Solvability of fractional boundary value problem with p -Laplacian via critical point theory

Taiyong Chen; Wenbin Liu

doi:10.1186/s13661-016-0583-x

Research
Open Access

Solvability of fractional boundary value problem with p-Laplacian via critical point theory

Taiyong Chen¹ and
Wenbin Liu¹Email author

Boundary Value Problems20162016:75

https://doi.org/10.1186/s13661-016-0583-x

Received: 5 January 2016
Accepted: 31 March 2016
Published: 5 April 2016

Abstract

In this paper, we discuss the fractional boundary value problem containing left and right fractional derivative operators and p-Laplacian. By using critical point theory we obtain some results on the existence of weak solutions of such a fractional boundary value problem.

Keywords

fractional differential equation
p-Laplacian
boundary value problem
critical point theory

MSC

26A33
34B15
58E05

1 Introduction

Fractional-order models can be found to be more adequate than integer-order models in some real-world problems since fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Indeed, we can find numerous applications in viscoelasticity, neurons, electrochemistry, control, porous media, electromagnetism, et cetera (see [1–6]). As a consequence, the subject of fractional differential equations is gaining more importance and attention.

In the past decade, many results on the existence and multiplicity of solutions to nonlinear fractional differential equations were obtained by using techniques of nonlinear analysis, such as fixed point theory [7], topological degree theory [8], and comparison method [9]. For more papers on fractional differential equations, see [10–18] and the references therein. Recently, the equations including both left and right fractional derivatives are also discussed [19–21]. Apart from their possible applications, equations with left and right fractional derivatives are an interesting and new field in fractional differential equations theory.

We should note that critical point theory has also turned out to be a very effective tool in determining the existence of solutions of integer-order differential equations with variational structures [22–26]. The idea behind is trying to find solutions of a given boundary value problem by looking for critical points of a suitable energy functional defined on an appropriate function space. We refer the readers to the books [27, 28] and the references therein.

In recent paper [20], for the first time, the authors showed that critical point theory is an effective approach to tackle the existence of weak solutions of fractional boundary value problems (FBVPs) of the form

$$\textstyle\begin{cases} {}_{t}D_{T}^{\alpha}({{}_{0}D_{t}^{\alpha}}u(t))=\nabla F(t,u(t)),& \mbox{a.e. } t\in [0,T],\\ u(0)=u(T)=0. \end{cases} $$

We note that it is not easy to use critical point theory to study FBVPs since it is often very difficult to establish suitable function spaces and variational functionals for FBVPs.

Motivated by the above-mentioned classical works, we want to contribute with the development of this new area on fractional differential equations theory. More precisely, we study the existence of weak solutions of the following FBVP:

$$\begin{aligned} \textstyle\begin{cases} {}_{t}D_{T}^{\alpha}\phi_{p}({{}_{0}D_{t}^{\alpha}}u(t))=f(t,u(t)),& t\in[0,T],\\ u(0)=u(T)=0, \end{cases}\displaystyle \end{aligned}$$

(1.1)

where ${}_{0}D_{t}^{\alpha}$ and ${}_{t}D_{T}^{\alpha}$ are the left and right Riemann-Liouville fractional derivatives of order $\alpha\in(0,1]$ respectively, $\phi_{p}:\mathbb{R}\rightarrow\mathbb{R}$ is the p-Laplacian defined by

$$\phi_{p}(s)=\vert s\vert ^{p-2}s\quad \mbox{if } s\neq0,\qquad \phi_{p}(0)=0,\quad p>1, $$

and $f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following condition:

($\mathrm{H}_{1}$):: $f\in C \bigl([0,T]\times\mathbb{R},\mathbb{R} \bigr)$.

Note that p-Laplacian $\phi_{p}$ introduced by Leibenson [29] often occurs in non-Newtonian fluid theory, nonlinear elastic mechanics, and so forth. Moreover, when $p=2$, the nonlinear operator ${}_{t}D_{T}^{\alpha}\phi_{p}({{}_{0}D_{t}^{\alpha}})$ reduces to the linear operator ${}_{t}D_{T}^{\alpha}{{}_{0}D_{t}^{\alpha}}$.

The rest of this paper is organized as follows. Section 2 contains some necessary notation, definitions, and properties of fractional calculus. In Section 3, we introduce the function space and energy functional for FBVP (1.1). In Section 4, based on some critical point theorems due to Mawhin and Willem [27] and Rabinowitz [28], we establish two theorems on the existence of weak solutions of FBVP (1.1).

2 Fractional calculus

For the convenience of the reader, we introduce some basic definitions and properties of the fractional calculus, which can be found, for instance, in [30, 31].

Definition 2.1

(Left and right Riemann-Liouville fractional integrals [30, 31])

Let x be a function defined on $[a,b]$. The left and right Riemann-Liouville fractional integrals of order $\gamma>0$ for x, denoted by ${}_{a}D_{t}^{-\gamma}x(t)$ and ${}_{t}D_{b}^{-\gamma }x(t)$, respectively, are defined by

$$\begin{aligned}& {}_{a}D_{t}^{-\gamma}x(t)=\frac{1}{\Gamma(\gamma)} \int_{a}^{t} (t-s)^{\gamma-1}x(s)\,ds, \\& {}_{t}D_{b}^{-\gamma}x(t)=\frac{1}{\Gamma(\gamma)} \int_{t}^{b} (s-t)^{\gamma-1}x(s)\,ds, \end{aligned}$$

provided that the right-hand side integrals are pointwise defined on $[a,b]$.

