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
Let
\(g\in C^{1}(\mathbf{R})\), \(\lim_{x\to\infty}g(x)=a\), \(\lim_{x\to+\infty}g(x)=b\), and
\(\lambda_{*}:=g(x_{0})=\min_{x\in\mathbf{R}}g(x)\). Here, a, b
may be +∞. Assume
\(g(x)>\lambda_{*}\)
for all
\(x\in\mathbf{R}\setminus\{x_{0}\}\)
and
\(\lambda _{*}<\min\{a,b\}\).

(1)
If one of the following conditions is satisfied, then problem (1.1) has no solution:

(1i)
\(h(t)\leq\lambda_{*}\)
and
\(h(t)\not\equiv\lambda_{*}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(1ii)
\(\max\{a,b\}<+\infty\), \(h(t)\geq\max\{a,b\}\)
for all
\(t\in [1,T]_{\mathbf{Z}}\)
and
\(\max\{a,b\}>g(x)\)
for all
\(x\in\mathbf{R}\).

(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:

(4i)
\(a\neq b\), \(\min\{a,b\}<+\infty\)
and
\(h(t)\equiv\min\{a,b\}\), \(t\in [1,T]_{\mathbf{Z}}\);

(4ii)
\(b< h(t)< a\), \(t\in[1,T]_{\mathbf{Z}}\);

