the literature, the notion of metric space is enlarged in a number of different ways (for instance, see [
26,
27,
28]). The definition of triple controlled metric type space is described below, which is used extensively in our main results.
Definition 6 ([
22]).
Let X be a non-empty set and .A function is called a triple controlled metric type if it satisfies:;
;
for all and for all distinct points , each distinct from w and v, respectively. The pair is called a triple controlled metric type space (in short, ).
The extension of the rectangular inequality is the most important feature of triple controlled metric type spaces. The recent research pertaining to
can be found at [
22,
29,
30,
31].
Example 1 ([
22]).
Let and Q is the set of all positive integers.Define : such that ’’ is symmetric and, for all Let be defined asand Therefore, is a .
Remark 3. By employing the same function(s), a triple controlled metric type becomes a controlled b-Branciari metric type space. In fact, the converse is not true (see Example 3.2 in [22]). Contractive mappings have aroused renewed interest, known for their ability to reduce the number of iterations required when working with numerical calculations of fixed point type problems. In the next section, we establish a fixed point theorem in the context of under a new contractive condition employing the controlled functions.
3. Fixed Point Theorem
Theorem 2. Let be a complete triple controlled metric type space and be a self mapping on X. Assume that for any and there exists such that For each and , , we haveandthen, as Moreover, ifexist and are finite, then A has a fixed point in X. Furthermore, A has a unique fixed point iffor any two fixed points of A. Proof. Let
and define an iterative sequence
by
By using a similar method, we obtain
It is indeed significant to note that if we take the limit of each of the inequalities above as
, we obtain
If
for some
, i.e., for
, we have
. Choose
and
, and then we obtain
, i.e.,
v is a periodic point of
A. Thus,
As , we obtain . Hence, v is a fixed point of A.
Assume that
for some
. To verify that
is a Cauchy sequence, we need to show
. Consider
with odd
m; we obtain the desired result by continuously applying the controlled rectangular inequality.
Through using Equation (
4) and Equation (
5), we obtain
and
Consequently,
and
are Cauchy sequences in
. Hence,
From the other end, if we consider
with even
m, and applying controlled rectangular inequality continuously, we obtain
By following the procedure above, we can deduce that
and
are Cauchy sequences in
. Therefore, we conclude that
Hence, in all the cases, we obtain
i.e.,
is a Cauchy sequence in
X. Therefore, by the above equation, as well as the completeness property of
, we obtain
for
. We will now illustrate that
w is a fixed point of
A. From inequalities (
6) and (
12), it is simple to prove
By Equation (
13) and the assumption of the theorem, we obtain
. Therefore,
, i.e.,
. Hence,
w is a fixed point of
A.
Finally, we show the uniqueness of the fixed point. Let
be two distinct fixed points of
A; then,
and
. By employing the inequality (
7),one can obtain
which is a contradiction. Thereby,
w is a unique fixed point of
A. □
One of the most significant applications of our result is to prove the existence and uniqueness of the Volterra–Fredholm integro-dynamic equation of the second kind, which is defined below.
4. Volterra–Fredholm Integro-Dynamic Equation of Second Kind
Let be a time scale with delta differential operator and forward jump operator , respectively.
Definition 7. A Volterra–Fredholm integro-dynamic equation of the second kind is given aswhere and are given functions, and u is the unknown function. In this work, we deal with the case where Equation (
3) has a first-order
-derivative, i.e., an equation of the type
Existence and Uniqueness Theorem
In this segment, we focus on an initial value problem concerned with the nonlinear Volterra–Fredholm integro-dynamic equations and examine whether its solution exists and is unique in the setting.
Let
be a time scale with delta differential operator
and forward jump operator
, respectively. Consider the initial value problem
where
,
,
are given functions. Let
be the space of continuous functions on
and
be defined as
Consequently,
is a complete triple controlled metric space with the below given controlled functions:
Firstly, we reveal that the initial value problem specified in Equation (
16) can be rewritten as
when both sides of
have their delta integrals taken, where
. It is evident that if the functions
and
G are delta integrable, then the right-hand side of Equation (
18) is a continuous function on
. Define the mapping
as
A solution to the problem mentioned in Equation (
16) (equivalently Equation (
18)) is, evidently, a fixed point of
A. The existence uniqueness theorem for the solution of Equation (
16) is presented in the following section.