Definition 2.2

(Left and right Riemann-Liouville fractional derivatives [30, 31])

Let x be a function defined on $[a,b]$. The left and right Riemann-Liouville fractional derivatives of order $\gamma>0$ for x, denoted by ${}_{a}D_{t}^{\gamma}x(t)$ and ${}_{t}D_{b}^{\gamma }x(t)$, respectively, are defined by

$$\begin{aligned}& {}_{a}D_{t}^{\gamma}x(t)=\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}{{}_{a}D_{t}^{\gamma-n}x(t)}, \\& {}_{t}D_{b}^{\gamma}x(t)=(-1)^{n} \frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}{{}_{t}D_{b}^{\gamma-n}x(t)}, \end{aligned}$$

where $n-1\leq\gamma< n$ and $n\in\mathbb{N}$.

Remark 2.1

(i)

For $n\in\mathbb{N}$, if γ becomes an integer $n-1$, according to Definition 2.2, we recover the usual definitions, namely
$$\begin{aligned}& {}_{a}D_{t}^{n-1}x(t)=x^{(n-1)}(t), \\& {}_{t}D_{b}^{n-1}x(t)=(-1)^{n-1}x^{(n-1)}(t), \end{aligned}$$
where $x^{(n-1)}$ is the usual derivative of order $n-1$.
(ii)

If $x\in AC^{n}([a,b],\mathbb{R})$, following [30], we know that the Riemann-Liouville fractional derivative of order $\gamma\in [n-1,n)$ exists a.e. on $[a,b]$, where $AC([a,b],\mathbb{R})$ is the space of functions that are absolutely continuous on $[a,b]$, and $AC^{k}([a,b],\mathbb{R})$ ($k=1,2,\ldots$) is the space of functions x such that $x\in C^{k-1}([a,b],\mathbb{R})$ and $x^{(k-1)}\in AC([a,b],\mathbb{R})$.

Definition 2.3

(Left and right Caputo fractional derivatives [30])

Let $\gamma\geq0$ and $n\in\mathbb{N}$. If $n-1<\gamma<n$ and $x\in AC^{n}([a,b],\mathbb{R})$, then the left and right Caputo fractional derivatives of order γ for x, denoted by ${}_{a}^{c}D_{t}^{\gamma}x(t)$ and ${}_{t}^{c}D_{b}^{\gamma}x(t)$, respectively, exist almost everywhere on $[a,b]$ and are represented by

$$\begin{aligned}& {}_{a}^{c}D_{t}^{\gamma}x(t)={{}_{a}D_{t}^{\gamma-n}x^{(n)}(t)}, \\& {}_{t}^{c}D_{b}^{\gamma}x(t)=(-1)^{n}{{}_{t}D_{b}^{\gamma-n}x^{(n)}(t)}. \end{aligned}$$

If $\gamma=n-1$ and $x\in AC^{n-1}([a,b],\mathbb{R})$, then ${}_{a}^{c}D_{t}^{n-1}x(t)$ and ${}_{t}^{c}D_{b}^{n-1}x(t)$ are represented by

$$\begin{aligned}& {}_{a}^{c}D_{t}^{n-1}x(t)=x^{(n-1)}(t), \\& {}_{t}^{c}D_{b}^{n-1}x(t)=(-1)^{n-1}x^{(n-1)}(t). \end{aligned}$$

The Riemann-Liouville fractional derivatives and the Caputo fractional derivatives are connected with each other by the following relations.

Property 2.1

([30, 31])

Let $n\in\mathbb{N}$ and $n-1<\gamma<n$. If x is a function defined on $[a,b]$ for which the Caputo fractional derivatives ${}_{a}^{c}D_{t}^{\gamma}x(t)$ and ${}_{t}^{c}D_{b}^{\gamma}x(t)$ of order γ exist together with the Riemann-Liouville fractional derivatives ${}_{a}D_{t}^{\gamma}x(t)$ and ${}_{t}D_{b}^{\gamma}x(t)$, then

$${}_{a}^{c}D_{t}^{\gamma}x(t)={{}_{a}D_{t}^{\gamma}x(t)} -\sum_{j=0}^{n-1}\frac{x^{(j)}(a)}{\Gamma(j-\gamma+1)}(t-a)^{j-\gamma} $$

and

$${}_{t}^{c}D_{b}^{\gamma}x(t)={{}_{t}D_{b}^{\gamma}x(t)} -\sum_{j=0}^{n-1}\frac{x^{(j)}(b)}{\Gamma(j-\gamma+1)}(b-t)^{j-\gamma}. $$

In particular, when $0<\gamma<1$, we get

$$\begin{aligned}& {}_{a}^{c}D_{t}^{\gamma}x(t)={{}_{a}D_{t}^{\gamma}x(t)} -\frac{x(a)}{\Gamma(1-\gamma)}(t-a)^{-\gamma}, \end{aligned}$$

(2.1)

$$\begin{aligned}& {}_{t}^{c}D_{b}^{\gamma}x(t)={{}_{t}D_{b}^{\gamma}x(t)} -\frac{x(b)}{\Gamma(1-\gamma)}(b-t)^{-\gamma}. \end{aligned}$$

(2.2)

Now we present the rule for fractional integration by parts.