(4iii)
\(a< h(t)< b\), \(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi }{4T}\)
on
\([x_{0}, \infty)\).
Proof
(1i) Since \(h(t)\leq\lambda_{*}\) and \(h(t)\not\equiv\lambda_{*}\) for \(t\in[1,T]_{\mathbf{Z}}\), we have \(\sum_{t=1}^{T} h(t)< T\lambda_{*}\). Assume problem (1.1) has a solution u. Since \(g(u(t))\geq\lambda_{*}\), \(t\in[1,T]_{\mathbf{Z}}\), \(\sum_{t=1}^{T} g(u(t))\geq T\lambda_{*}\). Summing both sides of the equation \(\Delta^{2} u(t1)+g(u(t))=h(t)\) from 1 to T, we have by the boundary conditions \(\Delta u(0)=\Delta u(T)=0\),
$$ T\lambda_{*}\leq\sum_{t=1}^{T} g\bigl(u(t) \bigr)=\sum_{t=1}^{T} h(t)< T\lambda_{*}, $$
which is a contradiction.
(1ii) 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 \(\Delta^{2} u(t1)=h(t)g(u(t))> 0\) 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 \(\Delta^{2} v(t1)\leq0\), \(t\in[1,T]_{\mathbf{Z}}\). 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 \(x\in\mathbf{R}\setminus\{x_{0}\}\).
(3) Without loss of generality, we assume \(b\leq a\). Since \(\lambda_{*}\leq h(t)< b\), \(h(t)\not\equiv\lambda_{*}\), \(t\in [1,T]_{\mathbf{Z}}\), we see by \(\lim_{x\to\infty}g(x)=a\) and \(\lim_{x\to+\infty }g(x)=b\) that there exist \(c_{1}< x_{0}< c_{2}\) such that
$$ g(c_{1})>h(t),\qquad g(c_{2})>h(t),\quad t\in [1,T]_{\mathbf{Z}}. $$
(4.1)
Set \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv x_{0}\), \(u_{3}(t)\equiv c_{2}\), \(t\in [0,T+1]_{\mathbf{Z}} \). We prove that problem (1.1) has at least two solutions \(u_{1}^{*}\), \(u_{2}^{*}\): \(u_{1}^{*}\in[u_{1}, u_{2}]\), \(u_{2}^{*}\in[u_{2}, u_{3}]\). Here, \([u_{1}, u_{2}]\) and \([u_{2}, u_{3}]\) denote order intervals.
First, we show that \(u_{2}(t)\) is not a solution of (1.1). In fact, if it is not true, then summing both sides of the equation of (1.1) yields a contradiction:
$$ T\lambda_{*}=\sum_{t=1}^{T} g \bigl(u_{2}(t)\bigr)=\sum_{t=1}^{T} h(t)>T\lambda_{*}. $$
Second, we prove that (1.1) has at least one solution \(u_{1}^{*}\in[u_{1}, u_{2}]\). Let \(E=\{u:[0,T+1]_{\mathbf{Z}}\to\mathbf{R}\} \) with the norm \(\u\=\max_{t\in[0,T+1]_{\mathbf{Z}}}u(t) \). Since \(g'\) is continuous on the bounded closed interval \([c_{1}, x_{0}]\), there exists \(L>0\) such that \(g'(x)>L\) for \(x\in[c_{1}, x_{0}]\). Consider the following problem:
$$ \left \{ \textstyle\begin{array}{l} \Delta^{2} u(t1)+Lu(t)=g(u(t))+Lu(t)h(t),\quad t\in [1,T]_{\mathbf {Z}}, \\ \Delta u(0)=\Delta u(T)=0. \end{array}\displaystyle \right . $$
(4.2)
By Lemma 2.2, (4.2) is equivalent to \(u(t)=Su(t)\). Here, \(S:E\to E\) is defined by
$$ Su(t)=\sum_{s=1}^{T} G_{2}(t,s)\bigl[g\bigl(u(s)\bigr)+Lu(s)h(s)\bigr], \quad t\in [0,T+1]_{\mathbf{Z}}. $$
(4.3)
It is easy to see that S is continuous. Since \(u_{1}(t)\) satisfies
$$ \left \{ \textstyle\begin{array}{l} \Delta^{2} u(t1)+Lu(t)=Lu_{1}(t), \quad t\in [1,T]_{\mathbf{Z}}, \\ \Delta u(0)=\Delta u(T)=0, \end{array}\displaystyle \right . $$
we see by (4.1) and the positivity of \(G_{2}(t,s)\) that
$$\begin{aligned} Su_{1}(t) =&\sum_{s=1}^{T} G_{2}(t,s)\bigl[g\bigl(u_{1}(s)\bigr)+Lu_{1}(s)h(s) \bigr] \\ >&\sum_{s=1}^{T} G_{2}(t,s)Lu_{1}(s)=u_{1}(t), \quad t\in[0,T+1]_{\mathbf{Z}}, \end{aligned}$$
which shows that \(u_{1}\) is a lower solution of the operator S. Similarly, one can check that \(Su_{2}(t)\leq u_{2}(t)\) and hence \(u_{2}\) is an upper solution of S. On the other hand, S is an increasing operator defined on \([u_{1}, u_{2}]\). In fact, for \(v_{1}, v_{2}\in[u_{1}, u_{2}]\) with \(v_{1}\leq v_{2}\), we have
$$\begin{aligned} \begin{aligned} Sv_{2}(t)Sv_{1}(t)&=\sum_{s=1}^{T} G_{2}(t,s)\bigl[g\bigl(v_{2}(s)\bigr)g\bigl(v_{1}(s) \bigr)+L\bigl(v_{2}(s)v_{1}(s)\bigr)\bigr] \\ &\geq 0,\quad t\in[0,T+1]_{\mathbf{Z}}, \end{aligned} \end{aligned}$$
by \(g'(x)>L\) for \(x\in[c_{1}, x_{0}]\). Therefore, S has at least one solution \(u_{1}^{*}\in[u_{1}, u_{2}]\) by the fixed point theorem of increasing operator in ordered Banach spaces due to Amann [31]. So, (4.2), and hence (1.1), has at least one solution \(u_{1}^{*}\in[u_{1}, u_{2}]\).
Now, we prove that (1.1) has at least one solution \(u_{2}^{*}\in[u_{2}, u_{3}]\). Since \(g'(x)<4\sin^{2} \frac{\pi}{4T}\) for \(x\in [x_{0},+\infty)\), there exists \(K>0\) such that
$$ g'(x)< K< 4\sin^{2} \frac{\pi}{4T}, \quad x\in[x_{0},c_{2}]. $$
(4.4)
Consider the following problem:
$$ \left \{ \textstyle\begin{array}{l} \Delta^{2} u(t1)+Ku(t)=h(t)g(u(t))+Ku(t),\quad t\in [1,T]_{\mathbf {Z}}, \\ \Delta u(0)=\Delta u(T)=0. \end{array}\displaystyle \right . $$
(4.5)
By Lemma 2.1, \(G_{1}(t,s)>0\), and (4.5) is equivalent to \(u(t)=Qu(t)\). Here, \(Q:E\to E\) is defined by
$$ Qu(t)=\sum_{s=1}^{T} G_{1}(t,s)\bigl[h(s)g\bigl(u(s)\bigr)+Ku(s)\bigr], \quad t\in [0,T+1]_{\mathbf{Z}}. $$
(4.6)
Similar to the discussion of the operator S, Q has at least one solution \(u_{2}^{*}\in[u_{2}, u_{3}]\), which is a solution of (1.1). Therefore, problem (1.1) has at least two solutions.
(4i) 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).
(4ii) 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).
(4iii) 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
Let
\(g\in C^{1}(\mathbf{R})\), \(\lim_{x\to\infty}g(x)=a\), \(\lim_{x\to+\infty}g(x)=b\), and
\(\lambda^{*}:=g(x_{0})=\max_{x\in\mathbf{R}}g(x)\). Here, a, b
may be −∞. Assume
\(g(x)<\lambda^{*}\)
for all
\(x\in\mathbf{R}\setminus\{x_{0}\}\)
and
\(\lambda ^{*}>\max\{a,b\}\).

