Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 498781, 21 pages

http://dx.doi.org/10.1155/2013/498781

Research Article

Existence Results for a Coupled System of Nonlinear Singular Fractional Differential Equations with Impulse Effects

Yuji Liu,¹ Juan J. Nieto,2,3 and Óscar Otero-Zarraquiños²

¹Department of Mathematics, Guangdong University of Business Studies, Guangzhou 510320, China

²Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

³Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 2 October 2012; Accepted 15 February 2013

Academic Editor: Jocelyn Sabatier

Copyright © 2013 Yuji Liu 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.

A boundary value problem for the singular fractional differential system with impulse effects is presented. By applying Schauder's fixed point theorem in a suitably Banach space, we obtain the existence of at least one solution for this problem. Two examples are presented to illustrate the main theorem.

1. Introduction

Fractional differential equations have received increasing attention during recent years since the behavior of many physical, chemical, and engineering processes can be properly described by using fractional differential equations theory; see the books [1–3], papers [4, 5] and references therein. For details on the geometric and physical interpretation of the derivatives of noninteger order, see, for example, [6–11]. For some recent works with applications to engineering we refer the reader to [12–15].

For an introduction of the basic theory of impulsive differential equation, we refer the reader to [16]. Among previous research, little is concerned with differential equations with fractional order with impulses [17]. Ahmad and Sivasundaram [18, 19] gave some existence results for two-point boundary value problems involving nonlinear impulsive hybrid differential equations of fractional order . Ahmad and Nieto in [20] establish sufficient conditions for the existence of solutions of the antiperiodic boundary value problem for impulsive differential equations with the Caputo derivative of order . Some recent results on impulsive initial value problems or boundary value problems for fractional differential equations on a finite interval can be found in [21–23] and references therein. The memory property of fractional calculus makes studies more complicated.

This paper is motivated by [24] in which the following boundary value problem for the fractional differential equation

was studied, where , , and , , , and are continuous functions, and is the Riemann-Liouville fractional derivative. An existence result was proved for BVP (1) in [24]. The growth assumptions imposed on and are sublinear cases (see [24, Theorem 2.1]); that is, there exist functions , nonnegative constants , and such that

In [25], the following boundary value problem for the fractional differential equation

was studied, where , , and , and are continuous functions, and is the Riemann-Liouville fractional derivative. The growth assumptions imposed on and are sublinear cases (see [25, Theorem 3.1]), that is, there exist functions , nonnegative constants , , and such that

or sublinear cases, that is, there exist nonnegative constants , and such that

We find that in the superlinear cases, BVP (3) has a pair of solutions without needing any other assumptions. Hence, these cases are trivial ones discussed in [25].

It is interesting to consider the solvability of BVP (1) when the growth assumptions imposed on are superlinear cases. Furthermore, the solvability of BVP (1) is not studied when or .

In this paper we consider the following nonlinear boundary value problem for the singular multiterm fractional differential equation with impulse effects whose boundary conditions are of integral form

where

(a), , and , is the Riemann-Liouville fractional derivative,(b) defined on ,(c) with , defined on ,(d),(e).

A pair of functions defined on is called a solution of BVP (1) and BVP (3), if , and , are continuous, there exists the limits

, and satisfies all equations in (6) and (7).

The novelty of this paper is as follows: first, the fractional differential equations in (6) are multiterm ones and their nonlinearities depend on the lower fractional derivatives; second, both and may be singular at and , that is, and may be not continuous functions on , the boundary conditions are integral boundary conditions, and we obtain the results on the existence of at least one solution of BVP (6)-(7); third, and are supposed; the growth assumptions imposed on and are allowed to be sublinear cases. Finally, two examples are given to illustrate the efficiency of the main theorem.

The remainder of this paper is as follows: in Section 2, we present preliminary results. In Section 3, the main theorem and its proof are given. In Section 4, two examples are given to illustrate the main results.

2. Preliminaries

In this section, we present some background definitions and preliminary results.

Definition 1 (see [1]). The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side exists.

Definition 2 (see [1]). The Riemann-Liouville fractional derivative of order of a continuous function is given by

where , provided that the right-hand side is pointwise defined on .

Definition 3. is called a -Caratheodory function if satisfies that

(i) is continuous on for every ;(ii) is continuous on for every ;(iii)for each there exists a constant such that , , .

