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Positive solutions of a discrete secondorder boundary value problems with fully nonlinear term
Advances in Difference Equations volume 2020, Article number: 583 (2020)
Abstract
In this paper, we mainly consider a kind of discrete secondorder boundary value problem with fully nonlinear term. By using the fixedpoint index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.
1 Introduction
Let a and b are two integers with \(a< b\) and \([a,b]_{\mathbb{Z}}=\{a, a+1,\ldots , b\}\). In this paper, we consider the existence of positive solutions of the following discrete problem:
where \(T>1\) is a positive integer, Δ is the forward difference operator with \(\Delta u(t)=u(t+1)u(t)\), \(f:[1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\times \mathbb{R}\to \mathbb{R}^{+}\) is a continuous function and \(\mathbb{R}^{+}=[0,\infty )\).
In the past few years, boundary value problems for difference equations have been deduced from different disciplines, such as the computer sciences, economics, mechanical engineering and control systems and so on; see, for instance, [1, 6, 23, 24]. Therefore, many scholars studied the discrete boundary value problems, including the linear discrete problems and nonlinear discrete problems [2–5, 8, 10–17, 19–21, 26, 28–30, 33–35]. In 1999, by using the upper and lower solution method, Agarwal and O’Regan [2] studied the existence of solutions and nonnegative solutions for the following discrete problem:
Thereafter, many authors focused on the existence of solutions and positive solutions of (1.2). In particular, since Zhou et al. [26] introduced the variation method to solve the discrete boundary condition, several excellent existence results of discrete boundary value problems have been obtained by using this method; see, for instance, [8, 11, 26, 33, 35] and the references therein. For example, by using the variation method, Bonanno et al. [8] studied the existence of multiple positive solutions of (1.2). Meanwhile, as a very important method, the bifurcation technique has also been introduced to discuss the discrete problem as (1.2). For example, by using the bifurcation technique, Gao et al. [12] studied the continuum of the positive and negative solutions of the boundary value problem (1.2) and they also obtained the existence of positive solutions and negative solutions of (1.2). Meanwhile, Ma et al. [27–29] and Gao [10] also used the same method to consider different discrete boundary value problems. Finally, another important method used to discuss the positive solutions of the discrete boundary value problems should be noted: fixedpoint theory in cones. In fact, since Merdivenci [31] introduced the fixedpoint theory in cones to consider the positive solutions of the twopoint discrete boundary value problems as (1.2), lots of interesting and excellent results have been obtained. For example, by using the fixedpoint theory in cones, Wong and Agarwal [34] considered the existence results of positive solutions for a boundary value problems of a higherorder difference equation, Ma and Raffoul [30] considered the existence of positive solutions of the discrete threepoint boundary value problems in 2004. Later, Henderson and Luca [19–21], Agarwal and Luca [2] considered the existence of positive solutions of the discrete multipoint systems.
However, it is noted that most of the above results focus on the problems as (1.2) which does not contain the damping term Δu in the nonlinear term f. As we know, the damping phenomenon exists widely in the real world. Therefore, it is interesting to consider such a problem which has the damping term in the nonlinear term; see, for instance, [7, 22, 32]. In [7], Anderson et al. considered the existence of the solutions of this kind of problems by using Schaefer’s theorem. In [22, 32], the method of lower and upper solutions are used to consider the existence of solutions a kind of discrete problems with the fully nonlinear term. Therefore, inspired by the above the results, we try our best to consider the existence of positive solutions of the discrete boundary value problem (1.1), which has a damping term Δu in the nonlinear term. Our main tools here are also some fixedpoint theories in a cone, called the fixedpoint index theories, we only briefly list them in Sect. 3 and we can find them in the references [9, 18] for more details. Furthermore, in the present paper, the superlinear and the sublinear conditions on the nonlinear term f at 0 and ∞ do not hold as the limitation form, but some weaker conditions hold at 0 and ∞; see Remarks 3.1 and 3.2. Finally, it is noted that the continuous problems with fully nonlinear terms have been studied by [25].
The rest of the present paper is organized as follows: In Sect. 2, we give some preliminaries, including the work space, the properties of the Green’s function and the spectral results of the linear eigenvalue problems. In Sect. 3, we give our main results and prove them.
2 Preliminaries
At first, let us introduce our work space. Let
with the maximum norm \(\u\_{E}=\max_{t\in [0,T+1]_{\mathbb{Z}}}u(t)\) and
with the maximum norm \(\u\_{Y}=\max_{t\in [0,T]_{\mathbb{Z}}}u(t)\).
Let \(j: E\to Y\) by
Then j is an isomorphism from E to Y. Furthermore, define
then P is a cone in E.
Now, let us consider the following linear boundary value problems:
Then the following results hold.
Lemma 2.1
Let \(h\in P\). Then the problem (2.1), (2.2) has a unique nonnegative solution
where \(G(t,s)\) is the Green’s function defined as
Proof
Summing Eq. (2.1) from \(s=1\) to \(s=t1\), we get
Then continuing to sum the above equation from \(s=1\) to \(s=t1\), we obtain
Combining this with the boundary condition \(u(T+1)=0\), we get
□
Lemma 2.2
The Green’s function \(G(t,s)\) satisfies the following properties:

