# Existence of solutions of two-point boundary value problems for fractional p-Laplace differential equations at resonance

## Authors

- First Online:

- Received:

DOI: 10.1007/s12190-012-0598-0

- Cite this article as:
- Tang, X., Yan, C. & Liu, Q. J. Appl. Math. Comput. (2013) 41: 119. doi:10.1007/s12190-012-0598-0

- 4 Citations
- 314 Views

## Abstract

*α*,

*β*≤1, 1<

*α*+

*β*≤2. By using the coincidence degree theory, a new result on the existence of solutions for above fractional boundary value problem is obtained. These results extend the corresponding ones of ordinary differential equations of integer order. Finally, an example is inserted to illustrate the validity and practicability of our main results.

### Keywords

Caputo fractional derivativep-Laplace differential equationTwo-point boundary value problemResonanceCoincidence degree theory### Mathematics Subject Classification

34A0834B15## 1 Introduction

*α*,

*β*≤1, 1<

*α*+

*β*≤2,

*f*is continuous functions. The boundary value problem (BVP for short) (1.1) is a resonance problem, because its associated homogeneous boundary value problem has a nontrivial solution

*u*(

*t*)=

*ct*

^{β},

*c*∈

*R*.

*α*<2,

*β*>0 are real numbers,

*α*−

*β*≥1. They established the existence results by the fixed point theorem in a cone.

*α*,

*β*≤1, 1<

*α*+

*β*≤2. They obtained the existence results by using the coincidence degree theory.

However, to the best of our knowledge, no paper has concerned the existence of solutions to the boundary value problem (1.1) at resonance. Motivated by the work above, in this paper, we consider the existence of solutions for two-point boundary value problem for fractional p-Laplace differential equation (1.1) at resonance. By using the coincidence degree theory, a new result on the existence of solutions for above fractional boundary value problem (1.1) is obtained. These results extend the corresponding ones of ordinary differential equations of integer order.

This paper is organized as follows. In Sect. 2, we introduce some basic definitions and preliminaries later used. In Sect. 3, the existence results of solutions for fractional p-Laplace differential equation (1.1) is discussed by using the coincidence degree theory. In Sect. 4, we give an example to illustrate our main results.

## 2 Preliminary results

The material in this section is basic in some sense. For the reader’s convenience, we present some definitions form fractional calculus theory and preliminary results.

### Definition 2.1

([18])

### Definition 2.2

([18])

### Lemma 2.1

([19])

### Lemma 2.2

([19])

*Given that u*∈*C*(0,1)∩*L*(0,1) *with a fractional derivative of order**α* (*α*>0) *that belongs to**C*(0,1)∩*L*(0,1). *Then*\(I^{\alpha}_{0^{+}}D^{\alpha}_{0^{+}}u(t)=u(t)+C_{0} +C_{1}t+\cdots+C_{n-1}t^{n-1}\), *for some**C*_{i}∈ℝ, *i*=0,2,…,*n*−1, *where**n**is the smallest integer greater than or equal to**α*.

Now, we briefly recall some notation and an abstract existence result due to Mawhin [11].

Let *X*, *Y* be two real Banach spaces and \(L: \operatorname{dom}L\subset X\rightarrow Y \) a linear operator which is a Fredholm map of index zero, *N*:*X*→*Y* is nonlinear continuous map. If \(\dim\ker L=\dim(Y/\operatorname{Im}L)<+\infty\), and \(\operatorname{Im}L\) is a closed set of *Y*, then *L* is a Fredholm map of index zero. If *L* is a Fredholm map of index zero, and *P*:*X*→*X*, *Q*:*Y*→*Y* be projectors such that \(\operatorname{Im}P= \operatorname{Ker} L\), \(\operatorname{Ker}Q= \operatorname{Im}L\), \(X= \operatorname{Ker} L\oplus \operatorname{Ker} P\), \(Y= \operatorname{Im}L \oplus \operatorname{Im}Q\). It follows that \(L_{p}=L|_{\operatorname{dom}L\cap\operatorname {Ker}P}:\operatorname{dom}L\cap\operatorname{Ker} P \rightarrow\operatorname{Im}L\) is invertible. We denote the inverse by *K*_{p}. Let *Ω* is an open bounded subset of *X*, and \(\operatorname{dom}L\cap\overline{\varOmega}\neq\emptyset\), the map *N*:*X*→*Y* will be called *L*-compact on \(\overline{\varOmega}\) if \(QN(\overline{\varOmega})\) is bounded and \(K(I-Q)N:\overline{\varOmega}\rightarrow X\) is compact.

