- Research
- Open Access
Exact multiplicity of solutions for discrete second order Neumann boundary value problems
- Dingyong Bai^{1, 2},
- Hairong Lian^{3} and
- Haiyan Wang^{4, 5, 6}Email author
- Received: 18 August 2015
- Accepted: 21 November 2015
- Published: 8 December 2015
Abstract
Our concern is the second order difference equation \(\Delta^{2} u(t-1)+g(u(t))=h(t)\) subject to the Neumann boundary conditions \(\Delta u(0)=\Delta u(T)=0\). Under convex/concave conditions imposed on g, some results on the exact numbers of solutions and positive solutions are established based on the discussions to the maximum and minimum numbers of (positive) solutions.
Keywords
- difference equation
- Neumann boundary value problem
- exact numbers of solutions and positive solutions
1 Introduction
In these last years, the existence and multiplicity of solutions for nonlinear discrete problems subject to various boundary value conditions have been widely studied by using different abstract methods such as critical point theory, fixed point theorems, lower and upper solutions method, and Brower degree (see, e.g., [1–17] and the references therein). All these results are about the unique solution, or the minimum amount of solutions, and positive solutions. To the best of our knowledge, there is no report on the exact number of solutions for discrete boundary value problems.
For BVPs of differential equations, there are many papers concerned with the bifurcation values and exact multiplicities of solutions and positive solutions by bifurcation theory, quadrature method, time-map analysis and otherwise. See [18–29] and the references therein. For difference equations, however, the loss of continuity puts some methods used well in differential equations, such as the quadrature method and its time-map analysis, out of action. Therefore, it is very meaningful to study the exact number of solutions for discrete boundary value problems. In this paper, based on the discussions to the maximum and minimum numbers of (positive) solutions, we establish some results on the exact number of (positive) solutions of (1.1) under convex/concave conditions.
The remaining part of this paper is organized as follows. In Section 2, under the Neumann boundary conditions \(\Delta u(0)=\Delta u(T)=0\), we show the Green’s functions of the linear difference operators \(-\Delta^{2} u(t-1)+L u(t)\) and \(\Delta^{2} u(t-1)+K u(t)\), where L and K are two positive constants with \(K<4\sin^{2}\frac{\pi}{2T}\). Then, in Section 3, we make some estimates on the maximum number of solutions of (1.1), and, in Section 4, we establish some results on the minimum numbers of solutions and positive solutions of (1.1). Finally, we give the results on the exact multiplicities of solutions and positive solutions of (1.1) in Section 5.
2 Green’s functions
In this section, we show the Green’s functions of the linear difference operators \(-\Delta^{2} u(t-1)+L u(t)\) and \(\Delta^{2} u(t-1)+K u(t)\) satisfying the Neumann boundary conditions \(\Delta u(0)=\Delta u(T)=0\).
Lemma 2.1
Proof
Since the corresponding homogeneous problem has only the trivial zero solution, problem (2.1) has a unique solution u, which is given by (2.1). Clearly, if \(0< K<4\sin^{2}\frac{\pi}{4T}\), then \(0<\theta<\frac{\pi}{2T}\) and hence \(G_{1}(t,s)>0\) for all \((t,s)\in[0,T+1]_{\mathbf{Z}}\times[1,T]_{\mathbf{Z}}\). The proof is complete. □
Lemma 2.2
Proof
It is easy to check that the corresponding homogeneous problem has only the trivial zero solution. So problem (2.4) has a unique solution u which is given by (2.5). Finally, since \(A>1\), we have, for all \((t,s)\in[0,T+1]_{\mathbf{Z}}\times[1,T]_{\mathbf{Z}}\), \(G_{2}(t,s)>0\). The proof is complete. □
3 Estimates on the maximum number of solutions
In this section, we make estimations on the maximum number of solutions of problem (1.1). First, we prove some lemmas for later use. The first lemma is the discrete Sobolev inequality.
