Skip to main content

Theory and Modern Applications

Nontrivial solutions for boundary value problems of a fourth order difference equation with sign-changing nonlinearity

Abstract

In this paper, using the topological degree theory, we establish two existence theorems for nontrivial solutions for boundary value problems of a fourth order difference equation with a sign-changing nonlinearity.

1 Introduction

For \(a,b\in\mathbb {Z}\), let \(\mathbb {T}_{a}^{b}=\{ a,a+1,a+2,\ldots,b\}\) with \(a< b\). In this paper we consider the existence of nontrivial solutions for boundary value problems of the following fourth order difference equation with a sign-changing nonlinearity

$$ \textstyle\begin{cases} \Delta^{4} u(t-2)=f(t,u(t)), \\ u(1)=u(T+1)=\Delta^{2} u(0)=\Delta^{2} u(T)=0, \end{cases} $$
(1.1)

where T is an integer with \(T\ge5\), and \(f:\mathbb {T}_{2}^{T}\times \mathbb {R}\to\mathbb {R}\) is a continuous function with \(\mathbb {T}_{2}^{T}=\{ 2,3,\ldots,T\}\) and \(\mathbb {R}=(-\infty,+\infty)\) (it is assumed to be continuous from the topological space \(\mathbb {T}_{2}^{T}\times\mathbb {R} \) into the topological space \(\mathbb {R}\), the topology on \(\mathbb {T}_{2}^{T}\) being the discrete topology).

Difference equations with discrete boundary value conditions have been widely studied in the literature; see, for example, [111] and the references therein. However, as mentioned in [6], very few results are available with sign-changing nonlinearities; see [611]. Other related work in this field can be found in [1245] and the references therein. In [7], C.S. Goodrich used the Krasnosel’skiĭ fixed point theorem to obtain the existence of at least one positive solution to the following discrete fractional semipositone boundary value problem

$$ \textstyle\begin{cases} \Delta^{\nu}y(t)=\lambda f(t+\nu-1,y(t+\nu-1)), \quad t\in[0,T]\cap \mathbb {Z}, \\ y(\nu-1)=y(\nu+T)+\sum_{i=1}^{N} F(t_{i}, y(t_{i})), \end{cases} $$
(1.2)

where \(\Delta^{\nu}\) is the νth fractional difference with \(\nu \in(0,1)\), f is continuous, bounded below (i.e., \(f+M\ge0\) for some \(M>0\)), and

$$ \lim_{y\to+\infty}\frac {f(t,y)}{y}=0 \quad \text{uniformly for }t\in[\nu-1,\nu+T]_{\mathbb {Z}_{\nu-1}}. $$
(1.3)

In [10], J. Xu and D. O’Regan used the fixed point index to obtain the existence of nontrivial solutions for (1.2) with weaker conditions than that of (1.3), and also in [11], J. Xu et al. considered the existence of positive solutions for system (1.2), with adopted convex and concave functions to depict the coupling behavior of nonlinearities. In [40], Y. Cui used the \(u_{0}\)-positive operator to study the uniqueness of solutions for the following nonlinear fractional boundary value problems:

$$ \textstyle\begin{cases} D^{p}x(t)+p(t)f(t,x(t))+q(t)=0,\quad t\in(0,1), \\ x(0)=x'(0)=0,\qquad x(1)=0, \end{cases} $$
(1.4)

where \(D^{p}\) is the Riemann–Liouville fractional derivative, and f is a Lipschitz continuous function, with the Lipschitz constant associated with the first eigenvalue for the relevant operator. Using similar methods, the authors in [12, 39, 41] obtained some existence and nonexistence theorems for their problems.

Motivated by the works mentioned above, we consider the existence of nontrivial solutions for (1.1) involving sign-changing nonlinearities. Using the topological degree theory of a completely continuous field, and conditions concerning the first eigenvalue corresponding to the relevant linear problem, two existence theorems are obtained.

2 Preliminaries

For convenience, we let \(\mathbb {T}_{1}^{T+1}=\{1,2,3,\ldots,T,T+1\}\), \(\mathbb {T}_{0}^{T+2}=\{0,1,2,3,\ldots,T+1,T+2\}\), \(\mathbb {T}_{2}^{T}=\{ 2,3,\ldots,T\}\). Then we define our space E as the collection of all maps from \(\mathbb {T}_{0}^{T+2}\) to \(\mathbb {R}\) equipped with the norm \(\| u\|=\max_{j\in\mathbb {T}_{0}^{T+2}}|u(j)|\). Consequently, E is a Banach space, and we let \(P=\{u\in E: u(t)\ge0, t\in\mathbb {T}_{1}^{T+1}\}\). Then P is a cone on E. Throughout our paper, we let \(B_{\rho}=\{u\in E:\|u\|<\rho\}\) for \(\rho>0\). Now \(\partial B_{\rho}=\{ u\in E: \|u\|=\rho\}\) and \(\overline{B}_{\rho}=\{u\in E: \|u\|\le\rho \}\).

In what follows, we establish the Green’s function for (1.1). As in [3, 4], we transform (1.1) into its equivalent sum equation

$$ u(t)= \sum_{s=2}^{T} H(t,s) \sum_{j=2}^{T} H(s,j) f\bigl(j,u(j)\bigr), \quad t\in\mathbb {T}_{1}^{T+1}, $$
(2.1)

where

$$ H(t,s)=\frac{1}{T} \textstyle\begin{cases} (t-1)(T+1-s), & 1\le t\le s\le T, \\ (s-1)(T+1-t), & 2\le s\le t\le T+1. \end{cases} $$
(2.2)

Lemma 2.1

Green’s function H has the following properties:

  1. (i)

    \(H(t,s)>0\) for \((t,s)\in\mathbb {T}_{2}^{T}\times\mathbb {T}_{2}^{T}\),

  2. (ii)

    \(\frac{1}{T}H(t,t)H(s,s)\le H(t,s)\le H(s,s)\) for \((t,s)\in \mathbb {T}_{2}^{T}\times\mathbb {T}_{1}^{T+1}\).