(i)
\(G(t,s)=G(s,t)\), for \(t,s\in [0,T+1]_{\mathbb{Z}}\times [0,T+1]_{\mathbb{Z}}\);

(ii)
\(G(0,s)=G(T+1,s)=0\) for \(s\in [0,T+1]_{\mathbb{Z}}\);

(iii)
\(G(t,s)>0\), for \(t,s\in [1,T]_{\mathbb{Z}}\times [1,T]_{\mathbb{Z}}\);

(iv)
\(G(t,s)\leq G(s,s)\), for \(t,s\in [0,T+1]_{\mathbb{Z}}\times [0,T+1]_{\mathbb{Z}}\);

(v)
\(G(t,s)\geq \frac{1}{T+1} G(t,t)G(s,s)\).
Proof
The properties (i)–(iv) are obvious. We only prove (v) here. In fact,
Therefore, (v) holds. □
Lemma 2.3
Let \(u\in P\) be a solution of (2.1), (2.2). Then u satisfies the following properties:

(i)
\(u(t)\geq \frac{1}{T+1}G(t,t)\u\_{E}\) for \(t\in [0,T+1]_{\mathbb{Z}}\);

(ii)
\(\u\_{E}\leq T\max_{t\in [0,T1]_{\mathbb{Z}}}\Delta u(t)\);

