1 Introduction
Functional differential equations with impulse effect occur in many applications, such as population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc., and can be expressed by functional differential equations with impulses, see [1‐5]. Functional differential equations with impulses are characterized by sudden changing of their state and by the fact that the processes under consideration depend on their prehistory at each moment of time. Therefore, the study of impulsive functional differential equations has gained prominence and it is a rapidly growing field, see Zhang and Feng [6], Nieto and Rodríguez-López [7], Yan and Shen [8], Li and Shen [9], Yang and Shen [10], YS Liu [11], YJ Liu [12], He and Yu [13] and Ding et al. [14] and the references therein. We note that the difficulties dealing with such problems are that they have deviating arguments and their states are discontinuous. Therefore, the results of impulsive functional differential equations, especially for higher order impulsive functional differential equations, are fewer than those of differential equations without impulses and deviating arguments.
Moreover, owing to its importance in modeling the stationary states of the deflection of an elastic beam, fourth order boundary value problems have attracted much attention from many authors; for example, see Sun and Wang [15], Yao [16], O’Regan [17], Yang [18], Zhang [19], Gupta [20], Agarwal [21], Bonanno and Bella [22] and Han and Xu [23]. In particular, we would like to mention some results of Zhang and Liu [24] and Feng [25]. In [24], Zhang and Liu studied the following fourth order four-point boundary value problem without impulsive effect:
where \(0<\xi,\eta<1\), \(0\leq a< b<1\). By using the upper and lower solution method, fixed point theorems and the properties of Green’s function \(G(t,s)\) and \(H(t,s)\), the authors give sufficient conditions for the existence of one positive solution.
$$\left \{ \begin{array}{l} (\phi_{p}(x''(t)))''=w(t)f(t,x(t)), \quad t\in[0,1], \\ x(0)=0, \qquad x(1)=ax(\xi), \\ x''(0)=0, \qquad x''(1)=bx''(\eta), \end{array} \right . $$
Anzeige
Recently, Feng [25] studied a fourth order boundary value problem with impulses and integral boundary conditions
Using a suitably constructed cone and fixed point theory for cones, the existence of multiple positive solutions was established. Furthermore, upper and lower bounds for these positive solutions were given.
$$\left \{ \begin{array}{l} (\phi_{p}(y''(t)))''=f(t,y(t)), \quad t\in J, t\neq t_{k}, k=1,2,\ldots,n, \\ \Delta y'|_{t=t_{k}}=-I_{k}(y(t_{k})),\quad k=1,2,\ldots,n, \\ y(0)=y(1)=\int_{0}^{1}g(s)y(s)\,ds, \\ \phi_{p}(y''(0))=\phi_{p}(y''(1))=\int_{0}^{1}h(s)\phi_{p}(y''(s))\,ds. \end{array} \right . $$
However, to the best of our knowledge, no paper has considered the existence, multiplicity and nonexistence of positive solutions for fourth order impulsive differential equations with one-dimensional m-Laplacian, multiple parameters and deviating arguments till now; for example, see [26‐30] and the references therein.
In this paper, we investigate a fourth order impulsive integral boundary value problem with one-dimensional m-Laplacian and deviating arguments
where \(\lambda>0\) and \(\mu>0\) are two parameters, \(a, b>0\), \(J=[0,1]\), \(\phi_{m}(s)\) is an m-Laplace operator, i.e., \(\phi_{m}(s)=|s|^{m-2}s\), \(m>1\), \((\phi_{m})^{-1}=\phi_{m^{*}}\), \(\frac{1}{m}+\frac{1}{m^{*}}=1\), \(t_{k}\) (\(k=1,2,\ldots,n\)) (where n is a fixed positive integer) are fixed points with \(0=t_{0}< t_{1}<t_{2}<\cdots <t_{k}<\cdots <t_{n}<t_{n+1}=1\), \(\Delta y' |_{t=t_{k}}=y'(t_{k}^{+})-x'(t_{k}^{-})\), where \(y'(t_{k}^{+})\) and \(y'(t_{k}^{-})\) represent the right-hand limit and the left-hand limit of \(y'(t)\) at \({t=t_{k}}\), respectively. In addition, ω, f, \(I_{k}\), g and h satisfy
$$ \left \{ \begin{array}{l} (\phi_{m}(y''(t)))''=\lambda\omega(t)f(t,y(\alpha(t))), \quad t\in J, t\neq t_{k}, k=1,2,\ldots,n, \\ \Delta y'|_{t=t_{k}}=-\mu I_{k}(t_{k},y(t_{k})), \quad k=1,2,\ldots ,n, \\ ay(0)-by'(0)=\int_{0}^{1}g(s)y(s)\,ds, \\ ay(1)+by'(1)=\int_{0}^{1}g(s)y(s)\,ds, \\ \phi_{p}(y''(0))=\phi_{p}(y''(1))=\int_{0}^{1}h(t)\phi_{p}(y''(t))\,dt, \end{array} \right . $$
(1.1)
(H1)
\(\omega\in L^{p}[0,1]\) for some \(1\leq p \leq+\infty\), and there exists \(\eta>0\) such that \(\omega(t)\geq\eta\) a.e. on J;
(H2)
\(f:J\times R_{+}\rightarrow R_{+}\) is continuous with \(f(t,y)>0\) for all \(t\in J\) and \(y>0, \alpha\in C(J,J)\) with \(R_{+}=[0,+\infty)\);
(H3)
\(I_{k}:J\times R_{+}\rightarrow R_{+}\) is continuous with \(I_{k}(t,y)>0\) (\(k=1,2,\ldots,n\)) for all t and \(y>0\);
(H4)
\(g,h\in L^{1}[0,1]\) are nonnegative and \(\xi\in[0,a)\), \(\nu \in[0,1)\), where
$$ \xi=\int_{0}^{1}g(t)\,dt, \qquad \nu=\int _{0}^{1}h(t)\,dt. $$
(1.2)
Some special cases of (1.1) have been investigated. For example, Jankowski [31] considered problem (1.1) with \(\lambda=1\), \(I_{k}=0\) and \(\omega\in C[0,1]\), not \(\omega\in L^{p}[0,1]\) for some \(1\leq p \leq+\infty\). By using a fixed point theorem for cones due to Avery and Peterson, the author proved the existence results of positive solutions for problem (1.1).
Anzeige
Motivated by the results mentioned above, in this paper we study the existence, multiplicity and nonexistence of positive solutions for problem (1.1) by using different methods from that of the proof of Theorem 2.1 and Theorem 2.2 in [30] to overcome difficulties arising from the appearances of \(\alpha(t)\not\equiv t\) and \(\omega(t)\) is \(L^{p}\)-integrable. The arguments are based upon a fixed point theorem due to Krasnoselskii which deals with fixed points of a cone-preserving operator defined on an ordered Banach space.
The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.1). In Section 3, we present some definitions and lemmas which are needed throughout this paper. In Section 4, we use a fixed point theorem to obtain the existence, multiplicity and nonexistence of positive solutions for problem (1.1) with advanced argument α. In Section 5, we formulate sufficient conditions under which delayed problem (1.1) has positive solutions. In particular, our results in these sections are new when \(\alpha(t)\equiv t\) on \(t\in J\). Finally, in Section 6, we offer some remarks and comments of the associated problem (1.1).
2 Expression and properties of Green’s function
We shall reduce problem (1.1) to an integral equation. To this goal, firstly by means of the transformation
we convert problem (1.1) into
and
$$ \phi_{m}\bigl(y''(t)\bigr)=-x(t), $$
(2.1)
$$ \left \{ \begin{array}{l} x''(t)+\lambda\omega(t)f(t,y(\alpha(t)))=0, \quad t\in J, \\ x(0)=x(1)=\int_{0}^{1}h(t)x(t)\,dt \end{array} \right . $$
(2.2)
$$ \left \{ \begin{array}{l} y''(t)=-\phi_{m^{*}}(x(t)), \quad t\in J, t\neq t_{k}, \\ \Delta y'|_{t=t_{k}}=-\mu I_{k}(t_{k},y(t_{k})), \quad k=1,2,\ldots ,n, \\ ay(0)-by'(0)=\int_{0}^{1}g(s)y(s)\,ds, \\ ay(1)+by'(1)=\int_{0}^{1}g(s)y(s)\,ds. \end{array} \right . $$
(2.3)
Theorem 2.1
If (H1), (H2) and (H4) hold, then problem (2.2) has a unique solution
x
given by
where
$$ x(t) =\lambda\int_{0}^{1}H(t,s)\omega(s)f \bigl(s,y\bigl(\alpha(s)\bigr)\bigr)\,ds, $$
(2.4)
$$\begin{aligned}& H(t,s)=G(t,s)+\frac{1}{1-\nu}\int_{0}^{1}G(s, \tau)h(\tau) \,d\tau, \end{aligned}$$
(2.5)
$$\begin{aligned}& G(t,s)=\left \{ \begin{array}{l@{\quad}l} t(1-s), & 0\leq t \leq s\leq1, \\ s(1-t), & 0\leq s \leq t \leq1. \end{array} \right . \end{aligned}$$
(2.6)
Write \(e(t)=t(1-t)\). Then from (2.5) and (2.6) we can prove that \(H(t,s)\) and \(G(t,s)\) have the following properties.
