Let \(\psi_{1}\) and \(\psi_{2}\) be the unique solution of the BVP
$$\left \{ \textstyle\begin{array}{lcl} \psi_{1}''(t)+a(t)\psi_{1}'(t)+b(t)\psi_{1}(t)=0, \\ \psi_{1}(0)=0,\qquad \psi_{1}(1)=1, \end{array}\displaystyle \right . $$
and
$$\left \{ \textstyle\begin{array}{l} \psi_{2}''(t)+a(t)\psi_{2}'(t)+b(t)\psi_{2}(t)=0, \\ \psi_{2}(0)=1, \qquad\psi_{2}(1)=0, \end{array}\displaystyle \right . $$
respectively. By [8], we know that \(\psi_{1}\), \(\psi_{2}\) are strictly increasing and strictly decreasing on J, respectively.
We adopt the following assumptions throughout this article.
-
\(\mathrm{(H_{1})}\)
:
-
\(a\in C(J)\), \(b\in C(J, R_{-})\);
-
\(\mathrm{(H_{2})}\)
:
-
\(g, h: J_{+}\to R^{+}\) are integrable, and \(k_{1}>0\), \(k_{4}>0\), \(k>0\), where
$$\begin{gathered} k_{1}=1- \int_{0}^{1}\psi_{2}(s)g(s)\,\mathrm{d}s, \qquad k_{2}= \int_{0}^{1}\psi_{1}(s)g(s)\,\mathrm{d}s, \\k_{3}= \int_{0}^{1}\psi_{2}(s)h(s)\,\mathrm{d}s, \qquad k_{4}=1- \int_{0}^{1}\psi_{1}(s)h(s)\,\mathrm{d}s , \\k=k_{1}k_{4}-k_{2}k_{3}; \end{gathered} $$
-
\(\mathrm{(H_{3})}\)
:
-
\(f\in C(J_{+}\times R_{+}, R^{+})\);
-
\(\mathrm{(H_{4})}\)
:
-
there exist three functions \(\widehat{a}, \widehat{b}\in C(J_{+}, R^{+})\), \(\widehat{g}\in C(R_{+}, R^{+})\) satisfying
$$f(t,u)\leq\widehat{a}(t)\widehat{g}(u)+\widehat{b}(t),\quad \forall t\in J_{+}, u \in R_{+}, $$
where, in addition,
$$\widehat{a}_{r}^{*}= \int_{0}^{1}\mathcal{H}(t)\widehat{a}(t)\widehat {g}_{r}(t)\,\mathrm{d}t< +\infty, \qquad \widehat{b}^{*}= \int_{0}^{1}\mathcal {H}(t)\widehat{b}(t) \,\mathrm{d}t< +\infty, $$
and
$$\widehat{g}_{r}(t)=\max\bigl\{ \widehat{g}(u): \gamma(t)r \leq u\leq r \bigr\} ,\quad \forall r>0, $$
here, \(\gamma(t)\) is defined in (6), \(\mathcal{H}(t)\) is defined in (7);
-
\(\mathrm{(H_{5})}\)
:
-
there exists a function \(\widehat{c}\in C(J_{+}, R^{+})\) satisfying
$$\frac{f(t,u)}{\widehat{c}(t)u}\to+\infty\quad \mbox{as } u\to +\infty $$
uniformly for \(t\in J_{+}\), and in addition,
$$\widehat{c}^{*}= \int_{0}^{1}\widehat{c}(t)\,\mathrm{d}t< +\infty; $$
-
\(\mathrm{(H_{6})}\)
:
-
there exists a function \(\widehat{d}\in C(J_{+}, R^{+})\) satisfying
$$\frac{f(t,u)}{\widehat{d}(t)}\to+\infty \quad\mbox{as } u\to0^{+} $$
uniformly for \(t\in J_{+}\), and in addition,
$$\widehat{d}^{*}= \int_{0}^{1}\widehat{d}(t)\,\mathrm{d}t< +\infty. $$
