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Positive solutions and eigenvalue intervals for a second order pLaplacian discrete system
Advances in Difference Equations volume 2018, Article number: 281 (2018)
Abstract
In this paper, we investigate the existence of at least one positive solution to a second order pLaplacian discrete system. As applications, we characterize the eigenvalue intervals for one typical ndimensional system. The proof is based on a wellknown fixed point theorem in cones.
Introduction
In this paper, we investigate the existence of at least one positive solution for the second order pLaplacian discrete boundary value system
where \(u=(u_{1},u_{2},\ldots,u_{n})\), \(\phi(s)=s^{p2}s\) (\(p>1\)), \(N = \{1,2,\ldots,T\}\), \(T\geq1\) is a fixed positive integer and \(\Delta u(k)=u(k+1)u(k)\) is the forward difference operator.
The discrete boundary value problems arise in different fields of research. For instance, they are widely used in discrete optimization, computer science, population genetics, and so on. Therefore, different types of discrete boundary value problems have been studied in the past three decades, here we refer the reader to [1, 2, 4, 5, 7, 8, 15, 19]. Among those, the existence of positive solutions for the discrete pLaplacian system has attracted special attention [4, 7, 19]. Some classical and wellknown tools, such as the variational methods [3, 17], the approach of upper and lower solutions [10, 11], and some fixed point theorems [5, 9], have been widely used.
This paper is mainly motivated by previous paper [6], in which Chu et al. [5, 6] stated and showed that, under adequate assumptions on the nonlinear term f, the discrete boundary value problems have at least one positive solution. In this paper, we will choose another adequate cone to prove the main results, which will improve and generalize those results obtained in [6].
As applications of our new results, we characterize the eigenvalue intervals for the ndimensional system
where \(\lambda>0\) is a positive parameter. We prove that system (2) has at least one positive solution for each λ in an explicit eigenvalue interval. Recently, several eigenvalue characterizations for different kinds of boundary value problems have appeared, and we refer the reader to [5, 12, 14, 16, 18].
The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminaries which are used to prove the main results. Some results on the existence of at least one positive solution to system (1) are established in Sect. 3. Finally, in Sect. 4 we study the existence of positive solutions for the following discrete system:
and characterize the eigenvalue intervals for system (2).
Preliminaries
First let us introduce some notation. For abbreviation, let \({\mathbb {R}}\) be the set of real numbers, \({\mathbb {R}}_{+}=[0,\infty )\), \({\mathbb {R}}^{n}_{+}= \prod^{n}_{i=1}{\mathbb {R}}_{+}\), and \(N^{+}=\{0,1,\ldots,T+1\}\). We will denote by \(C(N^{+},{\mathbb {R}})\) the set of all continuous functions on \(N^{+}\) (discrete topology) with supremum norm \(\u\= \max_{k \in N^{+}} u(k)\). Then \(C(N^{+},{\mathbb {R}})\) is a Banach space. Given \(u,v\in {\mathbb {R}}^{N}\), the usual inner product formula is denoted by
Take \(X=C(N^{+},{\mathbb {R}}) \times\cdots\times C(N^{+},{\mathbb {R}})\) (n copies). For any \(u=(u_{1}, \ldots, u_{N})\in X\), we define the vnorm by
where \(v\in {\mathbb {R}}_{+}^{N}\) is a fixed vector. It is easy to see that X is a Banach space.
Next we present some wellknown tools which will be used throughout the paper.
Lemma 2.1
([1])
Assume that there exists a function \(y \in C(N^{+},{\mathbb {R}})\) with \(y(k)\geq 0\) for all \(k \in N^{+}\). If \(u \in C(N^{+},{\mathbb {R}})\) satisfies
then \(u(k)\geq q(k)\u\\) for all \(k \in N^{+}\), where
Remark 2.2
From (4), it is easy to see that \(q(k)\geq\frac{1}{T+1}\) for any \(k \in N\).
Lemma 2.3
([9])
If \(u \in C(N^{+},{\mathbb {R}})\) satisfies
then \(u(k)\geq q(k)\u\\) for \(k \in N^{+}\).
Lemma 2.4
([9])
If \(u, y \in C(N^{+},{\mathbb {R}})\) satisfies
then \(u(k)\geq y(k)\) for \(k \in N^{+}\).
Our main tool is a wellknown fixed point theorem in cones established in [13], which we recall here for the convenience of the readers. If D is a subset of X, we write \(D_{K}=D\cap K\) and \(\partial _{K}D=(\partial D)\cap K\).
Theorem 2.5
Let X be a Banach space, and let K be a cone in X. Assume that \(\Omega^{1}\), \(\Omega^{2}\) are open bounded subsets of X with \(\Omega ^{1}_{K}\neq\emptyset\), \(\overline{\Omega^{1}}_{K} \subset\Omega^{2}_{K}\). Let
be a continuous and completely continuous operator such that