Theorem 2.2
Let
G
and
H
be given as in Theorem
2.1. If (H4) holds, then
where
$$\begin{aligned}& H(t,s)>0,\qquad G(t,s)>0,\quad \forall t,s\in(0,1), \end{aligned}$$
(2.7)
$$\begin{aligned}& H(t,s)\geq0,\qquad G(t,s)\geq0,\quad \forall t,s\in J, \end{aligned}$$
(2.8)
$$\begin{aligned}& e(t)e(s)\leq G(t,s)\leq G(t,t)= t(1-t)=e(t)\leq\bar{e}= \max _{t\in J}e(t)=\frac{1}{4},\quad \forall t,s\in J, \end{aligned}$$
(2.9)
$$\begin{aligned}& \rho e(s)\leq H(t,s)\leq\gamma s(1-s)=\gamma e(s)\leq\frac{1}{4}\gamma , \quad \forall t,s\in J, \end{aligned}$$
(2.10)
$$ \gamma=\frac{1}{1-\nu},\qquad \rho=\frac{\int_{0}^{1}e(\tau)h(\tau)\,d\tau }{1-\nu}. $$
(2.11)
Theorem 2.3
If (H1), (H3) and (H4) hold, then problem (2.3) has a unique solution
y
expressed in the form
where
and
$$ y(t) =\int_{0}^{1}H_{1}(t,s) \phi_{m^{*}}\bigl(x(s)\bigr)\,ds +\mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr), $$
(2.12)
$$\begin{aligned}& H_{1}(t,s)=G_{1}(t,s)+\frac{1}{a-\xi}\int _{0}^{1}G_{1}(s,\tau)g(\tau )\,d \tau, \end{aligned}$$
(2.13)
$$\begin{aligned}& G_{1}(t,s) =\frac{1}{d}\left \{ \begin{array}{l@{\quad}l} (b+as)(b+a(1-t)),& \textit{if } 0\leq s \leq t \leq1, \\ (b+at)(b+a(1-s)),& \textit{if } 0\leq t \leq s \leq1 \end{array} \right . \end{aligned}$$
(2.14)
$$d=a(2b+a). $$
From (2.13) and (2.14) we can prove that \(H_{1}(t,s)\) and \(G_{1}(t,s)\) have the following properties.
Theorem 2.4
Let
\(\zeta\in(0,t_{1})\), \(G_{1}\)
and
\(H_{1}\)
be given as in Theorem
2.3. If (H4) holds, then we have
where
$$\begin{aligned}& \frac{1}{d}b^{2}\leq G_{1}(t,s)\leq G_{1}(s,s)\leq\frac{1}{d}(b+a)^{2}, \quad \forall t,s \in J, \end{aligned}$$
(2.15)
$$\begin{aligned}& \rho_{1}\leq H_{1}(t,s)\leq\frac{a}{a-\xi}G_{1}(s,s) \leq\rho_{2}, \quad \forall t,s\in J, \end{aligned}$$
(2.16)
$$\begin{aligned}& G_{1}(t,s)\geq\delta G_{1}(s,s)\geq\frac{b^{2}\delta}{d}, \qquad H_{1}(t,s)\geq\frac{\delta a}{a-\xi} G_{1}(s,s)\geq\delta \rho_{1},\quad \forall t\in[\zeta,1], s\in J, \end{aligned}$$
(2.17)
$$ \delta=\frac{b}{a+b}, \qquad \rho_{1}=\frac{b^{2}\gamma_{1}}{a+2b}, \qquad \rho _{2}=\frac{\gamma_{1} (b+a)^{2}}{a+2b}, \qquad \gamma_{1}= \frac{1}{a-\xi }. $$
(2.18)
Proof
It follows from the definition of \(G_{1}(t,s)\) and \(H_{1}(t,s)\) that (2.15) and (2.16) hold. Now, we show that (2.17) also holds.
In fact, for \(t\in[\zeta,1]\) and \(s\in J\), we have that
This and (2.13) show that
This together with (2.13) and (2.15) finishes the proof of (2.17). □
$$\begin{aligned}& \frac{G_{1}(t,s)}{G_{1}(s,s)}=\frac{b+a(1-t)}{b+a(1-s)} \geq\frac{b}{b+a} \quad \mbox{for }s \leq t, \\& \frac{G_{1}(t,s)}{G_{1}(s,s)}=\frac{b+at}{b+as} \geq\frac{b+a\zeta}{b+a}\quad \mbox{for }t \leq s. \end{aligned}$$
$$\begin{aligned} H_{1}(t,s)&\geq\delta G_{1}(s,s) \biggl[1+\frac{1}{a-\xi} \int_{0}^{1}g(\tau)\,d\tau \biggr] \\ &\geq\frac{a\delta}{a-\xi}G_{1}(s,s), \quad t\in[\zeta,1], s\in J. \end{aligned}$$
Theorem 2.5
Assume that (H1)-(H4) hold. Then problem (1.1) has a unique solution
y
given by
$$\begin{aligned} y(t) =&\int_{0}^{1}H_{1}(t,s) \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\ &{}+ \mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr). \end{aligned}$$
3 Preliminaries for the case \(\alpha(t)\geq t\) on J
We begin by introducing the notations
We will also need the functions
and let
where \(k=1,2,\ldots,n\).
$$\begin{aligned}& f^{0}=\limsup_{y \rightarrow0^{+}}\max_{t\in J} \frac{f(t,y)}{\phi_{m}(y)},\qquad f^{\infty}=\limsup_{y \rightarrow\infty}\max _{t\in J} \frac{f(t,y)}{\phi_{m}(y)}, \\& f_{0}=\liminf _{y \rightarrow0^{+}}\min_{t\in J} \frac{f(t,y)}{\phi_{m}(y)}, \qquad f_{\infty}=\liminf_{y \rightarrow\infty}\min_{t\in J} \frac{f(t,y)}{\phi _{m}(y)}, \\& I^{0}(k)=\limsup_{y \rightarrow0^{+}}\max _{t\in J} \frac{I_{k}(t,y)}{y}, \qquad I^{\infty}(k)=\limsup _{y \rightarrow\infty}\max_{t\in J} \frac{I_{k}(t,y)}{y}, \\& I_{0}(k)=\liminf_{y \rightarrow0^{+}}\min_{t\in J} \frac{I_{k}(t,y)}{y},\qquad I_{\infty}(k)=\liminf_{y \rightarrow\infty}\min _{t\in J} \frac{I_{k}(t,y)}{y},\quad k=1,2,\ldots,n. \end{aligned}$$
$$f^{*}(u)=\max \Bigl\{ \max_{t\in J}f(t,y), y\in[0,u] \Bigr\} ,\qquad I_{k}^{*}(u)=\max \Bigl\{ \max _{t\in J}I_{k}(t,y), y\in[0,u] \Bigr\} , $$
$$\begin{aligned}& f_{0}^{*}=\lim_{u\rightarrow0^{+}}\frac{f^{*}(u)}{\phi _{m}(u)}, \qquad f_{\infty}^{ *}=\lim_{u\rightarrow\infty} \frac {f^{*}(u)}{\phi_{m}(u)}, \\& I_{0}^{*}(k)=\lim _{u\rightarrow 0^{+}}\frac{I_{k}^{*}(u)}{u}, \qquad I_{\infty}^{ *}(k)= \lim_{u\rightarrow\infty}\frac{I_{k}^{*}(u)}{u}, \end{aligned}$$
Our first lemma gives some relationships between the functions f and \(f^{*}\) and \(I_{k}^{*}\) and \(I_{k}\).
Lemma 3.1
(See [32])
Assume that (H1) holds. Then
where
\(k=1,2,\ldots,n\).
$$f_{0}^{*}=f^{0}, \qquad f_{\infty}^{ *}=f^{\infty}, \qquad I_{0}^{*}(k)=I^{0}(k),\qquad I_{\infty}^{ *}(k)=I^{\infty}(k), $$
The following fixed point theorem of a cone is crucial in the proofs of our results.
Lemma 3.2
(See [33])
Let
P
be a cone in a real Banach space
E. Assume that
\(\Omega_{1}\)
and
\(\Omega_{2}\)
are bounded open sets in
E
with
\(0 \in\Omega_{1}\), \(\bar{\Omega}_{1}\subset\Omega_{2}\). If
is completely continuous such that either
then
A
has at least one fixed point in
\(P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\).
$$A:P\cap(\bar{\Omega}_{2}\backslash\Omega_{1})\rightarrow P $$
(i)
\(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial \Omega_{1}\)
and
\(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\), or
(ii)
\(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial \Omega_{1}\)
and
\(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\),
Let \(J'=J\backslash\{t_{1},t_{2},\ldots,t_{m}\}\), and
Then \(PC^{1}[0,1]\) is a real Banach space with the norm
where \(\|y\|_{\infty}=\sup_{t\in J}|y(t)|\), \(\|y'\|_{\infty}=\sup_{t\in J}|y'(t)|\).
$$PC^{1}[0,1]= \bigl\{ y\in C[0,1]:y'|_{(t_{k},t_{k+1})}\in C(t_{k},t_{k+1}), y'\bigl(t_{k}^{-} \bigr), y'\bigl(t_{k}^{+}\bigr)\text{ exist}, k=1,2, \ldots,m \bigr\} . $$
$$ \|y\|_{PC^{1}}=\max \bigl\{ \|y\|_{\infty}, \bigl\Vert y'\bigr\Vert _{\infty} \bigr\} , $$
(3.1)
A function \(y\in PC^{1}[0,1]\cap C^{4}(J')\) with \(\varphi_{p}(y'')\in C^{2}(0,1)\) is called a solution of problem (1.1) if it satisfies (1.1).