Property 2.2

([30, 32])

We have the following property of fractional integration:

$$\begin{aligned}& \int_{a}^{b} \bigl({{}_{a}D_{t}^{-\gamma}x(t)} \bigr)y(t)\,dt= \int_{a}^{b}x(t){{}_{t}D_{b}^{-\gamma}y(t)}\,dt,\quad \gamma>0, \end{aligned}$$

provided that $x\in L^{p}([a,b],\mathbb{R})$, $y\in L^{q}([a,b],\mathbb {R})$, and $p\geq1$, $q\geq1$, $1/p+1/q\leq1+\gamma$ or $p\neq1$, $q\neq1$, $1/p+1/q=1+\gamma$.

3 Fractional derivative space and variational structure

In order to establish the variational framework that will enable us to reduce the existence of solutions of FBVP (1.1) to the problem of finding critical points of the corresponding functional, it is necessary to introduce an appropriate function space. In the following, we cite some results from [20].

Definition 3.1

([20])

Let $0<\alpha\leq1$ and $1< p<\infty $. The fractional derivative space $E_{0}^{\alpha,p}$ is defined by

$$\begin{aligned}& E_{0}^{\alpha,p}= \bigl\{ u\in L^{p} \bigl([0,T], \mathbb{R} \bigr) \vert{{}_{0}^{c}D_{t}^{\alpha}}u \in L^{p} \bigl([0,T],\mathbb{R} \bigr), u(0)=u(T)=0 \bigr\} \end{aligned}$$

with the norm

$$\begin{aligned}& \Vert u\Vert _{\alpha,p}= \bigl(\Vert u\Vert _{L^{p}}^{p}+ \bigl\Vert {{}_{0}^{c}D_{t}^{\alpha}}u \bigr\Vert _{L^{p}}^{p} \bigr)^{\frac{1}{p}},\quad \forall u\in E_{0}^{\alpha,p}, \end{aligned}$$

(3.1)

where $\Vert u\Vert _{L^{p}}= (\int_{0}^{T}\vert u(t)\vert ^{p}\,dt)^{1/p}$ is the norm of $L^{p}([0,T],\mathbb{R})$.

Remark 3.1

For any $u\in E_{0}^{\alpha,p}$, according to (2.1) and (2.2) and in view of $u(0)=u(T)=0$, we have ${{}_{0}^{c}D_{t}^{\alpha}}u(t)={{}_{0}D_{t}^{\alpha}}u(t)$, ${{}_{t}^{c}D_{T}^{\alpha}}u(t)={{}_{t}D_{T}^{\alpha}}u(t)$ for $t\in[0,T]$.

Lemma 3.1

([20])

Let $0<\alpha\leq1$ and $1< p<\infty$. The fractional derivative space $E_{0}^{\alpha,p}$ is a reflexive and separable Banach space.

Lemma 3.2

([20])

Let $0<\alpha\leq1$ and $1< p<\infty$. For $u\in E_{0}^{\alpha,p}$, we have

$$\begin{aligned}& \Vert u\Vert _{L^{p}}\leq\frac{T^{\alpha}}{\Gamma(\alpha +1)} \bigl\Vert {{}_{0}^{c}D_{t}^{\alpha}}u \bigr\Vert _{L^{p}}. \end{aligned}$$

(3.2)

Moreover, if $\alpha>1/p$ and $1/p+1/q=1$, then

$$\begin{aligned}& \Vert u\Vert _{\infty}\leq\frac{T^{\alpha-\frac{1}{p}}}{ \Gamma(\alpha)((\alpha-1)q+1)^{\frac{1}{q}}} \bigl\Vert {{}_{0}^{c}D_{t}^{\alpha}}u \bigr\Vert _{L^{p}}, \end{aligned}$$

(3.3)

where $\Vert u\Vert _{\infty}=\max_{t\in[0,T]}\vert u(t)\vert $ is the norm of $C([0,T],\mathbb{R})$.

Remark 3.2

According to (3.2), we know that the norm (3.1) is equivalent to the norm of the form

$$\begin{aligned}& \Vert u\Vert _{\alpha,p}= \bigl\Vert {{}_{0}^{c}D_{t}^{\alpha}}u \bigr\Vert _{L^{p}}. \end{aligned}$$

(3.4)

Hence, in what follows, we can consider $E_{0}^{\alpha,p}$ with the norm (3.4).