(iii)
\(\max_{t\in [0,T]_{\mathbb{Z}}}\Delta u(t)\leq \Delta u(0)\Delta u(T)\).
Proof
(i) For \(t\in [1,T]_{\mathbb{Z}}\), by the properties of \(G(t,s)\), we have
Therefore,
Furthermore, by the property (v) of \(G(t,s)\), we know that
(ii) By direct calculation, we know that
Then, for \(t\in [1,T]_{\mathbb{Z}}\), we get
Combining this with the fact that \(u(0)=u(T+1)=0\), we see that the assertion (ii) holds.
(iii) By Lemma 2.1 and the fact \(h\in P\), it is not difficult to see that \(\Delta u(0)\geq 0\) and \(\Delta u(T)\leq 0\). Moreover, \(\Delta ^{2} u(t1)=h(t)\geq 0\), we know that \(\Delta u(t)\) is an increasing function on \([0,T]_{\mathbb{Z}}\). Therefore,
□
Lemma 2.4
The linear eigenvalue problem
has T real and simple eigenvalues \(22\cos \frac{k\pi }{T+1}, k=1,2,\ldots , T\), and the corresponding eigenfunction is \(\varphi _{k}=\sin \frac{k\pi t}{T+1}\), \(k=1,2,\ldots , T\).
Proof
This result is the wellknown discrete Sturm–Liouville theory, we can find it in several classical book, like Kelly and Peterson [23]. To be complete, we give a brief proof here.
The characteristic equation of the equation in (2.4) is \(\mu ^{2}+{{(\lambda 2)}}\mu +1=0\). Then
If \(\lambda 2\geq 2\), then the general solution of the equation in (2.4) is
Combining the boundary condition \(u(0)=u(T+1)=0\), we know that \(c_{1}=c_{2}=0\). Therefore, the problem (2.4) has only a trivial solution in this case.
If \(\lambda 2< 2\), we could set \(2\lambda =2\cos \theta \). Then
Therefore, the general solution of the equation in (2.4) is
Combining the boundary condition \(u(0)=u(T+1)=0\), we know that
Let
Then we get the eigenvalue of the problem (2.4) is
and the corresponding eigenfunction is \(\varphi _{k}=\sin \frac{k\pi t}{T+1}\), \(k=1,2,\ldots , T\). The proof is complete. □
3 Main results
In this section, we try our best to find the nontrivial positive solution of the problem (1.1). Let
Then K is a positive cone in E. Define an operator \(A:K\to E\) by
Since \(f:[1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\times \mathbb{R}\to \mathbb{R}^{+}\) is a continuous function, it is not difficult to see that \(A:K\to K\) is a completely continuous mapping. Now, it suffices to find the nontrivial positive fixedpoint of A. To get it, let us recall some basic concepts and lemmas on the fixedpoint theory in a cone; see [9, 18].
Let E be a Banach space, \(K\subset E\) is a closed convex cone. Suppose that D is a bounded open subset of E with boundary ∂D, and \(K\cap D\neq \emptyset \). Then the following lemmas hold.
Lemma 3.1
Let D be a bounded open subset of E with \(\theta \in D\), and \(A : K \cap \bar{D} \to K\) a completely continuous mapping. If \(\mu Au \neq u\) for every \(u\in K \cap \partial D\) and \(0 < \mu < 1\), then \(i (A, K \cap D, K) = 1\).
Lemma 3.2
Let D be a bounded open subset of E and \(A : K \cap \bar{D} \to K\) a completely continuous mapping. If there exists \(v_{0}\in K \setminus \{\theta \}\) such that \(uAu\neq \tau v_{0}\) for every \(u\in K \cap \partial D\) and \(\tau \geq 0\), then \(i (A, K \cap D, K) = 0\).
Lemma 3.3
Let D be a bounded open subset of E, and \(A, A_{1} : K \cap \bar{D} \to K\) be two completely continuous mappings. If \((1t)Au + tA_{1}u \neq u\) for every \(u\in K \cap \partial D\) and \(0\leq t \leq 1\), then \(i (A, K \cap D, K) = i (A_{1}, K \cap D, K)\).
Now, let us introduce two notations. For \(r>0\), let
The first main result is as follows.
Theorem 3.1
Let \(f : [1,T]_{\mathbb{Z}} \times \mathbb{R}^{+} \times \mathbb{R} \to \mathbb{R}^{+}\) be a continuous function. Suppose that the conditions
 (H1):

there exist three positive constants \(a>0\), \(b>0\), \(\delta >0\) with \(a+2b< \frac{1}{T^{2}}\) such that
$$ f (t, u, v) \leq a u + b \vert v \vert , \quad (t, u, v) \in [1, T]_{\mathbb{Z}} \times [0, \delta ] \times [2\delta , 2\delta ], $$and
 (H2):