Define a cone in \(PC^{1}[0,1]\) by
where δ, \(\rho_{1}\) and \(\rho_{4}\) are defined in (2.18) and (3.8), respectively. For \(r>0\), let
$$ K= \biggl\{ y\in PC^{1}[0,1]:y(t)\geq0 \mbox{ on }J\mbox{ and } \min _{t\in [\zeta,1]} y(t) \geq\delta\frac{\rho_{1}}{\rho_{4}} \|y\|_{PC^{1}} \biggr\} , $$
(3.2)
$$ \Omega_{r}=\bigl\{ y\in K:\Vert y\Vert _{PC^{1}}< r\bigr\} . $$
(3.3)
Define an operator \(T_{\lambda}^{\mu}:K\rightarrow PC^{1}[0,1]\) by
$$\begin{aligned} \bigl(T_{\lambda}^{\mu}y\bigr) (t) =&\int_{0}^{1}H_{1}(t,s) \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\ &{}+\mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr). \end{aligned}$$
(3.4)
Lemma 3.3
Assume that (H1)-(H4) hold. Then
\(y\in K\)
is a positive fixed point of
\(T_{\lambda}^{\mu}\)
if and only if
y
is a positive solution of problem (1.1).
Lemma 3.4
Suppose that (H1)-(H4) hold. Then
\(T_{\lambda}^{\mu}(K) \subset K\)
and
\(T_{\lambda}^{\mu}: K\rightarrow K\)
is completely continuous.
Proof
For all \(u\in K\), then \(T_{\lambda}u\geq0\) on J and it follows from (2.16) and (3.4) that
where \(\rho_{3}=\frac{1}{d}a(a+b)\),
and
It follows from (3.5) and (3.6) that
where
$$\begin{aligned}& \bigl(T_{\lambda}^{\mu}y\bigr) (t)=\int_{0}^{1}H_{1}(t,s) \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\& \hphantom{\bigl(T_{\lambda}^{\mu}y\bigr) (t)=}{}+\mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr) \\& \hphantom{\bigl(T_{\lambda}^{\mu}y\bigr) (t)}\leq\rho_{2} \Biggl(\int_{0}^{1} \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds + \mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr), \\& \hphantom{\bigl(T_{\lambda}^{\mu}y\bigr) (t)={}}t \in J, \end{aligned}$$
(3.5)
$$\begin{aligned}& \bigl\vert \bigl(T_{\lambda}^{\mu}y\bigr)'(t)\bigr\vert \leq\int_{0}^{1}\bigl\vert H_{1t}'(t,s)\bigr\vert \phi _{m^{*}} \biggl( \lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl( \tau,y(\tau)\bigr) \,d\tau \biggr)\,ds \\& \hphantom{\bigl\vert \bigl(T_{\lambda}^{\mu}y\bigr)'(t)\bigr\vert \leq}{}+\mu\sum_{k=1}^{m}\bigl\vert H_{1t}'(t,t_{k})\bigr\vert I_{k}\bigl(t_{k},y(t_{k})\bigr) \\& \hphantom{\bigl\vert \bigl(T_{\lambda}^{\mu}y\bigr)'(t)\bigr\vert }\leq\rho_{3} \Biggl(\int_{0}^{1} \phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds +\mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr), \\& \hphantom{\bigl\vert \bigl(T_{\lambda}^{\mu}y\bigr)'(t)\bigr\vert \leq{}}t \in J, \end{aligned}$$
(3.6)
$$H_{1t}'(t,s)= G_{1t}'(t,s) = \frac{1}{d}\left \{ \begin{array}{l@{\quad}l} -a(b+as), &\mbox{if } 0\leq s \leq t \leq1, \\ a(b+a(1-s)), & \mbox{if } 0\leq t \leq s \leq1 \end{array} \right . $$
$$\max_{t,s\in J,t\neq s}\bigl\vert H_{1t}'(t,s) \bigr\vert =\max_{t,s\in J,t\neq s}\bigl\vert G_{1t}'(t,s) \bigr\vert =\frac{1}{d}a(b+a). $$
$$\begin{aligned} \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq& \rho_{4} \Biggl(\int_{0}^{1}\phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\ &{}+ \mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr), \end{aligned}$$
(3.7)
$$ \rho_{4}=\max\{\rho_{2},\rho_{3} \}. $$
(3.8)
Noticing (2.17), (3.4) and (3.7), we have
Thus, \(T_{\lambda}^{\mu}(K)\subset K\).
$$\begin{aligned}& \min_{t\in[\zeta,1]}\bigl(T_{\lambda}^{\mu}y\bigr) (t) \\& \quad = \min_{t\in [\zeta,1]} \Biggl( \int_{0}^{1}H_{1}(t,s) \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\& \qquad {}+\mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\& \quad \geq \delta\rho_{1} \Biggl(\int_{0}^{1} \phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds + \mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\& \quad = \delta\frac{\rho_{1}}{\rho_{4}}\rho_{4} \Biggl(\int_{0}^{1} \phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds +\mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\& \quad \geq \delta\frac{\rho _{1}}{\rho_{4}}\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}. \end{aligned}$$
Lemma 3.5
(Hölder)
Let
\(f\in L^{p}[a,b]\)
with
\(p>1\), \(g\in L^{q}[a,b]\)
with
\(q>1\), and
\(\frac{1}{p}+\frac{1}{q}=1\). Then
\(fg\in L^{1}[a,b]\)
and
Let
\(f\in L^{1}[a,b]\), \(g\in L^{\infty}[a,b]\). Then
\(fg\in L^{1}[a,b]\)
and
$$\|fg\|_{1}\leq\|f\|_{p}\|g\|_{q}. $$
$$\|fg\|_{1}\leq\|f\|_{1}\|g\|_{\infty}. $$
Next, we consider the following cases for \(\omega\in L^{p}[0,1]\): \(p>1\), \(p=1\), \(p=\infty\). Case \(p>1\) is treated in Lemma 3.6 and Lemma 3.8.
Lemma 3.6
Assume that (H1)-(H4) hold, \(\alpha(t)\geq t\)
on
J
and let
\(r>0\)
be given. If there exist
\(\varepsilon_{1} >0\)
and
\(\varepsilon_{2} >0\)
such that
\(f^{*}(r)\leq\varepsilon_{1}\phi _{m}(r) \)
and
\(I_{k}^{*}(r)\leq\varepsilon_{2} r\) (\(k=1,2,\ldots,n\)), then
$$ \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda \gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon _{2} \bigr)\|y \|_{PC^{1}}, \quad y\in\partial\Omega_{r}. $$
(3.9)
Proof
By the definition of \(f^{*}(r)\) and \(I_{k}^{*}\), if \(f^{*}(r)\leq\varepsilon_{1} \phi_{m}(r)\) and \(I_{k}^{*}(r)\leq \varepsilon_{2} r\) (\(k=1,2,\ldots,n\)), then
$$f(t,y)\leq\varepsilon_{1} \phi_{m}(r), \qquad I_{k}(t,y)\leq\varepsilon_{2} r \quad \mbox{for }t\in J \mbox{ and }0\leq y\leq r. $$
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(0\leq y(t)\leq r\) on J that \(0\leq y(\alpha(t))\leq r\).
Therefore, we have \(f(t,y(\alpha(t)))\leq\varepsilon_{1} \phi_{m}(r)\) for \(t\in J\), and it follows from (3.7) that
This completes the proof. □
$$\begin{aligned} \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq& \rho_{4} \Biggl(\int_{0}^{1}\phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds + \mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\int _{0}^{1}\gamma e(\tau )\omega(\tau)f\bigl(\tau,y \bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr) +\mu\sum _{k=1}^{n}I_{k}\bigl(t_{k},y(t_{k}) \bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\gamma\int _{0}^{1} e(\tau )\omega(\tau)\varepsilon_{1} \phi_{m}(r) \,d\tau \biggr) +\mu\sum_{k=1}^{n} \varepsilon_{2} r \Biggr) \\ =& \rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\gamma \varepsilon_{1} \phi _{m}(r)\int_{0}^{1} e(\tau)\omega(\tau) \,d\tau \biggr) +\mu\sum_{k=1}^{n} \varepsilon_{2} r \Biggr) \\ \leq&\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \phi _{m}(r)\Vert e\Vert _{q} \|\omega\|_{p} \bigr) +\mu n\varepsilon_{2} r \bigr) \\ =&\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon_{2} r \bigr)\|y \|_{PC^{1}}, \quad \forall y \in\partial\Omega_{r}. \end{aligned}$$
The following result deals with the case \(p=1\).
Corollary 3.1
Assume that (H1)-(H4) hold, \(\alpha (t)\geq t\)
on
J
and let
\(r>0\)
be given. If there exist
\(\varepsilon _{1} >0\)
and
\(\varepsilon_{2} >0\)
such that
\(f^{*}(r)\leq\varepsilon _{1}\phi_{m}(r) \)
and
\(I_{k}^{*}(r)\leq\varepsilon_{2} r\) (\(k=1,2,\ldots ,n\)), then
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda \gamma \varepsilon_{1} \Vert e\Vert _{\infty}\|\omega\|_{1} \bigr) +\mu n\varepsilon_{2} r \bigr)\|y\|_{PC^{1}}, \quad y\in \partial\Omega_{r}. $$
Proof
By Lemma 3.6, let \(\|e\|_{\infty}\|\omega\|_{1}\) replace \(\|e\|_{q}\|\omega\|_{p}\) and repeat the argument above. □
Finally we consider the case of \(p=\infty\).