Lemma 3.3

([20])

Let $0<\alpha\leq1$ and $1< p<\infty$. Assume that $\alpha>1/p$ and the sequence $\{u_{k}\}$ converges weakly to u in $E_{0}^{\alpha,p}$, that is, $u_{k}\rightharpoonup u$. Then $u_{k}\rightarrow u$ in $C([0,T],\mathbb{R})$, that is,

$$\begin{aligned}& \Vert u_{k}-u\Vert _{\infty}\rightarrow0,\quad k\rightarrow \infty. \end{aligned}$$

For $v\in E_{0}^{\alpha,p}$, by Remark 3.1 and Definition 2.2 we have

$$\begin{aligned} \int_{0}^{T} \bigl[{{}_{t}D_{T}^{\alpha}} \phi_{p} \bigl({{}_{0}D_{t}^{\alpha}}u(t) \bigr) \bigr]v(t)\,dt &= \int_{0}^{T} \bigl[{{}_{t}D_{T}^{\alpha}} \phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr]v(t)\,dt \\ &=- \int_{0}^{T}v(t)\,d\bigl[{{}_{t}D_{T}^{\alpha-1}} \phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr] \\ &= \int_{0}^{T} \bigl[{{}_{t}D_{T}^{\alpha-1}} \phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr]v'(t)\,dt. \end{aligned}$$

Thus, from Property 2.2 and Definition 2.3 we have

$$\begin{aligned} \int_{0}^{T} \bigl[{{}_{t}D_{T}^{\alpha}} \phi_{p} \bigl({{}_{0}D_{t}^{\alpha}}u(t) \bigr) \bigr]v(t)\,dt &= \int_{0}^{T}\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr){{}_{0}D_{t}^{\alpha-1}}v'(t)\,dt \\ &= \int_{0}^{T}\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr){{}_{0}^{c}D_{t}^{\alpha}}v(t)\,dt. \end{aligned}$$

So we can define the weak solutions of FBVP (1.1) as follows.

Definition 3.2

By a weak solution to FBVP (1.1) we mean a function $u\in E_{0}^{\alpha,p}$ such that $f(\cdot,u(\cdot))\in L^{1}([0,T],\mathbb{R})$ and the following equation holds:

$$\begin{aligned}& \int_{0}^{T}\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr){{}_{0}^{c}D_{t}^{\alpha}}v(t)\,dt = \int_{0}^{T}f \bigl(t,u(t) \bigr)v(t)\,dt,\quad \forall v \in E_{0}^{\alpha,p}. \end{aligned}$$

Next, we shall introduce a functional for FBVP (1.1) on $E_{0}^{\alpha,p}$. Also, we will show that the critical points of that functional are weak solutions of FBVP (1.1).

Define the functional $I:E_{0}^{\alpha,p}\rightarrow\mathbb{R}$ by

$$\begin{aligned}& I(u)=\frac{1}{p} \int_{0}^{T} \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert ^{p}\,dt - \int_{0}^{T}F \bigl(t,u(t) \bigr)\,dt, \end{aligned}$$

(3.5)

where $F(t,s)=\int_{0}^{s}f(t,\tau)\,d\tau$.

Remark 3.3

By Lemma 3.3 we get that the functional $u\rightarrow \int_{0}^{T}F(t,u(t))\,dt$ is weakly continuous on $E_{0}^{\alpha,p}$. Hence, as the sum of a convex continuous functional and a weakly continuous one, I is a weakly lower semicontinuous functional on $E_{0}^{\alpha,p}$ with $\alpha>1/p$. Moreover, following [28], we can show that $I\in C^{1}(E_{0}^{\alpha,p},\mathbb{R})$, and we have

$$\begin{aligned} \bigl\langle I'(u),v\bigr\rangle =& \int_{0}^{T}\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr){{}_{0}^{c}D_{t}^{\alpha}}v(t)\,dt \\ &{}- \int_{0}^{T}f \bigl(t,u(t) \bigr)v(t)\,dt,\quad \forall v \in E_{0}^{\alpha,p}. \end{aligned}$$

(3.6)

Remark 3.4

By Definition 3.2 and (3.6), if $u\in E_{0}^{\alpha,p}$ is a solution of the Euler equation $I'(u)=0$, then u is a weak solution of FBVP (1.1).

4 Existence of weak solutions of FBVP (1.1)

For finding the critical points of the functional I defined by (3.5), we need to use some critical point theorems, which can be found, for example, in [27, 28]. For the reader’s convenience, we present some necessary definitions and theorems of critical point theory.

Let X be a real Banach space, and let $C^{1}(X,\mathbb{R})$ denote the space of continuously Fréchet-differentiable functionals on X.

Definition A

Let $\varphi\in C^{1}(X,\mathbb{R})$. If any sequence $\{u_{k}\}\subset X$ for which $\{\varphi(u_{k})\}$ is bounded and $\varphi '(u_{k})\rightarrow0$ as $k\rightarrow\infty$ possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (P.S. condition for short).

Theorem A

([27])

Let X be a real reflexive Banach space. If the functional $\varphi:X\rightarrow\mathbb{R}$ is weakly lower semicontinuous and coercive, that is, $\lim_{\Vert z\Vert \rightarrow\infty}\varphi(z)=+\infty$, then there exists $z_{0}\in X$ such that $\varphi(z_{0})=\inf_{z\in X}\varphi(z)$. Moreover, if φ is also Fréchet differentiable on X, then $\varphi'(z_{0})=0$.