Proof

We only need to prove the first inequality of (ii). Indeed, for all \((t,s)\in\mathbb {T}_{2}^{T}\times\mathbb {T}_{1}^{T+1}\), from the definitions of \(H(t,s)\) and \(H(s,s)\) we have

$$\frac{H(t,s)}{H(s,s)}= \textstyle\begin{cases} \frac{t-1}{s-1}\ge\frac{t-1}{T}\ge\frac{t-1}{T} \frac{T+1-t}{T}= \frac{1}{T} H(t,t) , & 1\le t\le s\le T, \\ \frac{T+1-t}{T+1-s}\ge\frac{T+1-t}{T} \ge \frac{T+1-t}{T} \frac {t-1}{T}= \frac{1}{T} H(t,t), & 2\le s\le t\le T+1. \end{cases} $$

Then we have \(H(t,s)\ge \frac{1}{T} H(t,t)H(s,s) \) for \((t,s)\in \mathbb {T}_{2}^{T}\times\mathbb {T}_{1}^{T+1}\). This completes the proof. □

We define an operator \(A: E\to E\) as follows:

$$ (Au) (t)= \sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) f\bigl(j,u(j) \bigr),\quad t\in\mathbb {T}_{1}^{T+1}. $$
(2.3)

The existence of solutions for (1.1) is equivalent to that of fixed points of A.

From [4], we know that \(\sin\frac{\pi(t-1)}{T}:=\varphi _{0}(t)\), \(t\in\mathbb {T}_{2}^{T}\) is the eigenfunction related to the eigenvalue \(\frac{1}{16} \sin^{-4} \frac{\pi}{2T}\) of the eigenproblem

$$\textstyle\begin{cases} \Delta^{4} u(t-2)=\lambda u(t), \quad t\in\mathbb {T}_{2}^{T}, \\ u(1)=u(T+1)=\Delta^{2} u(0)=\Delta^{2} u(T)=0, \end{cases} $$

i.e., the following two equations hold:

$$\begin{aligned}& \sum_{s=2}^{T} \sum _{j=2}^{T} H(t,s) H(s,j)\sin\frac{\pi(j-1)}{T}= \frac{1}{16} \sin^{-4} \frac{\pi}{2T} \sin\frac{\pi(t-1)}{T} , \quad t\in\mathbb {T}_{2}^{T}, \end{aligned}$$
(2.4)
$$\begin{aligned}& \sum_{s=2}^{T} \sum _{t=2}^{T} H(t,s) H(s,j)\sin\frac{\pi(t-1)}{T}= \frac{1}{16} \sin^{-4} \frac{\pi}{2T} \sin\frac{\pi(j-1)}{T} , \quad t\in\mathbb {T}_{2}^{T}. \end{aligned}$$
(2.5)

Lemma 2.2

Let \(e(t)= \frac{1}{T} H(t,t) \) and \(P_{0}=\{u\in P: u(t)\ge e(t)\|u\|, t\in\mathbb {T}_{1}^{T+1}\}\). Then \(L(P)\subset P_{0}\), where

$$ (Lu) (t)=\sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) u(j), \quad t\in \mathbb {T}_{1}^{T+1}. $$
(2.6)

This is a direct result from Lemma 2.1(ii), so we omit its proof.

Now, we offer two basic theorems from the topological degree theory; for details we refer the reader to [46].

Lemma 2.3

Let E be a Banach space and Ω a bounded open set in E. Suppose that \(A: \Omega\to E\) is a continuous compact operator. If there exists \(u_{0}\in E\setminus \{0\}\) such that

$$u-Au\neq \mu u_{0}, \quad \forall u\in\partial\Omega, \mu\ge0, $$

then the topological degree \(\deg(I-A,\Omega,0)=0\).

Lemma 2.4

Let E be a Banach space and Ω a bounded open set in E with \(0\in\Omega\). Suppose that \(A: \Omega\to E\) is a continuous compact operator. If

$$Au\neq \mu u, \quad \forall u\in\partial\Omega, \mu\ge1, $$

then the topological degree \(\deg(I-A,\Omega,0)=1\).

3 Nontrivial solutions for (1.1)

Now we present some assumptions for our nonlinearity f.

  1. (H1)

    There exist two constants \(a>0\), \(b>0\) and a function \(k\in C(\mathbb {R}, \mathbb {R}^{+})\) such that

    $$f(t,u)\ge-a-bk(u),\quad \forall u\in\mathbb {R}, t\in\mathbb {T}_{2}^{T}. $$
  2. (H2)

    \(\lim_{|u|\to+\infty} \frac{k(u)}{|u|}=0\).

  3. (H3)

    \(\liminf_{|u|\to+\infty}\frac{f(t,u)}{|u|}>16 \sin^{4} \frac {\pi}{2T}\) uniformly on \(t\in\mathbb {T}_{2}^{T}\),

  4. (H4)

    \(\limsup_{|u|\to0}\frac{|f(t,u)|}{|u|}<16 \sin^{4} \frac{\pi }{2T}\) uniformly on \(t\in\mathbb {T}_{2}^{T}\),

  5. (H5)

    \(\liminf_{u\to0^{+}}\frac{f(t,u)}{u}>16 \sin^{4} \frac{\pi}{2T}\), \(\limsup_{u\to0^{-}}\frac{f(t,u)}{u}<16 \sin^{4} \frac{\pi}{2T}\), uniformly on \(t\in\mathbb {T}_{2}^{T}\),

  6. (H6)

    \(\limsup_{|u|\to+\infty}\frac{|f(t,u)|}{|u|}<16 \sin^{4} \frac{\pi}{2T}\) uniformly on \(t\in\mathbb {T}_{2}^{T}\).