there exist constants \(c > \lambda _{1}=22\cos \frac{\pi }{T+1}\) and \(H > 0\) such that
$$ f (t, u, v) \geq c u,\quad (t, u, v) \in [1, T]_{\mathbb{Z}} \times \mathbb{R}^{+}\times \mathbb{R}, \vert u \vert + \vert v \vert > H, $$hold.
Then the boundary value problem (1.1) has at least one positive solution in K.
Proof
Let \(r_{1}\in (0,\delta )\) small enough, where δ is the positive constant introduced by (H1). Then, by Lemma 3.1, we try to prove that, for any \(u\in K\cap \partial \Omega _{r_{1}}\) and \(0<\mu \leq 1\),
Suppose to the contrary that there exists \(u_{0}\in K\cap \partial \Omega _{r_{1}}\) and \(0<\mu _{0}\leq 1\) such that \(\mu _{0} Au_{0}= u_{0}\). This implies that \(u_{0}\) is a positive solution of the problem
Now, by (H1), we have
Therefore, combining this with the fact
we get
Combining this with Lemma 2.3 (ii) and (iii), we obtain
This contradicts the assumption \(a+2b<\frac{1}{T^{2}}\). Therefore, (3.1) holds. By Lemma 3.1, we get
Now, let \(L_{0}=\max \{f(t,u,v)cu:(t,u,v)\in [1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\times \mathbb{R}, u+v\leq H\}+1\). Then, the condition (H2) implies that
Define a operator \(A_{1}: K\to E\) by
Then \(A_{1}:K\to K\) is a completely continuous operator. Now, let \(r_{2}>\delta \), we show that
To get it, by Lemma 3.2, we only need to show that
where \(\varphi _{1}(t)=\sin \frac{\pi t}{T+1} / \\sin \frac{\pi t}{T+1} \_{E}\) is the eigenfunction of the linear eigenvalue problem (2.4), which corresponds to the first eigenvalue \(\lambda _{1}=22\cos \frac{\pi }{T+1}\). Then \(\\varphi _{1}\_{E}=1\) and \(\varphi _{1}(t)>0\) on \([1,T]_{\mathbb{Z}}\). Suppose to the contrary that there exist \(u_{1}\in K\cap \partial \Omega _{r_{2}}\) and \(\tau _{1}\geq 0\) such that \(u_{1}A_{1}u_{1}=\tau _{1}\varphi _{1}\). Combining this with the definition of \(A_{1}\), we know that \(u_{1}\) is a solution of the problem
Therefore, by (H2), we get
Multiplying this inequality by \({{\varphi _{1}(t)}}\) and summing from \(s=1\) to \(s=t\), we get
Now, if \(\sum_{{{t=1}}}^{T}u_{1}(t)\varphi _{1}(t)\neq 0\), then we get \(\lambda _{1}\geq c\). In fact, by Lemma 2.3 (i), for \(t\in [0,T+1]_{\mathbb{Z}}\), we get
This implies that
Therefore, \(\lambda _{1}\geq c\). However, this contradicts the condition (H2). So, we see that (3.4) holds and then (3.3) holds too.
Now, let us show that the operator A and \(A_{1}\) satisfy the condition of Lemma 3.3 for \(r_{2}>0\) large enough, i.e.,
Suppose to the contrary that there exist \(u_{2}\in K \cap \partial \Omega _{r_{2}}\) and \(0\leq t_{0}\leq 1\) such that
Therefore, by the definition of A and \(A_{2}\), we know that \(u_{2}\) is a solution of the problem
Therefore,
Multiplying both sides of this inequality by \(\varphi _{1}(t)\) and summing from \(s=1\) to \(s=T\), we get
This implies that
Furthermore, by Lemma 2.3 (i), we know that
So,
Combining this with (3.6), we obtain
Let
Now, let \(r_{2}=\max \{M,\delta \}\). Then, by the definition of \(\Omega _{r_{2}}\), we get \(\u_{2}\_{E}={{r_{2}}}>M\) if \(u\in K\cap \partial \Omega _{r_{2}}\). However, this contradicts (3.7). Therefore, (3.5) holds, which implies that the operator A and \(A_{1}\) satisfy the condition in Lemma 3.2. Therefore, by Lemma 3.2, we get
Combining this with (3.3), we get
Hence,
Therefore, A has a fixedpoint \(u\in K\cap (\Omega _{r_{2}}\setminus \overline{\Omega }_{r_{1}}) \). Furthermore, it is a positive solution of (1.1). □
Remark 3.1
In this remark, we try to show that our condition (H1) and (H2) are weaker than the usual limitation conditions. In fact, Let \(f:[1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\times \mathbb{R}\to \mathbb{R}^{+}\) is continuous and
Then it is not difficult to see that
imply the condition (H1) and the condition (H2) hold, respectively. In fact, if \(f^{0}<\frac{1}{2T^{2}}\), then there exist two positive constants \(\varepsilon _{1}>0\) and \(\delta >0\) small enough such that \(f^{0}+\varepsilon _{1}<\frac{1}{2T^{2}}\) and
Now, if we choose \(a=f^{0}+\varepsilon _{1}\) and \(b=\frac{f^{0}+\varepsilon _{1}}{2}\), then \(a+b<\frac{1}{T^{2}}\) and
This means that the condition (H1) holds. If \(f_{\infty }>\lambda _{1}\), then there exist a constant \(\varepsilon _{2}>0\) small enough and a positive constant \(H>0\) big enough such that \({{f_{\infty }\varepsilon _{2}}}>\lambda _{1}\) and
Now, let \(c=f_{\infty }\varepsilon _{2}\). Then (H2) holds.
Theorem 3.2
\(f : [1,T]_{\mathbb{Z}} \times \mathbb{R}^{+} \times \mathbb{R} \to \mathbb{R}^{+}\) be a continuous function. Suppose that the conditions
 (H3):

