In this installment, we list some definitions and lemmas which are needed throughout this paper.
Let \(J'=J\setminus \{t_{1},t_{2},\ldots,t_{m} \}\) and \(E=C[0,1]\). We define \(\mathit {PC}^{1}[0,1]\) in E by
$$ \mathit {PC}^{1}[0,1]= \bigl\{ x\in E:x'(t)\in C(t_{k}, t_{k+1}), \exists x' \bigl(t_{k}^{-} \bigr), x' \bigl(t_{k}^{+} \bigr), k=1,2, \ldots,m \bigr\} . $$
(2.1)
Then \(\mathit {PC}^{1}[0,1]\) is a real Banach space with norm
$$ \Vert x \Vert _{\mathit {PC}^{1}}=\max \bigl\{ \Vert x \Vert _{\infty}, \bigl\Vert x' \bigr\Vert _{\infty} \bigr\} , $$
(2.2)
where \(\Vert x \Vert _{\infty}=\sup_{t\in J} \vert x(t) \vert \), \(\Vert x' \Vert _{\infty}=\sup_{t\in J} \vert x'(t) \vert \).
Suppose that \(G(t,s)\) is the Green’s function of the boundary value problem
$$-x''(t)+Mu(t)=0,\quad\quad x'(0)=x'(1)=0, $$
then
$$ G(t,s)= \frac{1}{\gamma\sinh\gamma} \textstyle\begin{cases} \cosh\gamma(1-t)\cosh\gamma s,& 0\leq s\leq t\leq1,\\ \cosh\gamma(1-s)\cosh\gamma t,& 0\leq t\leq s\leq1. \end{cases} $$
(2.3)
Lemma 2.1
By the definition of
\(G(t,s)\)
and the properties of
\(sinhx\)
and
\(coshx\), we have the following results.
-
(a)
For any
\(t, s\in J\), there is
$$ A=\frac{1}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma}{\gamma \sinh\gamma}=B. $$
(2.4)
Then it follows from (2.4) that
$$A\leq G(t,s)\leq G(s,s)\leq B. $$
-
(b)
For any
\(\tau\in(0,\delta)\),
$$ \frac{D'_{k}}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma(1-\tau )\cosh\gamma\tau'_{k}}{\gamma\sinh\gamma}, \quad \forall t\in \bigl[ \tau,\tau '_{k} \bigr], s\in J, $$
(2.5)
where
$$\tau'_{k}=\max\{1-\tau, 1-t_{k}\},\quad\quad D'_{k}=\max \bigl\{ \cosh\gamma\tau, \cosh\gamma \bigl(1- \tau'_{k} \bigr) \bigr\} , \quad k=1,2,3,\ldots, m. $$
-
(c)
$$ G_{t}'(t,s)= \frac{1}{\sinh\gamma} \textstyle\begin{cases} -\sinh\gamma(1-t)\cosh\gamma s,& 0\leq s\leq t\leq1,\\ \sinh\gamma(1-s)\cosh\gamma t,& 0\leq t\leq s\leq1, \end{cases} $$
(2.6)
and
$$ \max_{t,s\in J,t\neq s} \bigl\vert G_{t}'(t,s) \bigr\vert \leq \sinh\gamma. $$
(2.7)
Proof
We can get equations (2.4)-(2.7) by the definition of \(G(t,s)\), so we omit it here. □
To establish the existence of positive solutions to problem (1.1), for a fixed \(\tau\in(0,\delta)\), we construct the cone \(K_{\tau}\) in \(\mathit {PC}^{1}[0,1]\) by
$$ K_{\tau}= \Bigl\{ x \in \mathit {PC}^{1}[0,1]: x(t) \geq0,t\in J, \min _{t\in[\tau ,\tau'_{k}]}x(t) \geq\sigma_{k} \Vert x \Vert _{\mathit {PC}^{1}} \Bigr\} , $$
(2.8)
where
$$\begin{aligned}& \sigma_{k}=\frac{D'_{k}}{ \rho\gamma\sinh\gamma}, \quad k=1,2,\ldots,m, \end{aligned}$$
(2.9)
$$\begin{aligned}& \rho=\max\{B, \sinh\gamma\}. \end{aligned}$$
(2.10)
It is easy to see \(K_{\tau}\) is a closed convex cone of \(\mathit {PC}^{1}[0,1]\).
Let \(\{\tau_{i}\}_{i=1}^{\infty}\) be such that \(t_{i+1}'<\tau _{i}<t_{i}'\), \(i=1,2,\dots\). Then for any \(i\in\mathrm{N}\), we define the cone \(K_{\tau_{i}}\) by
$$ K_{\tau_{i}}= \Bigl\{ x(t)\in \mathit {PC}^{1}[0,1]: x(t)\geq0,t\in J,\min _{t\in[\tau_{i},\tau'_{ik}]}x(t)\geq\sigma_{ik} \Vert x \Vert _{\mathit {PC}^{1}} \Bigr\} , $$
(2.11)
where
$$\begin{aligned}& \tau'_{ik}=\max\{1-\tau_{i},1-t_{k} \}, \quad\quad \sigma_{ik}=\frac{D'_{ik}}{ \rho\gamma\sinh\gamma}, \end{aligned}$$
(2.12)
$$\begin{aligned}& D'_{ik}=\max \bigl\{ \cosh\gamma\tau_{i}, \cosh \gamma \bigl(1-\tau'_{k} \bigr) \bigr\} , \quad i=1,2, \ldots, k=1,2,\ldots,m. \end{aligned}$$
(2.13)
It is easy to see \(K_{\tau_{i}}\) is a closed convex cone of \(\mathit {PC}^{1}[0,1]\).