Theorem 3.1

Suppose that (H1)(H4) hold. Then (1.1) has at least one nontrivial solution.

Proof

From (H3) there exist \(\varepsilon_{0}>0\) and \(X_{0}>0\) such that

$$ f(t,u)\ge \biggl(16 \sin^{4} \frac{\pi }{2T}+\varepsilon_{0} \biggr)|u|, \quad \forall t\in\mathbb {T}_{2}^{T}, |u|>X_{0}. $$
(3.1)

For any given ε with \(\varepsilon_{0} - b\varepsilon>0\), and from (H2), there exists \(X_{1}>X_{0}\) such that

$$ k(u)\le\varepsilon|u|,\quad \forall|u|>X_{1}. $$
(3.2)

Now since \(a>0\), \(b>0\) and k is a nonnegative function, we have

$$\begin{aligned} f(t,u)&\ge \biggl(16 \sin^{4} \frac{\pi }{2T}+ \varepsilon_{0} \biggr)|u|-a-bk(u) \\ & \ge \biggl(16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{0} \biggr)|u|-a-b \varepsilon|u|, \quad \forall|u|>X_{1}. \end{aligned}$$
(3.3)

Now we choose \(c_{1}= (16 \sin^{4} \frac{\pi}{2T}+\varepsilon _{0}-b \varepsilon )X_{1}+\max_{t\in\mathbb {T}_{2}^{T}, |u|\le X_{1}}|f(t,u)|\) and \(k^{*}=\max_{|u|\le X_{1}}k(u)\). Then we have

$$\begin{aligned} f(t,u)&\ge \biggl(16 \sin^{4} \frac{\pi }{2T}+\varepsilon_{0}-b \varepsilon \biggr)|u|-a-c_{1} \\ &= \biggl(16 \sin ^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon \biggr)|u|-c_{2}, \quad \forall t\in\mathbb {T}_{2}^{T}, u\in\mathbb {R}, \end{aligned}$$
(3.4)

where \(c_{2}=c_{1}+a\). Note that ε can be chosen arbitrarily small, and we let

$$\begin{aligned} R >& \max \biggl\{ \frac{(c_{2}+bk^{*}) [ (\varepsilon _{0}-b \varepsilon )\sum_{s=2}^{T} H(s,s)\sum_{j=2}^{T} H(s,j)+ (16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon ) \sum_{s=2}^{T} \sum_{j=2}^{T} H(s,j) ]}{\varepsilon_{0}-b \varepsilon-b\varepsilon [ (\varepsilon_{0}-b \varepsilon )\sum_{s=2}^{T} H(s,s)\sum_{j=2}^{T} H(s,j)+ (16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon ) \sum_{s=2}^{T} \sum_{j=2}^{T} H(s,j) ]}, \\ & \frac{\sum_{s=2}^{T} H(s,s)\sum_{j=2}^{T} H(s,j)(c_{2}+bk^{*})}{1-b\varepsilon\sum_{s=2}^{T} H(s,s)\sum_{j=2}^{T} H(s,j) }, 0 \biggr\} . \end{aligned}$$

Now we prove that

$$ u-Au\neq \mu\varphi_{0},\quad \forall u\in \partial B_{R}, \mu\ge0. $$
(3.5)

From (2.4) and Lemma 2.2, we have \(\varphi_{0}=16 \sin^{4} \frac{\pi}{2T}L\varphi_{0}\in P_{0} \). Indeed, if (3.5) isn’t true, then there exist \(u_{0}\in\partial B_{R}\) and \(\mu_{0}>0\) such that

$$ u_{0}-Au_{0}=\mu_{0} \varphi_{0}. $$
(3.6)

Let \(\tilde{u}(t)=\sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j)(a+bk(u_{0})+c_{1})\). Then

$$\begin{aligned} \tilde{u}(t)&\le\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl(c_{2}+b \varepsilon|u_{0}|+bk^{*}\bigr) \\ &\le\sum_{s=2}^{T} H(s,s)\sum_{j=2}^{T} H(s,j) \bigl(c_{2}+b\varepsilon\|u_{0}\|+bk^{*}\bigr). \end{aligned}$$

Therefore,

$$ \|\tilde{u}\|\le\sum_{s=2}^{T} H(s,s)\sum _{j=2}^{T} H(s,j) \bigl(c_{2}+b \varepsilon R+bk^{*}\bigr). $$
(3.7)

Then from \(L(P)\subset P_{0}\), \(\varphi_{0}\in P_{0}\), and

$$\begin{aligned} u_{0}(t)+\tilde{u}(t)&=\tilde{u}(t)+(Au_{0}) (t)+\mu _{0}\varphi_{0}(t) \\ & =\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl( f\bigl(j,u_{0}(j) \bigr)+bk\bigl(u_{0}(j)\bigr)+a+c_{1}\bigr)+ \mu_{0}\varphi_{0}(t), \end{aligned}$$

we have

$$u_{0}+\tilde{u}\in P_{0}. $$

As a result, we obtain

$$\begin{aligned} &(Au_{0}) (t)+\tilde{u}(t) \\ &\quad =\sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \bigl( f \bigl(j,u_{0}(j)\bigr)+bk\bigl(u_{0}(j)\bigr)+c_{2} \bigr) \\ &\quad \ge\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \biggl( \biggl(16 \sin ^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon \biggr) \bigl\vert u_{0}(j) \bigr\vert -c_{2}+bk \bigl(u_{0}(j)\bigr)+c_{2} \biggr) \\ &\quad \ge \biggl(16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon \biggr) \sum_{s=2}^{T} H(t,s) \sum_{j=2}^{T} H(s,j) \bigl\vert u_{0}(j) \bigr\vert \\ &\quad \ge \biggl(16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon \biggr) \sum_{s=2}^{T} H(t,s) \sum_{j=2}^{T} H(s,j) u_{0}(j). \end{aligned}$$
(3.8)

