International Journal of Differential Equations
Volume 2016 (2016), Article ID 7327319, 8 pages
http://dx.doi.org/10.1155/2016/7327319

Research Article

On Some Existence and Uniqueness Results for a Class of Equations of Order on Arbitrary Time Scales

Abdourazek Souahi,¹ Assia Guezane-Lakoud,² and Rabah Khaldi²

¹Laboratory of Applied Mathematics and Modeling, University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria
²Laboratory of Advanced Materials, University of Badji Mokhtar-Annaba, P.O. Box 12, 23000 Annaba, Algeria

Received 21 April 2016; Accepted 9 June 2016

Academic Editor: Patricia J. Y. Wong

Copyright © 2016 Abdourazek Souahi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper investigates the existence and uniqueness of solution for a class of nonlinear fractional differential equations of fractional order in arbitrary time scales. The results are established using extensions of Krasnoselskii-Krein, Rogers, and Kooi conditions.

1. Introduction

This work concerns the investigation of sufficient conditions for the existence and uniqueness of the solution of the following initial value problem with fractional derivative up to the first order on arbitrary time scales:

where is the (left) Riemann-Liouville fractional derivative of order on time scales , is the Riemann-Liouville fractional integral on time scales, and is an interval on . We assume that is a right-dense continuous function.

The theory of time scales calculus allows us to study the dynamic equations, which include both difference and differential equations, both of which are very important in implementing applications; for further information about the theoretical and potential applications of the theory of time scales, we refer the reader to [1–8] and the survey [9].

The quantitative behaviour of solutions to ordinary differential equations on time scales is currently undergoing active investigations. Many authors studied the existence and the uniqueness of the solutions of initial and boundary differential equations; see [8, 10–20] and the references cited therein. In the papers [21–25], several authors were interested by the existence and uniqueness of the first-order differential equations on time scales with initial or boundary conditions using diverse techniques and conditions. On the other hand, some existence results for the fractional order differential equations were obtained in [10].

Our ideas arise from the papers [26–34], especially [30, 31], where the authors used Nagumo and Krasnoselskii-Krein conditions on the nonlinear term , without satisfying Lipschitz assumption. Motivated greatly by the above works, under appropriate time scales versions of the Krasnoselskii-Krein conditions, we obtain the uniqueness and existence of solution for the following two classes of differential equations, namely, the first-order ODE

and the fractional order FDE:

The rest of the paper is organized as follows. In Section 2, we give some definitions and lemmas that will be used in our work. Section 3 is devoted to the main results; we first establish the uniqueness of the solution under Krasnoselskii-Krein conditions for the first-order problem; then we establish the convergence of the successive approximations to the unique solution. Later, we prove the uniqueness for the fractional order problem under some other conditions.

2. Preliminaries

In this section, we recall basic results and definitions in time scales calculus.

A time scale is a nonempty closed subset of . We assume that . The forward and backward jump operators , are, respectively, defined by

The point is left-dense, left-scattered, right-dense, and right scattered if , , , and , respectively.

We set whenever admits a left-scattered maximum, and otherwise. We denote . An interval of is defined by , where is an interval of .

Definition 1 (delta derivative [1]). Assume and let . We define

provided the limit exists. We call the delta derivative (or Hilger derivative) of at . Moreover, we say that is delta differentiable on provided exists for all . The function is called the (delta) derivative of on .

Definition 2 (see [10]). A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous function is denoted by . Similarly, a function is called ld-continuous provided it is continuous at left-dense points in and its right-sided limits exist (finite) at right-dense points in . The set of ld-continuous function is denoted by . For define . It is easy to see that is a Banach space with this norm.

Definition 3 (delta antiderivative [10]). A function is called a delta antiderivative of a function provided is continuous on , delta differentiable on , and for all . Then, we define the -integral of from to by

Lemma 4. Let be an increasing continuous function on the . We define the extension of to the real interval by

Then

Lemma 5. Let be continuous. Then the general solution of the differential equation

is given by

Proof. Lemma 5 is an immediate consequence of Theorem  4.1 [5].

Definition 6 (fractional integral on time scales [10]). Suppose is a time scale, is an interval of , and is an integrable function on . Let . Then the (left) fractional integral of order of is defined by

where is the gamma function.

Definition 7 (fractional Riemann-Liouville derivative on time scales [10]). Let be a time scale, , , and . Then the (left) Riemann-Liouville fractional derivative of order of is defined by

For the sake of simplicity, we use the following notation and instead of and , respectively, whenever .

Lemma 8 (see [10]). For any function integrable on one has the following:

Lemma 9 (see [10]). Let and . If , then

Lemma 10 (see [10]). Let and . The function is a solution of problem (2) if and only if it is a solution of the following integral equation:

Lemma 11 (see [31]). The solution of the equation

is given by

where and and is the fractional Riemann-Liouville derivative of order on the interval ; see [35].

3. Main Results

In the following, we denote , .

3.1. Uniqueness Results for First-Order ODE

Theorem 12 (Krasnoselskii-Krein conditions). Let be continuous in and for all satisfying

( H1), ,(H2), where and are positive constants; the real number is such that , and .

Then, the first-order initial value problem (2) has at most one solution on .