Remark 2.1
For any \(i=1,2,\ldots\) , \(k=1,2,\ldots,m\), it follows from the definition of \(\sigma_{k}\) and \(\sigma_{ik}\) that \(0<\sigma _{k},\sigma_{ik} <1\).
Lemma 2.2
If
\((H_{1})\)-\((H_{3})\)
hold, then problem (1.1) has a unique solution
x
given by
$$x(t) = \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr). $$
Proof
The proof is similar to that of Lemma 2.4 in [26]. □
Definition 2.1
A function \(x(t)\) is said to be a solution of problem (1.1) on J if:
-
(i)
\(x(t)\) is absolutely continuous on each interval \((0,t_{1}]\) and \((t_{k},t_{k+1}]\), \(k =1,2,\ldots,n\);
-
(ii)
for any \(k =1,2,\ldots,m\), \(x(t_{k}^{ +})\), \(x(t_{k}^{-})\) exist;
-
(iii)
\(x(t)\) satisfies (1).
Define an operator \(T: K_{\tau} \to \mathit {PC}^{1}[0,1]\) by
$$ (Tx) (t) = \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr). $$
(2.14)
From (2.14), we know that \(x(t)\in \mathit {PC}^{1}[0,1]\) is a solution of problem (1.1) if and only if x is a fixed point of the operator T. Also, for a positive number r, define \(\Omega_{r}\) by
$$\Omega_{r}= \bigl\{ x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}< r \bigr\} . $$
Note that \(\partial\Omega_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}=r \}\) and \(\bar{\Omega}_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}\leq r \}\).
Definition 2.2
An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.
Lemma 2.3
Assume that
\((H_{1})\)-\((H_{3})\)
hold. Then
\(T(K_{\tau })\subset K_{\tau} \)
and
\(T: K_{\tau} \to K_{\tau}\)
is a completely continuous.
Proof
For \(t\in J\), \(x\in K_{\tau}\), it follows from ((2.5)) and (2.14) that
$$ \begin{aligned}[b] (Tx) (t) &= \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ & \leq B \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds + \sum_{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr]. \end{aligned} $$
(2.15)
On the other hand, it follows from (2.6), (2.7) and (2.14) that
$$ \begin{aligned}[b] \bigl\vert (Tx)'(t) \bigr\vert &= \Biggl\vert \int _{0}^{1}G_{t}'(t,s) \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum_{k=1}^{m}G_{t}'(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr\vert \\ &\leq \int_{0}^{1} \bigl\vert G_{t}'(t,s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m} \bigl\vert G_{t}'(t,t_{k}) \bigr\vert I_{k} \bigl(t_{k},x(t_{k}) \bigr) \\ &\leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr]. \end{aligned} $$
(2.16)
For any \(t\in J\), combined with (2.15) and (2.16), we have
$$ \Vert Tx \Vert _{\mathit {PC}^{1}} \leq\rho \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr]. $$
(2.17)
Then, by (2.5), (2.8) and (2.17), we have
$$ \begin{aligned}[b] \min_{t\in[\tau,\tau_{k}]}(Tx) (t) &=\min _{t\in[\tau,\tau_{k}]} \Biggl[ \int_{0}^{1}G(t,s)\omega(s)f \bigl(s,x(s) \bigr) \,ds+ \sum_{k=1}^{m}G(t,t_{k})I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\geq\frac{D'_{k}}{\gamma\sinh\gamma} \Biggl[ \int_{0}^{1}\omega(s)f\bigl(s,x(s)\bigr)\,ds+\sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr] \\ &\geq\frac{D'_{k}}{\rho\gamma\sinh\gamma} \rho \Biggl[ \int_{0}^{1}\omega(s)f\bigl(s,x(s)\bigr)\,ds+\sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k})\bigr) \Biggr] \\ &\geq\sigma_{k} \Vert Tx \Vert _{\mathit {PC}^{1}}. \end{aligned} $$
(2.18)
Evidently, \(T(K_{\tau})\subset K_{\tau}\).
Next, we prove that the operator \(T: K_{\tau}\to K_{\tau}\) is a completely continuous.
It is obvious that T is continuous.