On the other hand, from the definition of L, we get

$$\begin{aligned}& \biggl(16 \sin^{4} \frac{\pi }{2T} +\varepsilon_{0}-b \varepsilon \biggr) \sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) u_{0}(j) \\& \quad = 16 \sin^{4} \frac{\pi}{2T} \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \bigl(u_{0}(j)+\tilde{u}(j)\bigr) \\& \qquad {}-16 \sin^{4} \frac{\pi}{2T} \sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \tilde{u}(j) \\& \qquad {}+ (\varepsilon_{0}-b \varepsilon ) \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) u_{0}(j) \\& \quad \ge16 \sin^{4} \frac{\pi}{2T} \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \bigl(u_{0}(j)+\tilde{u}(j)\bigr); \end{aligned}$$
(3.9)

in order to obtain the above inequality, we prove that

$$\begin{aligned}& -16 \sin^{4} \frac{\pi}{2T} \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \tilde{u}(j) \\& \quad {}+ (\varepsilon_{0}-b \varepsilon ) \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) u_{0}(j)\ge0. \end{aligned}$$
(3.10)

Indeed, since \(u_{0}+\tilde{u}\in P_{0}\), we have \(u_{0}(t)+\tilde {u}(t)\ge e(t)\|u_{0}+\tilde{u}\|\ge e(t) (\|u_{0}\|-\|\tilde{u}\| )\). Note that \(H(t,s)\) vanishes at \(t=1\) and \(t=T+1\), \(H(t,s)\) is symmetric on \(\mathbb {T}_{2}^{T}\), i.e., \(H(t,s)=H(s,t)\). Then

$$\begin{aligned}& (\varepsilon_{0} -b \varepsilon ) \sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \bigl( \tilde{u}(j)+ u_{0}(j)\bigr) \\& \qquad {}- \biggl(16 \sin ^{4} \frac{\pi}{2T}+\varepsilon_{0}-b \varepsilon \biggr) \sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \tilde{u}(j) \\& \quad \ge (\varepsilon_{0}-b \varepsilon ) \bigl(R-\|\tilde{u}\|\bigr) \sum _{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) e(j) \\& \qquad {}- \biggl(16 \sin^{4} \frac{\pi}{2T}+ \varepsilon_{0}-b \varepsilon \biggr) \sum _{s=2}^{T} H(t,s) \\& \qquad {}\times\sum_{j=2}^{T} H(s,j) e(j) \Biggl(\sum_{s=2}^{T} \sum _{j=2}^{T} H(s,j) \bigl(c_{2}+b \varepsilon R+bk^{*}\bigr) \Biggr) \\& \quad \ge0. \end{aligned}$$

Combining (3.8), (3.9) and (3.10), we have

$$\begin{aligned} (Au_{0}) (t)+\tilde{u}(t)&\ge16 \sin^{4} \frac{\pi }{2T} \sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl(u_{0}(j)+\tilde {u}(j)\bigr) \\ &=16 \sin^{4} \frac{\pi}{2T} \bigl(L(u_{0}+ \tilde{u})\bigr) (t). \end{aligned}$$
(3.11)

Using (3.6) we obtain

$$ u_{0}+\tilde{u}=Au_{0}+\tilde{u}+\mu_{0} \varphi _{0}\ge16 \sin^{4} \frac{\pi}{2T} L(u_{0}+\tilde{u})+\mu_{0} \varphi_{0}\ge \mu_{0} \varphi_{0}. $$
(3.12)

Define

$$\mu^{*}=\sup\{\mu>0:u_{0}+\tilde{u}\ge\mu\varphi_{0}\}. $$

Note that \(\mu_{0}\in\{\mu>0:u_{0}+\tilde{u}\ge\mu\varphi_{0}\} \), and then \(\mu^{*}\ge\mu_{0}\), \(u_{0}+\tilde{u}\ge\mu^{*} \varphi_{0}\). From (2.4) we have

$$16 \sin^{4} \frac{\pi}{2T} L(u_{0}+\tilde{u})\ge\mu^{*} 16 \sin ^{4} \frac{\pi}{2T} L \varphi_{0}=\mu^{*} \varphi_{0} , $$

and hence

$$u_{0}+\tilde{u}\ge16 \sin^{4} \frac{\pi}{2T} L(u_{0}+\tilde {u})+\mu_{0} \varphi_{0}\ge\bigl( \mu_{0}+\mu^{*}\bigr)\varphi_{0}, $$

which contradicts the definition of \(\mu^{*}\). Therefore, (3.5) holds, and from Lemma 2.3 we obtain

$$ \deg(I-A,B_{R},0)=0. $$
(3.13)

On the other hand, from (H4), there exist \(\varepsilon_{1}\in(0,16 \sin ^{4} \frac{\pi}{2T})\) and \(r\in(0,R)\) such that

$$ \bigl\vert f(t,u) \bigr\vert \le{ \biggl(16 \sin^{4} \frac {\pi}{2T}-\varepsilon_{1} \biggr)|u|}, \quad \forall t\in\mathbb {T}_{2}^{T}, |u|< r. $$
(3.14)

Now for this r, we show that

$$ Au\neq \mu u,\quad u\in\partial B_{r}, \mu\ge1. $$
(3.15)

Otherwise, there would exist \(u_{1}\in\partial B_{r}\), \(\mu_{1}\ge1\) such that

$$\begin{aligned} \bigl\vert u_{1}(t) \bigr\vert &=\frac{1}{\mu_{1}} \bigl\vert (Au_{1}) (t) \bigr\vert \le \bigl\vert (Au_{1}) (t) \bigr\vert \\ &= \Biggl\vert \sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) f \bigl(j,u_{1}(j)\bigr) \Biggr\vert \\ & \le\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl\vert f\bigl(j,u_{1}(j) \bigr) \bigr\vert \\ & \le{ \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{1} \biggr) } \sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl\vert u_{1}(j) \bigr\vert . \end{aligned}$$