### Theorem 2.1

([20])

*Let*

*X*,

*Y*

*be real Banach spaces*, \(L:\operatorname{dom}L\subset X\rightarrow Y\)

*be a Fredholm operator of index zero and*

*N*:

*X*→

*Y*

*be*

*L*-

*compact on*\(\overline{\varOmega}\).

*Assume that the following conditions are satisfied*

- (1)
*Lx*≠*λNx*, \(\forall(x, \lambda)\in[(\operatorname{dom}L\backslash\operatorname{Ker} L)\cap \partial\varOmega]\times(0, 1)\); - (2)
\(Nx\notin\operatorname{Im}L\), \(\forall x\in\operatorname {Ker} L\cap\partial\varOmega\);

- (3)
\(\deg(QN|_{\operatorname{Ker}L}, \varOmega\cap \operatorname{Ker} L, 0)\neq0\),

*where*,*Q*:*Y*→*Y**is a projection such that*\(\operatorname{Im}L= \operatorname{Ker} Q\).

*Then the equation*

*Lx*=

*Nx*

*has at least one solution in*\(\operatorname{dom}L\cap\overline{\varOmega}\).

In this paper, we take *Y*=*C*[0,1], with the norm ∥*y*∥_{Y}=max_{t∈[0,1]}|*y*(*t*)|, and \(X=\{x|x,D^{\beta}_{0^{+}}x\in Y\}\), with the norm \(\|x\|_{X}=\max\{\|x\|_{\infty}, \|D^{\beta}_{0^{+}}x\|_{\infty}\}\), where ∥*x*∥_{∞}=max_{t∈[0,1]}|*x*(*t*)|. By means of the linear functional analysis theory, we can prove that *X* is a Banach space (see [21, Lemma 3.2]).

## 3 Some lemmas

In this section, we need the following auxiliary lemmas to prove the existence of solutions to (1.1).

### Lemma 3.1

### Proof

*x*(0)=0, one has that (3.1) holds.

On the other hand, suppose *y*∈*Y*, and satisfies (3.3). Let \(x(t)=I^{\beta}_{0^{+}}\phi_{q}(I^{\alpha}_{0^{+}}y(t))\), then \(x\in\operatorname{dom}L\), and \(Lx(t)=D^{\alpha}_{0^{+}}\phi_{p}(D^{\beta}_{0^{+}}x(t))=y(t)\). So that, \(y\in\operatorname{Im}L\). The proof is complete. □

### Lemma 3.2

*Let*

*L*

*be defined by*(2.1);

*then*

*L*

*is a Fredholm operator of index zero*,

*and the linear continuous projector operators*

*P*:

*X*→

*X*

*and*

*Q*:

*Y*→

*Y*

*can be defined as*

*Furthermore*,

*the operator*\(K_{p}: \operatorname{Im}L\rightarrow\operatorname{dom}L\cap\operatorname {Ker}_{P}\)

*can be written by*

### Proof

*P*

^{2}

*x*=

*Px*. It follows from

*x*=(

*x*−

*Px*)+

*Px*that \(X= \operatorname{Ker} P+ \operatorname{Ker} L\). By simple calculation, we can get that \(\operatorname{Ker}P\cap\operatorname{Ker} L=\{0\}\). Then we get For any

*y*∈

*Y*, we have Let

*y*

_{1}=

*y*−

*Qy*, then we get from (3.4) that which implies \(y_{1}\in\operatorname{Im}L\). Hence \(Y= \operatorname {Im}L+ \operatorname{Im}Q\). Since \(\operatorname{Im}L\cap\operatorname{Im}Q=\{0\}\), we have \(Y= \operatorname{Im}L \oplus \operatorname{Im}Q\). Thus This means that

*L*is a Fredholm operator of index zero.