Theorem B

(Mountain pass theorem [28])

Let X be a real Banach space, and $\varphi\in C^{1}(X,\mathbb{R})$ satisfy the P.S. condition. Suppose that

($\mathrm{C}_{1}$):: $\varphi(0)=0$,

($\mathrm{C}_{2}$):: there exist $\rho>0$ and $\sigma>0$ such that $\varphi(z)\geq \sigma$ for all $z\in X$ with $\Vert z\Vert =\rho$,

($\mathrm{C}_{3}$):: there exists $z_{1}\in X$ with $\Vert z_{1}\Vert \geq\rho$ such that $\varphi(z_{1})<\sigma$.

Then φ possesses a critical value $c\geq\sigma$. Moreover, c can be characterized as

$$\begin{aligned}& c=\inf_{g\in\Omega}\max_{z\in g([0,1])}\varphi(z), \end{aligned}$$

where $\Omega=\{g\in C([0,1],X)\vert g(0)=0,g(1)=z_{1}\}$.

First, we use Theorem A to consider the existence of weak solutions of FBVP (1.1).

Theorem 4.1

Let $1/p<\alpha\leq1$ and ($\mathrm{H}_{1}$) be satisfied. Assume that

($\mathrm{H}_{2}$):: there exist $a\in(0,(\Gamma(\alpha+1))^{p}/pT^{\alpha p})$ and $b\in L^{1}([0,T],\mathbb{R}^{+})$ such that
$$\begin{aligned}& \bigl\vert F(t,x) \bigr\vert \leq a\vert x\vert ^{p}+b(t),\quad \forall t\in[0,T],x\in\mathbb{R}. \end{aligned}$$

Then FBVP (1.1) has at least one weak solution that minimizes I on $E_{0}^{\alpha,p}$.

Proof

According to Lemma 3.1, Remark 3.3, and Theorem A, we only need to prove that I is coercive on $E_{0}^{\alpha,p}$.

For $u\in E_{0}^{\alpha,p}$, it follows from ($\mathrm{H}_{2}$) that

$$\begin{aligned} I(u) =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}-a \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt- \int_{0}^{T}b(t)\,dt \\ =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}-a\Vert u \Vert _{L^{p}}^{p}-\Vert b\Vert _{L^{1}}, \end{aligned}$$

which, together with (3.2), implies

$$\begin{aligned}& I(u)\geq\biggl[\frac{1}{p}-\frac{aT^{\alpha p}}{(\Gamma(\alpha+1))^{p}} \biggr] \Vert u\Vert _{\alpha,p}^{p}-\Vert b\Vert _{L^{1}}. \end{aligned}$$

Thus, noting that $a\in(0,(\Gamma(\alpha+1))^{p}/pT^{\alpha p})$, we have

$$\begin{aligned}& \lim_{\Vert u\Vert _{\alpha,p}\rightarrow\infty}I(u)=+\infty, \end{aligned}$$

that is, I is coercive. The proof is complete. □

Next, we use Theorem B to discuss the existence of mountain pass solutions of FBVP (1.1).

Theorem 4.2

Let $1/p<\alpha\leq1$ and ($\mathrm{H}_{1}$) be satisfied. Assume that

($\mathrm{H}_{3}$):: there exist constants $\mu\in(0,1/p)$ and $M>0$ such that
$$\begin{aligned}& 0< F(t,x)\leq\mu xf(t,x),\quad \forall t\in[0,T],x\in\mathbb{R} \textit{ with } \vert x \vert \geq M, \end{aligned}$$

($\mathrm{H}_{4}$):: for $t\in[0,T]$ and $x\in\mathbb{R}$, we have
$$\begin{aligned}& \limsup_{\vert x\vert \rightarrow0}\frac{F(t,x)}{\vert x\vert ^{p}} < \frac{(\Gamma(\alpha+1))^{p}}{pT^{\alpha p}}. \end{aligned}$$

Then FBVP (1.1) has at least one nontrivial weak solution on $E_{0}^{\alpha,p}$.

Proof

We will verify that I satisfies all the conditions of Theorem B.

First, we show that I satisfies the P.S. condition. Since $F(t,x)-\mu xf(t,x)$ is continuous, there exists $c\in\mathbb{R^{+}}$ such that

$$\begin{aligned}& F(t,x)\leq\mu xf(t,x)+c,\quad t\in[0,T],\vert x\vert \leq M. \end{aligned}$$

Thus, from ($\mathrm{H}_{3}$) we get

$$\begin{aligned}& F(t,x)\leq\mu xf(t,x)+c,\quad t\in[0,T],x\in\mathbb{R}. \end{aligned}$$

(4.1)

Let $\{u_{k}\}\subset E_{0}^{\alpha,p}$ be such that

$$\begin{aligned}& \bigl\vert I(u_{k}) \bigr\vert \leq K,\quad I'(u_{k}) \rightarrow0\quad \mbox{as } {k\rightarrow\infty}. \end{aligned}$$