Multiplying both sides of the above inequality by \(\sin\frac{\pi (t-1)}{T}\), then summing from 2 to T, and using (2.5), we obtain

$$\begin{aligned} &\sum_{t=2}^{T} \bigl\vert u_{1}(t) \bigr\vert \sin\frac{\pi(t-1)}{T} \\ &\quad \le{ \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{1} \biggr)} \sum _{t=2}^{T} \Biggl[\sum _{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) \bigl\vert u_{1}(j) \bigr\vert \Biggr]\sin \frac {\pi(t-1)}{T} \\ &\quad =\frac{16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{1}}{16 \sin^{4} \frac{\pi}{2T}} \sum_{t=2}^{T} \bigl\vert u_{1}(t) \bigr\vert \sin\frac{\pi (t-1)}{T}. \end{aligned}$$

This implies that \(\sum_{t=2}^{T}|u_{1}(t)|\sin\frac{\pi(t-1)}{T}=0\), and whence \(u_{1}(t)\equiv0\), which contradicts \(u_{1}\in\partial B_{r}\). Hence, (3.15) holds, and from Lemma 2.4 we obtain

$$ \deg(I-A,B_{r},0)=1. $$
(3.16)

This, together with (3.13), implies that

$$\deg(I-A,B_{R}\setminus \overline{B}_{r},0)= \deg(I-A,B_{R},0)-\deg(I-A,B_{r},0)=-1. $$

Therefore, the operator A has at least one fixed point in \(B_{R}\setminus \overline{B}_{r}\), and (1.1) has at least one nontrivial solution. This completes the proof. □

Theorem 3.2

Suppose that (H5)(H6) hold. Then (1.1) has at least one nontrivial solution.

Proof

From (H5), there are \(\varepsilon_{2}\in(0, 16 \sin^{4} \frac{\pi}{2T})\) and \(r>0\) such that

$$f(t,u)\ge \biggl(16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{2} \biggr)u,\quad \forall u\in[0,r], t\in\mathbb {T}_{2}^{T}, $$

and

$$f(t,u)\ge \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{2} \biggr)u,\quad \forall u\in[-r,0], t\in\mathbb {T}_{2}^{T}. $$

The above two inequalities enable us to obtain

$$\begin{aligned}& f(t,u)\ge \biggl(16 \sin^{4} \frac{\pi }{2T}+ \varepsilon_{2} \biggr)u, \quad \forall u\in[-r,r], t\in\mathbb {T}_{2}^{T}, \end{aligned}$$
(3.17)
$$\begin{aligned}& f(t,u)\ge \biggl(16 \sin^{4} \frac{\pi }{2T}- \varepsilon_{2} \biggr)u,\quad \forall u\in[-r,r], t\in\mathbb {T}_{2}^{T}. \end{aligned}$$
(3.18)

Define a cone \(P_{1}\) as follows:

$$P_{1}= \Biggl\{ u\in P: \sum_{t=2}^{T} u(t)\sin\frac{\pi(t-1)}{T}\ge \delta\|u\| \Biggr\} , $$

where \(\delta=\sum_{t=2}^{T} e(t) \sin\frac{\pi(t-1)}{T} \). Then we claim

$$ L(P)\subset P_{1}. $$
(3.19)

Indeed, for \(u\in P\), from Lemma 2.1 we have

$$\begin{aligned} \sum_{t=2}^{T} (Lu) (t)\sin \frac{\pi(t-1)}{T}&= \sum_{t=2}^{T} \sum _{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) u(j) \sin\frac{\pi (t-1)}{T} \\ & \ge\sum_{t=2}^{T} \sum _{s=2}^{T} e(t) H(\tau,s)\sum _{j=2}^{T} H(s,j) u(j) \sin\frac{\pi(t-1)}{T} \\ & = \delta(Lu) (\tau),\quad \forall\tau\in\mathbb {T}_{2}^{T}, \end{aligned}$$

and thus

$$\sum_{t=2}^{T} (Lu) (t)\sin \frac{\pi(t-1)}{T}\ge\delta\|Lu\|. $$

Moreover, \(\varphi_{0}\in P_{1}\) since \(\varphi_{0}=16 \sin^{4} \frac{\pi }{2T}L\varphi_{0}\in P_{1} \). Now we claim that

$$ u-Au\neq \mu\varphi_{0}, \quad \forall u\in \partial B_{r}, \mu\ge0. $$
(3.20)

If the claim is false, then there exist \(u_{2}\in\partial B_{r}\) and \(\mu _{2}\ge0\) such that

$$ u_{2}-Au_{2} = \mu_{2} \varphi_{0}. $$
(3.21)

From (3.17) we have \(Au_{2}\ge(16 \sin^{4} \frac{\pi }{2T}+\varepsilon_{2}) Lu_{2}\) and so \(u_{2}\ge(16 \sin^{4} \frac{\pi }{2T}+\varepsilon_{2}) Lu_{2}\), i.e.,

$$u_{2}(t)\ge \biggl(16 \sin^{4} \frac{\pi}{2T}+ \varepsilon_{2} \biggr)\sum_{s=2}^{T} H(t,s)\sum_{j=2}^{T} H(s,j) u_{2}(j). $$

Multiplying both sides of the above inequality by \(\sin\frac{\pi (t-1)}{T}\), then summing from 2 to T, and using (2.5), we obtain

$$\begin{aligned} &\sum_{t=2}^{T} u_{2}(t)\sin \frac{\pi(t-1)}{T} \\ &\quad \ge{ \biggl(16 \sin^{4} \frac{\pi}{2T}+ \varepsilon_{2} \biggr)} \sum_{t=2}^{T} \Biggl[\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) u_{2}(j) \Biggr]\sin \frac{\pi (t-1)}{T} \\ &\quad = \frac{16 \sin^{4} \frac{\pi}{2T}+\varepsilon_{2}}{16 \sin^{4} \frac{\pi}{2T}} \sum_{t=2}^{T} u_{2}(t)\sin\frac{\pi (t-1)}{T}, \end{aligned}$$

which implies that

$$ \sum_{t=2}^{T} u_{2}(t)\sin\frac{\pi (t-1)}{T}\le0. $$
(3.22)