Theorem 3. Let be a time scale and be a finite interval for some . Assume that the following conditions are satisfied:
1. The functions and are delta integrable on .
2.
The functions F and G are delta integrable on and satisfyfor some
.
for some .
Hence, the map A specified in Equation (19) has a unique fixed point, implying that the integral equation in Equation (18) has a unique solution in . Proof. According to the definition of the map A defined in Equation (
19) and the Cauchy–Schwarz inequality for integrals on time scales [
2], we have
Taking the supremum over
and in accordance with the metric’s definition given in Equation (
17), we obtain
at which
and
are the time scale monomials described in Equation (
1). Hence, Equation (
22) can be modified as
where
and
, respectively. It is obvious that
for all
. Consider
As a result, we observe that
is strictly decreasing for all
and a sequence bounded below; hence, it converges to some L. As
is a monotone sequence, it is known from Dini theorem that
converges to some
Observe that
In a similar manner, we can verify the inequalities (
5), (
6) and (
7), respectively. Thereby, A fulfils all the hypotheses of the Theorem 2, the map A described in Equation (
19) has a unique fixed point, and the integral Equation (
16) has a unique solution in
. □
5. Application
Theorem 3 is applied to an illustration of Volterra–Fredholm integro-dynamic equations of the second kind in this section.
Example 2. Let . Take into consideration the following nonlinear Volterra–Fredholm integro-dynamic equation:along with the initial condition . Take note of the fact that In actuality, is an immediate solution to the specified nonlinear Volterra integro-dynamic Equation (23). Indeed, Let and The delta derivatives of and can then be determined as follows:and Henceforth, the right-hand side of the nonlinear Volterra–Fredholm integro-dynamic Equation (23) is therefore The map A given in Equation (19) is defined by so that and . The Cauchy–Schwarz inequality and the definition of the map A in Equation (24) lead to the conclusion that Consequently, Equation (25) becomes Thereafter, by evaluating the delta integral and utilizing the fact that , we acquirewhere and . When the supremum over is combined with the metric’s definition given in Equation (17), one obtains By Theorem 3, we havewhere and , i.e., and , then and . Hence, the equation is further transformed intowhere and . Thereby, the map A specified in Equation (24) has a unique fixed point by Theorem 3, i.e., the integral equation stated in Equation (23) has a unique solution in . 6. Conclusions
This article explores the existence and uniqueness of solutions for a class of nonlinear Volterra–Fredholm integro-dynamic equations of the second kind on time scales. We approach the problem in the context of triple controlled metric type spaces, which adopts a different perspective on its solution. Eventually, we demonstrate an example to ensure the existence of a unique solution to an integro-dynamic problem. In future studies, we can extend this technique to higher-order delta derivatives with fixed point theorems of different types to acquire further general conditions for the existence and uniqueness of IVPs correlated with Volterra–Fredholm integro-dynamic equations on time scales.
Author Contributions
Conceptualization, K.G., S.T.Z. and T.A.; methodology, S.T.Z., T.A. and N.M.; validation, T.A. and N.M.; investigation, T.A. and N.M.; writing—original draft preparation, K.G. and S.T.Z.; writing—review and editing, K.G. and S.T.Z.; supervision, K.G.; funding acquisition, T.A. and N.M. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The authors Kalpana Gopalan and Sumaiya Tasneem Zubair would like to thank the Management of Sri Sivasubramaniya Nadar College of Engineering for their continuous support and encouragement to carry out this research work. The authors Thabet Abdeljawad and Nabil Mlaiki would like to thank Prince Sultan University for paying the APC and for the support through the TAS research lab.
Conflicts of Interest
The authors declare no conflict of interest.
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