Proof. Suppose and are two solutions of (2) in . We will show that . Let us define and by

such that is the extension of to the real interval . It follows from condition (H2) that

On the other hand, since , for , and , for every we deduce from (18) and (19) that

Multiplying both sides of this inequality by and then integrating the resulting inequality, we obtain

It immediately follows that

Moreover, if we define , we get

It follows that the exponent of in the above inequality is positive, since . Hence, . Therefore, if we define , then the function is rd-continuous in .
Now, to prove that , we prove by absurdity that on . Assume that does not vanish at some points ; that is, on ; then there exists a maximum reached when is equal to some : such that , for . But from condition (H1), we have

which is a contradiction. Thus, the uniqueness of the solution is established.

Theorem 13 (Kooi’s conditions). Let be continuous in and satisfying for all

(I1), ,(I2), where and are positive constants; the real numbers , are such that , and .

Then, the first-order initial value problem (2) has at most one solution on .

Proof. The proof is similar to that of Theorem 12; thus we omit it.

3.2. Existence of the Solution under Krasnoselskii-Krein Conditions on Time Scales

Theorem 14. Assume that conditions (H1) and (H2) are satisfied; then the successive approximations given by

converge uniformly to the unique solution of (2) on , where , and is the bound for on .

Proof. With the uniqueness of the solution being proved in Theorem 12, we prove the existence of the solution using Arzela-Ascoli Theorem.
Step 1. The successive approximations , given by (25) are well defined and continuous. Indeed,

This yields for

By induction, the sequence is well defined and uniformly bounded on .
Step 2. We prove that is a continuous function in , where is defined by

For , we have

In fact,

The right-hand side in inequality (29) is at most for large if provided that . Since is arbitrary and , can be interchangeable, we get

This implies that is continuous on . Using condition (H2) and the definition of successive approximations, we obtain

The sequence is equicontinuous: that is, for each function and any , if there exists such that , then

All of the Arzela-Ascoli Theorem conditions are fulfilled for the family in . Hence, there exists a subsequence converging uniformly on as .
Let us note

Further, if as , then the limit of any subsequence is the unique solution of (25). It follows that a selection of subsequences is unnecessary and that the entire sequence converges uniformly to . For that, it suffices to show that which will lead to being null.
Setting

and by defining , we show that .
We prove by absurdity that . Assume that at any point in ; then there exists such that . Hence, from condition (H1), we obtain

We end up with a contradiction. So . Therefore, the Picard iterates (25) converge uniformly to the unique solution of (2) on .

3.3. Uniqueness Results for Fractional Order ODE

In this section, we denote and .

Theorem 15 (Krasnoselskii-Krein conditions). Let be continuous in and satisfying for all

(J1), ,(J2), where , , are positive constants such that , , and , and the real number is such that .

Then, the fractional order initial value problem (3) has at most one solution on .

Proof. Suppose and are two solutions of (3) in . We will show that . Let us define and by

such that is the extension of to the real interval . It follows from condition (J2) that

On the other hand, , for , and , for every . Now from relations (37) and (38) and using Lemma 11, we obtain for every

where and are defined as in Lemma 11. Moreover, if we define , we get

It follows that the exponent of in the above inequality is positive, since . Hence, . Therefore, if we define , then the function is rd-continuous in .
Now, to show that , we prove by absurdity that on . Assume that does not vanish at some points ; that is, on ; then there exists a maximum reached when is equal to some such that , for . But from condition (J1), we have

which is a contradiction. Thus, the uniqueness of the solution is established.

Theorem 16 (Kooi’s conditions). Let be continuous in and satisfying for all

(K1), ,(K2), where , , and are positive constants; the positive real numbers , , , are such that , and , and .

Then, the first-order initial value problem (3) has at most one solution on .

Proof. The proof is similar to that of Theorem 15; thus, we omit it.

3.4. Existence of Solutions under Krasnoselskii-Krein Conditions on Time Scales

Theorem 17. Assume that conditions (J1) and (J2) are satisfied; then the successive approximations given by

converge uniformly to the unique solution of (3) on , where

and is the bound for on .

Proof. With the uniqueness of the solution being proved in Theorem 15, we prove the existence of the solution using Arzela-Ascoli Theorem.
Step 1. The successive approximations , given by (42) are well defined and continuous. Indeed,

This yields, for ,

By induction, the sequence is well defined and uniformly bounded on .
Step 2. We prove that is a continuous function in , where is defined by

For , we have

In fact,

The right-hand side in inequality (47) is at most for large if provided that . Since is arbitrary and , can be interchangeable, we get

This implies that is continuous on . Using condition (J2) and the definition of successive approximations, we obtain

The sequence is equicontinuous: that is, for each function and any , if there exists such that ; then

where we used a similar argument as in (48).
All of the Arzela-Ascoli Theorem conditions are fulfilled for the family in . Hence, there exists a subsequence converging uniformly on as .
Let us note

Further, if as , then the limit of any subsequence is the unique solution of (42). It follows that a selection of subsequences is unnecessary and that the entire sequence converges uniformly to . For that, it is sufficient to show that which will lead to being null.
Setting

and defining and then using Lemma 11, we obtain that . Which yields that .
We prove by absurdity that . Assume that at any point in ; then there exists such that . Hence, from condition (J1), we obtain

We end up with a contradiction. So . Therefore, the Picard iterates (42) converge uniformly to the unique solution of (2) on .

Remark 18. For the case , Theorem 15 is reduced to [31, Theorem  2.1].

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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