In this section, we establish the existence of triple positive solutions for problem (1.1). We consider the following three cases for \(w\in L^{p}[0,1]\): \(p> 1\), \(p=1\), and \(p=\infty\). The case \(p>1\) is treated in the following theorem.
For convenience, we introduce the following notation:
$$\begin{aligned} &\rho_{3}=\max \biggl\{ \frac{a}{a-\mu}, \frac{a}{b} \biggr\} , \qquad D=\rho_{3}\Vert G\Vert _{q}\Vert \omega \Vert _{p}, \qquad D_{1}=n\rho_{3}\biggl(1+ \frac{b}{a}\biggr), \qquad \delta^{*}=\frac{\delta\rho_{1}}{\rho_{2}}, \\ &f^{\infty}=\limsup_{u\rightarrow\infty}\max_{t\in J} \frac{f(t,u)}{u}, \qquad I^{\infty}(k)=\limsup_{u\rightarrow\infty} \frac{I_{k}(u)}{u}, \quad k=1, 2, \ldots, n. \end{aligned}$$
Theorem 4.1
Assume that (\(H_{1}\))-(\(H_{3}\)) hold. Furthermore, suppose that there exist constants
\(0< m< c< \frac{c}{\delta^{*}}\le l \)
such that
- (\(H_{4}\)):
-
\(f^{\infty}<\frac{1}{2D}\), \(I^{\infty}(k)<\frac{1}{2D_{1}}\), \(k=1, 2, \ldots, n\);
- (\(H_{5}\)):
-
\(f(t,u)\ge\frac{2c}{\xi\delta\rho_{1} N}\), \(\forall (t,u)\in[0,\xi] \times[c, \frac{c}{\delta^{*}}]\);
- (\(H_{6}\)):
-
\(f(t,u)< \frac{m}{2D}\), \(I_{k}(u)<\frac{m}{2D_{1}}\), \(\forall (t,u)\in J \times[0,m]\), \(k=1, 2, \ldots, n\).
Then problem (1.1) has at least three positive solutions
\(u_{1}\), \(u_{2}\), and
\(u_{3}\)
satisfying
$$\Vert u_{1}\Vert _{PC^{1}}< m,\qquad c< \beta(u_{2}), \qquad \Vert u_{3}\Vert _{PC^{1}}>m,\quad \textit{and} \quad \beta(u_{3})< c. $$
Proof
Let \(\beta(u)=\min_{0\le t\le\xi}u(t)\). It is clear that \(\beta(u)\) is a nonnegative continuous concave functional on the cone K satisfying \(\beta(u)\leq\|u\|_{PC^{1}}\) for all \(u\in K\). By (\(H_{4}\)) there exist \(0<\gamma<\frac{1}{2D}\), \(0<\gamma_{1}<\frac {1}{2D_{1}}\), and \(\rho^{\prime}>0\) such that
$$f(t,u)\leq\gamma u, \qquad I_{k}(u)\le\gamma_{1} u, \quad k=1, 2, \ldots, n, \forall t \in J, u \ge\rho^{\prime}. $$
Let
$$\eta=\max_{(t,u)\in[0,1]\times[0,\rho^{\prime}]}f(t,u),\qquad \eta _{1}=\max _{u\in[0,\rho^{\prime}]}I_{k}(u), \quad k=1, 2, \ldots, n. $$
Then
$$ f\bigl(t,u(t)\bigr)\leq\gamma u(t)+\eta, \qquad I_{k}(u)\leq\gamma_{1} u+\eta_{1}, \quad \forall t \in J, u\ge0. $$
(4.1)
Since \(0\le\alpha(t)\le t \le1\) on J, it follows from \(u(t)\ge\rho^{\prime}\) on J that
$$ u\bigl(\alpha(t)\bigr)\ge\rho^{\prime},\quad \forall t \in J. $$
(4.2)
Set \(l>\max\{\frac{2D\eta}{1-2D\gamma}, \frac{2D_{1}\eta _{1}}{1-2D_{1}\gamma_{1}}, \frac{c}{\delta^{*}}\}\).