On the other hand, from (3.21) we have

$$\begin{aligned} &u_{2}(t)-{ \biggl(16 \sin^{4} \frac{\pi}{2T}- \varepsilon _{2} \biggr)}(Lu_{2}) (t) \\ &\quad =(Au_{2}) (t)-{ \biggl(16 \sin^{4} \frac{\pi }{2T}-\varepsilon_{2} \biggr)}(Lu_{2}) (t)+\mu_{2} \varphi_{0}(t) \\ &\quad =\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \biggl[f\bigl(j,u_{2}(j) \bigr)- { \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{2} \biggr)} u_{2}(j) \biggr]+\mu_{2} \varphi_{0}(t). \end{aligned}$$

Then (3.18), (3.19) and \(\varphi_{0}\in P_{1}\) enable us to find \(u_{2}-(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{2})Lu_{2}\in P_{1}\), and thus

$$\begin{aligned} &\biggl\Vert u_{2}-{ \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon _{2} \biggr)}Lu_{2} \biggr\Vert \\ &\quad \le\frac{1}{\delta} \sum_{t=2}^{T} \biggl[u_{2}(t)- \biggl(16 \sin^{4} \frac{\pi}{2T}- \varepsilon_{2} \biggr) (Lu_{2}) (t) \biggr] \sin \frac{\pi(t-1)}{T} \\ &\quad =\frac{\varepsilon_{2}}{\delta16 \sin^{4} \frac{\pi}{2T}}\sum_{t=2}^{T} u_{2}(t)\sin\frac{\pi(t-1)}{T}\le0. \end{aligned}$$

Note that \((16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{2})r(L)<1\), where \(r(L)\) is the spectral radius of L. Hence, we have \(u_{2}=0\), contradicting \(u_{2}\in\partial B_{r}\). This implies that (3.20) holds, and from Lemma 2.3 we have

$$ \deg(I-A,B_{r},0)=0. $$
(3.23)

On the other hand, from (H6) there exist \(\varepsilon_{3}\in(0,16 \sin ^{4} \frac{\pi}{2T})\) and \(c_{3}>0\) such that

$$ \bigl\vert f(t,u) \bigr\vert \le \biggl(16 \sin^{4} \frac{\pi }{2T}-\varepsilon_{3} \biggr) \vert u \vert +c_{3},\quad \forall t\in\mathbb {T}_{2}^{T}, u\in \mathbb {R}. $$
(3.24)

Let \(\mathcal {M}=\{u\in E: u=\lambda Au, \lambda\in[0,1]\}\). Then we prove that \(\mathcal {M}\) is bounded in E. If \(u\in\mathcal {M}\), then from (3.24) we have

$$\begin{aligned} \bigl\vert u(t) \bigr\vert &=\lambda \bigl\vert (Au) (t) \bigr\vert \le \sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \bigl\vert f\bigl(j,u(j)\bigr) \bigr\vert \\ &\le\sum_{s=2}^{T} H(t,s)\sum _{j=2}^{T} H(s,j) \biggl[ \biggl(16 \sin^{4} \frac{\pi}{2T}-\varepsilon_{3} \biggr) \bigl\vert u(j) \bigr\vert +c_{3} \biggr]. \end{aligned}$$

Multiplying both sides of the above inequality by \(\sin\frac{\pi (t-1)}{T}\), then summing from 2 to T, and using (2.5), we obtain

$$\sum_{t=2}^{T} \bigl\vert u(t) \bigr\vert \sin\frac{\pi(t-1)}{T}\le\frac{1}{16 \sin^{4} \frac{\pi}{2T}} \sum _{t=2}^{T} \biggl[ \biggl(16 \sin^{4} \frac{\pi }{2T}-\varepsilon_{3} \biggr) \bigl\vert u(t) \bigr\vert +c_{3} \biggr] \sin\frac{\pi(t-1)}{T}, $$

and then

$$\sum_{t=2}^{T} \bigl|u(t)\bigr|\sin\frac{\pi(t-1)}{T} \le c_{3}\varepsilon _{3}^{-1}\sum _{t=2}^{T} \sin\frac{\pi(t-1)}{T}. $$

We know that there is a \(t_{0}\in\mathbb {T}_{2}^{T}\) such that \(\|u\| =|u(t_{0})|\), and thus

$$\bigl\vert u(t_{0}) \bigr\vert \sin\frac{\pi(t_{0}-1)}{T}\le\sum _{t=2}^{T} \bigl\vert u(t) \bigr\vert \sin\frac {\pi(t-1)}{T}. $$

This implies that

$$\|u\|\le c_{3}\varepsilon_{3}^{-1} \sin^{-1}\frac{\pi(t_{0}-1)}{T}\sum_{t=2}^{T} \sin\frac{\pi(t-1)}{T}, $$

proving the boundedness of \(\mathcal {M}\). Choose \(R>\max\{\sup_{u\in \mathcal {M}} \|u\|, r\} \) (r is defined by (3.17)), then

$$ \lambda Au\neq u,\quad u\in\partial B_{R}, \lambda \in[0, 1]. $$
(3.25)

Lemma 2.4 implies that

$$ \deg(I-A,B_{R},0)=1. $$
(3.26)

This, together with (3.23), implies that

$$\deg(I-A,B_{R}\setminus \overline{B}_{r},0)= \deg(I-A,B_{R},0)-\deg(I-A,B_{r},0)=1. $$

Therefore, the operator A has at least one fixed point in \(B_{R}\setminus \overline{B}_{r}\), and (1.1) has at least one nontrivial solution. This completes the proof. □

Example 3.3

Let \(f(t,x)= a|x|-bk(x)\), \(k(x)=\ln(|x|+1)\), \(x\in\mathbb {R}\), where \(a\in(16 \sin^{4} \frac{\pi}{2T}, +\infty)\) and \(b\in(0, a+16 \sin^{4} \frac{\pi }{2T})\). Then \(\lim_{|x|\to+\infty}\frac{k(x)}{|x|}=0\), and \(\lim_{|x|\to+\infty} \frac{a|x|-b\ln(|x|+1)}{|x|}=a>16 \sin^{4} \frac {\pi}{2T}\), \(\lim_{|x|\to0} \frac{|a|x|-b\ln(|x|+1)|}{|x|} =|a-b|<16 \sin^{4} \frac{\pi}{2T} \). Therefore, (H1)–(H4) hold.