Lemma 3.1
Proof
Definition 3.1
We say that \(u:[0,T+1]_{\mathbf{Z}}\to\mathbf{R}\) has a generalized zero at \(t_{0}\in[0,T+1]_{\mathbf{Z}}\) provided that \(u(t_{0})=0\) if \(t_{0}=0\) and if \(t_{0}\in[1,T+1]_{\mathbf{Z}}\) either \(u(t_{0})=0\) or \(u(t_{0}-1)u(t_{0})<0\).
Lemma 3.2
Proof
Lemma 3.3
Proof
Lemma 3.4
Assume \(g\in C^{1}(\mathbf{R})\) and \(g'(x)<\frac{4}{T^{2}-1}\) for all \(x\in\mathbf{R}\). Let \(u_{1}\) and \(u_{2}\) are two distinct solutions of (1.1), then \(u_{1}(t)-u_{2}(t)\) has no generalized zero on \([0,T+1]_{\mathbf{Z}}\) and hence \(u_{1}\) and \(u_{2}\) are strictly ordered.
Proof
Now, we show our main results of this section. First, if g is strictly convex on R, we have the following result.
Theorem 3.5
Assume \(g\in C^{1}(\mathbf{R})\), \(g'(x)<\frac{4}{T^{2}-1}\) for all \(x\in \mathbf{R}\) and \(g'\) is strictly increasing. Then problem (1.1) has at most two solutions.
Proof
Similarly, if g is strictly concave on R, we have the following result.
Theorem 3.6
Assume \(g\in C^{1}(\mathbf{R})\), \(g'(x)<\frac{4}{T^{2}-1}\) for all \(x\in \mathbf{R}\) and \(g'\) is strictly decreasing. Then problem (1.1) has at most two solutions.
4 On the minimum number of solutions and positive solutions
In this section, we consider the minimum numbers of solution and positive solutions of (1.1). First, we consider the nonexistence, existence and multiplicity of solutions of (1.1) when \(g\in C^{1}(\mathbf{R})\) and g has a unique global minimum value on R.
Theorem 4.1
- (1)If one of the following conditions is satisfied, then problem (1.1) has no solution:
- (1-i)
\(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (1-ii)
\(\max\{a,b\}<+\infty\), \(h(t)\geq\max\{a,b\}\) for all \(t\in [1,T]_{\mathbf{Z}}\) and \(\max\{a,b\}>g(x)\) for all \(x\in\mathbf{R}\).
- (1-i)
- (2)
If \(h(t)\equiv\lambda_{*}\) on \([1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one solution.
- (3)
If \(\lambda_{*}\leq h(t)<\min\{a, b\}\), \(h(t)\not\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \([x_{0}, \infty)\), then problem (1.1) has at least two solutions.
- (4)If one of the following conditions is satisfied, then problem (1.1) has at least one solution:
- (4-i)
\(a\neq b\), \(\min\{a,b\}<+\infty\) and \(h(t)\equiv\min\{a,b\}\), \(t\in [1,T]_{\mathbf{Z}}\);
- (4-ii)
\(b< h(t)< a\), \(t\in[1,T]_{\mathbf{Z}}\);
- (4-iii)
\(a< h(t)< b\), \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi }{4T}\) on \([x_{0}, \infty)\).
- (4-i)
Proof
(1-ii) Suppose (1.1) has a solution u. Then \(g(u(t))<\max\{ a,b\}\) for all \(t\in[1,T]_{\mathbf{Z}}\), which implies that \(\Delta^{2} u(t-1)=h(t)-g(u(t))> 0\) for \(t\in [1,T]_{\mathbf{Z}}\). Thus, we have the contradiction that \(\Delta u(T)>\Delta u(0)\).