Consequently, for any \(t \in J\) and \(u \in\bar{K}_{l}\), (3.2) and (3.9) imply
$$\begin{aligned} &\begin{aligned}[b] (Tu) (t)&= \int_{0}^{1}H(t,s)\omega(s)f\bigl(s,u\bigl(\alpha(s) \bigr)\bigr)\,ds+\sum_{k=1}^{n}H(t,t_{k})I_{k} \bigl(u(t_{k})\bigr) \\ &\le\frac{a}{a-\mu} \Biggl( \int_{0}^{1}G(s,s)\omega (s)f\bigl(s,u\bigl( \alpha(s)\bigr)\bigr)\,ds+\sum_{k=1}^{n}G(t,t_{k})I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ &\le\frac{a}{a-\mu} \Biggl(\Vert G\Vert _{q}\Vert \omega \Vert _{p} \int_{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\biggl(1+\frac{b}{a}\biggr)\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr), \end{aligned} \end{aligned}$$
(4.3)
$$\begin{aligned} &\begin{aligned}[b] \bigl\vert (Tu)^{\prime}(t) \bigr\vert &\le \int_{0}^{1}\bigl\vert H_{t}^{\prime}(t,s) \bigr\vert \omega(s)f\bigl(s,u\bigl(\alpha (s)\bigr)\bigr)\,ds+\sum _{k=1}^{n}\bigl\vert H_{t}^{\prime}(t,t_{k}) \bigr\vert I_{k}\bigl(u(t_{k})\bigr) \\ &\le \int_{0}^{1}\omega(s)f\bigl(s,u\bigl(\alpha (s) \bigr)\bigr)\,ds+\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ &= \int_{0}^{1}\frac{1}{G(s,s)} G(s,s)\omega (s)f \bigl(s,u\bigl(\alpha(s)\bigr)\bigr)\,ds+\frac{1}{G(s,s)}\sum _{k=1}^{n}G(s,s)I_{k}\bigl(u(t_{k}) \bigr) \\ &\le\frac{a}{b} \Biggl( \int _{0}^{1}G(s,s)\omega(s)f\bigl(s,u\bigl( \alpha(s)\bigr)\bigr)\,ds+\sum_{k=1}^{n}G(s,s)I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ &\le\frac{a}{b} \Biggl(\Vert G\Vert _{q}\Vert \omega \Vert _{p} \int_{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\biggl(1+\frac{b}{a}\biggr)\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr). \end{aligned} \end{aligned}$$
(4.4)
It follows from (4.3) and (4.4) that
$$ \begin{aligned}[b] \Vert Tu\Vert _{PC^{1}}& \le\rho_{3} \Biggl(\Vert G\Vert _{q}\Vert \omega \Vert _{p} \int _{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\biggl(1+\frac{b}{a}\biggr)\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ &\le\rho_{3} \Vert G\Vert _{q}\Vert \omega \Vert _{p} \bigl(\gamma \Vert u\Vert _{PC^{1}}+\eta\bigr)+ \rho_{3}\biggl(1+\frac{b}{a}\biggr)n\bigl(\gamma_{1} \Vert u\Vert _{PC^{1}}+\eta_{1}\bigr) \\ &\le\rho_{3}\Vert G\Vert _{q}\Vert \omega \Vert _{p} (\gamma l+\eta)+ \rho_{3}\biggl(1+\frac{b}{a} \biggr)n(\gamma_{1} l+\eta_{1}) \\ & < \frac{l}{2}+\frac{l}{2} = l, \end{aligned} $$
(4.5)
which implies that \(Tu\in K_{l}\).
Hence, we have shown that if (\(H_{4}\)) holds, then the operator \(T:\bar{K}_{l}\to\bar{K}_{l}\) is completely continuous.
