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Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance
Advances in Difference Equations volume 2013, Article number: 312 (2013)
In this paper, by using the extension of Mawhin’s continuation theorem due to Ge, we study the existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. A new result on the existence of solutions for a fractional boundary value problem is obtained.
In the recent years, fractional differential equations have played an important role in many fields such as physics, electrical circuits, biology, control theory, etc. (see [1–9]). Recently, many scholars have paid more attention to boundary value problems for fractional differential equations (see [10–25]).
In , by means of a fixed point theorem on a cone, Agarwal et al. considered a two-point boundary value problem at nonresonance given by
where , are real numbers, and is the Riemann-Liouville fractional derivative.
By using the coincidence degree theory, Bai (see ) considered m-point fractional boundary value problems at resonance in the form
where is a real number, , are given constants such that , and , are the Riemann-Liouville differentiation and integration.
In , relying on Schauder’s fixed point theorem, Ahmad et al. considered a three-point boundary value problem for a coupled system of nonlinear fractional differential equations at nonresonance given by
where , , , , , , , D is the standard Riemann-Liouville differentiation and are given continuous functions.
In , by using the coincidence degree theory due to Mawhin, Jiang discussed the existence of solutions to a coupled system of fractional differential equations at resonance
where , , , , , .
The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see ) introduced the p-Laplacian equation as follows:
where , . Obviously, is invertible and its inverse operator is , where is a constant such that .
In the past few decades, many important results relative to a p-Laplacian equation with certain boundary value conditions have been obtained. We refer the reader to [35–38] and the references cited therein. We noticed that is a quasi-linear operator. So, Mawhin’s continuation theorem is not suitable for a p-Laplacian operator. In , Ge and Ren extended Mawhin’s continuation theorem, which is used to deal with more general abstract operator equations.
Motivated by all the works above, in this paper, we consider the following boundary value problem (BVP for short) for a coupled system of fractional p-Laplacian equations given by
where , , , are the standard Caputo fractional derivatives, , , and is continuous.
The rest of this paper is organized as follows. Section 2 contains some necessary notations, definitions and lemmas. In Section 3, we establish a theorem on the existence of solutions for BVP (1.1) under nonlinear growth restriction of f and g, based on the extension of Mawhin’s continuation theorem due to Ge (see ). Finally, in Section 4, an example is given to illustrate the main result.
In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper.
Definition 2.1 Let X and Y be two Banach spaces with norms and , respectively. A continuous operator
is said to be quasi-linear if
is a closed subset of Y,
is linearly homeomorphic to , .
Definition 2.2 Let X be a real Banach space and . The operator is said to be a projector provided , for , . The operator is said to be a semi-projector provided .
Definition 2.3 ()
Let and be the complement space of in X, then . On the other hand, suppose that is a subspace of Y and is the complement space of in Y so that . Let be a projector and be a semi-projector, and let be an open and bounded set with origin , where θ is the origin of a linear space.
Suppose that , is a continuous operator. Denote by N. Let . is said to be M-compact in if there is with and an operator continuous and compact such that for ,
Lemma 2.1 (, Ge-Mawhin’s continuation theorem)
Let X and Y be two Banach spaces with norms and , respectively. is an open and bounded nonempty set. Suppose that
is a quasi-linear operator and
is M-compact in . In addition, if
(C1) , ,
(C2) , for ,
where and is a homeomorphism with , then the equation has at least one solution in .
Definition 2.4 The Riemann-Liouville fractional integral operator of order of a function x is given by
provided that the right-hand side integral is pointwise defined on .
Definition 2.5 The Caputo fractional derivative of order of a continuous function x is given by
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral is pointwise defined on .
Lemma 2.2 
Assume that , . Then
where , , here n is the smallest integer greater than or equal to α.
Lemma 2.3 
Assume that and . Then
In this paper, we denote with the norm , with the norm and with the norm , where . Then we denote with the norm and with the norm . Obviously, both and are Banach spaces.
Define the operator by
Define the operator by
Define the operator by
Define the operator by
Then BVP (1.1) is equivalent to the operator equation
3 Main result
In this section, a theorem on the existence of solutions for BVP (1.1) will be given.
Theorem 3.1 Let be continuous. Assume that
(H1) there exist nonnegative functions () with
such that for all , ,
where , , ();
(H2) there exists a constant such that for all , , either
Then BVP (1.1) has at least one solution.