(i)
\(u\neq\lambda Tu\) for \(\lambda\in[0,1)\) and \(u \in \partial_{K}\Omega^{1}\), and

(ii)
there exists \(e\in K\setminus \{0\}\) such that \(u \neq Tu+\lambda e\) for all \(u \in \partial_{K}\Omega^{2}\) and all \(\lambda> 0\).
Then T has a fixed point in \(\overline{\Omega^{2}}_{K} \setminus \Omega^{1}_{K}\).
Main results
In this section, we establish an existence result to the discrete pLaplacian system (1). Define a cone K in X by
In addition, the operator \(S:K \rightarrow X\) is defined by \(S=(S_{1},S_{2},\ldots,S_{n})\) and
where \(\tau_{i}\) is a solution of the equation
Lemma 3.1
Suppose that \(f^{i}: N^{+} \times {\mathbb {R}}^{n}_{+} \rightarrow {\mathbb {R}}_{+}\) is continuous for each \(i=1,2,\ldots,n\). Then the operator S is continuous and completely continuous. Furthermore, \(S: K \rightarrow K\) is well defined.
Proof
Since \(\phi^{1}\) is a continuous, strictly increasing function on \({\mathbb {R}}\) and \(\phi^{1}({\mathbb {R}}) = {\mathbb {R}}\), then \(\tau_{i}\) is a unique solution of equation (6). Therefore, the operator S is well defined. It follows from [6, Lemma 6.1] that \(S:K\rightarrow X\) is continuous and completely continuous. Next, we show that S maps K into K.
Since \(f^{i}(k,u(k)) \geq0\) for \(k\in N\), by Lemma 2.3, it is clear that
Therefore
which shows that \(S(K) \subset K\). □
In order to apply Theorem 2.5, we need to select adequate open sets. Define
Lemma 3.2
\(\Omega^{r}\), \(B^{r}\) defined above have the following properties:

(a)
\(\Omega^{r}_{K}\), \(B^{r}_{K}\) are open relative to K.

(b)
\(B^{r/T+1}_{K} \subset\Omega^{r}_{K} \subset B^{r}_{K}\).

(c)
\(u\in\partial{_{K}}\Omega^{r}\) if and only if \(u \in K\) and \(\min_{k \in N}\langle v, u(k)\rangle=\frac{r}{T+1}\).