Next, we verify that \(\{u\mid u\in K(\beta, c, \frac{c}{\delta ^{*}}), \beta(u)> c \}\ne{\O}\) and \(\beta(Tu)>c\) for all \(u\in K(\beta, c, \frac{c}{\delta^{*}})\).
Take \(u_{0}(t)=\frac{\delta^{*} +1}{2\delta^{*}}c\), for \(t\in J\). Then
$$u_{0}\in \biggl\{ u\mid u\in K\biggl(\beta,c, \frac{c}{\delta^{*}} \biggr), \beta (u)>c \biggr\} , $$
which shows that
$$\biggl\{ u\mid u\in K\biggl(\beta, c, \frac{c}{\delta^{*}}\biggr), \beta(u)> c \biggr\} \ne{\O}. $$
Since \(0\le\alpha(t)\le t\le\xi\) on \([0,\xi]\), it follows from \(c\le u(t)\le\frac{c}{\delta^{*}}\) on J that
$$c\le u\bigl(\alpha(t)\bigr)\le\frac{c}{\delta^{*}},\quad \forall t\in[0,\xi]. $$
Then, it follows from (\(H_{5}\)) that
$$\begin{aligned} \beta(Tu) = &\min_{t\in[0, \xi]}(Tu) \\ =&\min_{t\in[0,\xi]} \int_{0}^{1}H(t,s)\omega (s)f\bigl(s,u\bigl( \alpha(s)\bigr)\bigr)\,ds+\sum_{k=1}^{n}H(t,t_{k})I_{k} \bigl(u(t_{k})\bigr) \\ \ge&\min_{t\in[0,\xi]} \int_{0}^{1}H(t,s)\omega (s)f\bigl(s,u\bigl( \alpha(s)\bigr)\bigr)\,ds \\ \ge&\delta\rho_{1} \int_{0}^{1}\omega(s)f\bigl(s,u\bigl(\alpha (s) \bigr)\bigr)\,ds \\ >&\frac{1}{2}\delta\rho_{1} \int_{0}^{1}\omega (s)f\bigl(s,u\bigl(\alpha(s) \bigr)\bigr)\,ds \\ \ge&\frac{1}{2}\delta\rho_{1} \int_{0}^{\xi}N\frac {2c}{\xi\delta\rho_{1} N}\,ds \\ = &c. \end{aligned}$$
(4.6)
This implies that condition (i) of Lemma 3.4 holds.
Since \(0\le\alpha(t)\le t \le\xi\) on \([0,1]\), it follows from \(0\le\|u(t)\|_{PC^{1}}\le m\) on J that
$$0\le\bigl\Vert u\bigl(\alpha(t)\bigr)\bigr\Vert _{PC^{1}}\le m,\quad \forall t \in[0,1]. $$
Then, for \(u \in\bar{K}_{m}\), it follows from (\(H_{6}\)) and (4.5) that
$$ \begin{aligned}[b] \Vert Tu\Vert _{PC^{1}}& \le\rho_{3} \Biggl( \int_{0}^{1} \Vert G\Vert _{q}\Vert \omega \Vert _{p}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr)\,ds+\biggl(1+ \frac{b}{a}\biggr)\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ & =\rho_{3} \Vert G\Vert _{q}\Vert \omega \Vert _{p} \int _{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\rho_{1}\biggl(1+\frac{b}{a}\biggr)\sum _{k=1}^{n}I_{k}\bigl(u(t_{k}) \bigr) \\ & < \rho_{3} \Vert G\Vert _{q}\Vert \omega \Vert _{p} \int _{0}^{1}\frac{m}{2D}\,ds+ \rho_{3}\biggl(1+\frac{b}{a}\biggr)\sum _{k=1}^{n}\frac {m}{2D_{1}} \\ & = \frac{m}{2}+\frac{m}{2} = m. \end{aligned} $$
(4.7)
This implies that condition (ii) of Lemma 3.4 holds.
