Existence of solutions for a coupled system of fractional p-Laplacian equations at resonance
© Hu et al.; licensee Springer. 2013
Received: 24 May 2013
Accepted: 2 October 2013
Published: 8 November 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.
Keywordsfractional p-Laplacian equation coupled system boundary value problem degree theory resonance
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]).
where , are real numbers, and is the Riemann-Liouville fractional derivative.
where is a real number, , are given constants such that , and , are the Riemann-Liouville differentiation and integration.
where , , , , , , , D is the standard Riemann-Liouville differentiation and are given continuous functions.
where , , , , , .
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.
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.
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.
Lemma 2.1 (, Ge-Mawhin’s continuation theorem)
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 .
provided that the right-hand side integral is pointwise defined on .
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 
where , , here n is the smallest integer greater than or equal to α.
Lemma 2.3 
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.
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
where , , ();
Then BVP (1.1) has at least one solution.
In order to prove Theorem 3.1, we need to prove some lemmas below.
and M is a quasi-linear operator.
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 .
where and .
Similar proof can show that . Thus, 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).
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.
It is easy to verify that is the zero operator. Similarly, we can get and is the zero operator. So (2.3) holds.
Similarly, we have . So, (2.4) holds. Then we have that is M-compact in . The proof is complete. □
Then, by the integral mean value theorem, there exist constants such that and . So, from (H2), we get and .
So, is bounded. The proof is complete. □
Hence, is bounded. The proof is complete. □
which contradicts (3.11) or (3.12). Therefore, is bounded. The proof is complete. □
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 .
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. □
Then (H1) and the first part of (H2) hold.
By Theorem 3.1, we obtain that BVP (4.1) has at least one solution.
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).
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