Let \(B_{d}=\{x\in \mathit {PC}^{1}[0,1] \mid \Vert x \Vert _{\mathit {PC}^{1}}\le d\}\) be bounded set. Then, for all \(x\in B_{d}\), by the definition of \(\Vert Tx \Vert _{\infty}\), \(\Vert Tx' \Vert _{\infty}\), \(\Vert Tx \Vert _{\mathit {PC}^{1}}\), we have
$$\begin{aligned}& \begin{aligned} \Vert Tx \Vert _{\infty}&=\sup _{t\in J} \bigl\vert Tx(t) \bigr\vert \\ &\le B \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\le B \bigl( \Vert \omega \Vert _{1}L+mL^{*} \bigr) \\ &=\Gamma_{0}, \end{aligned} \\& \begin{aligned} \bigl\Vert Tx' \bigr\Vert _{\infty}&=\sup_{t\in J} \bigl\vert Tx'(t) \bigr\vert \\ &\leq \sinh\gamma \Biggl[ \int_{0}^{1}\omega(s)f \bigl(s,x(s) \bigr)\,ds+ \sum _{k=1}^{m}I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr] \\ &\le\sinh\gamma \bigl( \Vert \omega \Vert _{1}L+mL^{*} \bigr) \\ &=\Gamma_{1}, \end{aligned} \end{aligned}$$
and
$$\Vert Tx \Vert _{\mathit {PC}^{1}}=\max \bigl\{ \Vert Tx \Vert _{\infty}, \bigl\Vert Tx' \bigr\Vert _{\infty} \bigr\} \le\max \{\Gamma_{0}, \Gamma_{1} \}, $$
where
$$\begin{aligned}& L=\max_{t\in J, x\in K_{\tau}, \Vert x \Vert _{\mathit {PC}^{1}}\le d}f(t,x), \quad\quad L^{*}=\max \{L_{k}, k=1,2, \ldots,m\}, \\& L_{k}=\max_{t\in J, x\in K_{\tau}, \Vert x \Vert _{\mathit {PC}^{1}}\le d}I_{k} \bigl(t_{k},x(t_{k}) \bigr). \end{aligned}$$
Therefore \(T(B_{d})\) is uniformly bounded.
On the other hand, for all \(t_{1}, t_{2}\in J_{k}\) with \(t_{1}< t_{2}\), we have
$$\bigl\vert (Tx) (t_{1})-(Tx) (t_{2}) \bigr\vert = \biggl\vert \int _{t_{1}}^{t_{2}}(Tx)'(t)\,dt \biggr\vert \le\Gamma_{1} \vert t_{1}-t_{2} \vert \rightarrow0 \quad (t_{1}\rightarrow t_{2}). $$
Noting (2.7), we know that \(G'(t,s)\) is a constant and
$$\begin{aligned} \bigl\vert (Tx)'(t_{1})-(Tx)'(t_{2}) \bigr\vert &= \Biggl\vert \int _{0}^{1} \bigl[G_{t}'(t_{1},s)-G_{t}'(t_{2},s) \bigr]\omega(s)f \bigl(s,x(s) \bigr)\,ds \\ &\quad{} +\sum_{k=1}^{n} \bigl[G_{t}'(t_{1},t_{k})-G_{t}'(t_{2},t_{k}) \bigr]I_{k} \bigl(t_{k},x(t_{k}) \bigr) \Biggr\vert \\ & \leq \int_{0}^{1} \bigl\vert G_{t}'(t_{1},s)-G_{t}'(t_{2},s) \bigr\vert \omega(s)f \bigl(s,x(s) \bigr)\,ds \\ &\quad{} +\sum_{k=1}^{n} \bigl\vert G_{t}'(t_{1},t_{k})-G_{t}'(t_{2},t_{k}) \bigr\vert I_{k} \bigl(t_{k},x(t_{k}) \bigr) \rightarrow0 \quad (t_{1}\rightarrow t_{2}), \end{aligned}$$
which shows that \(T(B_{d})\) is equicontinuous. The Arzelà-Ascoli theorem implies that T is completely continuous, and the lemma is proved. □
Lemma 2.4
Hölder
Let
\(e\in L^{p}[a,b]\)
with
\(p>1\), \(h\in L^{q}[a,b]\)
with
\(q>1\)
and
\(\frac{1}{p}+\frac{1}{q}=1\). Then
\(eh\in L^{1}[a,b]\)
and
$$\Vert eh \Vert _{1}\le \Vert e \Vert _{p} \Vert h \Vert _{q}. $$
Let
\(e\in L^{1}[a,b]\), \(h\in L^{\infty}[a,b]\). Then
\(eh\in L^{1}[a,b]\)
and
$$\Vert eh \Vert _{1}\le \Vert e \Vert _{1} \Vert h \Vert _{\infty}. $$
Lemma 2.5
See [28]; fixed point theorem of cone expansion and compression of norm type
Let
E
be a Banach space, P
be a cone in
E. Assume that
\(\Omega _{1}\), \(\Omega_{2}\)
are bounded open subsets in
E
with
\(\theta \in\Omega_{1}\)
and
\(\bar{\Omega}_{1}\subset\Omega_{2}\), where
θ
denotes zero operator. Suppose
\(A : P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})\rightarrow P \)
is completely continuous such that either
-
(i)
\(\Vert Ax \Vert \leq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{1}\); \(\Vert Ax \Vert \geq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{2} \);
-
(ii)
\(\Vert Ax \Vert \leq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{2}\); \(\Vert Ax \Vert \geq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{1} \).
Then
A
has a fixed point in
\(P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})\).