Finally, we assert that if \(u\in K(\beta,c, l)\) and \(\|Tu\|_{PC^{1}}>\frac{c}{\delta^{*}}\), then \(\beta(Tu)>c\).
Suppose that \(u\in K(\beta,c, l)\) and \(\|Tu\|_{PC^{1}}>\frac{c}{\delta^{*}}\). Then it follows form (3.7) that
$$ \begin{aligned}[b] \beta(Tu) &= \min _{t\in[0, \xi]}(Tu) \\ & =\min_{t\in[0,\xi]} \Biggl[ \int _{0}^{1}H(t,s)\omega(s)f\bigl(s,u\bigl( \alpha(s)\bigr)\bigr)\,ds+ \sum_{k=1}^{n}H(t,t_{k})I_{k} \bigl(u(t_{k})\bigr) \Biggr] \\ & \ge\frac{\delta\rho_{1}}{\rho_{2}}\Vert Tu\Vert _{PC^{1}} \\ & =\delta^{*}\Vert Tu\Vert _{PC^{1}} >c. \end{aligned} $$
(4.8)
This implies that condition (iii) of Lemma 3.4 holds.
To sum up, the hypotheses of Lemma 3.4 hold. Therefore, an application of Lemma 3.4 implies that problem (1.1) has at least three positive solutions \(u_{1}\), \(u_{2}\), and \(u_{3}\) satisfying
$$\Vert u_{1}\Vert _{PC^{1}}< m,\qquad c< \beta(u_{2}), \qquad \Vert u_{3}\Vert _{PC^{1}}>m\quad \mbox{and} \quad \beta(u_{3})< c. $$
The following results deal with the case \(p=\infty\). □
Corollary 4.1
Assume that (\(H_{1}\))-(\(H_{6}\)) hold. Then problem (1.1) has at least three positive solutions
\(u_{1}\), \(u_{2}\), and
\(u_{3}\)
satisfying
$$\Vert u_{1}\Vert _{PC^{1}}< m,\qquad c< \beta(u_{2}), \qquad \Vert u_{3}\Vert _{PC^{1}}>m,\quad \textit{and} \quad \beta(u_{3})< c. $$
Proof
Let \(\|G\|_{1}\| \omega\|_{\infty}\) replace \(\|G\|_{q}\| \omega\|_{p}\) and repeat the previous argument.
Finally, we consider the case of \(p=1\). Let
- (\(H_{4}^{\prime}\)):
-
\(f^{\infty}< \frac{1}{2D^{\prime}}\), \(I^{\infty}(k)<\frac {1}{2D_{1}^{\prime}}\), \(k=1, 2, \ldots, n\);
- (\(H_{6}^{\prime}\)):
-
\(f(t,u)<\frac{m}{2D^{\prime}}\), \(I_{k}(u)<\frac {m}{2D_{1}^{\prime}}\), \(\forall (t,u)\in J \times[0,m]\), \(k=1, 2, \ldots , n\), where \(D^{\prime}=\rho_{2}\|\omega\|_{1}\), \(D_{1}^{\prime}=\rho_{2}n\).