According to (3.6), we have

$$\begin{aligned}& \bigl\langle I'(u_{k}),u_{k}\bigr\rangle = \Vert u_{k} \Vert _{\alpha,p}^{p}- \int_{0}^{T}f \bigl(t,u_{k}(t) \bigr)u_{k}(t)\,dt, \end{aligned}$$

which, together with (4.1), yields

$$\begin{aligned} K \geq&I(u_{k}) \\ =&\frac{1}{p}\Vert u_{k}\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u_{k}(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u_{k}\Vert _{\alpha,p}^{p}- \mu \int_{0}^{T}f \bigl(t,u_{k}(t) \bigr)u_{k}(t)\,dt-cT \\ =& \biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}+\mu\bigl\langle I'(u_{k}),u_{k} \bigr\rangle -cT \\ \geq& \biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}-\mu\bigl\Vert I'(u_{k}) \bigr\Vert _{-\alpha,q} \Vert u_{k}\Vert _{\alpha,p}-cT, \end{aligned}$$

where q is a constant such that $1/p+1/q=1$. Since $I'(u_{k})\rightarrow 0$, there exists $N_{0}\in\mathbb{N}$ such that

$$\begin{aligned}& K\geq\biggl(\frac{1}{p}-\mu\biggr)\Vert u_{k}\Vert _{\alpha,p}^{p}-\Vert u_{k}\Vert _{\alpha,p}-cT,\quad k>N_{0}. \end{aligned}$$

It follows from $\mu\in(0,1/p)$ that $\{u_{k}\}$ is bounded in $E_{0}^{\alpha,p}$. Since $E_{0}^{\alpha,p}$ is a reflexive space, up to a subsequence, we can assume that $u_{k}\rightharpoonup u$ in $E_{0}^{\alpha,p}$. Hence, we have

$$\begin{aligned} & \bigl\langle I'(u_{k})-I'(u),u_{k}-u \bigr\rangle \\ &\quad =\bigl\langle I'(u_{k}),u_{k}-u\bigr\rangle -\bigl\langle I'(u),u_{k}-u\bigr\rangle \\ &\quad \leq \bigl\Vert I'(u_{k}) \bigr\Vert _{-\alpha,q} \Vert u_{k}-u\Vert _{\alpha,p}-\bigl\langle I'(u),u_{k}-u\bigr\rangle \\ &\quad \rightarrow 0, \quad k\rightarrow\infty. \end{aligned}$$

(4.2)

Moreover, by (3.3) and Lemma 3.3 we obtain that $u_{k}$ is bounded in $C([0,T],\mathbb{R})$ and $\Vert u_{k}-u\Vert _{\infty}\rightarrow0$ as $k\rightarrow\infty$. Then we get

$$\begin{aligned}& \int_{0}^{T} \bigl(f \bigl(t,u_{k}(t) \bigr)-f \bigl(t,u(t) \bigr) \bigr) \bigl(u_{k}(t)-u(t) \bigr)\,dt \rightarrow0, \quad k\rightarrow\infty. \end{aligned}$$

(4.3)

Note that

$$\begin{aligned} & \bigl\langle I'(u_{k})-I'(u),u_{k}-u \bigr\rangle \\ &\quad = \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ & \qquad {}- \int_{0}^{T} \bigl(f \bigl(t,u_{k}(t) \bigr)-f \bigl(t,u(t) \bigr) \bigr) \bigl(u_{k}(t)-u(t) \bigr)\,dt. \end{aligned}$$

Thus, from (4.2) and (4.3) we have

$$\begin{aligned}& \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \rightarrow0 \end{aligned}$$

(4.4)

as $k\rightarrow\infty$.

Following [33], we obtain that there exist $c_{1},c_{2}>0$ such that

$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq \textstyle\begin{cases} c_{1}\int_{0}^{T}\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{p}\,dt,& p\geq2,\\ c_{2}\int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{(\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt,& 1< p< 2. \end{cases}\displaystyle \end{aligned}$$

(4.5)

When $1< p<2$, we have

$$\begin{aligned} & \int_{0}^{T} \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert ^{p}\,dt \\ &\quad \leq \biggl( \int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{ (\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt \biggr)^{\frac{p}{2}} \\ &\qquad {}\cdot\biggl( \int_{0}^{T} \bigl( \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr\vert + \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert \bigr)^{p}\,dt \biggr) ^{\frac{2-p}{2}}. \end{aligned}$$

Thus, noting that $(s_{1}+s_{2})^{\gamma}\leq2^{\gamma-1}(s_{1}^{\gamma}+s_{2}^{\gamma})$ where $s_{1},s_{2}\geq0$, $\gamma\geq1$ (see [34]), we have

$$\begin{aligned} & \int_{0}^{T} \bigl\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr\vert ^{p}\,dt \\ &\quad \leq c_{3} \bigl(\Vert u_{k}\Vert _{\alpha,p}^{p}+ \Vert u\Vert _{\alpha,p}^{p} \bigr)^{\frac{2-p}{2}} \\ &\qquad {}\cdot\biggl( \int_{0}^{T}\frac{\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert ^{2}}{ (\vert {{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)\vert +\vert {{}_{0}^{c}D_{t}^{\alpha}}u(t)\vert )^{2-p}}\,dt \biggr)^{\frac{p}{2}}, \end{aligned}$$

where $c_{3}=2^{(p-1)(2-p)/2}$, which, together with (4.5), implies

$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq c_{2}c_{3}^{-\frac{2}{p}} \bigl(\Vert u_{k}\Vert _{\alpha,p}^{p}+\Vert u\Vert _{\alpha,p}^{p} \bigr)^{\frac{p-2}{p}} \Vert u_{k}-u \Vert _{\alpha,p}^{2},\quad 1< p< 2. \end{aligned}$$