(2) It is easy to see that \(u(t)\equiv x_{0}\) is a solution of (1.1). Assume that v is also a solution of (1.1). Since \(g(v(t))\geq\lambda_{*}\), \(t\in[1,T]_{\mathbf{Z}}\), we have \(\Delta^{2} v(t-1)\leq0\), \(t\in[1,T]_{\mathbf{Z}}\). It follows by the boundary conditions \(\Delta v(0)=\Delta v(T)=0\) that \(\Delta v(t)\equiv0\), \(t\in[0,T]_{\mathbf{Z}}\). Thus, \(g(v(t))\equiv g(x_{0})\). Therefore, \(v(t)\equiv x_{0}\), \(t\in [0,T+1]_{\mathbf{Z}}\), since \(g(x)>g(x_{0})\) for all \(x\in\mathbf{R}\setminus\{x_{0}\}\).
(4-i) Without loss of generality, we assume \(b< a\). Since \(h(t)\equiv b\), \(t\in [1,T]_{\mathbf{Z}}\), there exists \(c_{1}< x_{0}\) such that \(g(c_{1})=b\). Obviously, \(u(t)\equiv c_{1}\) is a solution of (1.1).
(4-ii) Since \(b< h(t)< a\), \(t\in [1,T]_{\mathbf{Z}}\), there exist \(c_{1}< c_{2}< x_{0}\) such that \(g(c_{1})>h(t)\), \(g(c_{2})< h(t)\), \(t\in [1,T]_{\mathbf{Z}}\). By the continuity of \(g'\), there exists \(L>0\) such that \(g'(x)>-L\), \(x\in[c_{1}, c_{2}]\). Let \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv c_{2}\). Considering the operator S defined as (4.3) on \([u_{1},u_{2}]\), one can see that S has a fixed point \(u^{*}\in[u_{1}, u_{2}]\), which is a solution of (1.1).
(4-iii) Since \(a< h(t)< b\), \(t\in [1,T]_{\mathbf{Z}}\), there exist \(x_{0}< c_{1}< c_{2}\) such that \(g(c_{1})< h(t)\), \(g(c_{2})>h(t)\), \(t\in [1,T]_{\mathbf{Z}}\). By \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) for \(x\geq x_{0}\), there exists \(K>0\) such that \(g'(x)< K<4\sin^{2} \frac{\pi}{4T}\), \(x\in[c_{1},c_{2}]\). Let \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv c_{2}\). Considering the operator Q defined as (4.5) on \([u_{1},u_{2}]\), one can see that Q has a fixed point \(u^{*}\in[u_{1}, u_{2}]\), which is a solution of (1.1).
The proof is complete. □
Similarly, if \(g\in C^{1}(\mathbf{R})\) and g has a unique global maximum value on R, we have the following result.
Theorem 4.2
- (1)If one of the following conditions is satisfied, then problem (1.1) has no solution:
- (1-i)
\(h(t)\geq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (1-ii)
\(\min\{a,b\}>-\infty\), \(h(t)\leq\min\{a,b\}\) for \(t\in[1,T]_{\mathbf {Z}}\) and \(\min\{a,b\}< g(x)\) for \(x\in\mathbf{R}\).
- (1-i)
- (2)
If \(h(t)\equiv\lambda^{*}\) on \([1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one solution.
- (3)
If \(\max\{a,b\}< h(t)\leq\lambda^{*}\), \(h(t)\not\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\); and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \((-\infty,x_{0}]\), then problem (1.1) has at least two solutions.
- (4)If one of the following conditions is satisfied, then problem (1.1) has at least one solution:
- (4-i)
\(a\neq b\), \(\max\{a,b\}>-\infty\) and \(h(t)\equiv\max\{a,b\}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (4-ii)
\(a< h(t)< b\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi }{4T}\) on \((-\infty,x_{0}]\);
- (4-iii)
\(b< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (4-i)
Now, we consider the positive solutions of (1.1). Let \(\langle0,+\infty)\) denote \((0,+\infty)\) or \([0,\infty)\). First, if \(g\in C^{1}(\langle0,+\infty), \mathbf{R})\) and g has a unique global minimum value on \(\langle0,+\infty)\), we have the following result.