Example 3.4

Let \(f(t,x)=\scriptsize{ \bigl \{ \begin{array}{l@{\quad}l} ax+b \sin x,& x\ge0, \\ ax-be^{x}+b, &x\le0, \end{array} \bigr .} \) where \(a,b>0\) with \(a<16 \sin^{4} \frac{\pi}{2T}\), \(a+b> 16 \sin^{4} \frac{\pi}{2T} \) and \(a-b<16 \sin^{4} \frac{\pi}{2T}\). Then \(\lim_{x\to0^{+}} \frac{ax+b \sin x}{x}=a+b\), \(\lim_{x\to 0^{-}}\frac{ax-be^{x}+b}{x}=a-b\), \(\lim_{x\to+\infty} \vert \frac{ax+b \sin x}{x} \vert =a\), and \(\lim_{x\to-\infty} \vert \frac{ax-be^{x}+b}{x} \vert =a\). Therefore, (H5)–(H6) hold.

4 Conclusions

In this paper, we established the existence of nontrivial solutions for the boundary value problems of the fourth order difference equation (1.1) with sign-changing nonlinearity using the topological degree theory. Under some conditions concerning the first eigenvalue corresponding to the relevant linear problem, the results here improve and generalize those obtained in [111].

References

  1. Goodrich, C.S.: On discrete sequential fractional boundary value problems. J. Math. Anal. Appl. 385, 111–124 (2012)

    Article  MathSciNet  Google Scholar 

  2. Lv, Z., Gong, Y., Chen, Y.: Multiplicity and uniqueness for a class of discrete fractional boundary value problems. Appl. Math. 59(6), 673–695 (2014)

    Article  MathSciNet  Google Scholar 

  3. Ma, R., Xu, Y.: Existence of positive solution for nonlinear fourth-order difference equations. Comput. Math. Appl. 59, 3770–3777 (2010)

    Article  MathSciNet  Google Scholar 

  4. Xu, J.: Positive solutions for a fourth order discrete p-Laplacian boundary value problem. Math. Methods Appl. Sci. 36, 2467–2475 (2013)

    Article  MathSciNet  Google Scholar 

  5. Rehman, M., Iqbal, F., Seemab, A.: On existence of positive solutions for a class of discrete fractional boundary value problems. Positivity 21, 1173–1187 (2017)

    Article  MathSciNet  Google Scholar 

  6. Bai, D., Henderson, J., Zeng, Y.: Positive solutions of discrete Neumann boundary value problems with sign-changing nonlinearities. Bound. Value Probl. 2015, Article ID 231 (2015)

    Article  MathSciNet  Google Scholar 

  7. Goodrich, C.S.: On a first-order semipositone discrete fractional boundary value problem. Arch. Math. 99, 509–518 (2012)

    Article  MathSciNet  Google Scholar 

  8. Goodrich, C.S.: On semipositone discrete fractional boundary value problems with non-local boundary conditions. J. Differ. Equ. Appl. 19(11), 1758–1780 (2013)

    Article  MathSciNet  Google Scholar 

  9. Dahal, R., Duncan, D., Goodrich, C.S.: Systems of semipositone discrete fractional boundary value problems. J. Differ. Equ. Appl. 20(3), 473–491 (2014)

    Article  MathSciNet  Google Scholar 

  10. Xu, J., O’Regan, D.: Existence and uniqueness of solutions for a first-order discrete fractional boundary value problem. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (in press)

  11. Xu, J., Goodrich, C.S., Cui, Y.: Positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(4), 1005–1016 (2018)

    Article  Google Scholar 

  12. Wang, Y., Liu, L.: Positive solutions for a class of fractional infinite-point boundary value problems. Bound. Value Probl. 2018, Article ID 118 (2018)

    Article  MathSciNet  Google Scholar 

  13. Wang, Y., Liu, L., Wu, Y.: Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity. Nonlinear Anal., Theory Methods Appl. 74(17), 6434–6441 (2011)

    Article  MathSciNet  Google Scholar 

  14. Wang, Y., Liu, L., Zhang, X., Wu, Y.: Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection. Appl. Math. Comput. 258, 312–324 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, X., Liu, L., Zou, Y.: Fixed-point theorems for systems of operator equations and their applications to the fractional differential equations. J. Funct. Spaces 2018, Article ID 7469868 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Hao, X., Wang, H., Liu, L., Cui, Y.: Positive solutions for a system of nonlinear fractional nonlocal boundary value problems with parameters and p-Laplacian operator. Bound. Value Probl. 2017, Article ID 182 (2017)

    Article  MathSciNet  Google Scholar 

  17. Zhao, Z.: Positive solutions of semi-positone Hammerstein integral equations and applications. Appl. Math. Comput. 219(5), 2789–2797 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Liu, J., Zhao, Z.: Multiple positive solutions for second-order three-point boundary-value problems with sign changing nonlinearities. Electron. J. Differ. Equ. 2012, 152 (2012)

    Article  MathSciNet  Google Scholar 

  19. Zhao, Z.: Existence of positive solutions for 2nth-order singular semipositone differential equations with Sturm–Liouville boundary conditions. Nonlinear Anal., Theory Methods Appl. 72(3–4), 1348–1357 (2010)