Corollary 3.2
Assume that (H1)-(H4) hold, \(\alpha (t)\geq t\)
on
J
and let
\(r>0\)
be given. If there exist
\(\varepsilon _{1} >0\)
and
\(\varepsilon_{2} >0\)
such that
\(f^{*}(r)\leq\varepsilon _{1}\phi_{m}(r) \)
and
\(I_{k}^{*}(r)\leq\varepsilon_{2} r\) (\(k=1,2,\ldots ,n\)), then
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda \gamma \varepsilon_{1} \Vert e\Vert _{1}\|\omega\|_{\infty} \bigr) +\mu n\varepsilon_{2} r \bigr)\|y\|_{PC^{1}}, \quad y\in \partial\Omega_{r}. $$
Proof
By Lemma 3.6, let \(\|e\|_{1}\|\omega\|_{\infty}\) replace \(\|e\|_{q}\|\omega\|_{p}\) and repeat the argument above. □
Lemma 3.7
Assume that (H1)-(H4) hold, \(\alpha(t)\geq t\)
on
J
and let
\(l_{1}>0\)
and
\(l_{2}\)
be given. If
\(f(t,y)\geq l_{1}\phi_{m}(y)\)
and
\(I_{k}(t,y)\geq l_{2}y\) (\(k=1,2,\ldots,n\)) for
\(t\in J\)
and
\(y\in K\), then
where
\(\beta=\phi_{m^{*}}(\int_{\zeta}^{1}e(\tau)\,d\tau)\).
$$ \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\geq \delta\frac{\rho_{1}^{2}}{\rho_{4}} \bigl((l_{1}\lambda\rho\eta)^{m^{*}-1} \beta+\mu nl_{2} \bigr)\|y\| _{PC^{1}}, $$
(3.10)
Proof
By the definition of K, if \(y\in K\), then we have
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(y(t)\geq0\) on J that \(y(\alpha(t))\geq0\).
$$y(t)\geq0 \quad \mbox{on }J \quad \mbox{and}\quad \min_{t\in[\zeta,1]}y(t) \geq\delta\frac {\rho_{1}}{\rho_{4}}\|y\|_{PC^{1}}. $$
Similarly, since \(\zeta\leq t\leq\alpha(t)\leq1\) on \([\zeta,1]\), it follows from \(\min_{t\in[\zeta,1]}y(t)\geq\delta\frac{\rho _{1}}{\rho_{4}}\|y\|_{PC^{1}}\) that
$$\min_{t\in[\zeta,1]}y\bigl(\alpha(t)\bigr)\geq\delta\frac{\rho_{1}}{\rho _{4}} \|y\|_{PC^{1}}. $$
Therefore, \(f(t,y(\alpha(t)))\geq l_{1}\phi_{m}(y)\) for \(t\in J\), and it follows from the definition of \(T_{\lambda}^{\mu}\) and (2.17) that
This completes the proof. □
$$\begin{aligned} \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \geq& \rho_{1}\phi_{m^{*}} \biggl(\lambda\int_{0}^{1} \rho e(\tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \\ \geq&\rho_{1}(\lambda\rho\eta)^{m^{*}-1}\phi _{m^{*}} \biggl(\int_{0}^{1} e(\tau)f\bigl(\tau,y\bigl( \alpha(\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}\sum _{k=1}^{n}I_{k}\bigl(t_{k},y(t_{k}) \bigr) \\ \geq&\rho_{1}(\lambda\rho\eta)^{m^{*}-1}\phi _{m^{*}} \biggl(\int_{0}^{1} e(\tau)l_{1} \phi_{m}\bigl(y\bigl(\alpha(\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}\sum_{k=1}^{n}l_{2}y(t_{k}) \\ \geq&\rho_{1}(l_{1}\lambda\rho\eta )^{m^{*}-1} \phi_{m^{*}} \biggl(\int_{\zeta}^{1} e(\tau) \phi_{m}\bigl(y\bigl(\alpha (\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}l_{2}\sum_{k=1}^{n}y(t_{k}) \\ \geq&\rho_{1}(l_{1}\lambda\rho\eta )^{m^{*}-1} \phi_{m^{*}} \biggl(\int_{\zeta}^{1} e(\tau) \phi_{m}\biggl(\delta \frac{\rho_{1}}{\rho_{4}}\|y\|_{PC^{1}}\biggr)\,d \tau \biggr)+\mu\rho_{1}l_{2}\sum _{k=1}^{n}\delta\frac{\rho_{1}}{\rho_{4}}\|y \|_{PC^{1}} \\ =&\delta\frac{\rho_{1}^{2}}{\rho_{4}} \bigl((l_{1}\lambda\rho \eta)^{m^{*}-1}\beta+\mu nl_{2} \bigr)\|y\|_{PC^{1}}. \end{aligned}$$
Lemma 3.8
Assume that (H1)-(H4) hold and
\(\alpha (t)\geq t\)
on
J. If
\(y\in\partial\Omega_{r}\), \(r>0\), then
where
$$ \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma M_{r} \Vert e\Vert _{q}\|\omega\|_{p} \bigr) +\mu n M^{*} \bigr), $$
(3.11)
$$\begin{aligned}& M_{r}=\max_{t\in J, 0\leq y\leq r}\bigl\{ f(t,y)\bigr\} >0,\qquad M^{*}=\max\bigl\{ M_{k}^{*}, k=1,2,\ldots,n\bigr\} >0, \\& M_{k}^{*}=\max_{t\in J, 0\leq y\leq r}\bigl\{ I_{k}(t,y)\bigr\} ,\quad k=1,2,\ldots,n. \end{aligned}$$
Proof
If \(y\in\partial\Omega_{r}\), then \(0\leq y(t)\leq r\) for \(t\in J\).
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(0\leq y\leq r\) that
$$0\leq y\bigl(\alpha(t)\bigr)\leq r. $$
Therefore, from \(f(t,y)\leq{M_{r}}\) for \(t\in J\) and \(y\in\partial \Omega_{r}\), we have
$$f\bigl(t,y\bigl(\alpha(t)\bigr)\bigr)\leq{M_{r}}. $$
So, for \(y\in\partial\Omega_{r}\), we have
This gives the proof. □
$$\begin{aligned} \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq& \rho_{4} \Biggl(\int_{0}^{1}\phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds + \mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\int _{0}^{1}\gamma e(\tau )\omega(\tau)f\bigl(\tau,y \bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr) +\mu\sum _{k=1}^{n}I_{k}\bigl(t_{k},y(t_{k}) \bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\gamma\int _{0}^{1} e(\tau )\omega(\tau)M_{r} \,d \tau \biggr) +\mu\sum_{k=1}^{n}M^{*} \Biggr) \\ =& \rho_{4} \biggl(\phi_{m^{*}} \biggl(\lambda\gamma M_{r}\int_{0}^{1} e(\tau )\omega(\tau) \,d\tau \biggr) +\mu n M^{*} \biggr) \\ \leq&\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma M_{r}\Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n M^{*} \bigr). \end{aligned}$$
Corollary 3.3
When
\(p=1\), assume that (H1)-(H4) hold and
\(\alpha(t)\geq t\)
on
J. If
\(y\in\partial\Omega_{r}\), \(r>0\), then
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma M_{r} \Vert e\Vert _{\infty}\|\omega\|_{1} \bigr) +\mu n M^{*} \bigr). $$
Proof
By Lemma 3.8, let \(\|e\|_{\infty}\|\omega\|_{1}\) replace \(\|e\|_{q}\|\omega\|_{p}\) and repeat the argument above. □
Corollary 3.4
When
\(p=\infty\), assume that (H1)-(H4) hold and
\(\alpha(t)\geq t\)
on
J. If
\(y\in\partial\Omega_{r}\), \(r>0\), then
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma M_{r} \Vert e\Vert _{1}\|\omega\|_{\infty} \bigr) +\mu n M^{*} \bigr). $$
Proof
By Lemma 3.8, let \(\|e\|_{1}\|\omega_{\infty}\|\) replace \(\|e\|_{q}\|\omega_{p}\|\) and repeat the argument above. □
Lemma 3.9
Assume that (H1)-(H4) hold and
\(\alpha (t)\geq t\)
on
J. If
\(y\in\partial\Omega_{r}\), \(r>0\), then
where
$$ \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \geq \rho_{1}(\sigma_{r}\lambda\rho\eta )^{m^{*}-1}\beta+\mu \rho_{1}n\sigma^{*}_{r}, $$
(3.12)
$$\begin{aligned}& \sigma_{r}=\min_{t\in J, 0\leq y\leq r}\bigl\{ f(t,y)\bigr\} >0,\qquad \sigma ^{*}=\min\{m_{k}, k=1,2,\ldots,n\}>0, \\& \sigma_{k}=\min_{t\in J, 0\leq y\leq r}\bigl\{ I_{k}(t,y) \bigr\} ,\quad k=1,2,\ldots,n. \end{aligned}$$
Proof
If \(y\in\partial\Omega_{r}\), then \(0\leq y(t)\leq r\) for \(t\in J\).