(d)
If \(u\in\partial{_{K}}\Omega^{r}\), then \(\frac{r}{T+1} \leq\langle v, u(k)\rangle\leq r\) for \(k \in N\) and \(u_{v}\leq r\).
Proof
Since \(\min_{k \in N}\langle v, u(k)\rangle\) is continuous (discrete topology), we check at once that (a) is true. (c) is clear since, for each \(u \in K\), we have
Now let us prove (d). From (c), we have
Thus \(u_{v} \leq r\) and \(\frac{r}{T+1} \leq\langle v, u(k)\rangle \leq r\) for \(k\in N\).
Next we prove (b). Let \(u \in B^{r/T+1}_{K}\), then we have \(u_{v} < \frac{r}{T+1}\), so \(\min_{k \in N}\langle v, u(k)\rangle< \frac{r}{T+1}\) and \(u \in\Omega^{r}_{K}\). Since \(u \in\Omega^{r}_{K}\), for all \(k\in N\), we have \(\min_{k \in N}\langle v, u(k)\rangle< \frac{r}{T+1}\) and \(\langle v, u(k)\rangle\geq q(k)u_{v} \geq\frac{1}{T+1}u_{v}\). Hence \(u_{v}< r\), that is, \(\Omega^{r}_{K}\subset B^{r}_{K}\). □
It follows from the above properties that, for each \(\delta> r\),
Theorem 3.3
Suppose that \(f^{i}: N^{+} \times {\mathbb {R}}^{n}_{+} \rightarrow {\mathbb {R}}_{+}\) is continuous. Moreover, suppose further that there exist \(\alpha, \beta >0\) such that
 (\(\mathrm{D}_{1}\)):

For each \(i=1,2,\ldots, n\), there exists a continuous function \(\psi_{i}: N \rightarrow(0,\infty)\) such that
$$f^{i}(j,u) \geq\phi\biggl(\frac{1}{T+1}\alpha\biggr) \psi_{i}(j),\quad \textit{for all }j \in N,\qquad \frac{\alpha}{T+1} \leq\bigl\langle v, u(j)\bigr\rangle \leq\alpha $$and
$$\min_{k \in N}\bigl\langle v, P(k) \bigr\rangle \geq1, $$where \(P(k)=(P_{1}(k),P_{2}(k),\ldots,P_{n}(k))\) is the unique solution of
$$\begin{aligned} \textstyle\begin{cases} \Delta(\phi(\Delta P(k1)))+\psi(k)=0,\quad k\in {\mathbb {N}}, \\ P(0)=0,\qquad P(T+1)=0 \end{cases}\displaystyle \end{aligned}$$(7)with \(\psi=(\psi_{1},\psi_{2},\ldots,\psi_{n})\).
 (\(\mathrm{D}_{2}\)):

For each \(i=1,2,\ldots, n\), there exists a continuous function \(\chi_{i}: N \rightarrow(0,\infty)\) such that
$$f^{i}(j,u) \rangle\leq\phi(\beta)\chi_{i}(j),\quad \textit{for all }j \in N,\qquad 0< \bigl\langle v, u(j)\bigr\rangle \leq\beta $$and
$$\max_{k \in N}\bigl\langle v, Q(k) \bigr\rangle \leq1, $$where \(Q(k)=(Q_{1}(k),Q_{2}(k),\ldots,Q_{n}(k))\) is the unique solution of
$$\begin{aligned} \textstyle\begin{cases} \Delta(\phi(\Delta Q(k1)))+\chi(k)=0,\quad k\in {\mathbb {N}}, \\ Q(0)=0,\qquad Q(T+1)=0 \end{cases}\displaystyle \end{aligned}$$(8)with \(\chi=(\chi_{1},\chi_{2},\ldots,\chi_{n})\).

(a)
If \(\beta< \frac{\alpha}{T+1}\), then system (1) has at least one positive solution u with
$$\beta\lequ_{v} \leq\alpha; $$ 
(b)
If \(\alpha< \beta\), then system (1) has at least one positive solution u with
$$\frac{\alpha}{T+1}\lequ_{v} \leq\beta. $$
Proof
To obtain the desired result, we claim that

(i)
\(u\neq\lambda Su\) for \(\lambda\in[0,1)\) and \(u \in \partial_{K}B^{\beta}\), and

