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Advances in Mathematical Physics
Volume 2015 (2015), Article ID 174156, 9 pages
http://dx.doi.org/10.1155/2015/174156
Research Article

On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative

1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2School of Mathematics, South China University of Technology, Guangzhou 510641, China

Received 3 August 2015; Accepted 27 October 2015

Academic Editor: Ivan Avramidi

Copyright © 2015 Hailong Ye and Rui Huang. 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

The purpose of this paper is to investigate the existence of solutions to the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivative , , , , where , are Caputo fractional derivatives, , , , and . Local existence of solutions is established by employing Schauder fixed point theorem. Then a growth condition imposed to guarantees not only the global existence of solutions on the interval , but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to . Three illustrative examples are also presented. Existence results for initial value problems of ordinary differential equations with -Laplacian on the half-axis follow as a special case of our results.

1. Introduction

This paper deals with the following initial value problem for nonlinear fractional differential equation involving Caputo sequential fractional derivative:where , are Caputo fractional derivatives, , , , and , and is continuous on , . When , the equation in (1) becomes a sequential fractional differential equation. Here, we follow the definition of sequential fractional derivative presented by Podlubny [1]:where the symbol () means the Riemann-Liouville derivative or the Caputo derivative. It is easy to see that (2) is a generalized expression presented by Miller and Ross in [2].

Fractional differential equations have been of great interest for the past three decades; see the monographs [14] and the papers of [58]. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of pure mathematics, fractional differential equations can be used in modeling of various fields of science and engineering such as rheology, self-similar dynamical processes, porous media, fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many other branches of science. For details, see [912] and the references therein.

Recently, we note that the investigation for the existence of solutions of sequential fractional differential equations associated with a variety of initial and boundary value conditions has attracted the considerable attention of researchers. Here, we mention some works on them. In [13, 14], the authors investigated a class of Riemann-Liouville sequential fractional differential equation and obtained existence, nonexistence, and asymptotic property of the solutions by some nonlinear analysis methods. In [15, 16], the authors considered the existence and uniqueness of the solutions for initial value problems involving Riemann-Liouville sequential fractional derivative by using monotone iterative method and fixed point method. As for sequential fractional differential equations associated with boundary value conditions, we refer the reader to [1722]. For example, Chen and Tang [21] considered the existence of the solutions for the following Caputo sequential fractional differential equation: with -point boundary value conditions by the coincidence degree continuation theorem. After that, in 2015, Jiang [22] investigated the following Riemann-Liouville sequential fractional differential equations with -Laplacian at resonance: where , . By the extension of the continuous theorem in [23] and constructing suitable operators, they obtained the existence of solutions satisfied integral boundary value conditions.

In view of the facts that the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives and that the Caputo derivative of a constant is 0, sequential fractional differential equations involving the Caputo fractional derivative have more clear physical interpretations than those involving the Riemann-Liouville fractional derivative (see [3, 4]).

To the best of our knowledge, there is no paper dealing with the existence of solutions of Caputo sequential fractional differential equations with initial value conditions. In our latest paper [24], by virtue of uniform Lipschitz continuity of on for every , we proved the existence and uniqueness of solutions of problem (1). Now, in this paper, we are concerned with the initial value problem (1) without uniform Lipschitz continuity of . By fractional Taylor expansion theorem, we first obtain an integral equation equivalent to the initial value problem (1), to which local existence of solutions is established utilizing Schauder fixed point theorem. Then a growth condition imposed to guarantees not only the global existence of solutions on the interval , but also the fact that the intervals of existence of solutions with any fixed initial value can be extended to . In addition, existence results for initial value problems of ordinary differential equations with -Laplacian on the half-axis follow as a special case of our results.

The paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used in our discussions. The main results and their proofs are given in Section 3. In Section 4, we will give three examples to illustrate our results.

2. Preliminaries

In this section, we introduce some basic definitions and notations (see the monographs [1, 2] for further details) and give several useful preliminary results which are used throughout this paper.

Definition 1. Let . The Riemann-Liouville fractional integral of a function of order is given by provided that the right-hand side is pointwise defined on . Here and in what follows is the Gamma function.

The fractional integration operator has that the following semigroup property holds for all , .

Definition 2. Let and let be the smallest integer that exceeds . The Riemann-Liouville fractional derivative of a continuous function of order is given by provided that the right-hand side is pointwise defined on .

Obviously, the Riemann-Liouville fractional differentiation is the left inverse of the Riemann-Liouville fractional integration for continuous function in the following sense: for . However, it is not the right inverse. More precisely, we have the following fractional Taylor expansion theorem.

Theorem 3. Let . Assume that is such that is absolutely continuous. Then

Definition 4. Let and let be the smallest integer that exceeds . The Caputo fractional derivative of a continuous function of order is given by provided that the right-hand side is pointwise defined on .

The definition of solutions of the initial value problem (1) is given as follows.

Definition 5. Let ; a function is called a solution of (1) on , if (i),  , ;(ii) satisfies problem (1) on .

Definition 6. A function is called a solution of (1) on , if for any , is a solution of problem (1) on .

In order to study the existence of solutions of (1), we should transform problem (1) into an equivalent integral equation. We need the following two lemmas.

Lemma 7. Let ; then one has

Lemma 8. Let . If is a continuous function defined on , then is continuous with respect to in .

Proof. Let and take . For , we haveSince is continuous and bounded in the neighborhood of , we conclude thatwhere . Combining (13), (14) with (15), we arrive at In addition, it is easy to see that Therefore, is continuous with respect to in .

Remark 9. If , then is continuous on . According to Lemma 8, the function is also continuous in and .