(1)
If one of the following conditions is satisfied, then problem (1.1) has no solution:

(1i)
\(h(t)\geq\lambda^{*}\)
and
\(h(t)\not\equiv\lambda^{*}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(1ii)
\(\min\{a,b\}>\infty\), \(h(t)\leq\min\{a,b\}\)
for
\(t\in[1,T]_{\mathbf {Z}}\)
and
\(\min\{a,b\}< g(x)\)
for
\(x\in\mathbf{R}\).

(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:

(4i)
\(a\neq b\), \(\max\{a,b\}>\infty\)
and
\(h(t)\equiv\max\{a,b\}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(4ii)
\(a< h(t)< b\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi }{4T}\)
on
\((\infty,x_{0}]\);

(4iii)
\(b< h(t)< a\)
for
\(t\in[1,T]_{\mathbf{Z}}\).
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
Let
\(g\in C^{1}(\langle0,+\infty), \mathbf{R})\), \(\lim_{x\to 0^{+}}g(x)=a\), \(\lim_{x\to+\infty}g(x)=b\). Here, a, b
may be +∞. Assume that there exists
\(x_{0}>0\)
such that
$$ \lambda_{*}:=g(x_{0})=\min_{x\in\langle0,+\infty)}g(x), $$
and
\(g(x)>\lambda_{*}\)
for all
\(x\in\langle0,+\infty)\setminus\{x_{0}\}\), \(\lambda_{*}<\min\{a,b\}\).
(I) Suppose
\(\lambda_{*}\geq0\).

(1)
If one of the following conditions is satisfied, then problem (1.1) has no positive solution:

(1i)
\(h(t)\leq\lambda_{*}\)
and
\(h(t)\not\equiv\lambda_{*}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(1ii)
\(\max\{a,b\}<+\infty\), \(h(t)\geq\max\{a,b\}\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g(x)<\max\{a,b\}\)
for all
\(x\in(0,+\infty)\).

(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:

(4i)
\(a\neq b\), \(\min\{a,b\}<+\infty\), \(h(t)\equiv\min\{a,b\}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(4ii)
\(b< h(t)< a\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(4iii)
\(a< h(t)< b\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi }{4T}\)
on
\([x_{0}, \infty)\).
(II) Suppose
\(\lambda_{*}< 0\).

(5)
If one of the following conditions is satisfied, then problem (1.1) has no positive solution:

(5i)
\(h(t)\leq\lambda_{*}\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(5ii)
\(\max\{a,b\}<+\infty\)
and
\(h(t)\geq\max\{a,b\}\)
for
\(t\in [1,T]_{\mathbf{Z}}\), and
\(g(x)<\max\{a,b\}\)
for all
\(x\in( 0,+\infty)\).

(6)
If one of the following conditions is satisfied, then problem (1.1) has at least two positive solutions:

(6i)
\(a>0\), \(b>0\), \(0< h(t)< \min\{a,b\}\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi}{4T}\)
on
\([x_{0}, \infty)\);

(6ii)
\(a>0\), \(b>0\), and
\(h(t)\equiv0\)
for
\(t\in[1,T]_{\mathbf{Z}}\).