    Article  Google Scholar 

  20. Lin, X., Zhao, Z.: Sign-changing solution for a third-order boundary-value problem in ordered Banach space with lattice structure. Bound. Value Probl. 2014, Article ID 132 (2014)

    Article  MathSciNet  Google Scholar 

  21. Wu, Y., Zhao, Z.: Positive solutions for third-order boundary value problems with change of signs. Appl. Math. Comput. 218(6), 2744–2749 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Zhang, K.: On a sign-changing solution for some fractional differential equations. Bound. Value Probl. 2017, Article ID 59 (2017)

    Article  MathSciNet  Google Scholar 

  23. Zhang, K.: Nontrivial solutions of fourth-order singular boundary value problems with sign-changing nonlinear terms. Topol. Methods Nonlinear Anal. 40(1), 53–70 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Fan, W., Hao, X., Liu, L., Wu, Y.: Nontrivial solutions of singular fourth-order Sturm–Liouville boundary value problems with a sign-changing nonlinear term. Appl. Math. Comput. 217(15), 6700–6708 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Hao, X., Zuo, M., Liu, L.: Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities. Appl. Math. Lett. 82, 24–31 (2018)

    Article  MathSciNet  Google Scholar 

  26. Liu, L., Liu, B., Wu, Y.: Nontrivial solutions for higher-order m-point boundary value problem with a sign-changing nonlinear term. Appl. Math. Comput. 217(8), 3792–3800 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Guo, Y.: Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. Bull. Korean Math. Soc. 47(1), 81–87 (2010)

    Article  MathSciNet  Google Scholar 

  28. Guo, Y.: Nontrivial periodic solutions of nonlinear functional differential systems with feedback control. Turk. J. Math. 34(1), 35–44 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Guo, Y.: Positive solutions of second-order semipositone singular three-point boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2009, 5 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Qiu, X., Xu, J., O’Regan, D., Cui, Y.: Positive solutions for a system of nonlinear semipositone boundary value problems with Riemann–Liouville fractional derivatives. J. Funct. Spaces 2018, Article ID 7351653 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Pu, R., Zhang, X., Cui, Y., Li, P., Wang, W.: Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions. J. Funct. Spaces 2017, Article ID 5892616 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Chen, C., Xu, J., O’Regan, D., Fu, Z.: Positive solutions for a system of semipositone fractional difference boundary value problems. J. Funct. Spaces 2018, Article ID 6835028 (2018)

    MathSciNet  Google Scholar 

  33. Cheng, W., Xu, J., Cui, Y.: Positive solutions for a system of nonlinear semipositone fractional q-difference equations with q-integral boundary conditions. J. Nonlinear Sci. Appl. 10(8), 4430–4440 (2017)

    Article  MathSciNet  Google Scholar 

  34. Li, H., Sun, J.: Positive solutions of sublinear Sturm–Liouville problems with changing sign nonlinearity. Comput. Math. Appl. 58(9), 1808–1815 (2009)

    Article  MathSciNet  Google Scholar 

  35. Li, H., Sun, J.: Positive solutions of superlinear semipositone nonlinear boundary value problems. Comput. Math. Appl. 61(9), 2806–2815 (2011)

    Article  MathSciNet  Google Scholar 

  36. Wu, J., Zhang, X., Liu, L., Wu, Y., Cui, Y.: The convergence analysis and error estimation for unique solution of a p-Laplacian fractional differential equation with singular decreasing nonlinearity. Bound. Value Probl. 2018, Article ID 82 (2018)

    Article  MathSciNet  Google Scholar 

  37. Zuo, M., Hao, X., Liu, L., Cui, Y.: Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditions. Bound. Value Probl. 2017, Article ID 161 (2017)

    Article  MathSciNet  Google Scholar 

  38. Cui, Y.: Computation of topological degree in ordered Banach spaces with lattice structure and applications. Appl. Math. 58(6), 689–702 (2013)

    Article  MathSciNet  Google Scholar 

  39. Cui, Y., Ma, W., Sun, Q., Su, X.: New uniqueness results for boundary value problem of fractional differential equation. Nonlinear Anal., Model. Control 23(1), 31–39 (2018)

    Article  MathSciNet  Google Scholar 

  40. Cui, Y.: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 48–54 (2016)

    Article  MathSciNet  Google Scholar 

  41. Zou, Y., He, G.: On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 74, 68–73 (2017)

    Article  MathSciNet  Google Scholar 

  42. Zhang, X., Liu, L., Wu, Y., Zou, Y.: Existence and uniqueness of solutions for systems of fractional differential equations with Riemann–Stieltjes integral boundary condition. Adv. Differ. Equ. 2018, Article ID 204 (2018)

    Article  MathSciNet  Google Scholar 

  43. Bai, Z., Dong, X., Yin, C.: Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions. Bound. Value Probl. 2016, Article ID 63 (2016)

    Article  MathSciNet  Google Scholar 

  44. Bai, Z., Zhang, Y.: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 218(5), 1719–1725 (2011)

    MathSciNet  MATH  Google Scholar 

  45. Zhang, Y., Bai, Z., Feng, T.: Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl. 61(4), 1032–1047 (2011)

    Article  MathSciNet  Google Scholar 

  46. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their valuable suggestions and comments.

Availability of data and materials

Not applicable.

Funding

This work is supported by Natural Science Foundation of Shandong Province (ZR2018MA009, ZR2015AM014).

Author information

Authors and Affiliations

Authors

Contributions

This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Keyu Zhang.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, K., O’Regan, D. & Fu, Z. Nontrivial solutions for boundary value problems of a fourth order difference equation with sign-changing nonlinearity. Adv Differ Equ 2018, 370 (2018). https://doi.org/10.1186/s13662-018-1840-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13662-018-1840-3

Keywords