Lemma 1
[4]
Assume that
\((\mathrm{H_{1}})\)
and
\((\mathrm{H_{2}})\)
hold. Then, for any
\(y\in C(J_{+})\cap L^{1}(J)\), the BVP
$$ \left \{ \textstyle\begin{array}{l} u''(t)+a(t)u'(t)+b(t)u(t)+y(t)=0,\quad t\in J_{+},\\ u(0)= \int_{0}^{1}g(s)u(s)\,\mathrm{d}s, \qquad u(1)=\int _{0}^{1}h(s)u(s)\,\mathrm{d}s, \end{array}\displaystyle \right . $$
(2)
has a unique solution
u
that can be expressed in the form
$$u(t)= \int_{0}^{1}H(t,s)y(s)\,\mathrm{d}s, \quad t\in J, $$
where
$$\begin{aligned}& \begin{aligned}[b] H(t,s)={}& G(t,s)p(s)+\frac{\psi_{1}(t)k_{3}+\psi _{2}(t)k_{4}}{k}\int_{0}^{1}G(\tau,s)p(s)g(\tau)\,\mathrm{d}\tau\\ & +\frac{\psi_{1}(t)k_{1}+\psi_{2}(t)k_{2}}{k}\int_{0}^{1}G(\tau ,s)p(s)h(\tau)\,\mathrm{d}\tau, \end{aligned} \end{aligned}$$
(3)
$$\begin{aligned}& p(t)=\exp \biggl( \int_{0}^{t}a(s)\,\mathrm{d}s \biggr), \\& G(t,s)=\frac{1}{\rho} \left \{ \textstyle\begin{array}{l@{\quad}l} \psi_{1}(t)\psi_{2}(s), & 0\leq t\leq s\leq1,\\ \psi_{1}(s)\psi_{2}(t), & 0\leq s\leq t\leq1, \end{array}\displaystyle \right .\quad \rho=\psi_{1}'(0). \end{aligned}$$
(4)
Moreover, \(u(t)\geq0\)
on
J
provided
\(y\geq0\).
By Remark 2.1 in [4], we have
Lemma 2
[4] Suppose that
\((\mathrm{H_{1}})\)
and
\((\mathrm{H_{2}})\)
hold, then, for any
\(t,s \in J\), we have
$$\begin{aligned}& 0\leq G(t,s)\leq G(s,s), \qquad 0\leq H(t,s)\leq\mathcal {H}(s), \end{aligned}$$
(5)
$$\begin{aligned}& H(t,s)\geq\gamma(t)\mathcal{H}(s), \end{aligned}$$
(6)
where
\(\gamma(t)=\min\{\psi_{1}(t), \psi_{2}(t)\}\), \(t\in J\)
and
$$ \mathcal{H}(s)=G(s,s)p(s)+\frac{k_{3}+k_{4}}{k} \int _{0}^{1}G(\tau,s)p(s)g(\tau)\,\mathrm{d}\tau+ \frac{k_{1}+k_{2}}{k} \int _{0}^{1}G(\tau,s)p(s)h(\tau)\,\mathrm{d} \tau. $$
(7)
Let \(E=C(J)\) be the standard Banach space with the maximum norm and P be the typical cone of nonnegative continuous functions in the form
$$ P=\bigl\{ u\in E:u(t)\geq\gamma(t)\|u\|, t\in J\bigr\} . $$
Let \(P_{mn}=\{u\in P, m\leq\|u\|\leq n\}\), \(P_{r}=\{u\in P: \|u\|\leq r\}\) for \(n>m>0\), \(r>0\).
First, we give an operator \(T:P\setminus\{0\}\to C(J)\) as follows:
$$ (Tu) (t)= \int_{0}^{1}H(t,s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s, \quad t \in J. $$
(8)
Lemma 3
If conditions
\((\mathrm{H_{1}})\)
and
\((\mathrm{H_{2}})\)
are satisfied, then, for any
\(n>m>0\), \(T:P_{mn}\to P\)
is a completely continuous operator.