there exist constants \(c > \lambda _{1}\) and \(\eta > 0 \) such that
$$ f (t, u, v) \geq c u,\quad (t, u, v) \in [1,T]_{\mathbb{Z}}\times [0, \eta ] \times [2\eta ,2\eta ]; $$  (H4):

there exist three positive constants \(a>0\), \(b>0\) and \(H>0\) with \(a + 2b < \frac{1}{T^{2}}\), such that
$$ f (t, u, v) \leq a u + b \vert v \vert ,\quad (t, u, v) \in [1, T]_{\mathbb{Z}} \times \mathbb{R}^{+}\times \mathbb{R}, \vert u \vert + \vert v \vert > H, $$hold.
Then the boundary value problem (1.1) has at least one positive solution in K.
Proof
Let \(r_{3}\in (0,\eta )\), where η is the constant in (H3). Then we will show that
To get it, we choose \(v_{0}=\varphi _{1}(t)\) and verify the condition of Lemma 3.2 holds, that is,
Suppose to the contrary that there exist \(u_{3}\in K\cap \partial \Omega _{r_{3}}\) and \(\tau _{3}\geq 0\) such that
Then, by the definition of A, we know that \(u_{3}\) is a solution of the problem
Furthermore, since \(u_{3}\in K\cap \partial \Omega _{r_{3}}\), we know that \(0\leq u_{3}(t)\leq \u_{3}\_{E}={{r_{3}}}<\eta \) and \(0\leq \Delta u_{3}(t)\leq 2\u_{3}\_{E}=2{{r_{3}}}<2 \eta \). Therefore, by (H4), we get
Now, similar to the proof of (3.4), we get a contradiction. Therefore, (3.8) holds and then, by Lemma 3.2,
Next, let \(r_{4}>\delta \) large enough. Then, by Lemma 3.1, we only need to show that
Suppose to the contrary that there exist \(u_{4}\in K\cap \partial \Omega _{r_{4}}\) and \(\mu _{4}\in (0,1]\) such that
Then, by the definition of A, we know that \(u_{4}\) is a solution of the problem
Now, choose \(L_{1}=\max \{f(t,u,v)(au+bv):(t,u,v)\in [1,T]_{\mathbb{Z}}\times \mathbb{R}^{+}\times \mathbb{R}, u+v\leq H\}+1\). Then the condition (H4) implies that
Then, by the facts that \(u_{4}\in K\cap \partial \Omega _{r_{4}}\) and \(\mu _{4}\in (0,1]\), we obtain
Summing both sides of the above inequality from \(s=1\) to \(s=T\), then we get
Furthermore, by Lemma 2.3 (ii) and (iii),
Combining this with (3.11), we obtain
Let \(r_{4}>\max \{M_{1},\delta \}\). Since \(u_{4}\in K\cap \partial \Omega _{r_{4}}\), we know that \(\u_{4}\_{E}=r_{4}>M_{1}\). However, this contradicts (3.12). Therefore, (3.10) holds. Now, by Lemma 3.1, we get
Combining (3.9) with (3.13), we obtain
Therefore, A has a fixedpoint \(u\in K\cap (\Omega _{r_{4}}\setminus \overline{\Omega }_{r_{3}}) \). Furthermore, it is a positive solution of (1.1). □
Remark 3.2
Similar to Remark 3.1, it is not difficult to see that the condition (H3) and (H4) are also weaker than the usual limitation conditions.
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The second author is supported by Natural Science Foundation of Liaoning Provice (Grant No. 2019MS109) and HSSF of Chinese Ministry of Education (Grant No. 20YJA790049).
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Jin, L., Luo, H. Positive solutions of a discrete secondorder boundary value problems with fully nonlinear term. Adv Differ Equ 2020, 583 (2020). https://doi.org/10.1186/s13662020030394
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DOI: https://doi.org/10.1186/s13662020030394