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(0\leq y(t)\leq r\) for \(t\in J\) that
$$0\leq y\bigl(\alpha(t)\bigr)\leq r. $$
Therefore, from \(f(t,y)\geq\sigma_{r}\) for \(t\in J\) and \(0\leq y \leq r\), we have
$$f\bigl(t,y\bigl(\alpha(t)\bigr)\bigr)\geq\sigma_{r}. $$
So, for \(y\in\partial\Omega_{r}\), we have
This finishes the proof. □
$$\begin{aligned} \bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \geq& \rho_{1}\phi_{m^{*}} \biggl(\lambda\int_{0}^{1} \rho e(\tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \\ \geq&\rho_{1}(\lambda\rho\eta)^{m^{*}-1}\phi _{m^{*}} \biggl(\int_{0}^{1} e(\tau)f\bigl(\tau,y\bigl( \alpha(\tau)\bigr)\bigr)\,d\tau \biggr)+\mu \rho_{1}\sum _{k=1}^{n}I_{k}\bigl(t_{k},y(t_{k}) \bigr) \\ \geq&\rho_{1}(\lambda\rho\eta)^{m^{*}-1}\phi _{m^{*}} \biggl(\int_{0}^{1} e(\tau)\sigma_{r}\,d \tau \biggr)+\mu\rho_{1}\sum_{k=1}^{n} \sigma^{*}_{r} \\ \geq&\rho_{1}(\sigma_{r}\lambda\rho\eta )^{m^{*}-1} \phi_{m^{*}} \biggl(\int_{\zeta}^{1} e(\tau)\,d \tau \biggr)+\mu\rho _{1}n\sigma^{*}_{r} \\ =&\rho_{1}(\sigma_{r}\lambda\rho\eta )^{m^{*}-1} \beta+\mu\rho_{1}n\sigma^{*}_{r}. \end{aligned}$$
4 Main results for the case \(\alpha(t)\geq t\) on J
In this section, we apply Lemma 3.2 to establish the existence, multiplicity and nonexistence of positive solutions for problem (1.1). We consider the following three cases for \(\omega\in L^{p}[0,1]\): \(p> 1\), \(p=1\) and \(p=\infty\). Case \(p>1\) is treated in the following theorem.
Theorem 4.1
Assume that (H1)-(H4) hold and
\(\alpha (t)\geq t\)
on
J. Then:
(a)
If
\(f^{0}=0\)
and
\(I^{0}(k)=0\)
or
\(f^{\infty}=0\)
and
\(I^{\infty }(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has a positive solution for
\(\lambda>\lambda_{0}\)
and
\(\mu >\mu_{0}\).
(b)
If
\(f_{0}=\infty\)
and
\(I_{0}(k)=\infty\)
or
\(f_{\infty}=\infty\)
and
\(I_{\infty}(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu _{0}>0\)
such that problem (1.1) has a positive solution for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(c)
If
\(f^{0}=f^{\infty}=0\)
and
\(I^{0}(k)=I^{\infty}(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(\lambda> \lambda_{0}\)
and
\(\mu>\mu _{0}\).
(d)
If
\(f_{0}=f_{\infty}=\infty\)
and
\(I_{\infty}(k)=I_{\infty }(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(e)
If
\(f^{0}<\infty\), \(I^{0}(k)<\infty\), \(f^{\infty}<\infty\)
and
\(I^{\infty}<\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(0<\lambda<\lambda _{0}\)
and
\(0<\mu<\mu_{0}\).
(f)
If
\(f_{0}>0\), \(I_{0}(k)>0\), \(I_{\infty}>0\)
and
\(f_{\infty}>0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(\lambda> \lambda_{0}\)
and
\(\mu_{0}>0\).
Proof
Part (a). Choose a number \(r_{1}>0\). By Lemma 3.9, we have \(\|T_{\lambda}^{\mu}y\|_{PC^{1}}>\|y\|_{PC^{1}}\) for \(y\in\partial \Omega_{r_{1}}\), \(\lambda> \lambda_{0}\) and \(\mu>\mu_{0}\), where
$$\lambda_{0}= \biggl(\frac{1}{2\rho_{1}\beta}r_{1} \biggr)^{m-1}(\sigma _{r}\rho\eta)^{-1}>0,\qquad \mu_{0}=\frac{1}{2\rho_{1}n\sigma_{r_{1}}^{*}}r>0. $$
If \(f^{0}=0\) and \(I^{0}(k)=0\), then from Lemma 3.1 we have \(f_{0}^{*}=0\) and \(I_{0}^{*}(k)=0\), and so we can choose \(r_{2}\in(0, r_{1})\) so that \(f^{*}(r_{2})\leq\varepsilon_{1} r_{2}\) and \(I_{k}^{*}(r_{2})\leq \varepsilon_{2} r_{2}\), where \(\varepsilon_{1} >0\) and \(\varepsilon_{2} >0\) respectively satisfy
Then Lemma 3.6 shows that
$$ 2\rho_{4}\phi_{m^{*}} \bigl(\lambda\gamma\varepsilon_{1} \Vert e\Vert _{q}\| \omega\|_{p} \bigr)< 1, \qquad 2 \rho_{4}\mu n\varepsilon_{2}<1. $$
(4.1)
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon_{2} \bigr)\|y \|_{PC^{1}}< \|y\|_{PC^{1}} \quad \mbox{for } y\in \partial \Omega_{r_{2}}. $$
If \(f^{\infty}=0\) and \(I^{\infty}(k)=0\), then from Lemma 3.1, \(f_{\infty }^{*}=0\) and \(I_{\infty}^{*}(k)=0\). Hence, there exists \(r_{3}\in (2r_{1},\infty)\) such that \(f^{*}(r_{3})\leq\varepsilon_{1} r_{3}\) and \(I_{k}^{*}(r_{3})\leq\varepsilon_{2} r_{4}\), where \(\varepsilon_{1} >0\) and \(\varepsilon_{2} >0\) satisfies (4.1). Thus
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon_{2} \bigr)\|y \|_{PC^{1}}< \|y\|_{PC^{1}} \quad \mbox{for } y\in \partial \Omega_{r_{3}}. $$
Therefore, it follows from Lemma 3.2 that \(T_{\lambda}^{\mu}\) has a fixed point in \(\bar{\Omega}_{r_{1}}\backslash\Omega_{r_{2}} \) or \(\bar {\Omega}_{r_{3}}\backslash\Omega_{r_{1}}\), according to whether \(f^{0}=0\) and \(I^{0}(k)=0\) or \(f^{\infty}=0\) and \(I^{\infty}(k)=0\), respectively. Consequently, problem (1.1) has a positive solution for \(\lambda> \lambda_{0}\) and \(\mu>\mu_{0}\).
Part (b). Choose a number \(r_{1}>0\). By Lemma 3.8, there exists \(\lambda _{0}>0 \) such that
where
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}< \|y \|_{PC^{1}} \quad \mbox{for } y\in\partial\Omega _{r_{1}}, 0<\lambda< \lambda_{0}\mbox{ and }0<\mu<\mu_{0}, $$
$$\lambda_{0}= \biggl(\frac{1}{2\rho_{4}} \biggr)^{m-1} \bigl( \gamma M_{r}\Vert e\Vert _{q}\|\omega\|_{p} \bigr)^{-1}, \qquad \mu_{0}=\frac{1}{2\rho_{4} n M^{*}}. $$
If \(f_{0}=\infty\) and \(I_{0}(k)=\infty\), there exists \(r_{2} \in (0,r_{1})\) such that \(f(t,y)\geq l_{1}\phi_{m}(y)\) and \(I_{k}(t,y)\geq l_{2} y\) for \(t\in J\) and \(0\leq y \leq r_{2}\), where \(l_{1}>0\) and \(l_{2}>0\) are chosen so that
$$ 2\delta\frac{\rho_{1}^{2}}{\rho_{4}}(l_{1}\lambda\rho\eta)^{m^{*}-1}\beta >1, \qquad 2\delta\frac{\rho_{1}^{2}}{\rho_{4}}\mu nl_{2}>1. $$
(4.2)
Obviously,
Then, from Lemma 3.7, we can obtain
$$f(t,y)\geq l_{1}\phi_{m}(y), \qquad I_{k}(t,y) \geq l_{2} y\quad (k=1,2,\ldots,n)\mbox{ for }y\in\partial \Omega_{r_{2}} \mbox{ and }t \in J. $$
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \geq \delta\frac{\rho_{1}^{2}}{\rho _{4}} \bigl((l_{1}\lambda\rho\eta)^{m^{*}-1} \beta+\mu nl_{2} \bigr)\|y\| _{PC^{1}}>\|y\|_{PC^{1}}\quad \mbox{for }y\in\partial\Omega_{r_{3}}. $$
If \(f_{\infty}=\infty\) and \(I_{\infty}(k)=\infty\), then there exists \(\hat{N}>0\) such that \(f(t,y)\geq l_{1}\phi_{m}(y)\), \(I_{k}(t,y)\geq l_{2} y\) (\(k=1,2,\ldots,n\)) for \(t\in J\) and \(y\geq\hat{N}\), and \(l_{1} >0\) and \(l_{2}>0\) satisfy (4.2).
Let \(r_{3}=\max\{2r_{1},\hat{N}\rho_{4}/ \delta\rho_{1}\}\). If \(y\in \partial\Omega_{r_{3}}\), then
So,
From Lemma 3.7, we can get
$$\min_{t\in[\zeta,1]}y(t)\geq\frac{\delta\rho_{1}}{\rho_{4}} \| y\|_{PC^{1}}\geq \hat{N}. $$
$$f(t,y)\geq l_{1}\phi_{m}(y), \qquad I_{k}(t,y) \geq l_{2} y\quad (k=1,2,\ldots,n)\mbox{ for }t\in J\mbox{ and }y\in\partial \Omega_{r_{3}}. $$
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \geq \delta\frac{\rho_{1}^{2}}{\rho _{4}} \bigl((l_{1}\lambda\rho\eta)^{m^{*}-1} \beta+\mu nl_{2} \bigr)\|y\| _{PC^{1}}>\|y\|_{PC^{1}}\quad \mbox{for }y\in\partial\Omega_{r_{3}}. $$
Therefore, it follows from Lemma 3.2 that \(T_{\lambda}^{\mu}\) has a fixed point in \(\bar{\Omega}_{r_{1}}\backslash\Omega_{r_{2}} \) or \(\bar{\Omega}_{r_{3}}\backslash\Omega_{r_{1}}\), according to whether \(f_{0}=\infty\) and \(I_{0}(k)=\infty\) or \(f_{\infty}=\infty\) and \(I_{\infty}(k)=\infty\), respectively. Consequently, problem (1.1) has a positive solution for \(0<\lambda<\lambda_{0}\) and \(0<\mu<\mu_{0}\).