*P*,

*K*

_{P}, it is easy to see that the generalized inverse of

*L*

_{P}is

*K*

_{P}. In fact, for \(y\in\operatorname{Im}L\), we have Moreover, for \(x\in\operatorname{dom}L\cap\operatorname{Ker} P\), we get \(x(0)=D^{\beta}_{0^{+}}x(0)=D^{\beta}_{0^{+}}x(1)=0\). By Lemma 2.1, we obtain that which together with \(D^{\beta}_{0^{+}}x(0)=0\), yields that Thus, we have which together with

*x*(0)=0, yields that Combining (3.5) with (3.6), we know that

*K*

_{P}is the inverse of

*L*

_{P}. The proof is complete. □

### Lemma 3.3

*Assume**Ω*⊂*X**is an open bounded subset such that*\(\operatorname{dom}L\cap\overline{\varOmega}\neq\emptyset\), *then**N**is**L*-*compact on*\(\overline{\varOmega}\).

### Proof

For convenience, denote *K*_{P,Q}=*K*_{P}(*I*−*Q*)*N*. By the continuity of *f*, we can get that \(QN(\overline{\varOmega})\) and \(K_{P,Q}(\overline{\varOmega})\) are bounded. Moreover, there exists a constant *M*>0 such that \(|I^{\alpha}_{0^{+}}(I-Q)Nx|\leq M\), \(\forall x\in\overline{\varOmega}\), *t*∈[0,1]. Thus, in view of the Arzela-Ascoli theorem, we need only prove that \(K_{P,Q}(\overline{\varOmega})\subset X\) is equicontinuous.

*t*

_{1}<

*t*

_{2}≤1, \(x\in\overline{\varOmega}\), we have Since

*t*

^{β}is uniformly continuous on [0, 1], we can obtain that \(K_{P,Q}(\overline{\varOmega})\subset C[0,1]\) is equicontinuous. Similar proof can show that \(I^{\alpha}_{0^{+}}(I-Q)N(\overline{\varOmega})\subset C[0,1]\) is equicontinuous. This, together with the uniformly continuity of

*ϕ*(

*s*) on [−

*T*,

*T*], yields that \(D^{\beta}_{0^{+}}(K_{P,Q})(\overline{\varOmega}) =\phi_{q}(I^{\alpha}_{0^{+}}(I-Q)N)(\overline{\varOmega})\subset C[0,1]\) is also equicontinuous. Thus, we get that \(K_{P}(I-Q)N: \overline{\varOmega}\rightarrow X\) is compact. The proof is complete. □

### Lemma 3.4

*Suppose*(H

_{1})

*and*(H

_{2})

*hold*,

*then the set*

*is bounded*. (H

_{1})

*and*(H

_{2})

*will be given in Sect*. 4.

### Proof

*x*∈

*Ω*

_{1}, then

*Lx*=

*λNx*and \(Nx\in\operatorname {Im}L\). By (3.2), we have Then, by the integral mean value theorem, there exists a constant

*ξ*∈(0,1) such that So, from (H

_{2}), we get \(|D^{\beta}_{0^{+}}x(\xi)|\leq D\).

*x*(0)=0. Therefore That is By

*Lx*=

*λNx*, we have Take

*t*=

*ξ*, we get Together with \(|D^{\beta}_{0^{+}}x(\xi)|\leq D\), (H

_{1}) and (3.6), we have So, we have Thus, from

*Γ*(

*α*+1)−2(∥

*b*∥

_{∞}+∥

*c*∥

_{∞})>0, we obtain that and Combining (3.10) with (3.11), we have Therefore,

*Ω*

_{1}is bounded. The proof is complete. □

### Lemma 3.5

### Proof

### Lemma 3.6

### Proof

The proof divided into two cases by the relations of *f*(*t*,*u*,*v*) and *v*.

*D*

^{∗}>0 such that for all

*c*∈

*R*with |

*c*|>

*D*

^{∗}, In this case, \(\varOmega_{3}=\{x|x\in\operatorname{Ker} L, \lambda x+(1-\lambda)QNx=0,\lambda\in[0,1]\}\).

*x*∈

*Ω*

_{3}, we have

*x*(

*t*)=

*ct*

^{β},

*c*∈

*R*, and If

*λ*=0, then |

*c*|≤

*D*

^{∗}. If

*λ*∈(0,1], we can also obtain |

*c*|≤

*D*

^{∗}. Otherwise, if |

*c*|>

*D*

^{∗}, by (3.12), one has which contradicts to (3.13). Therefore,

*Ω*

_{3}is bounded.

*D*

^{∗}>0 such that for all

*c*∈

*R*with |

*c*|>

*D*

^{∗}, In this case, \(\varOmega_{3}=\{x|x\in\operatorname{Ker} L, -\lambda x+(1-\lambda)QNx=0,\lambda\in[0,1]\}\).