In order to prove Theorem 3.1, we need to prove some lemmas below.
Lemma 3.1 Let M be defined by (2.5), then
and M is a quasi-linear operator.
Proof By Lemma 2.2, has the solution
Combining with the boundary value condition , we have
For , there exists such that . By Lemma 2.2, we have
From the condition , one has . By the condition , we obtain that
On the other hand, suppose that and satisfies . Let , then . By Lemma 2.3, we have . So that . Then we have
Then we have and closed. Therefore, is a quasi-linear operator. Similarly, we can get
and is a quasi-linear operator. Then the proof is complete. □
Lemma 3.2 Let be an open and bounded set, then is M-compact in .
Proof Define the continuous projector and the semi-projector
where and .
Obviously, and . It follows from that . By a simple calculation, we can get that . Then we get
For , we have
By the definition of , we can get
Similar proof can show that . Thus, we have .
Let , where , . It follows from and that . Then we have
Let be an open and bounded set with . For each , we can get . Thus, . Take any in the type . Since , we can get . So (2.1) holds. It is easy to verify (2.2).
Furthermore, define by
By the continuity of f and g, it is easy to get that is continuous on . Moreover, for all , there exists a constant such that , so we can easily obtain that is uniformly bounded. By the Arzela-Ascoli theorem, we just need to prove that is equicontinuous. Furthermore, for , , we have
By , we have
Since is uniformly continuous on , so is equicontinuous. Similarly, we can get is equicontinuous. Considering that is uniformly continuous on , we have is also equicontinuous. So, we can obtain that is compact.
Similarly, we can get that is compact. So, we can obtain that is compact.
For each , we have . Thus,
which together with yields that
It is easy to verify that is the zero operator. Similarly, we can get and is the zero operator. So (2.3) holds.
On the other hand,
Similarly, we have . So, (2.4) holds. Then we have that is M-compact in . The proof is complete. □
Lemma 3.3 Suppose that (H1), (H2) hold, then the set
Proof Take , then . By (3.3), we have
Then, by the integral mean value theorem, there exist constants such that and . So, from (H2), we get and .
By Lemma 2.2,
Take , we have
Then we have
So, we get
Similarly, we can get
By and , we get
So, from (H1), we have
which together with and (3.5) yields that
Similarly, we can get
Then from (3.1), (3.7) and (3.8), we can see that there exists a constant such that
Thus, from (3.5) and (3.6), we get
Combining (3.9) and (3.10), we have
So, is bounded. The proof is complete. □
Lemma 3.4 Suppose that (H3) holds, then the set
Proof For , we have . Then, from , we get
which together with (H2) implies . Thus, we have
Hence, is bounded. The proof is complete. □
Lemma 3.5 Suppose that the first part of (H2) holds, then the set
is bounded, where is a homeomorphism defined by
Proof For , we have and
If , then . For , we can obtain . Otherwise, if or , in view of the first part of (H2), one has
which contradicts (3.11) or (3.12). Therefore, is bounded. The proof is complete. □
Remark 3.1 If the second part of (H2) holds, then the set
Proof of Theorem 3.1 Set . It follows from Lemmas 3.1 and 3.2 that M is a quasi-linear operator and is M-compact on . By Lemmas 3.3 and 3.4, we get that the following two conditions are satisfied:
(C1) , ,
(C2) , for .
According to Lemma 3.5 (or Remark 3.1), we know that for . Therefore
So, condition (C3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that has at least one solution in . Therefore BVP (1.1) has at least one solution. The proof is complete. □
Example 4.1 Consider the following BVP:
Corresponding to BVP (1.1), we have that , , , , and
Choose , , , , . Then we have , , , . By a simple calculation, we get
Then (H1) and the first part of (H2) hold.
By Theorem 3.1, we obtain that BVP (4.1) has at least one solution.
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The authors are grateful to those who gave useful suggestions about the original manuscript. This research was supported by the Fundamental Research Funds for the Central Universities (2013QNA33).
The authors declare that they have no competing interests.
The authors contributed equally in this article. All authors read and approved the final manuscript.
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Hu, Z., Liu, W. & Liu, J. Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance. Adv Differ Equ 2013, 312 (2013). https://doi.org/10.1186/1687-1847-2013-312
- fractional p-Laplacian equation
- coupled system
- boundary value problem
- degree theory