Definition 4. is called a -Caratheodory function if satisfies that

(i) is continuous on for every ;(ii) is continuous on for every ;(iii)for each there exists a constant such that , , .

Lemma 5 (the Leray-Schauder nonlinear alternative [23]). Let be a Banach space and be a completely continuous operator. Suppose is a nonempty open subset of centered at zero. Then either there exists and such that or there exists such that .

Let the gamma and beta functions and be defined by

Choose

For , define the norm by

It is easy to show that is a real Banach space. For , define the norm by

It is easy to show that is a real Banach space. Thus, is a Banach space with the norm defined by .

In this paper, we suppose the following:

(A)  satisfies that there exist constants , , such that , , and for all ; satisfies that there exist constants , , such that , , and for all .(B)  are -Caratheodory functions and are -Caratheodory functions.

Remark 6. Suppose that is a -Caratheodory function. For example, , , choose , and , then , such that , , and for all . It is easy to see that is singular at and .

Lemma 7. Suppose that , and (a)–(e), (A)-(B) hold. Then is a solution of

if and only if satisfies the integral equation

where

Proof. If is a solution of BVP (15), then

and satisfies all equations in (31) From (B), is a -Caratheodory function, then there exists such that

Similarly we get that there exist constants such that

It follows from (15) that, for , there exist constants such that

From , we get

From , we get

From , we get

From , we get

It follows that

Then

So

Hence, for , we have

And for , we have

Hence, satisfies (16).

On the other hand, if and is a solution of (16), then we can prove that is a solution of BVP (6)-(7). The proof is completed.

Lemma 8. Suppose that , and (a)–(e), (A)-(B) hold. Then is a solution of

if and only if satisfies the integral equation

where

Proof. The proof is similar to that of the proof of Lemma 7 and is omitted.

Now, we define the operator on by with

Remark 9. By Lemmas 7 and 8, is a solution of BVP (6)-(7) if and only if is a fixed point of the operator .

Lemma 10. Suppose that (a)–(e) and (A)-(B) hold. Then is well defined and is completely continuous.

Proof. The proof is very long, so we list the steps. First, we prove that is well defined; second, we prove that is continuous, and, finally, we prove that is compact. So is completely continuous. Thus, the proof is divided into three steps.

Step  1. Prove that is well defined.

For , we have . Then

From (B), are -Caratheodory functions, then there exist constants such that

Hence,

From (34), (37), and (38), we see that is defined on , continuous on and , respectively. One sees that

and there exits the limit .

On the other hand, we have

It is easy to see that

From (37) and (41), we see that is defined on , continuous on and , respectively. One sees that

and there exits the limit .

From the above discussion, we have . Similarly, we can show that . Hence, . Then is well defined.

Step  2. We prove that is continuous. Let with as . We will show that as , that is, prove that and as .

In fact, we have such that . Then

From (B), are -Caratheodory functions, then there exist constants such that

as . We have

From the Lebesgue dominated convergence theorem, we get

as . Similarly, we can show that

as . It follows from (46) and (47) that is continuous.

Step  3. We prove that is compact, that is, for each nonempty open bounded subset of , prove that is relatively compact. We must prove that is uniformly bounded, equicontinuous on each subinterval , is equiconvergent as , and equiconvergent as .

Let be a bounded open subset of . We have such that

From (B), are -Caratheodory functions, then there exist constants such that

Substep  3.1. Prove that is uniformly bounded.

In fact, for , use (49), we have

Similarly, we can get for that

Furthermore, we have for that

and for that

Hence,

Similarly, we can show that

It is easy to see that is uniformly bounded.

Substep  3.2. Prove that is equicontinuous on each subinterval .

For each , and with , we have

Note that for all and .

Since

It follows that

For , and with , we can prove similarly that

On the other hand, for , and with , we have

It is easy to see that

For the third term, if , use , then

If , use , then

For , and with , we can prove similarly that

Similarly, we can show that for each , and with , we have

For , and with , we can prove similarly that

For each , and with , we have

For , and with , we can prove similarly that

So is equicontinuous on each subinterval .

Substep  3.3. Prove that is equiconvergent as , and equiconvergent as .

We have

It follows that

Similarly, we can show that is equiconvergent at . On the other hand, we have

It follows that

Similarly, we can show that is equiconvergent at .

Similarly we can prove that

and is equiconvergent at , both and are equiconvergent at .