(4.6)

When $p\geq2$, by (4.5) we get

$$\begin{aligned} & \int_{0}^{T} \bigl(\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t) \bigr)-\phi_{p} \bigl({{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr) \bigr) \bigl({{}_{0}^{c}D_{t}^{\alpha}}u_{k}(t)-{{}_{0}^{c}D_{t}^{\alpha}}u(t) \bigr)\,dt \\ &\quad \geq c_{1}\Vert u_{k}-u\Vert _{\alpha,p}^{p},\quad p\geq2. \end{aligned}$$

(4.7)

It follows from (4.4), (4.6), and (4.7) that

$$\begin{aligned}& \Vert u_{k}-u\Vert _{\alpha,p}\rightarrow0,\quad k\rightarrow \infty, \end{aligned}$$

that is, $\{u_{k}\}$ converges strongly to u in $E_{0}^{\alpha,p}$.

Now we show that I satisfies the geometry conditions of mountain pass theorem.

By ($\mathrm{H}_{4}$) there exist $\varepsilon\in(0,1)$ and $\delta>0$ such that

$$\begin{aligned}& F(t,x)\leq\frac{(1-\varepsilon)(\Gamma(\alpha +1))^{p}}{pT^{\alpha p}}\vert x\vert ^{p},\quad t \in[0,T],x\in\mathbb{R} \mbox{ with } \vert x\vert \leq\delta. \end{aligned}$$

(4.8)

Let $\rho=\frac{\Gamma(\alpha)((\alpha-1)q+1)^{1/q}}{T^{\alpha -1/p}}\delta>0$ and $\sigma=\varepsilon\rho^{p}/p>0$. Then, by (3.3) we have

$$\begin{aligned}& \Vert u\Vert _{\infty}\leq\frac{T^{\alpha-\frac{1}{p}}}{ \Gamma(\alpha)((\alpha-1)q+1)^{\frac{1}{q}}}\Vert u\Vert _{\alpha,p}=\delta,\quad u\in E_{0}^{\alpha,p} \mbox{ with } \Vert u\Vert _{\alpha,p}=\rho, \end{aligned}$$

which, together with (3.2) and (4.8), implies

$$\begin{aligned} I(u) =&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,u(t) \bigr)\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p} - \frac{(1-\varepsilon)(\Gamma(\alpha+1))^{p}}{pT^{\alpha p}} \int_{0}^{T} \bigl\vert u(t) \bigr\vert ^{p}\,dt \\ \geq&\frac{1}{p}\Vert u\Vert _{\alpha,p}^{p}- \frac{1-\varepsilon}{p} \Vert u\Vert _{\alpha,p}^{p} \\ =&\frac{\varepsilon}{p} \Vert u\Vert _{\alpha,p}^{p} \\ =&\sigma,\quad \forall u\in E_{0}^{\alpha,p} \mbox{ with } \Vert u \Vert _{\alpha,p}=\rho. \end{aligned}$$

Hence, condition ($\mathrm{C}_{2}$) in Theorem B is satisfied.

By ($\mathrm{H}_{3}$) a simple argument using the very definition of derivative shows that there exist $c_{4},c_{5}>0$ such that

$$\begin{aligned}& F(t,x)\geq c_{4}\vert x\vert ^{\frac{1}{\mu}}-c_{5},\quad t \in[0,T],x\in\mathbb{R}. \end{aligned}$$

For any $u\in E_{0}^{\alpha,p}\setminus\{0\}$, $\xi\in\mathbb{R}^{+}$, noting that $\mu\in(0,1/p)$, we get

$$\begin{aligned} I(\xi u) =&\frac{1}{p}\Vert \xi u\Vert _{\alpha,p}^{p}- \int_{0}^{T}F \bigl(t,\xi u(t) \bigr)\,dt \\ \leq&\frac{\xi^{p}}{p}\Vert u\Vert _{\alpha,p}^{p} -c_{4} \int_{0}^{T} \bigl\vert \xi u(t) \bigr\vert ^{\frac{1}{\mu}}\,dt+c_{5}T \\ =&\frac{\xi^{p}}{p}\Vert u\Vert _{\alpha,p}^{p} -c_{4}\xi^{\frac{1}{\mu}} \Vert u\Vert _{L^{\frac{1}{\mu}}}^{\frac {1}{\mu}}+c_{5}T \\ \rightarrow&-\infty, \quad \xi\rightarrow\infty. \end{aligned}$$

Thus, taking $\xi_{0}$ large enough and letting $e=\xi_{0}u$, we have $I(e)\leq0$. Therefore, condition ($\mathrm{C}_{3}$) in Theorem B is also satisfied.