Part (c). Choose two numbers \(0< r_{3}<r_{4}\). By Lemma 3.9, there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that
Since \(f^{0}=f^{\infty}=0\) and \(I^{0}(k)=I^{\infty}(k)=0\), from the proof of part (a), it follows that we can choose \(r_{1}\in(0,r_{3}/2)\) and \(r_{2}\in(2r_{4},\infty)\) such that
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}>\|y \|_{PC^{1}}\quad \mbox{for }y\in\partial\Omega _{r_{i}}, i=3,4. $$
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}< \|y \|_{PC^{1}} \quad \mbox{for } y\in\partial\Omega _{r_{i}}, i=1,2. $$
It follows from Lemma 3.2 that \(T_{\lambda}^{\mu}\) has two fixed points \(y_{1}\) and \(y_{2}\) such that \(y_{1}\in\bar{\Omega}_{r_{3}}\backslash \Omega_{r_{1}}\) and \(y_{2}\in\bar{\Omega}_{r_{2}}\backslash\Omega _{r_{4}}\). These are the desired distinct positive solutions of problem (1.1) for \(\lambda> \lambda_{0}\) and \(\mu>\mu_{0}\) satisfying
$$ r_{1}\leq\|y_{1}\|_{PC^{1}}\leq r_{3}< r_{4} \leq\|y_{2}\|_{PC^{1}}\leq r_{2}. $$
(4.3)
Part (d). Choose two numbers \(0< r_{3}<r_{4}\). By Lemma 3.8, there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that
Since \(f_{0}=\infty\) and \(f_{\infty}=\infty\) and \(I_{\infty}(k)=I_{\infty }(k)=\infty\), from the proof of part (b), we know that we can choose \(r_{1}\in(0,r_{3}/2)\) and \(r_{2}\in(2r_{4},\infty)\) such that
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}< \|y \|_{PC^{1}}\quad \mbox{for } 0 < \lambda< \lambda _{0}, 0<\mu< \mu_{0} \mbox{ and }y\in\partial\Omega_{r_{i}}, i=3,4. $$
$$\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}>\|y \|_{PC^{1}} \quad \mbox{for }y\in\partial\Omega _{r_{i}}, i=1,2. $$
It follows from Lemma 3.2 that \(T_{\lambda}^{\mu}\) has two fixed points \(y_{1}\) and \(y_{2}\) such that \(y_{1}\in\bar{\Omega}_{r_{3}}\backslash \Omega_{r_{1}}\) and \(y_{2}\in\bar{\Omega}_{r_{2}}\backslash\Omega _{r_{4}}\). These are the desired distinct positive solutions of problem (1.1) for \(0<\lambda< \lambda_{0}\) and \(0<\mu<\mu_{0}\) satisfying (4.3).
Part (e). Since \(f^{0}<\infty\), \(I^{0}(k)<\infty\), \(f^{\infty}<\infty\) and \(I^{\infty}(k)<\infty\), there exist positive numbers \(l_{i}>0\) (\(i=1,2,3,4\)), \(h_{1}>0\) and \(h_{2}>0\) such that \(h_{1}< h_{2}\) and for \(t \in J\), \(0< y\leq h_{1}\), we have
and for \(t \in J\), \(y\geq h_{2}\), we have
$$f(t,y)\leq l_{1}\phi_{m}(y), \qquad I_{k}(t,y) \leq l_{2}y $$
$$f(t,y)\leq l_{3}\phi_{m}(y), \qquad I_{k}(t,y) \leq l_{4}y. $$
Let
Thus, we have
$$\begin{aligned}& l=\max \biggl\{ l_{1}, l_{3}, \max \biggl\{ \frac{f(t,y)}{\phi_{m}(y)}:t \in J, h_{1}\leq y \leq h_{2} \biggr\} \biggr\} >0, \\& l^{*}=\max \biggl\{ l_{2}, l_{4}, \max \biggl\{ \frac{I_{k}(t,y)}{y}:t \in J, h_{1}\leq y \leq h_{2} \biggr\} \biggr\} >0. \end{aligned}$$
$$f(t,y)\leq l \phi_{m}(y), \qquad I_{k}(t,y)\leq l^{*}y \quad \mbox{for } t \in J\mbox{ and }y \in[0,\infty). $$
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(0\leq y(t)\leq h_{1}\), \(y(t)\geq h_{2}\) and \(h_{1}\leq y(t)\leq h_{2}\) on J that \(0\leq y(\alpha(t))\leq h_{1}\), \(y(\alpha(t))\geq h_{2}\) and \(h_{1}\leq y(\alpha(t))\leq h_{2}\) on J, respectively.
Assume that y is a positive solution of problem (1.1). We will show that this leads to a contradiction for
and
$$0< \lambda< \lambda_{0}= \biggl(\frac{1}{2\rho_{4}} \biggr)^{m-1} \bigl(\gamma l\Vert e\Vert _{q}\|\omega \|_{p} \bigr)^{-1} $$
$$0< \mu< \mu_{0}=\frac{1}{2\rho_{4}nl^{*}}. $$
Since \((T_{\lambda}^{\mu}y)(t)=y(t)\) for \(t\in J\), by Lemma 3.6 we have that
which is a contradiction.
$$\|y\|_{PC^{1}}=\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\leq\rho_{4} \bigl(\phi _{m^{*}} \bigl(\lambda \gamma l \Vert e\Vert _{q}\|\omega\|_{p} \bigr) +\mu nl^{*} \bigr)\|y\|_{PC^{1}}< \|y\|_{PC^{1}}, $$
Part (f). Since \(f_{0}>0\), \(I_{0}(k)>0\), \(I_{\infty}>0\) and \(f_{\infty }>0\), there exist positive numbers \(l_{i}>0\) (\(i=5,6,7,8\)), \(h_{3}>0\) and \(h_{4}>0\) such that \(h_{3}< h_{4}\) and for \(t \in J\), \(0\leq y\leq h_{3}\), we have
and for \(t \in J\), \(y\geq h_{4}\), we have
$$f(t,y)\geq l_{5}\phi_{m}(y), \qquad I_{k}(t,y) \geq l_{6} y, $$
$$f(t,y)\geq l_{7}\phi_{m}(y), \qquad I_{k}(t,y) \geq l_{8} y. $$
Let
Then
$$\begin{aligned}& l^{**}=\min \biggl\{ l_{5}, l_{7}, \min \biggl\{ \frac{f(t,y)}{\phi_{m}(y)}:t \in J, h_{3}\leq y \leq h_{4} \biggr\} \biggr\} >0, \\& l^{***}=\min \biggl\{ l_{6}, l_{8}, \min \biggl\{ \frac{I_{k}(t,y)}{y}:t \in J, h_{3}\leq y \leq h_{4} \biggr\} \biggr\} >0. \end{aligned}$$
$$f(t,y)\geq l^{**} \phi_{m}(y),\qquad I_{k}(t,y) \geq l^{***}y\quad \mbox{for }t \in J \mbox{ and }y \in[0,\infty). $$
Assume that y is a positive solution of problem (1.1). We will show that this leads to a contradiction for
$$\begin{aligned}& \lambda> \lambda_{0}= \biggl(\frac{\rho_{4}}{2\delta\rho_{1}^{2}\beta} \biggr)^{m-1} \bigl(l^{**}\rho\eta \bigr)^{-1}, \\& \mu> \mu_{0}=\frac{\rho_{4}}{2\delta\rho_{1}^{2}n l^{***}}. \end{aligned}$$
Since \((T_{\lambda}^{\mu}y)(t)=y(t)\) for \(t\in J\), by Lemma 3.7 we have that
which is a contradiction. □
$$\|y\|_{PC^{1}}=\bigl\Vert T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\geq\delta\frac{\rho _{1}^{2}}{\rho_{4}} \bigl(\bigl(l^{**}\lambda \rho\eta\bigr)^{m^{*}-1}\beta+\mu nl^{***} \bigr)\|y \|_{PC^{1}}>\|y\|_{PC^{1}}, $$
The results of the following theorem deal with the case \(p=1\).
Corollary 4.1
Assume that (H1)-(H4) hold and
\(\alpha (t)\geq t\)
on
J. Then:
(a)
If
\(f^{0}=0\)
and
\(I^{0}(k)=0\)
or
\(f^{\infty}=0\)
and
\(I^{\infty }(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has a positive solution for
\(\lambda>\lambda_{0}\)
and
\(\mu >\mu_{0}\).
(b)
If
\(f_{0}=\infty\)
and
\(I_{0}(k)=\infty\)
or
\(f_{\infty}=\infty\)
and
\(I_{\infty}(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu _{0}>0\)
such that problem (1.1) has a positive solution for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(c)
If
\(f^{0}=f^{\infty}=0\)
and
\(I^{0}(k)=I^{\infty}(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(\lambda> \lambda_{0}\)
and
\(\mu>\mu _{0}\).
(d)
If
\(f_{0}=f_{\infty}=\infty\)
and
\(I_{\infty}(k)=I_{\infty }(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(e)
If
\(f^{0}<\infty\), \(I^{0}(k)<\infty\), \(f^{\infty}<\infty\)
and
\(I^{\infty}<\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(0<\lambda<\lambda _{0}\)
and
\(0<\mu<\mu_{0}\).
(f)
If
\(f_{0}>0\), \(I_{0}(k)>0\), \(I_{\infty}>0\)
and
\(f_{\infty}>0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(\lambda> \lambda_{0}\)
and
\(\mu_{0}>0\).
