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.
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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
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, [1–11] and the references therein. However, as mentioned in [6], very few results are available with sign-changing nonlinearities; see [6–11]. Other related work in this field can be found in [12–45] 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
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
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:
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
where
Lemma 2.1
Green’s function H has the following properties:
-
(i)
\(H(t,s)>0\) for \((t,s)\in\mathbb {T}_{2}^{T}\times\mathbb {T}_{2}^{T}\),
-
(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
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:
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
i.e., the following two equations hold:
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
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
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
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.
-
(H1)
There exist two constants \(a>0\), \(b>0\) and a function \(k\in C(\mathbb {R}, \mathbb {R}^{+})\) such that
$$f(t,u)\ge-a-bk(u),\quad \forall u\in\mathbb {R}, t\in\mathbb {T}_{2}^{T}. $$ -
(H2)
\(\lim_{|u|\to+\infty} \frac{k(u)}{|u|}=0\).
-
(H3)
\(\liminf_{|u|\to+\infty}\frac{f(t,u)}{|u|}>16 \sin^{4} \frac {\pi}{2T}\) uniformly on \(t\in\mathbb {T}_{2}^{T}\),
-
(H4)
\(\limsup_{|u|\to0}\frac{|f(t,u)|}{|u|}<16 \sin^{4} \frac{\pi }{2T}\) uniformly on \(t\in\mathbb {T}_{2}^{T}\),
-
(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 \(t\in\mathbb {T}_{2}^{T}\),
-
(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
For any given ε with \(\varepsilon_{0} - b\varepsilon>0\), and from (H2), there exists \(X_{1}>X_{0}\) such that
Now since \(a>0\), \(b>0\) and k is a nonnegative function, we have
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
where \(c_{2}=c_{1}+a\). Note that ε can be chosen arbitrarily small, and we let
Now we prove that
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
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
Therefore,
Then from \(L(P)\subset P_{0}\), \(\varphi_{0}\in P_{0}\), and
we have
As a result, we obtain
On the other hand, from the definition of L, we get
in order to obtain the above inequality, we prove that
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
Combining (3.8), (3.9) and (3.10), we have
Using (3.6) we obtain
Define
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
and hence
which contradicts the definition of \(\mu^{*}\). Therefore, (3.5) holds, and from Lemma 2.3 we obtain
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
Now for this r, we show that
Otherwise, there would exist \(u_{1}\in\partial B_{r}\), \(\mu_{1}\ge1\) such that
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
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
This, together with (3.13), implies that
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
and
The above two inequalities enable us to obtain
Define a cone \(P_{1}\) as follows:
where \(\delta=\sum_{t=2}^{T} e(t) \sin\frac{\pi(t-1)}{T} \). Then we claim
Indeed, for \(u\in P\), from Lemma 2.1 we have
and thus
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
If the claim is false, then there exist \(u_{2}\in\partial B_{r}\) and \(\mu _{2}\ge0\) such that
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.,
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
which implies that
On the other hand, from (3.21) we have
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
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
On the other hand, from (H6) there exist \(\varepsilon_{3}\in(0,16 \sin ^{4} \frac{\pi}{2T})\) and \(c_{3}>0\) such that
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
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
and then
We know that there is a \(t_{0}\in\mathbb {T}_{2}^{T}\) such that \(\|u\| =|u(t_{0})|\), and thus
This implies that
proving the boundedness of \(\mathcal {M}\). Choose \(R>\max\{\sup_{u\in \mathcal {M}} \|u\|, r\} \) (r is defined by (3.17)), then
Lemma 2.4 implies that
This, together with (3.23), implies that
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 [1–11].
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This work is supported by Natural Science Foundation of Shandong Province (ZR2018MA009, ZR2015AM014).
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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
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DOI: https://doi.org/10.1186/s13662-018-1840-3