(ii)
there exists \(e\in K\setminus \{0\}\) such that \(u \neq Su+\lambda e\) for each \(u \in \partial_{K}\Omega^{\alpha}\) and all \(\lambda> 0\).
We start with (i). Suppose that there exist \(u \in\partial _{K}B^{\beta}\) and \(\lambda\in[0,1)\) such that \(u=\lambda Su\). Since \(u \in\partial_{K}B^{\beta}\), we have \(\frac {1}{T+1}\beta\leq\langle v,u(k)\rangle\leq\beta\) for all \(k\in N\) and \(\langle v,u(k^{*})\rangle= \beta\) for some \(k^{*}\in N\). From (\(\mathrm{D}_{2}\)) we get
By Lemma 2.4, we obtain
which is a contradiction. Then (i) is proved.
Next we consider claim (ii). Let \(e(t)\equiv1\). Then \(e\in K\setminus \{0\}\). Suppose, contrary to our claim, that there exist \(u\in\partial _{K}\Omega^{\alpha}\) and \(\lambda> 0\) such that \(u=Su+\lambda e\). By Lemma 3.2(d), for any \(u\in\partial_{K}\Omega^{\alpha}\), we have \(\frac{1}{T+1}\alpha\leq\langle v,u(k)\rangle\leq\alpha\) for \(k \in N\).
It follows from (\(\mathrm{D}_{1}\)) that
We conclude from Lemma 2.4 that
hence that \(\min_{k \in N}\langle v, u(k)\rangle >\frac{\alpha}{T+1}\), and finally that this is a contradiction to Lemma 3.2(c).
According to Lemma 3.2, if \(\beta< \frac{\alpha}{T+1}\), one has \(\overline{B^{\beta}}_{K}\subset B^{\alpha/T+1}_{K} \subset\Omega ^{\alpha}_{K}\). Now Theorem 2.5 guarantees the existence of at least one fixed point \(u\in\overline{\Omega^{\alpha}}_{K} \setminus B^{\beta}_{K}\) of S, so we have \(u_{v}\geq\beta\) and \(\frac{1}{T+1}\beta\leq \min_{k \in N}\langle v, u(k)\rangle\leq\frac{1}{T+1}\alpha\). Moreover, \(\frac{1}{T+1}u_{v} \leq \min_{k \in N}\langle v, u(k)\rangle\leq\frac{1}{T+1}\alpha\). We thus get \(u_{v} \leq\alpha\).
On the other hand, if \(\alpha< \beta\), one has \(\overline{\Omega ^{\alpha}}_{K}\subset B^{\beta}_{K}\). Similarly, we can obtain that S has at least one fixed point \(u\in\overline{B^{\beta}}_{K} \setminus \Omega^{\alpha}_{K}\) by Theorem 2.5, which implies that \(\frac{1}{T+1}\alpha\lequ_{v} \leq\beta\). □
Remark 3.4
Take \(p=2\) and the original system (1) is transformed into a common second order discrete boundary value system
then system (9) has at least one positive solution.
Eigenvalue intervals of (2)
In this section, we employ Theorem 3.3 to establish one existence result for system (3), and then characterize the eigenvalue intervals of system (2). First we assume that
 (\(\mathrm{H}_{1}\)):

\(g^{i}\): \({\mathbb {R}}^{n}_{+} \rightarrow {\mathbb {R}}_{+}\) is continuous with \(g^{i}(u) >0\) for \(u_{v} >0\);
 (\(\mathrm{H}_{2}\)):

\(\widetilde{H}(k)>0\) for \(k\in N\), where \(\widetilde {H}(k)=(\widetilde{H}_{1}(k),\widetilde{H}_{2}(k),\ldots,\widetilde{H}_{n}(k))\) is the unique solution of
$$\begin{aligned} \textstyle\begin{cases} \Delta(\phi(\Delta u(k1)))+h(k)=0,\quad k\in {\mathbb {N}}, \\ u(0)=0,\qquad u(T+1)=0, \end{cases}\displaystyle \end{aligned}$$with \(h(k)=(h_{1}(k),h_{2}(k),\ldots,h_{n}(k))\).
Theorem 4.1
Assume that (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)) hold. If one of the following conditions holds, then system (3) has at least one positive solution u satisfying \(\langle v, u(k)\rangle\neq0\) for \(k\in N\).
 (\(\mathrm{h}_{1}\)):