Hence, is equiconvergent as and is equiconvergent as .

So is relatively compact. Then is completely continuous. The proofs are completed.

3. Main Result

In this section, we will establish the existence of at least one solution of BVP (6)-(7).

Definition 11 (see [26]). An odd homeomorphism of the real line onto itself is called a pseudo-sub-multiplicative function if there exists a homeomorphism of onto itself which supports in the sense that for all we have .   is called the supporting function of .

Remark 12. Note that any submultiplicative function is a pseudo-submultiplicative function. Also any function of the form is pseudo-sup-multiplicative, provided that . Here, a supporting function is defined by , .

Remark 13. It is clear that a pseudo-submultiplicative function and any corresponding supporting function are increasing functions vanishing at zero; moreover, their inverses and , respectively, are increasing and for all , we have .

Theorem 14. Suppose that (a)–(e) and (A)-(B) hold, is a submultiplicative-like function with the supporting function , its inverse function is denoted by with the supporting function . Furthermore, suppose that

(i)there exist nonnegative numbers , and such that

holds for all , .(ii)there exist nonnegative numbers , and such that

hold for all , .(iii)there exist the nonnegative numbers , and such that

hold for all .(iv)there exist the nonnegative numbers , and such that

hold for all .

Then BVP (6)-(7) has at least one solution if

where

Proof. To apply Lemma 5, we should define an open bounded subset of centered at zero such that assumptions in Lemma 5 hold.

Let for some . We prove that is bounded. For , we get . It follows that and .

For , we obtain .

For ,

It follows that

Similarly, we have for that

and for

It follows that

Hence,

Similar to the above discussion we can prove that

Case  1. Consider (,   , .

With out loss of generality, suppose that

Then use Remark 13, and the previous inequalities to get

It follows that there exists a constant such that . Thus

Then

It follows that is bounded.

Case  2. Consider ,,   .

Without loss of generality, suppose that

Then using Remark 12 and the previous inequalities, we get

It follows that there exists a constant such that . We get

Then

It follows that is bounded.

To apply Lemma 5, let be a nonempty open bounded subset of such that centered at zero.

It is easy to see from Lemma 8 that is a completely continuous operator. One can see that

Thus, from Lemma 5, has at least one solution . So is a pair of solutions of BVP (3) and BVP (6). The proof of Theorem 14 is complete.

4. Two Examples

To illustrate the usefulness of our main result, we present two examples that Theorem 14 can readily apply.

Example 15. Consider the following impulsive boundary value problem:

where are constants.

Corresponding to BVP (1), we have

(a), , ,(b), and + defined on ,(c), , , , = ,(d),(e), , , .

It is easy to show that

(A)  satisfies , , and for all with and ;  satisfies , , and for all with and ;(B)  are -Caratheodory functions and are -Caratheodory functions.

Furthermore, we have and with and . It is easy to see that

(i)the inequalities

hold for all , with , and ;(ii)the inequalities

hold for all , with , , , ;(iii)the inequalities

hold for all with , , , ;(iv)the inequalities

hold for all with , , , .

By direct computation, we know that

Then Theorem 14 implies that the existence of at least one solution if

Example 16. Consider the following boundary value problem without impulse effects:

where , , , , , and are constants.

Corresponding to BVP (1), we have

(a), , and ,(b), , defined on , + and ,(c), ,(d)there exists no impulse point,(e).

It is easy to show that

(A)  satisfies , , for all with , ;  satisfies , , and for all with , ;(B)  are -Caratheodory functions and are -Caratheodory functions.

Furthermore, and , we have and , and

(i)the inequalities

hold for all , with , , and ;(ii)the inequalities

hold for all , with , , and ;(iii)the inequalities

hold for all with ;(iv)there exist the nonnegative numbers , such that

hold for all with .

By direct computation, we know that

Then Theorem 14 implies the existence of at least one solution if

Remark 17. It is easy to see that the previous boundary value problems have at least one solution for sufficiently small and , and . They cannot be solved by the theorems in [24, 25].

Acknowledgments

This research is partially supported by the Natural Science Foundation of Guangdong province (no. S2011010001900) and the Guangdong Higher Education Foundation for High-Level Talents. This research is partially supported by Ministerio de Economía y Competitividad and EC fund FEDER, Project no. MTM2010-15314, Spain.

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