Lastly, noting that $I(0)=0$, we get a critical point u such that $I(u)\geq\sigma>0$. Hence, u is a nontrivial weak solution of FBVP (1.1). The proof is complete. □

Declarations

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments, which have improved the presentation and quality of the manuscript. This work is supported by the National Natural Science Foundation of China (11271364) and the Nature Science Foundation of Jiangsu Province (BK20130170).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)

Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R. China

References

Diethelm, K, Freed, AD: On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In: Keil, F, Mackens, W, Voss, H, Werther, J (eds.) Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, pp. 217-224. Springer, Heidelberg (1999) View ArticleGoogle Scholar
Glockle, WG, Nonnenmacher, TF: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46-53 (1995) View ArticleGoogle Scholar
Hilfer, R: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) View ArticleMATHGoogle Scholar
Kirchner, JW, Feng, X, Neal, C: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, 524-526 (2000) View ArticleGoogle Scholar
Lundstrom, BN, Higgs, MH, Spain, WJ, Fairhall, AL: Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci. 11, 1335-1342 (2008) View ArticleGoogle Scholar
Mainardi, F: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A, Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291-348. Springer, Wien (1997) View ArticleGoogle Scholar
Bai, Z, Lü, H: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495-505 (2005) MathSciNetView ArticleMATHGoogle Scholar
Jiang, W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal. TMA 74, 1987-1994 (2011) MathSciNetView ArticleMATHGoogle Scholar
Zhang, S: Existence of a solution for the fractional differential equation with nonlinear boundary conditions. Comput. Math. Appl. 61, 1202-1208 (2011) MathSciNetView ArticleMATHGoogle Scholar
Agarwal, RP, O’Regan, D, Stanek, S: Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations. J. Math. Anal. Appl. 371, 57-68 (2010) MathSciNetView ArticleMATHGoogle Scholar
Babakhani, A, Gejji, VD: Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278, 434-442 (2003) MathSciNetView ArticleMATHGoogle Scholar
Benchohra, M, Hamani, S, Ntouyas, SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391-2396 (2009) MathSciNetView ArticleMATHGoogle Scholar
Bisci, GM, Repovs, D: Higher nonlocal problems with bounded potential. J. Math. Anal. Appl. 420, 167-176 (2014) MathSciNetView ArticleMATHGoogle Scholar
Chen, T, Liu, W: An anti-periodic boundary value problem for the fractional differential equation with a p-Laplacian operator. Appl. Math. Lett. 25, 1671-1675 (2012) MathSciNetView ArticleMATHGoogle Scholar
Darwish, MA, Ntouyas, SK: On initial and boundary value problems for fractional order mixed type functional differential inclusions. Comput. Math. Appl. 59, 1253-1265 (2010) MathSciNetView ArticleMATHGoogle Scholar
El-Shahed, M, Nieto, JJ: Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl. 59, 3438-3443 (2010) MathSciNetView ArticleMATHGoogle Scholar
Kosmatov, N: A boundary value problem of fractional order at resonance. Electron. J. Differ. Equ. 2010, 135 (2010) MathSciNetMATHGoogle Scholar
Su, X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64-69 (2009) MathSciNetView ArticleMATHGoogle Scholar
Jiao, F, Zhou, Y: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181-1199 (2011) MathSciNetView ArticleMATHGoogle Scholar
Jiao, F, Zhou, Y: Existence results for fractional boundary value problem via critical point theory. Int. J. Bifurc. Chaos Appl. Sci. Eng. 22, Article ID 1250086 (2012) MathSciNetView ArticleMATHGoogle Scholar
Torres, C: Mountain pass solution for a fractional boundary value problem. J. Fract. Calc. Appl. 5(1), 1-10 (2014) MathSciNetGoogle Scholar
Bisci, GM, Repovs, D: On some variational algebraic problems. Adv. Nonlinear Anal. 2, 127-146 (2013) MathSciNetMATHGoogle Scholar
Li, C, Agarwal, RP, Tang, C: Infinitely many periodic solutions for ordinary p-Laplacian systems. Adv. Nonlinear Anal. 4, 251-261 (2015) MathSciNetMATHGoogle Scholar
Pucci, P, Radulescu, V: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 9, 543-582 (2010) MathSciNetMATHGoogle Scholar
Radulescu, V: Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities. In: Handbook of Differential Equations: Stationary Partial Differential Equations, vol. IV, pp. 485-593. Elsevier, Amsterdam (2007) Google Scholar
Radulescu, V: Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods. Contemporary Mathematics and Its Applications, vol. 6. Hindawi Publishing Corporation, New York (2008) View ArticleMATHGoogle Scholar
Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) View ArticleMATHGoogle Scholar
Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65. Am. Math. Soc., Providence (1986) MATHGoogle Scholar
Leibenson, LS: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSR, Ser. Biol. Nauk 9, 7-10 (1983) MathSciNetGoogle Scholar
Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
Podlubny, I: Fractional Differential Equation. Academic Press, San Diego (1999) MATHGoogle Scholar
Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993) MATHGoogle Scholar
Simon, J: Régularité de la solution d’un problème aux limites non linéaires. Ann. Fac. Sci. Toulouse 3, 247-274 (1981) View ArticleMATHGoogle Scholar
Lian, H: Boundary value problems for nonlinear ordinary differential equations on Infinite intervals. Doctoral thesis (2007) Google Scholar