\(0 \leq g^{i}_{0}< (\frac{1}{M})^{p1}\) and \((\frac {1}{m})^{p1} < g^{i}_{\infty}\leq\infty\), \(i=1,2, \ldots, n\);
 (\(\mathrm{h}_{2}\)):

\(0 \leq g^{i}_{\infty} < (\frac{1}{M})^{p1}\) and \((\frac{1}{m})^{p1} < g^{i}_{0}\leq\infty\), \(i=1,2, \ldots, n\);
Proof
For this purpose, we set \(f^{i}(k,u)=h_{i}(k)g^{i}(u)\), \(i=1,2,\ldots,n\), and suppose that (\(\mathrm{h}_{1}\)) holds. In addition, the case when (\(\mathrm{h}_{2}\)) holds is similar.
From the first part of (\(\mathrm{h}_{1}\)), there exists \(\beta> 0\) such that \(g^{i}(u)\leq(\frac{1}{M})^{p1}\beta^{p1}\) for \(0< u_{v} \leq\beta\). Take \(\chi_{i}(k)=(\frac{1}{M})^{p1}h_{i}(k)\). When \(0< \langle v, u(k)\rangle\leq\beta\), then
and
Hence (\(\mathrm{D}_{2}\)) holds.
From the second part of (\(\mathrm{h}_{1}\)), there exists \(\alpha> 0\) such that \(\frac{1}{T+1} \alpha> \beta\) and \(g^{i}(u) \geq(\frac{1}{m})^{p1}(\frac{1}{T+1} \alpha)^{p1}\) for \(u_{v} \geq\frac{1}{T+1} \alpha\). Take \(\psi_{i}(k)=(\frac {1}{m})^{p1}h_{i}(k)\). When \(\frac{1}{T+1}\alpha\leq\langle v, u(k)\rangle\leq\alpha\), we have, for \(k\in N\), that
and
This implies that (\(\mathrm{D}_{1}\)) holds. Now the rest of the proof runs as Theorem 3.3. □
Next we employ Theorem 4.1 to characterize the eigenvalue intervals for system (2). As it is easy to prove, we only present the results here.
Theorem 4.2
Assume that (\(\mathrm{H}_{1}\)) and (\(\mathrm{H}_{2}\)) hold. If \(\frac{1}{m^{p1} \min_{i=1,2,\ldots,n}\{g^{i}_{\infty}\} }<\frac{1}{M^{p1} \max_{i=1,2,\ldots,n}\{g^{i}_{0}\}}\), then system (2) has at least one positive solution for each
If \(\frac{1}{m^{p1} \min_{i=1,2,\ldots,n}\{g^{i}_{0}\}}<\frac {1}{M^{p1} \max_{i=1,2,\ldots,n}\{g^{i}_{\infty}\}}\), the same result remains valid for each
Conclusions
In this paper, we established the existence of positive solutions for the second order pLaplacian discrete system by a wellknown fixed point theorem in cones. In this paper, we will choose another adequate cone to prove the main results, which will improve and generalize those results obtained in [6]. Moreover, we take advantage of a new cone to characterize the eigenvalue intervals in a simple way.
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Acknowledgements
The author is grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper. At the same time, the author would like to thank Professor Chu Jifeng for his valuable comments.
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This research is partially supported by the National Natural Science Foundation of China (Grant No. 11671118).
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Zhao, J. Positive solutions and eigenvalue intervals for a second order pLaplacian discrete system. Adv Differ Equ 2018, 281 (2018). https://doi.org/10.1186/s1366201817442
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MSC
 34B16
 39A10
Keywords
 Positive solution
 Discrete system
 pLaplacian
 ndimensional
 Eigenvalue intervals
 Fixed point theorem in cones