*x*∈

*Ω*

_{3}, we have

*x*(

*t*)=

*ct*

^{β},

*c*∈

*R*, and If

*λ*=0, then |

*c*|≤

*D*

^{∗}. If

*λ*∈(0,1], we can also obtain |

*c*|≤

*D*

^{∗}. Otherwise, if |

*c*|>

*D*

^{∗}, by (3.14), one has which contradicts to (3.13). Therefore,

*Ω*

_{3}is bounded.

From the above Case 1 and Case 2, we can know that *Ω*_{3} is bounded. The proof is complete. □

## 4 Main result and example

In this section, we firstly investigate the existence of solutions for two-point boundary value problem for fractional p-Laplace differential equation (1.1) at resonance, which is based on the coincidence degree theory. Then, we will give an example to illustrate the validity and practicability of our main results.

Now, we begin with some theorems below.

### Theorem 4.1

*Let*

*f*:[0,1]×

*R*

^{2}→

*R*

*be continuous*.

*Assume that*

- (H
_{1}) - (H
_{2}) - (H
_{3})

*Then BVP*(1.1)

*has at least one solution*,

*provided that*\(\frac{2(\|b\|_{\infty}+\|c\|_{\infty})}{\varGamma(\alpha+1)}<1\).

### Proof

*Ω*

_{1}∪

*Ω*

_{2}∪

*Ω*

_{3}⊂

*Ω*. It follows from Lemmas 3.2 and 3.3 that

*L*(defined by (2.1)) is a Fredholm operator of index zero and

*N*(defined by (2.2)) is

*L*-compact on \(\overline{\varOmega}\). By Lemmas 3.4 and 3.5, we get that the following two conditions are satisfied

- (1)
*Lx*≠*λNx*, \(\forall(x, \lambda)\in[(\operatorname{dom} L\backslash\operatorname{Ker} L)\cap\partial \varOmega]\times(0, 1)\); - (2)
\(Nx\notin\operatorname{Im}L\), \(\forall x\in\operatorname {Ker} L\cap\partial\varOmega\).

Consequently, by using Theorem 2.1, the operator equation *Lx*=*Nx* has at least one solution in dom\(L\cap\overline{\varOmega}\). Namely, BVP (1.1) has at least one solution in *X*. The proof is complete. □

### Theorem 4.2

*Let*

*f*:[0,1]×

*R*

^{2}→

*R*

*be continuous*.

*Supposed that the conditions*(H

_{2})

*and*(H

_{3})

*hold*.

*Further*,

*assume that*

*Then BVP*(1.1)

*has at least one solution*,

*provided that*\(\frac{4\|r\|_{\infty}}{\varGamma(\alpha+1)}<1\).

### Proof

Note that \(\frac{4\|r\|_{\infty}}{\varGamma(\alpha+1)}<1\), then there is a constant *ε*>0 such that \(\frac{4(\|r\|_{\infty}+\varepsilon)}{\varGamma(\alpha+1)}<1\).

_{4}), there exists

*H*>0 such that Let

*M*

^{∗}=max

_{t∈[0,1],|u|+|v|≤H}|

*f*(

*t*,

*u*,

*v*)|, From Theorem 4.1, BVP (1.1) has at least one solution in

*X*. The proof is complete. □

### Corollary 4.1

*Let*

*f*:[0,1]×

*R*

^{2}→

*R*

*be continuous*.

*Supposed that the conditions*(H

_{2})

*and*(H

_{3})

*hold*.

*Further*,

*assume that*

*Then BVP*(1.1)

*has at least one solution*,

*provided that*\(\frac{2\|r\|_{\infty}}{\varGamma(\alpha+1)}<1\).

Now, we will give an example to illustrate our main result.

### Example 4.1

*p*=3 and Choose

*a*(

*t*)=7,

*b*(

*t*)=0, \(c(t)=\frac{1}{4}\),

*D*=5. By simple calculation, we can get that ∥

*b*∥

_{∞}=0, \(\|c\|_{\infty}=\frac{1}{4}\), and Obviously, BVP (4.1) satisfies all conditions of Theorem 4.1. Hence, it has at least one solution.

## Acknowledgements

The author would like to thank the referee for his or her careful reading and some comments on improving the presentation of this paper.