(7)
If one of the following conditions is satisfied, then problem (1.1) has at least one positive solution:

(7i)
\(0<\min\{a,b\}<+\infty\), \(a\neq b\), and
\(h(t)\equiv\min\{a,b\}\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(7ii)
\(0< b< a\)
and
\(b< h(t)< a\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(7iii)
\(0< a< b\), \(a< h(t)< b\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi}{4T}\)
on
\([x_{0}, \infty)\);

(7iv)
\(a\leq0\), \(b>0\), and
\(h(t)\equiv0\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(7v)
\(a\leq0\), \(b>0\), \(0< h(t)< b\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi}{4T}\)
on
\([x_{0}, \infty)\);

(7vi)
\(a>0\), \(b<0\), and
\(0< h(t)< a\)
for
\(t\in[1,T]_{\mathbf{Z}}\).
Proof
We only show the proofs of (3), (6), and (7iv)(7vi).
(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).
(6) By \(a>0\), \(b>0\), and \(\lambda_{*}<0\), there exist \(x_{1}\) and \(x_{2}\) satisfying \(0< x_{1}< x_{0}< x_{2}\) such that \(g(x_{1})=g(x_{2})=0\). Therefore, if \(h(t)\equiv0\) on \([1,T]_{\mathbf{Z}}\), then \(u(t)\equiv x_{1}\) and \(u(t)\equiv x_{2}\) are two positive solutions of (1.1). If \(0< h(t)<\min\{a,b\}\), we know that there exist \(0< c_{1}< c_{2}< x_{1}\) and \(x_{2}< c_{3}< c_{4}\) such that
$$ g(c_{1})>h(t),\qquad g(c_{2})< h(t),\qquad g(c_{3})< h(t), \qquad g(c_{4})>h(t). $$
Let \(u_{i}(t)\equiv c_{i}\) (\(i=1,2,3,4\)). Similar to the proof of Theorem 4.1(3), consider the operator S defined as (4.3) on \([u_{1}, u_{2}]\) and the operator Q defined as (4.5) on \([u_{3}, u_{4}]\), one can find that S has a fixed point in \([u_{1}, u_{2}]\) and Q has a fixed point in \([u_{3}, u_{4}]\).
(7iv) 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 \(t\in[1,T]_{\mathbf{Z}}\).
(7v) 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 \(u_{1}(t)\equiv c_{1}\), \(u_{2}(t)\equiv c_{2}\), \(t\in[1,T]_{\mathbf{Z}}\).
(7vi) 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
Let
\(g\in C^{1}(\langle0,+\infty), \mathbf{R})\), \(\lim_{x\to 0^{+}}g(x)=a\), \(\lim_{x\to+\infty}g(x)=b\). Here, a, b
may be −∞. Assume that there exists
\(x_{0}>0\)
such that
$$ \lambda^{*}:=g(x_{0})=\max_{x\in\langle0,+\infty)}g(x) $$
and
\(g(x)<\lambda^{*}\)
for all
\(x\in\langle0,+\infty)\setminus\{x_{0}\}\), \(\lambda^{*}>0\), and
\(\lambda^{*}>\min\{a,b\}\).

(1)
If one of the following conditions is satisfied, then problem (1.1) has no positive solution:

(1i)
\(h(t)\geq\lambda^{*}\)
and
\(h(t)\not\equiv\lambda^{*}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(1ii)
\(\min\{a,b\}\geq0\), \(h(t)\leq\min\{a,b\}\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g(x)>\min\{a,b\}\)
for all
\(x\in(0,+\infty)\).

(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:

(3i)
\(\max\{a,b,0\}< h(t)\leq\lambda^{*}\), \(h(t)\not\equiv\lambda_{*}\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi}{4T}\)
on
\((0, x_{0})\);

(3ii)
\(a<0\), \(b<0\), and
\(h(t)\equiv0\)
for
\(t\in[1,T]_{\mathbf{Z}}\).

(4)
If one of the following conditions is satisfied, then problem (1.1) has at least one positive solution:

(4i)
\(\max\{a,b\}>0\), \(a\neq b\), and
\(h(t)\equiv\max\{a,b\}\)
for
\(t\in [1,T]_{\mathbf{Z}}\);

(4ii)
\(a>0\), \(b<0\), and
\(0< h(t)< a\)
for
\(t\in[1,T]_{\mathbf{Z}}\);

(4iii)
\(a<0\), \(b>0\), \(0< h(t)< b\)
for
\(t\in[1,T]_{\mathbf{Z}}\), and
\(g'(x)<4\sin^{2} \frac{\pi}{4T}\)
on
\((0, x_{0})\);

(4iv)
either
\(a\geq0\), \(b<0\)
or
\(a<0\), \(b\geq0\), and
\(h(t)\equiv0\)
for
\(t\in [1,T]_{\mathbf{Z}}\).