Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 945347, 7 pages
http://dx.doi.org/10.1155/2015/945347

Research Article

On the Asymptotic Behavior of Positive Solutions of Certain Fractional Differential Equations

Said R. Grace

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12613, Egypt

Received 8 July 2015; Accepted 25 August 2015

Academic Editor: Zhen-Lai Han

Copyright © 2015 Said R. Grace. 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 deals with the asymptotic behavior of positive solutions of certain forced fractional differential equations of the form , , , where , , and is a real constant. From the obtained results, we derive a technique which can be applied to some related fractional differential equations.

1. Introduction

Consider the forced fractional differential equation

where , , and is a real constant.

In the sequel we assume that

(i) is a continuous function;(ii) is continuous and assume that there exists a continuous function;  and a real number, , and is a real number such that

We only consider those solutions of (1) that are continuable and nontrivial in any neighborhood of . Such solution is said to be oscillatory if there exists a sequence , such that , and it is nonoscillatory otherwise.

In the last few decades, integral, integrodifferential, and fractional differential equations have gained considerably more attention due to their applications in many engineering and scientific disciplines as the mathematical models for systems and processes in fields of physics, mechanics, chemistry, aerodynamics, and the electrodynamics of complex media. For more details one can refer to [1–8].

Oscillation and asymptotic behavior results for integral, integrodifferential, and fractional differential equations are scarce; some results can be found in [5, 9–13]. It seems that there are no such results for forced fractional differential equations of type (1). The main objective of this paper is to establish some new criteria on the asymptotic behavior of positive solutions of (1) which is equivalent to the Volterra type equation:

From the obtained results, we derive a technique which can be applied to some related fractional differential equations.

We note that

is the Caputo derivative of the order of a -scalar valued function defined on the interval , . For the case when , this definition has been given by Caputo [14]. For the definition of the Caputo derivative of order , , see [1, 15, 16].

2. Main Results

To obtain our main results of this paper, we need the following two lemmas.

Lemma 1 (see [5, 7]). Let , , and be positive constants such that , . Then

where , , , and .

Lemma 2 (see [17]). If and are nonnegative, then

where equality hold if and only if .

In what follows, we let

and , for some , where is a continuous function.

Now we give sufficient conditions under which any solution of (1) satisfies as .

Theorem 3. Let and suppose that , , and , , , and is bounded on ,

where the function is defined as in (7) for any . If is a positive solution of (1), then .

Proof. Let be an eventually positive solution of (1). We may assume that for for some We let . In view of (i) and (ii) we may then write

Applying (6) of Lemma 2 to with

we have

and hence we obtain

Integrating inequality (15) from to and interchanging the order of integration, one can easily obtain

Interchanging the order of integration in second integral, we have

where is the upper bound of the function .
Integrating (17) from to and interchanging the order of integration in the last integral, we find

Now, one can easily see that

where . From the hypotheses of the theorem, we see that

with as the upper bound of the function

Applying Holder’s inequality and Lemma 1, we obtain

where and and so

Thus, inequality (20) becomes

Using this inequality and the elementary inequality

we obtain from (24)

If we denote , that is, , , and , then

The conclusions follow from Gromwell’s inequality and conclude that

This completes the proof.

Remark 4. Condition (10) can be replaced by

and the result remains valid.

The following example is illustrative.

Example 5. Let , , , and . Clearly

Let the functions and be as in (i) and (ii) with being a bounded function and let , , and , where and is a continuous function with ,

Condition (10) is also fulfilled. Thus, all conditions of Theorem 3 are satisfied and hence every positive solution of (1) satisfies .
Next, we consider the fractional differential equation

where , , and is a real constant.
Now we give sufficient conditions under which any positive solution of (32) satisfies .

Theorem 6. Let and suppose that , , and , , and ,

for any . If is a positive solution of (32), then .

Proof. Let be an eventually positive solution of (32). We may assume that for for some We let . In view of (ii) we may then write

Proceeding as in the proof of Theorem 3, we obtain

Integrating inequality (37) from to and interchanging the order of integration, one can easily obtain

Interchanging the order of integration in second integral we have

The rest of the proof is similar to that of Theorem 3 and hence is omitted.

Example 7. Let , , , and . Clearly

Let the functions and , where and is a continuous function with ,

Condition (34) is also fulfilled. Thus, all conditions of Theorem 8 are satisfied and hence every positive solution of (32) satisfies .
Finally, we consider the fractional differential equation

where and is a real constant.
Now, we give sufficient conditions for the boundedness of any positive solution of (42).

Theorem 8. Let and suppose that , , and , , and ,

for any . If is a positive solution of (42), then is bounded.

Proof. Let be an eventually positive solution of (42). We may assume that for for some We let . In view of (ii) we may then write

Proceeding as in the proof of Theorem 3, we obtain

or

The rest of the proof is similar to that of Theorem 3 and hence is omitted.

Example 9. Let , , , and . Clearly

Let the functions and , where and is a continuous function with ,

Condition (34) is also fulfilled. Thus, all conditions of Theorem 8 are satisfied and hence every positive solution of (42) is bounded.
Similar reasoning to that in the sublinear case guarantees the following theorems for the integrodifferential equation (1) when

Theorem 10. Let and the hypotheses of Theorems 3–8 hold with .

Then the conclusion of Theorems 3–8 holds.

General Remarks

(1)The results of this paper are presented in a form which is essentially new and it can also be obtained for higher fractional differential equations (1) of order , .(2)It would be of interest to study (1) when satisfies condition (ii) with

Conflict of Interests

The author declares that they have no competing interests.

Acknowledgments

The author is extremely grateful to the reviewers and the handling editor for many helpful comments and suggestions, which have contributed much to the improvement of this paper.

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