On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order.

In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fractional constant order. The obtained new results are based on the Schauder's fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam-Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results.

1. Introduction

Integral equations are an important part of the field of nonlinear analysis since many problems studied using the nonlinear analysis methods are often expressed as differential or integro-differential equations and then converted to integral equations to facilitate their study [1,2,3,4,5]. In addition, integral equations are used as models of real-world processes in biology, ecology, population dynamics and medicine [6,7,8].

The study of nonlinear integral equations in general has aroused great interest for researchers during the last two centuries, and in particular the Uryshon-type (or Friedholm-type) integral equations, which appeared in many applied problems. These equations are defined as where and g, K are given functions. For more details regarding this type of equations, see [5,9].

In recent years, we can find many applications of integral and differential equations of fractional order in physics, electricity, mechanics, engineering, economics and biology (see [10,11,12,13,14,15,16]). In addition, the concept of entropy in the framework of fractional calculus has attracted research interest recently. See, for example, ref. [17] and the references therein. However, this new concept is not sufficiently developed. Any new contribution in the area of fractional differential and integral operators can offer new opportunities for future progress and expansion of the theory of fractional-order entropies. It will also contribute to the progress of the entropy methods in applied sciences such as mathematical biology and artificial neural networks that used integral equations as modeling tools [18,19,20].

There are many papers that have studied the problem of the existence of solutions of functional integral or differential equations of the fractional constant order. We will refer to some very recent publications [21,22] and the references therein. On the contrary, only a few papers have investigated the existence of solutions of such equations of fractional variable order. For example, the authors in [23] applied the Darbo’s fixed point theorem combined with Kuratowski measure of noncompactness to analyze the existence properties of a Riemann–Liouville fractional differential equation of variable order. The existence and Ulam–Hyers stability for a variable-order Caputo-type fractional differential equation have been investigated in [24]. The existence, uniqueness and stability of solutions of a Hadamard-type fractional differential equations of variable fractional order are established in [25]. A study of the existence and uniqueness of solutions of a class of Hadamard fractional differential equations of variable order has been proposed in [26]. The authors applied piece-wise constant functions, the Krasnoselskii fixed-point theorem and the Banach contraction principle. A Caputo fractional differential equation of variable order is studied in [27] and criteria for the existence of its solutions are proposed. Some new criteria for the existence and stability of solutions of a Hadamard-type fractional differential equations of variable order have been examined in [28].

However, similar research for the Urysoh-type equations is not presented in the existing literature. This is the main motivation for our study which makes the results on the topic interesting and worthwhile. Given the importance of such equations for the theory and applications, it is meaningful to consider their extension to the fractional variable order and investigate their fundamental and qualitative properties. In fact, considering the variable fractional order is challenging since the research of such types of fractional models is still in its infancy and their properties are different from the corresponding properties of systems with a constant fractional order, including the semigroup property.

Stimulated by the above discussion, in this study, we introduce a quadratic integral equation of the Urysohn type of fractional variable order of the following type where is a continuous function, is a given function, is the Gamma function, and is a proper operator. In the above, the notation represents a Banach space of all continuous functions with the norm

To the introduced new model (2) of fractional variable order, we will apply the Schauder’s fixed-point theorem and the Banach contraction principle to investigate the existence and uniqueness of its solutions. In fact, although both methods are very powerful, they are not applied in the study of the fundamental properties of the quadratic integral equation of the Urysohn type to the fractional variable order (2).

The main contributions of our research are:

We generalize and extend the existing quadratic integral equation of Urysohn type to the fractional variable order in the form of a piece-wise constant function;

Efficient existence and uniqueness criteria for the extended model are proposed;

The obtained fundamental results are applied in the study of the Ulam–Hyers stability of the solution;

An example is elaborated to demonstrate our results.

The Riemann–Liouville fractional integral approach of variable-order is adopted in our research. The manuscript is organized according to the following plan. Some definitions and properties of the Riemann–Liouville fractional integral of variable fractional order are stated in Section 2. The concepts of generalized interval, partition and piece-wise constant functions are also defined. Section 3 is devoted to our main existence and uniqueness results for the introduced integral equation of the Urysohn type of fractional variable order. Three theorems are proved by the use of the Schauder’s fixed point theorem and the Banach contraction principle. In Section 4, in order to demonstrate the applicability of the proposed existence and uniqueness results, the Ulam–Hyers stability of the solution is considered. In Section 5 an example is derived to demonstrate the new results for the proposed integral model of fractional variable order. Section 6 presents our concluding remarks and future directions.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that are used throughout this paper.

We consider the mapping . Then, the left Riemann–Liouville fractional integral (RLFI) of variable-order for a function h is defined as [29,30,31]

In the case when is a constant, then RLFI coincides with the classical Riemann–Liouville fractional integral of a constant order, see, e.g., [29,30,32].

Note that, the semigroup property does not hold for arbitrary functions

If

The variable order fractional integral

Let .

The interval J will be called a generalized interval if it is either

A partition of J is a finite set

A function

In the proof of our main results we will also use the following Schauder fixed point theorem.

Assume that E is a Banach space and

The Equation (2) is Ulam–Hyers stable if there exists there exists a solution

3. Main Existence and Uniqueness Results

We will prove our existence and uniqueness criteria under the following assumption: where are constants, and is the index of the interval , (with ) defined as

For there exists a partition of the interval J defined as and a piece-wise constant function with respect to such that

The symbol indicates the Banach space of all continuous functions with the norm where .