Theorem 4.3
- (1)If one of the following conditions is satisfied, then problem (1.1) has no positive solution:
- (1-i)
\(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (1-ii)
\(\max\{a,b\}<+\infty\), \(h(t)\geq\max\{a,b\}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g(x)<\max\{a,b\}\) for all \(x\in(0,+\infty)\).
- (1-i)
- (2)
If \(h(t)\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one positive solution.
- (3)
If \(\lambda_{*}\leq h(t)< \min\{a,b\}\), \(h(t)\not\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \([x_{0}, +\infty)\), then problem (1.1) has at least two positive solutions.
- (4)If one of the following conditions is satisfied, then problem (1.1) has at least one positive solution:
- (4-i)
\(a\neq b\), \(\min\{a,b\}<+\infty\), \(h(t)\equiv\min\{a,b\}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (4-ii)
\(b< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (4-iii)
\(a< h(t)< b\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi }{4T}\) on \([x_{0}, \infty)\).
- (4-i)
- (5)If one of the following conditions is satisfied, then problem (1.1) has no positive solution:
- (5-i)
\(h(t)\leq\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (5-ii)
\(\max\{a,b\}<+\infty\) and \(h(t)\geq\max\{a,b\}\) for \(t\in [1,T]_{\mathbf{Z}}\), and \(g(x)<\max\{a,b\}\) for all \(x\in( 0,+\infty)\).
- (5-i)
- (6)If one of the following conditions is satisfied, then problem (1.1) has at least two positive solutions:
- (6-i)
\(a>0\), \(b>0\), \(0< h(t)< \min\{a,b\}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \([x_{0}, \infty)\);
- (6-ii)
\(a>0\), \(b>0\), and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (6-i)
- (7)If one of the following conditions is satisfied, then problem (1.1) has at least one positive solution:
- (7-i)
\(0<\min\{a,b\}<+\infty\), \(a\neq b\), and \(h(t)\equiv\min\{a,b\}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (7-ii)
\(0< b< a\) and \(b< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (7-iii)
\(0< a< b\), \(a< h(t)< b\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \([x_{0}, \infty)\);
- (7-iv)
\(a\leq0\), \(b>0\), and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (7-v)
\(a\leq0\), \(b>0\), \(0< h(t)< b\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \([x_{0}, \infty)\);
- (7-vi)
\(a>0\), \(b<0\), and \(0< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (7-i)
Proof
We only show the proofs of (3), (6), and (7-iv)-(7-vi).
(3) Without loss of generality, we assume \(b\leq a\). By the proof of Theorem 4.1(3), we can choose \(0< c_{1}< x_{0}< c_{2}\) and \(L>0\), \(K>0\), such that S defined as (4.3) has a fixed point \(u_{1}^{*}\) satisfying \(c_{1}\leq u_{1}^{*}(t)\leq x_{0}\) and Q defined as (4.5) has a fixed point \(u_{2}^{*}\) satisfying \(x_{0}\leq u_{2}^{*}(t)\leq c_{2}\). Clearly, \(u_{1}^{*}(t)\) and \(u_{2}^{*}(t)\) are two positive solutions of (1.1).
(7-iv) By \(a\leq0\), \(b>0\), then there exists \(x_{1}>x_{0}\) by \(\lambda_{*}<0\) such that \(g(x_{1})=0\). It is easy to see that \(u(t)\equiv x_{1}\) is a positive solution of (1.1) since \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
(7-v) By \(a\leq0\), \(b>0\), there exists \(x_{1}>0\) such that \(g(x_{1})=0\). Thus, \(0< h(t)< b\), \(t\in[1,T]_{\mathbf{Z}}\), implies that there exist \(c_{2}>c_{1}>x_{1}\) such that \(g(c_{1})< h(t)\) and \(g(c_{2})>h(t)\) for \(t\in[1,T]_{\mathbf{Z}}\). Similar to the proof of Theorem 4.1(3), Q defined on \([u_{1}, u_{2}]\) has a fixed point in \([u_{1}, u_{2}]\), where \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv c_{2}\), \(t\in[1,T]_{\mathbf{Z}}\).