Proof
For any given \(u\in P_{mn}\), we have \(m\leq\|u\|\leq n\). From the construction of P, we have
$$ \gamma(t)m\leq u(t)\leq n, \quad\forall t\in J. $$
(9)
Clearly, for any \(n>m>0\), condition \((\mathrm{H_{4}})\) means that
$$ \widehat{a}^{*}_{mn}= \int_{0}^{1}\mathcal{H}(t)\widehat {a}(t) \widehat{g}_{mn}(t)\,\mathrm{d}t< +\infty, $$
(10)
where
$$ \widehat{g}_{mn}(t)=\max\bigl\{ \widehat{g}(u):\gamma (t)m\leq u\leq n \bigr\} . $$
(11)
It follows from (8), \((\mathrm{H_{3}})\), \((\mathrm{H_{4}})\) and Lemma 2 that
$$ f\bigl(t, u(t)\bigr)\leq\widehat{a}(t)\widehat {g}_{mn}(t)+ \widehat{b}(t), \quad\forall t\in J_{+}, u\in P_{mn}, $$
(12)
and
$$ \begin{aligned}[b] (Tu)(t)&= \int_{0}^{1}H(t,s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s\\ &\leq \int_{0}^{1}\mathcal{H}(s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s\\ &\leq \int_{0}^{1}\mathcal{H}(s)\bigl[\widehat{a}(s)\widehat {g}_{mn}(s)+\widehat{b}(s)\bigr]\,\mathrm{d}s\\ &= \widehat{a}_{mn}^{*}+\widehat{b}^{*}, \quad\forall t\in J, \end{aligned} $$
(13)
which shows that T makes sense. According to Lemma 2, we have for any \(t\in J\)
$$\begin{aligned} (Tu)(t)&= \int_{0}^{1}H(t,s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s\\ &\leq \int_{0}^{1}\mathcal {H}(s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s. \end{aligned} $$
Hence,
$$ \|Tu\|\leq \int_{0}^{1}\mathcal{H}(s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s. $$
(14)
At the same time, by Lemma 2 and (14), we get
$$ \begin{aligned}[b] (Tu)(t)&= \int_{0}^{1}H(t,s)f\bigl(s,u(s)\bigr)\,\mathrm{d}s\\ &\geq \gamma(t)\int_{0}^{1}\mathcal{H}(s)f\bigl(s,u(s)\bigr)\,\mathrm {d}s\\ &\geq \gamma(t)\|Tu\|, \quad\forall t\in J. \end{aligned} $$
(15)
This indicates that T maps \(P_{mn}\) into P.
Next, we shall show the complete continuity of the operator T. Let \(u_{n}, \bar{u}\in P_{mn}\), with \(\|u_{n}-\bar{u}\|\to0\) (\(n\to\infty \)); then \(\lim _{n\to\infty}u_{n}(t)=\bar{u}(t)\), \(t\in J\). Let
$$\begin{gathered} (T_{1}u) (t)=f\bigl(t,u(t)\bigr),\quad \mbox{for any } t\in J_{+}, u\in P_{mn}, \\(T_{2}u) (t)= \int_{0}^{1}H(t,s)u(s)\,\mathrm{d}s, \quad\mbox{for any } t\in J_{+}, u\in L^{1}(J). \end{gathered}$$
By \((\mathrm{H_{1}})\),
$$ f\bigl(t,u_{n}(t)\bigr)\to f\bigl(t,\bar{u}(t)\bigr) \quad(n\to+\infty), \mbox{for any } t\in J_{+}. $$
(16)
Similar to (12), for \(u_{n}, \bar{u}\in P_{mn}\), one has
$$ f\bigl(t, u_{n}(t)\bigr)\leq\widehat{a}(t)\widehat {g}_{mn}(t)+\widehat{b}(t),\qquad f\bigl(t, \bar{u}(t)\bigr)\leq\widehat {a}(t)\widehat{g}_{mn}(t)+\widehat{b}(t), \quad\forall t\in J_{+}. $$
Then one gets
$$ \big|f\bigl(t, u_{n}(t)\bigr)-f\bigl(t, \bar{u}(t)\bigr)\big|\leq2\widehat {a}(t)\widehat{g}_{mn}(t)+2\widehat{b}(t)\stackrel{\Delta}{=}\sigma (t)\in L^{1}(J). $$
(17)
The Lebesgue dominated convergence theorem together with (16) and (17) generates
$$ \lim_{n\to\infty} \int_{0}^{1}\big|(T_{1}u_{n}) (t)- \bigl(T_{1}\bar {u}(t)\bigr)\big|\,\mathrm{d}t=0. $$
That is to say \(T_{1}: P_{mn}\to L^{1}(J)\) is continuous. Furthermore, the complete continuity of the operator \(T_{2}:L^{1}(J)\to C(J)\) can easily be verified by the Arzela-Ascoli theorem and a standard discussion. Hence, by the property of compound operators we see that \(T=T_{2}\circ T_{1}: P_{mn}\to C(J)\) is completely continuous. □
Lemma 4
[23]
Let
E
be a Banach space, \(P\subset E\)
a cone in
E. For
\(r>0\), define
\(P_{r}=\{u\in P: \|u\|\leq r\}\). Assume that
\(T:P_{r}\to P\)
is a compact map such that
\(Tu\neq u\)
for
\(u\in\partial P_{r}=\{u\in P: \|u\| =r\}\).
-
(i)
If
\(\|u\|\leq\|Tu\|\), \(\forall u\in \partial P_{r}\), then
-
(ii)
If
\(\|u\|\geq\|Tu\|\), \(\forall u\in \partial P_{r}\), then