We will first analyze the equation defined in (2). For any , the RLFI of variable order for the function , defined by (3), could be presented as a sum of left Riemann–Liouville fractional integrals of constant-orders .

Thus, according to (5), for any , Equation (2) can be written as

Let the function be a solution of the integral Equation (6) such that on . Then, (6) is reduced to

Now, we will study Equation (7) assuming that for all the following assumptions are satisfied:

There exists such that for each and .

There exist non-negative constants and such that for each and .

The function is continuous on and nondecreasing with respect to its three variables, separately, and there exist constants , such that for all and .

There exist a constant and continuous nondecreasing functions and such that for each and we have

Our first existence result is based on Theorem 1.

Suppose that where

Then, Equation (7) has at least one solution in

Consider the operator defined by

Define the set Clearly, is nonempty, bounded, closed and convex subset of .

Now, we will demonstrate that S satisfies the assumptions of Theorem 1.

STEP 1: Claim:

For , we have which means that .

STEP 2: Claim: S is continuous.

We presume that the sequence converges to y in and . Then, i.e., we obtain

The above relation shows that the operator S is continuous on .

STEP 3: Claim: S is compact.

In order to show that S is compact it is enough to prove that is relatively compact. By Step 1, we have that is uniformly bounded or . Thus, for each we have which means that is bounded. It remains to indicate that is equicontinuous.

For and , we have

By , the function is uniformly continuous on . Then, we have uniformly in and . Hence, we have By using the continuity of and together with (9) we conclude that as . This implies that is equicontinuous.

Hence, all conditions of Theorem 1 are satisfied and thus, Equation (7) has at least one solution . Since , the assertion of Theorem 2 is proved. □

The Banach contraction principle will be used in the proof of the next result.

Let the conditions of Theorem 2 be satisfied, and the inequality holds. Then, Equation (

We will show that is a contraction operator.

Let , and , we have

Therefore,

Ergo, by (10), we conclude that the operator S composes a contraction. Therefore, by the Banach’s contraction principle, S has a unique fixed point in , which is the unique solution of Equation (7). This proves Theorem 2. □

The existence result for Equation (2) will be proved in the next theorem.

Suppose that assumptions (A1)–(A5) and inequalities (8), (10) are satisfied for all

Then, Equation (2) has a unique solution in

According to Theorem 3 the fractional integral equation of a constant order (7) has a unique solution for any . We construct the function defined for . It is clear that is a solution of the integral Equation (6) for Then, the function is the unique solution of Equation (2) in . □

Since the quadratic integral equations of the Urysohn type are widely used in theory and applications, the proposed generalization extends the opportunities for its application. In addition, the established existence and uniqueness results open the door for the study of the qualitative properties of these types of equations such as stability, periodicity, asymptotic behavior, etc.

One of the states that is of great importance to researchers of the Urysohn type and related models are the so called “steady” or equilibrium states. Another solution of interest to applied sciences is the periodic solution. The results proposed in this paper can be applied to such specific solutions of interest. Hence, in the case, when

For

Let the conditions of Theorem 4 be satisfied, for all

Let be an arbitrary number and the function from satisfy the inequality (4)

For , we define the functions and for :

According to Theorem 4, the integral Equation (2) has a unique solution defined by for , where and is a unique solution of the integral Equation (7)

Let . Then, we obtain

Then,

Thus, for each we obtain Therefore, the integral Equation (2) is Ulam–Hyers stable. □

In this example, we deal with the following quadratic integral equation of the Urysohn type of fractional variable order where Let

By (14), according to (7) we consider the following two auxiliary equations and

We will show that assumptions (A1)–(A5) and inequalities (8), (10) hold.

For Then, (A2) is satisfied with Then, assumption (A3) holds for Hence, assumption (A4) is satisfied with

We have that Then, (A5) holds with is satisfied for each . Hence, condition ( is satisfied for each

It follows from Theorem 3 that the Equation (15) has a unique solution

For Then, (A2) is satisfied with Then, (A3) holds with Hence, (A4) is satisfied with Then, (H5) holds with is satisfied for each is satisfied for each

Then, by Theorem 4, the Equation (13) has a unique solution where

In addition, according to Theorem 5, the integral Equation (13) is Ulam–Hyers stable.

The proposed example shows the feasibility of our fundamental results. Since the obtained criteria are in the form of algebraic inequalities, they can be easily applied.

Example 1 also demonstrated that the proposed existence and uniqueness results can be used in the study of the qualitative properties of the solutions of the introduced model of fractional variable order.

6. Conclusions

In this paper, we present results about the existence and uniqueness of solutions for a quadratic integral equation of the Urysohn type of fractional variable order , where is a piece-wise constant function. All our results are based on the Schauder’s fixed-point theorem and the Banach contraction principle. The theoretical findings are also illustrated by an numerical example. Since fractional integral operators of variable order are applied in different models of real-world phenomena, we expect that the proposed results will be of interest to numerous audiences of researchers in mathematics, engineering and applied sciences. Moreover, the outcome of this research paper will benefit the investigations on integral equations of the Urysohn type in other spaces such as Frechet space. Furthermore, one could study the proposed integral equation with different fractional integrals, such as Caputo type, Hadamard type, Hilfer type and some others. It is possible to extend the results to the uncertain case since a real system always involves uncertainties due to some disturbances in system, inaccuracy in model parameter measurements or noises from external inputs, and the analysis of models with uncertainties is essential for theory and applications. A future direction of our investigations is also related to the study of the impact of continuous and impulsive controllers on the qualitative behavior of the introduced model.