(7-vi) By \(a>0\), \(b<0\), there exists \(x_{1}>0\) such that \(g(x_{1})=0\). Since \(0< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\), there exist \(0< c_{1}< c_{2}< x_{1}\) such that \(g(c_{1})>h(t)\) and \(g(c_{2})< h(t)\) for \(t\in[1,T]_{\mathbf{Z}}\). Similar to the proof of Theorem 4.1(3), S defined on \([u_{1}, u_{2}]\) has a fixed point in \([u_{1}, u_{2}]\), where \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv c_{2}\), \(t\in[1,T]_{\mathbf{Z}}\).
The proof is complete. □
Finally, if \(g\in C^{1}(\langle0,+\infty), \mathbf{R})\) and g has a unique global maximum value on \(\langle0,+\infty)\), we have the following result.
Theorem 4.4
- (1)If one of the following conditions is satisfied, then problem (1.1) has no positive solution:
- (1-i)
\(h(t)\geq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (1-ii)
\(\min\{a,b\}\geq0\), \(h(t)\leq\min\{a,b\}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g(x)>\min\{a,b\}\) for all \(x\in(0,+\infty)\).
- (1-i)
- (2)
If \(h(t)\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one positive solution.
- (3)If one of the following conditions is satisfied, then problem (1.1) has at least two positive solutions:
- (3-i)
\(\max\{a,b,0\}< h(t)\leq\lambda^{*}\), \(h(t)\not\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \((0, x_{0})\);
- (3-ii)
\(a<0\), \(b<0\), and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (3-i)
- (4)If one of the following conditions is satisfied, then problem (1.1) has at least one positive solution:
- (4-i)
\(\max\{a,b\}>0\), \(a\neq b\), and \(h(t)\equiv\max\{a,b\}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (4-ii)
\(a>0\), \(b<0\), and \(0< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (4-iii)
\(a<0\), \(b>0\), \(0< h(t)< b\) for \(t\in[1,T]_{\mathbf{Z}}\), and \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) on \((0, x_{0})\);
- (4-iv)
either \(a\geq0\), \(b<0\) or \(a<0\), \(b\geq0\), and \(h(t)\equiv0\) for \(t\in [1,T]_{\mathbf{Z}}\).
- (4-i)
5 Exact numbers of solutions and positive solutions
In this section, we establish the results of exact multiplicities of solutions and positive solutions for problem (1.1). First, we consider the exact number of solutions of (1.1).
Note that \(4\sin^{2} \frac{\pi}{4T}<\frac{4}{T^{2}-1}\) for \(T>3\).
If \(g\in C^{1}(\mathbf{R})\) and g is strictly convex on R, we have the following result by Theorem 3.5 and Theorem 4.1.
Theorem 5.1
- (1)
If \(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no solution.
- (2)
If \(h(t)\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one solution.
- (3)
If \(h(t)\geq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly two solutions.
If \(g\in C^{1}(\mathbf{R})\) and g is strictly concave on R, then, by Theorem 3.6 and Theorem 4.2, we have the following result.
Theorem 5.2
- (1)
If \(h(t)\geq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no solution.
- (2)
If \(h(t)\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one solution.
- (3)
If \(h(t)\leq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly two solutions.
Now, we consider the exact number of positive solutions of problem (1.1). First, we consider the case that \(g\in C^{1}(\langle0,+\infty),\mathbf {R})\) and g is strictly convex on \(\langle0,+\infty)\). If \(g\in C^{1}((0,\infty),\mathbf{R})\), we have the following result by Theorem 3.5 and Theorem 4.3.