Finally we consider the case of \(p=\infty\).
Corollary 4.2
Assume that (H1)-(H4) hold and
\(\alpha (t)\geq t\)
on
J. Then:
(a)
If
\(f^{0}=0\)
and
\(I^{0}(k)=0\)
or
\(f^{\infty}=0\)
and
\(I^{\infty }(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has a positive solution for
\(\lambda>\lambda_{0}\)
and
\(\mu >\mu_{0}\).
(b)
If
\(f_{0}=\infty\)
and
\(I_{0}(k)=\infty\)
or
\(f_{\infty}=\infty\)
and
\(I_{\infty}(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu _{0}>0\)
such that problem (1.1) has a positive solution for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(c)
If
\(f^{0}=f^{\infty}=0\)
and
\(I^{0}(k)=I^{\infty}(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(\lambda> \lambda_{0}\)
and
\(\mu>\mu _{0}\).
(d)
If
\(f_{0}=f_{\infty}=\infty\)
and
\(I_{\infty}(k)=I_{\infty }(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(e)
If
\(f^{0}<\infty\), \(I^{0}(k)<\infty\), \(f^{\infty}<\infty\)
and
\(I^{\infty}<\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(0<\lambda<\lambda _{0}\)
and
\(0<\mu<\mu_{0}\).
(f)
If
\(f_{0}>0\), \(I_{0}(k)>0\), \(I_{\infty}>0\)
and
\(f_{\infty}>0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(\lambda> \lambda_{0}\)
and
\(\mu_{0}>0\).
5 Positive solutions of problem (1.1) for the case of \(\alpha (t)\leq t\) on J
Now we deal with problem (1.1) for the case of \(\alpha(t)\leq t\) on J. Similarly as Theorem 2.4 and Lemmas 3.3-3.9, we can prove the following results.
Lemma 5.1
Let
\(\zeta^{*}\in(t_{n},1)\), \(G_{1}\)
and
\(H_{1}\)
be given as in Theorem
2.3. If (H4) holds, then we have
where
d
is defined in Theorem
2.3, δ
and
\(\rho_{1}\)
are defined in (2.8).
$$ G_{1}(t,s)\geq\delta G_{1}(s,s)\geq\frac{b^{2}\delta}{d},\qquad H_{1}(t,s)\geq\frac{\delta a}{a-\xi} G_{1}(s,s)\geq\delta \rho_{1},\quad \forall t\in\bigl[0,\zeta^{*}\bigr], s\in J, $$
(5.1)
Proof
In fact, for \(t\in[0,\zeta^{*}]\) and \(s\in J\), we have that
This and (2.13) show that
This together with (2.13) and (2.15) finishes the proof of (5.1). □
$$\begin{aligned}& \frac{G_{1}(t,s)}{G_{1}(s,s)}=\frac{b+a(1-t)}{b+a(1-s)} \geq\frac{b+a(1-\zeta^{*})}{b+a}\quad \mbox{for }s \leq t, \\& \frac{G_{1}(t,s)}{G_{1}(s,s)}=\frac{b+at}{b+as} \geq\frac{b}{b+a} \quad \mbox{for }t \leq s. \end{aligned}$$
$$\begin{aligned} H_{1}(t,s)&\geq\delta G_{1}(s,s) \biggl[1+\frac{1}{a-\xi} \int_{0}^{1}g(\tau)\,d\tau \biggr] \\ &\geq\frac{a\delta}{a-\xi}G_{1}(s,s),\quad t\in\bigl[0, \zeta^{*}\bigr], s\in J. \end{aligned}$$
Let \(PC^{1}[0,1]\) be as defined in Section 3. We define a cone \(K^{*}\) in \(PC^{1}[0,1]\) by
where δ, \(\rho_{1}\) and \(\rho_{4}\) are defined in (2.18) and (3.8), respectively. It is easy to see that \(K^{*}\) is a closed convex cone of \(PC^{1}[0,1]\).
$$K^{*}= \biggl\{ y\in PC^{1}[0,1]\Big|y(t)\geq0\mbox{ on }J \mbox{ and }\min_{t\in [0,\zeta^{*}]}\geq\delta\frac{\rho_{1}}{\rho_{4}} \|y \|_{PC^{1}} \biggr\} , $$
Define \({}^{*}T_{\lambda}^{\mu}: K^{*}\rightarrow PC^{1}[0,1]\) by
$$\begin{aligned} \bigl(^{*}T_{\lambda}^{\mu}y\bigr) (t) =&\int _{0}^{1}H_{1}(t,s)\phi_{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f \bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds \\ &{}+\mu\sum_{k=1}^{n}H_{1}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr). \end{aligned}$$
(5.2)
It is clear that y is a positive solution of problem (1.1) if and only of y is a fixed point of \({}^{*}T_{\lambda}^{\mu}\).
Lemma 5.2
Assume that (H1)-(H4) hold. Then
\(y\in K^{*}\)
is a positive fixed point of
\({}^{*}T_{\lambda}^{\mu}\)
if and only if
y
is a positive solution of problem (1.1).
Lemma 5.3
Assume that (H1)-(H4) hold. Then
\({}^{*}T_{\lambda}^{\mu}(K^{*})\subset K^{*}\)
and
\({}^{*}T_{\lambda}^{\mu }:K^{*}\rightarrow K^{*}\)
is completely continuous.
Let \(f^{*}\) and \(I_{k}^{*}\) be defined as in Section 3. Similar to the proof of that in Lemmas 3.6-3.9, we have the following results. Here, we only consider the case \(m>1\) and only give the proof of Lemma 5.4.
Lemma 5.4
Assume that (H1)-(H4) hold, \(\alpha(t)\leq t\)
on
J
and let
\(r>0\)
be given. If there exist
\(\varepsilon_{1} >0\)
and
\(\varepsilon_{2} >0\)
such that
\(f^{*}(r)\leq\varepsilon_{1}\phi _{m}(r) \)
and
\(I_{k}^{*}(r)\leq\varepsilon_{2} r\) (\(k=1,2,\ldots,n\)), then
$$ \bigl\Vert {}^{*}T_{\lambda}^{\mu}y\bigr\Vert \leq \rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda \gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon _{2} \bigr)\|y \|_{PC^{1}},\quad y\in\partial\Omega_{r}. $$
(5.3)
Proof
By the definition of \(f^{*}(r)\) and \(I_{k}^{*}\), if \(f^{*}(r)\leq\varepsilon_{1} \phi_{m}(r)\) and \(I_{k}^{*}(r)\leq \varepsilon_{2} r\) (\(k=1,2,\ldots,n\)), then
$$f(t,y)\leq\varepsilon_{1} \phi_{m}(r),\qquad I_{k}(t,y)\leq\varepsilon_{2} r\quad \mbox{for }t\in J \mbox{ and }0\leq y\leq r. $$
Since \(0\leq\alpha(t)\leq t\leq1\) on J, it follows from \(0\leq y(t)\leq r\) on J that \(0\leq y(\alpha(t))\leq r\).
Therefore, we have \(f(t,y(\alpha(t)))\leq\varepsilon_{1} \phi_{m}(r)\) for \(t\in J\), and it follows from (3.7) and (5.2) that
This completes the proof. □
$$\begin{aligned} \bigl\Vert {}^{*}T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq&\rho_{4} \Biggl(\int_{0}^{1} \phi _{m^{*}} \biggl(\lambda\int_{0}^{1}H(s, \tau)\omega(\tau)f\bigl(\tau,y\bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr)\,ds +\mu\sum_{k=1}^{n}I_{k} \bigl(t_{k},y(t_{k})\bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\int _{0}^{1}\gamma e(\tau )\omega(\tau)f\bigl(\tau,y \bigl(\alpha(\tau)\bigr)\bigr) \,d\tau \biggr) +\mu\sum _{k=1}^{n}I_{k}\bigl(t_{k},y(t_{k}) \bigr) \Biggr) \\ \leq&\rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\gamma\int _{0}^{1} e(\tau )\omega(\tau)\varepsilon_{1} \phi_{m}(r) \,d\tau \biggr) +\mu\sum_{k=1}^{n} \varepsilon_{2} r \Biggr) \\ =& \rho_{4} \Biggl(\phi_{m^{*}} \biggl(\lambda\gamma \varepsilon_{1} \phi _{m}(r)\int_{0}^{1} e(\tau)\omega(\tau) \,d\tau \biggr) +\mu\sum_{k=1}^{n} \varepsilon_{2} r \Biggr) \\ \leq&\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \phi _{m}(r)\Vert e\Vert _{q} \|\omega\|_{p} \bigr) +\mu n\varepsilon_{2} r \bigr) \\ =&\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda\gamma \varepsilon_{1} \Vert e\Vert _{q}\|\omega \|_{p} \bigr) +\mu n\varepsilon_{2} r \bigr)\|y \|_{PC^{1}}, \quad \forall y \in\partial\Omega_{r}. \end{aligned}$$
Lemma 5.5
Assume that (H1)-(H4) hold, \(\alpha(t)\leq t\)
on
J
and let
\(l_{1}>0\)
and
\(l_{2}\)
be given. If
\(f(t,y)\geq l_{1}\phi_{m}(y)\)
and
\(I_{k}(t,y)\geq l_{2}y\)
for
\(t\in J\)
and
\(y\in K^{*}\), then
$$\bigl\Vert {}^{*}T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}}\geq\delta\frac{\rho_{1}^{2}}{\rho _{4}} \bigl((l_{1}\lambda\rho \eta)^{m^{*}-1}\beta+\mu nl_{2} \bigr)\|y\|_{PC^{1}}. $$
Let \(\hat{\Omega}_{r}=\{y\in K^{*}:\|y\|_{PC^{1}}< r\}\).