Now we are ready to transform problem (1) into an equivalent integral equation. For the reader’s convenience, we list two special notations that will be used in the following paper: and for .

Proposition 10. A function defined in is a solution of problem (1) if and only if it satisfies the following integral equation on :where

Proof. First we prove the necessity. Let be a solution of problem (1) and define and then and . According to Definition 4 and Theorem 3, the differential equation of problem (1) can be transformed into the following form: Obviously, is absolutely continuous on . Combining with Theorem 3, we have That is, Applying Definition 4 and Theorem 3 again, we have Obviously, . Combining with Theorem 3, we have Therefore, satisfies the integral equation (19).
Next, we prove the sufficiency. Let be a solution of the integral equation (19). Combining with Definition 1, (19) reduces toFrom Remark 9, we see that and . That is, and . Applying the operator to both sides of (27), we obtain that Then we have and . By virtue of , we transform the above equation into the following form:Similarly, applying the operator to both sides of (29), we arrive at Therefore, is a solution of (1) on . Summing up, we complete the proof of Proposition 10.

Since can be chosen arbitrarily large in Proposition 10, according to Definition 6, we have the following result.

Corollary 11. Let ; then is a solution of problem (1) if and only if it satisfies the integral equation (19) on .

By Corollary 11, we obtain the following two existence results.

Remark 12. If , and for , then the constant function is a solution of problem (1).

Remark 13. If for , then problem (1) has a solution on .

The following fixed point lemma is the main tool in the proofs of our results.

Lemma 14 (Schauder fixed point theorem). Let be a closed, convex, and nonempty subset of a Banach space , and let be a mapping such that is a relatively compact subset in . Then has at least one fixed point in .

3. Main Results

In this section, we will give and prove our main results in this paper.

Theorem 15. For any fixed initial values and , there exists a sufficiently small constant such that problem (1) has a solution on .

Proof. For any given positive constant , choose sufficiently small which will be determined later. Let Obviously, is a closed, convex, and nonempty subset of . On this set we define the operator : where According to Proposition 10, in what follows, it suffices to show that the operator has a fixed point in .
Firstly, we will show that for any . To this end we begin by noting that for any , by Remark 9. Then we obtain that . Furthermore, for we have where Now we can choose so small thatwhich means that ; that is, maps the set to itself.
Secondly, we will also show that the family of functions is a relatively compact set. That is to say, we need to show that is uniformly bounded and equicontinuous on . The uniform boundedness follows from the definition of . As for the equicontinuity, for and , we see that where is independent of , , and . Therefore is uniformly bounded and equicontinuous on , and thus is a relatively compact subset of . By Lemma 14 there exists such that and is a solution of problem (1) on . The proof is completed.

We have proved the local existence of solutions of problem (1) in Theorem 15. However, maximal intervals of existence of those solutions are not necessarily , which change according to initial values and . We give an example to illustrate it.

Example 16. Consider the following initial value problem for second-order ordinary differential equationwhere is a constant. When , it is easy to see that the function is a solution of (38) on and is the maximum interval of existence. When , however, the function is a solution of (38) on and is the maximum interval of existence.

Under certain growth condition on the nonlinearity in (1), we will show that any solutions obtained in Theorem 15 can be extended to the interval as solutions of problem (1).

Theorem 17. Suppose that there exist constant and two nonnegative continuous functions , defined on such thatThen maximum intervals of existence of any solutions of (1) obtained in Theorem 15 are .

Proof. Let be a solution of (1) on where . By Definition 6, it suffices to prove that for any fixed , there exists a function as a solution of (1) on such that for . The proof will be completed by applying Lemma 14. Choose which will be determined later and denote Obviously, is a closed, convex, and nonempty subset of . On this set we define the operator : where It is easy to see that for any . Further, by virtue of (40) we have where Since , that is, , we can choose so large that Then maps the set to itself. To apply Lemma 14, we also need to show that the family of functions is a relatively compact set. That is to say, we need to show that is uniformly bounded and equicontinuous on . The uniform boundedness follows from the definition of . As for the equicontinuity, we see that, for and , where is independent of , , and . Therefore is uniformly bounded and equicontinuous on , and thus is a relatively compact subset in . By Lemma 14 there exists such that . Noting that , we define From the definition of , we see that is a solution of (1) on , which completes the proof.

In a special case, an estimate of increasing rate for solutions as is made.

Corollary 18. Suppose that is bounded on . Let be a solution of (1) on ; then   .

Proof. According to Corollary 11, we have Note that and we obtain where is independent of and . Since , we conclude that

4. Example

To illustrate our main results, we will present three examples.

Example 19. Consider the initial value problem for sequential fractional differential equationwhere . Here , , , and . It is easy to see that for . According to Theorem 17, problem (53) has at least one positive solution on for any fixed . Let be a positive solution of (53) on . Since is bounded on when , we further have an estimate of increasing rate for by Corollary 18:

Example 20. Consider the initial value problem for nonlinear fractional differential equationwhere . Here , , and . Choosing , we see that for , where . According to Theorem 17, problem (56) has at least one solution on . Let be a solution of (56) on . By simple computation similar to that in the proof of Corollary 18, we further have an estimate of increasing rate for :

Example 21. Consider the initial value problems for ordinary differential equations with -Laplacianwhere . Here and . Choosing , we see that for . According to Theorem 17, problem (59) has at least one solution on . Let be a solution of (59) on . By simple computation similar to that in the proof of Corollary 18, we further have an estimate of increasing rate for :

Conflict of Interests

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

Acknowledgments

The second author was supported in part by National Natural Science Foundation of China (no. 11071099), CPSF (no. 2015M572301), and the Fundamental Research Funds for the Central Universities (no. 2015ZM191).

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