Theorem 5.3
- (1)
If \(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no positive solution.
- (2)
If \(h(t)\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one positive solution.
- (3)If one of the following conditions is satisfied, then problem (1.1) has exactly two positive solutions:
- (i)
\(\lambda_{*}\geq0\), \(h(t)\geq\lambda_{*}\), and \(h(t)\not\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (ii)
\(\lambda_{*}<0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (iii)
\(\lambda_{*}<0\) and \(h(t)> 0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (i)
If \(g\in C^{1}([0,\infty),\mathbf{R})\), we have the following result.
Theorem 5.4
- (1)
If \(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no positive solution.
- (2)If one of the following conditions is satisfied, then problem (1.1) has exactly one positive solution:
- (2-i)
\(\lambda_{*}< a\) and \(h(t)\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (2-ii)
\(\lambda_{*}< a\) and \(h(t)\equiv a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (2-iii)
\(h(t)>a\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (2-i)
- (3)
If \(\lambda_{*}\leq h(t)< a\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly two positive solutions.
- (4)If one of the following conditions is satisfied, then problem (1.1) has exactly one positive solution:
- (4-i)
\(a>0\) and \(h(t)\equiv a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (4-ii)
\(a>0\) and \(h(t)> a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (4-iii)
\(a\leq0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (4-iv)
\(a\leq0\) and \(h(t)> 0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (4-i)
- (5)If one of the following conditions is satisfied, then problem (1.1) has exactly two positive solutions:
- (5-i)
\(a>0\) and \(0< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (5-ii)
\(a>0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (5-i)
Proof
Let \(\lambda_{*}=g(x_{0})\). We only show the proofs of (2-ii) and (2-iii).
(2-ii) By \(g'\) is strictly increasing, there exists a unique \(x_{1}>x_{0}\) such that \(g(x_{1})=a\) and \(g(x)>a\) for all \(x>x_{1}\). Clearly, \(u(t)\equiv x_{1}\) is a positive solution of (1.1). Assume that \(v(t)\) is another positive solution, then u and v are strictly ordered by Lemma 3.4. If \(x_{1}\equiv u(t)< v(t)\), \(t\in[1,T]_{\mathbf {Z}}\), then \(g(u(t))\equiv a< g(v(t))\), \(t\in[1,T]_{\mathbf{Z}}\), by the fact that \(g(x)\) is strictly increasing on \([x_{1},+\infty)\). Thus, \(\Delta^{2} v(t-1)<0\), \(t\in[1,T]_{\mathbf{Z}}\), which implies \(\Delta v(T)<\Delta v(0)\), a contradiction. If \(v(t)< u(t)\equiv x_{1}\), then the fact that \(g(x)< a\) for \(x\in (0,x_{1})\) shows that \(g(v(t))< g(u(t))\equiv a\) and hence \(\Delta^{2} v(t-1)>0\), \(t\in [1,T]_{\mathbf{Z}}\). It follows that \(\Delta v(T)>\Delta v(0)\), a contradiction.
(2-iii) We distinguish two cases to finish the proof.
Case 2: \(\lambda_{*}< a\). By \(g'\) is strictly increasing, there exists a unique \(x_{1}>x_{0}\) such that \(g(x_{1})=a\) and \(g(x)>a\) for all \(x>x_{1}\). Since \(h(t)>a\), there exist \(c_{1}\) and \(c_{2}\): \(x_{1}< c_{1}< c_{2}\) such that \(g(c_{1})< h(t)\), \(g(c_{2})>h(t)\), \(t\in[1,T]_{\mathbf{Z}}\). By the proof of Theorem 4.1(4-iii), we know that (1.1) has a positive solution \(u_{1}\) with \(c_{1}\leq u_{1}(t)\leq c_{2}\), \(t\in[1,T]_{\mathbf{Z}}\). Now, we assume that \(u_{2}\) is also a positive solution of (1.1). Then \(u_{1}\) and \(u_{2}\) are strictly ordered by Lemma 3.4.