□
Corollary 4.2
Assume that (\(H_{1}\))-(\(H_{3}\)), (\(H_{4}^{\prime}\)), (\(H_{5}\)), and (\(H_{6}^{\prime}\)) hold. Then problem (1.1) has at least three positive solutions
\(u_{1}\), \(u_{2}\), and
\(u_{3}\)
satisfying
$$\Vert u_{1}\Vert _{PC^{1}}< m,\qquad c< \beta(u_{2}), \qquad \Vert u_{3}\Vert _{PC^{1}}>m,\quad \textit{and} \quad \beta(u_{3})< c. $$
Proof
Set \(l^{\prime}>\max\{\frac{2D^{\prime}\eta}{1-2D^{\prime}\gamma^{\prime}}, \frac{2D_{1}^{\prime}\eta_{1}}{1-D_{1}^{\prime}\gamma_{1}}, \frac{c}{\delta^{*}}\}\), where \(0<\gamma^{\prime}<\frac{1}{2D^{\prime}}\). If \(u \in\bar{K}_{l^{\prime}}\), then, by assumption (\(H_{4}^{\prime}\)), from (4.1) and (4.2) we obtain
$$f\bigl(t,u\bigl(\alpha(t)\bigr)\bigr)\le\gamma^{\prime}u\bigl(\alpha(t) \bigr) +\eta. $$
Then, for \(u \in\bar{K}_{l^{\prime}}\), it follows from (3.6) and (3.9) that
$$\begin{aligned}[b] \Vert Tu\Vert _{PC^{1}}&\le \rho_{2} \Biggl( \int_{0}^{1}\omega(s)f\bigl(s,u\bigl(\alpha (s) \bigr)\bigr)\,ds+\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ &\le\rho_{2} \Vert \omega \Vert _{1} \int _{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\rho_{2}\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ &\le\rho_{2} \Vert \omega \Vert _{1} \int_{0}^{1}\bigl(\gamma ^{\prime} u\bigl( \alpha(s)\bigr)+\eta\bigr)\,ds+\rho_{2} n(\gamma_{1} u+ \eta_{1}) \\ &\le\rho_{2} \Vert \omega \Vert _{1} \bigl( \gamma^{\prime} \Vert u\Vert _{PC^{1}}+\eta\bigr)+ \rho_{2}n\bigl(\gamma_{1} \Vert u\Vert _{PC^{1}}+ \eta_{1}\bigr) \\ &\le\rho_{2} \Vert \omega \Vert _{1} \bigl( \gamma^{\prime} l^{\prime}+\eta\bigr)+\rho_{2}n\bigl( \gamma_{1} l^{\prime}+\eta_{1}\bigr) \\ & < \frac{l^{\prime}}{2}+\frac{l^{\prime}}{2} \\ & = l^{\prime}, \end{aligned} $$
which implies that \(Tu\in K_{l^{\prime}}\).
Hence, we have shown that if \((H_{4}^{\prime})\) holds, then the operator \(T:\bar{K}_{l^{\prime}}\to\bar{K}_{l^{\prime}}\) is completely continuous.
If \(u \in\bar{K}_{m}\), then it follows from (3.6) and (\(H_{6}^{\prime}\)) that
$$\begin{aligned}[b] \Vert Tu\Vert _{PC^{1}}&\le \rho_{2} \Biggl( \int_{0}^{1}\omega(s)f\bigl(s,u\bigl(\alpha (s) \bigr)\bigr)\,ds+\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \Biggr) \\ &\le\rho_{2} \Vert \omega \Vert _{1} \int _{0}^{1}f\bigl(s,u\bigl(\alpha(s)\bigr)\bigr) \,ds+\rho_{2}\sum_{k=1}^{n}I_{k} \bigl(u(t_{k})\bigr) \\ & < \rho_{2} \Vert \omega \Vert _{1} \int_{0}^{1}\frac {m}{2D^{\prime}}\,ds+ \rho_{2}\sum_{k=1}^{n} \frac{m}{2D_{1}^{\prime}} \\ & = \frac{m}{2}+\frac{m}{2} = m. \end{aligned} $$
Similarly to the proof of Theorem 4.1, we can get Corollary 4.2. □
Remark 4.1
Comparing with Jankowski [23], the main features of this paper are as follows.
-
(i)
A Green function, especially, a positive Green function, is available.
-
(ii)
We consider integral boundary conditions.
-
(iii)
\(\omega(t)\) is \(L^{p}\)-integrable, not only \(\omega(t)\in C[0,1]\) on \(t\in J\).