Lemma 5.6
Assume that (H1)-(H4) hold and
\(\alpha (t)\leq t\)
on
J. If
\(y\in\partial\hat{\Omega}_{r}\), \(r>0\), then
where
\(M_{r}\)
and
\(M^{*}\)
are defined in Lemma
3.8.
$$\bigl\Vert {}^{*}T_{\lambda}^{\mu}y\bigr\Vert _{PC^{1}} \leq\rho_{4} \bigl(\phi_{m^{*}} \bigl(\lambda \gamma M_{r}\Vert e\Vert _{q}\|\omega\|_{p} \bigr) +\mu n M^{*} \bigr), $$
Lemma 5.7
Assume that (H1)-(H4) hold and
\(\alpha (t)\leq t\)
on
J. If
\(y\in\partial\hat{\Omega}_{r}\), \(r>0\), then
where
\(\sigma_{r}\)
and
\(\sigma^{*}\)
are defined in Lemma
3.9.
$$\bigl\Vert {}^{*}T_{\lambda}u\bigr\Vert _{PC^{1}} \geq \rho_{1}(\sigma_{r}\lambda\rho\eta )^{m^{*}-1}\beta+\mu \rho_{1}n\sigma^{*}_{r}, $$
Let \(f^{0}\), \(f^{\infty}\), \(f_{0}\), \(f_{\infty}\), \(I^{0}(k)\), \(I^{\infty }(k)\), \(I_{0}(k)\) and \(I_{\infty}(k)\) be defined as in Section 3. Similar to the proof of Theorem 4.1, we have the following results.
Theorem 5.1
Assume that (H1)-(H4) hold and
\(\alpha (t)\leq t\)
on
J. Then:
(a)
If
\(f^{0}=0\)
and
\(I^{0}(k)=0\)
or
\(f^{\infty}=0\)
and
\(I^{\infty }(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has a positive solution for
\(\lambda>\lambda_{0}\)
and
\(\mu >\mu_{0}\).
(b)
If
\(f_{0}=\infty\)
and
\(I_{0}(k)=\infty\)
or
\(f_{\infty}=\infty\)
and
\(I_{\infty}(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu _{0}>0\)
such that problem (1.1) has a positive solution for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(c)
If
\(f^{0}=f^{\infty}=0\)
and
\(I^{0}(k)=I^{\infty}(k)=0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(\lambda> \lambda_{0}\)
and
\(\mu>\mu _{0}\).
(d)
If
\(f_{0}=f_{\infty}=\infty\)
and
\(I_{\infty}(k)=I_{\infty }(k)=\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has at least two positive solutions for
\(0<\lambda <\lambda_{0}\)
and
\(0<\mu<\mu_{0}\).
(e)
If
\(f^{0}<\infty\), \(I^{0}(k)<\infty\), \(f^{\infty}<\infty\)
and
\(I^{\infty}<\infty\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(0<\lambda<\lambda _{0}\)
and
\(0<\mu<\mu_{0}\).
(f)
If
\(f_{0}>0\), \(I_{0}(k)>0\), \(I_{\infty}>0\)
and
\(f_{\infty}>0\), then there exist
\(\lambda_{0}>0\)
and
\(\mu_{0}>0\)
such that problem (1.1) has no positive solution for
\(\lambda> \lambda_{0}\)
and
\(\mu_{0}>0\).
6 Remarks and comments
In this section, we offer some remarks and comments of the associated problem (1.1).
Remark 6.1
Remark 6.2
For example, we consider the following problems:
Here \(\lambda>0\) and \(\mu>0\) are two parameters, \(a, b>0\), \(J=[0,1]\), \(\phi_{m}(s)\) is an m-Laplace operator, i.e., \(\phi_{m}(s)=|s|^{m-2}s\), \(m>1\), \((\phi_{m})^{-1}=\phi_{m^{*}}\), \(\frac{1}{m}+\frac{1}{m^{*}}=1\), \(t_{k}\) (\(k=1,2,\ldots,n\)) (where n is a fixed positive integer) are fixed points with \(0=t_{0}< t_{1}<t_{2}<\cdots <t_{k}<\cdots <t_{n}<t_{n+1}=1\), \(\Delta y' |_{t=t_{k}}=y'(t_{k}^{+})-x'(t_{k}^{-})\), where \(y'(t_{k}^{+})\) and \(y'(t_{k}^{-})\) represent the right-hand limit and the left-hand limit of \(y'(t)\) at \({t=t_{k}}\), respectively.
$$ \left \{ \begin{array}{l} (\phi_{m}(y''(t)))''=\lambda\omega(t)f(t,y(\alpha(t))),\quad t\in J, t\neq t_{k}, k=1,2,\ldots,n, \\ \Delta y'|_{t=t_{k}}=-\mu I_{k}(t_{k},y(t_{k})), \quad k=1,2,\ldots ,n, \\ y(0)=0, \qquad y(1)=\int_{0}^{1}g(t)y(t)\,dt, \\ \phi_{p}(y''(0))=0, \qquad \phi_{p}(y''(1))=\int_{0}^{1}h(t)\phi _{p}(y''(t))\,dt. \end{array} \right . $$
(6.1)
By means of transformation (2.2), we can convert problem (4.1) into
and
$$ \left \{ \begin{array}{l} x''(t)+\lambda\omega(t)f(t,y(\alpha(t)))=0, \quad t\in J, \\ x(0)=0, \qquad x(1)=\int_{0}^{1}h(t)x(t)\,dt \end{array} \right . $$
(6.2)
$$ \left \{ \begin{array}{l} y''(t)=-\phi_{m^{*}}(x(t)), \quad t\in J, t\neq t_{k}, \\ \Delta y'|_{t=t_{k}}=-\mu I_{k}(t_{k}, y(t_{k})), \quad k=1,2,\ldots ,n, \\ y(0)=0, \qquad y(1)=\int_{0}^{1}g(t)y(t)\,dt. \end{array} \right . $$
(6.3)
Using a similar proof to that of Theorem 2.1 and Theorem 2.2, we can obtain the following results. In addition, if we replace ξ, ν by \(\xi^{*}\), \(\nu^{*}\) in (H2), respectively, then we obtain (\(\mathrm{H}^{*}_{2}\)), where
$$\xi^{*}=\int_{0}^{1}sg(s)\,ds, \qquad \nu^{*}=\int_{0}^{1}sh(s)\,ds. $$
Lemma 6.1
If (H1), (\(\mathrm{H}^{*}_{2}\)) and (H4) hold, then problem (6.2) has a unique solution
x
expressed by
where
\(G(t,s)\)
is defined in (2.6).
$$ x(t) =\lambda\int_{0}^{1}H^{*}(t,s)f \bigl(s,y(s)\bigr)\,ds, $$
(6.4)
$$ H^{*}(t,s)=G(t,s)+\frac{t}{1-\int_{0}^{1}sh(s)\,ds}\int_{0}^{1} G(\tau,s)h(\tau)\,d\tau, $$
(6.5)
Lemma 6.2
If (H1), (H3) and (H4) hold, then problem (6.3) has a unique solution
y
given by
where
$$ y(t) =\int_{0}^{1}H_{1}^{*}(t,s) \phi_{m^{*}}\bigl(x(s)\bigr)\,ds +\sum_{k=1}^{n}H_{1}^{*}(t,t_{k})I_{k} \bigl(t_{k},y(t_{k})\bigr), $$
(6.6)
$$ H_{1}^{*}(t,s)=G(t,s)+\frac{t}{1-\int_{0}^{1}sg(s)\,ds}\int _{0}^{1} G(s,\tau)g(\tau)\,d\tau. $$
(6.7)
It is not difficult to prove that \(H^{*}(t,s)\) and \(H_{1}^{*}(t,s)\) have similar properties to those of \(H(t,s)\) and \(H_{1}(t,s)\). However, there does not exist a positive number \(\delta\in(0,1)\) such that
or there does not exist a positive number \(\delta^{*}\in(0,1)\) such that
where \(\zeta\in(0,t_{1})\) and \(\zeta^{*}\in(t_{n},1)\). This implies that we cannot study the existence of positive solutions for problem (6.1) when the deviating arguments are delayed and advanced.
$$G(t,s)\geq\delta G(s,s)\quad \mbox{for }t\in[\zeta,1], s\in J; $$
$$G(t,s)\geq\delta^{*} G(s,s)\quad \mbox{for }t\in\bigl[0, \zeta^{*}\bigr], s\in J, $$
Remark 6.3
There are many functions \(\alpha(t)\) satisfying \(\alpha(t)\geq t\) or \(\alpha(t)\leq t\) on J. For example,
where n is a positive integral number.
$$\begin{aligned}& \mbox{if }\alpha(t)=t^{\frac{1}{n}},\quad \mbox{then }\alpha(t)\geq t \mbox{ on }J; \\& \mbox{if }\alpha(t)=t^{n},\quad \mbox{then }\alpha(t)\leq t \mbox{ on }J, \end{aligned}$$
Acknowledgements
We wish to express our thanks to Prof. Xuemei Zhang, Department of Mathematics and Physics, North China Electric Power University, Beijing, P.R. China, for her kind help, careful reading, and making useful comments on the earlier version of this paper. The authors also thank the project NSFC (11171032) and the improving project of graduate education of Beijing Information Science and Technology University (YJT201416) for their support.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MF checked the proofs and verified the calculation. JQ completed the main study and carried out the results of this article. All the authors read and approved the final manuscript.