If \(u_{2}(t)< u_{1}(t)\), \(t\in[1,T]_{\mathbf{Z}}\), then \(g(u_{2}(t))< g(u_{1}(t))\), \(t\in[1,T]_{\mathbf{Z}}\). In fact, For any given t, if \(x_{0}< u_{2}(t)< u_{1}(t)\), then \(g(u_{2}(t))< g(u_{1}(t))\) since g is strictly increasing on \([x_{0}, +\infty)\). If \(0< u_{2}(t)\leq x_{0}\), then we also have \(g(u_{2}(t))\leq a=g(x_{1})< g(c_{1})\leq g(u_{1}(t))\), \(t\in[1,T]_{\mathbf{Z}}\). Thus, \(\Delta^{2} (u_{1}-u_{2})(t-1)=g(u_{2}(t))-g(u_{1}(t))<0\), \(t\in[1,T]_{\mathbf{Z}}\). It follows that \(\Delta u_{1}(T)-\Delta u_{2}(T)<\Delta u_{1}(0)-\Delta u_{2}(0)\), a contradiction.
If \(u_{2}(t)>u_{1}(t)\), \(t\in[1,T]_{\mathbf{Z}}\), then, by the monotony of g on \([x_{0}, +\infty)\), \(g(u_{1}(t))< g(u_{2}(t))\), \(t\in[1,T]_{\mathbf{Z}}\), and consequently, \(\Delta^{2} (u_{1}-u_{2})(t-1)=g(u_{2}(t))-g(u_{1}(t))>0\), \(t\in[1,T]_{\mathbf {Z}}\), which also yields the contradiction that \(\Delta u_{1}(T)-\Delta u_{2}(T)>\Delta u_{1}(0)-\Delta u_{2}(0)\).
The proof is complete. □
Finally, we consider the case that \(g\in C^{1}(\langle0,+\infty),\mathbf {R})\) and g is strictly concave on \(\langle0,+\infty)\). If \(g\in C^{1}((0,+\infty),\mathbf{R})\), we have the following result by Theorem 3.6 and Theorem 4.4.
Theorem 5.5
- (1)
If \(h(t)\geq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no positive solution.
- (2)
If \(h(t)\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), then problem (1.1) has exactly one positive solution.
- (3)If one of the following conditions is satisfied, then problem (1.1) has exactly two positive solutions:
- (3-i)
\(0< h(t)\leq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\);
- (3-ii)
\(\lambda^{*}>0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (3-i)
If \(g\in C^{1}([0,+\infty),\mathbf{R})\), we have the following result.
Theorem 5.6
- (1)
If \(h(t)\geq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in [1,T]_{\mathbf{Z}}\), then problem (1.1) has no positive solution.
- (2)If one of the following conditions is satisfied, then problem (1.1) has exactly one positive solution:
- (2-i)
\(h(t)\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (2-ii)
\(a>0\) and \(h(t)\equiv a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (2-iii)
\(a>0\) and \(0< h(t)< a\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (2-iv)
\(a\geq0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (2-i)
- (3)If one of the following conditions is satisfied, then problem (1.1) has exactly two positive solutions:
- (3-i)
\(\max\{0,a\}< h(t)\leq\lambda^{*}\) and \(h(t)\not\equiv\lambda^{*}\) for \(t\in[1,T]_{\mathbf{Z}}\);
- (3-ii)
\(a<0\) and \(h(t)\equiv0\) for \(t\in[1,T]_{\mathbf{Z}}\).
- (3-i)
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was done when Bai was visiting the School of Mathematical and Natural Sciences, Arizona State University. He would like to thank the school for its hospitality. The research is supported partially by the Research Funds for the Doctoral Program of Higher Education of China (No. 20124410110001), by PCSIRT of China (No. IRT1226) and by the Natural Science Fund of China (No. 11371107).
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
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