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Boundary Value Problems
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The existence of solutions for impulsive p-Laplacian boundary value problems at resonance on the half-line

Boundary Value Problems20152015:39

https://doi.org/10.1186/s13661-015-0299-3

©  Jiang; licensee Springer. 2015

  • Received: 1 October 2014
  • Accepted: 28 January 2015
  • Published:

Abstract

By using the continuous theorem of Ge and Ren and constructing suitable Banach spaces and operators, we investigate the existence of solutions for an impulsive p-Laplacian boundary value problem with integral boundary condition at resonance on the half-line. An example is given to illustrate our main results.

Keywords

  • impulsive
  • p-Laplacian operator
  • boundary value problem
  • integral boundary condition
  • resonance

MSC

  • 34B40

1 Introduction

Boundary value problems on the half-line arise in various applications such as in the study of the unsteady flow of a gas through semi-infinite porous medium, in analyzing the heat transfer in radial flow between circular disks, in the study of plasma physics, in an analysis of the mass transfer on a rotating disk in a non-Newtonian fluid, etc. [1]

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. For some general and recent works on the theory of impulsive differential equations we refer the reader to [2–4]. Impulsive differential equations occur in biology, medicine, mechanics, engineering, chaos theory, etc. [5–9]. Impulsive boundary value problems have been studied by many papers; see [10–15]. For example, in [14], the authors studied the existence of solutions for the problem
$$\left \{ \begin{array}{@{}l} (p(t)u'(t))'=f(t,u(t)),\quad t\in(0,\infty){\backslash} \{t_{1},t_{2},\ldots,t_{n}\},\\ \triangle u'(t_{k})=I_{k}(u(t_{k})),\quad k=1,2,\ldots,n,\\ \alpha u(0)-\beta\lim_{t\rightarrow0^{+}}p(t)u'(t)=0,\\ \gamma\lim_{t\rightarrow \infty}u(t)+\delta\lim_{t\rightarrow\infty}p(t)u'(t)=0. \end{array} \right . $$
In [15], the impulsive boundary value problem on the half-line
$$\left \{ \begin{array}{@{}l} \frac{1}{p(t)}(p(t)x'(t))'=f(t,x_{t}),\quad t\in(0,\infty){\backslash} \{t_{1},t_{2},\ldots,t_{n}\},\\ \triangle x'(t_{k})=I_{k}(x_{t_{k}}),\quad k=1,2,\ldots,m,\\ \lambda x(0)-\beta\lim_{t\rightarrow0^{+}}p(t)x'(t)=a,\\ \gamma x(\infty)+\delta\lim_{t\rightarrow \infty}p(t)x'(t)=b \end{array} \right . $$
was studied.
A boundary value problem is said to be a resonance one if the corresponding homogeneous boundary value problem has a non-trivial solution. The boundary value problems at resonance have been studied by many papers; see [16–22]. In [22], the author gave the existence of solutions for the p-Laplacian boundary value problem at resonance on the half-line
$$\left \{ \begin{array}{@{}l} (\varphi_{p}(u') )'(t)=\psi(t)f (t,u(t),u'(t) ),\quad t\in [0,+\infty),\\ u'(+\infty)=0,\qquad u(0)=\int_{0}^{+\infty}h(t)u(t)\,dt, \end{array} \right . $$
where \(\varphi_{p}(s)=|s|^{p-2}s\), \(p>1\).
As far as we know, the impulsive p-Laplacian boundary value problems at resonance on the half-line have not been investigated. In this paper, we will discuss the existence of solutions for the problem
$$ \left \{ \begin{array}{@{}l} (\varphi_{p}(u'(t)))'+f(t,u(t),u'(t))=0,\quad t\in[0,\infty){\backslash} \{t_{1},t_{2},\ldots,t_{k}\},\\ \triangle\varphi_{p}(u'(t_{i}))=I_{i}(u(t_{i}),u'(t_{i})),\quad i=1,2,\ldots,k,\\ u(0)=0, \qquad\varphi_{p}(u'(+\infty))=\int_{0}^{+\infty}h(t)\varphi_{p}(u'(t))\,dt, \end{array} \right . $$
(1.1)
where \(0< t_{1}<t_{2}<\cdots<t_{k}<+\infty\), \(\triangle \varphi_{p}(u'(t_{i}))=\varphi_{p}(u'(t_{i}+0))-\varphi_{p}(u'(t_{i}-0))\).
In this paper, we will always suppose that the following conditions hold.
(H1): 

\(h(t)\geq0\), \(t\in[0,+\infty)\), \(\int_{0}^{+\infty}h(t)\,dt=1\), \(f:[0,+\infty)\times\mathbb{R}^{2}\rightarrow\mathbb{R}\), and \(I_{i}: \mathbb{R}^{2}\rightarrow\mathbb{R}\), \(i=1,2,\ldots,k\) are continuous.

(H2): 

For any constant \(r>0\), there exist a function \(h_{r}\in L[0,+\infty)\) and a constant \(M_{r}>0\), such that \(|f(t,(1+t)u,v)|\leq h_{r}(t)\), \(t\in[0,+\infty)\), \(|u|< r\), \(|v|< r\), \(|I_{i}(u,v)|\leq M_{r}\), \(i=1,2,\ldots,k\), \(|u|\leq r(1+t_{k})\), \(|v|\leq r\).

2 Preliminaries

For convenience, we introduce some notations and a theorem. For more details see [23].

Definition 2.1

[23]

Let X and Y be two Banach spaces with norms \(\|\cdot\|_{X}\), \(\|\cdot\|_{Y}\), respectively. A continuous operator \(M: X\cap \operatorname{dom}M\rightarrow Y\) is said to be quasi-linear if
  1. (i)

    \(\operatorname{Im}M:=M(X \cap \operatorname{dom}M)\) is a closed subset of Y,

     
  2. (ii)

    \(\operatorname{Ker}M:=\{x\in X \cap \operatorname{dom} M: Mx=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\), \(n<\infty\),

     
where domM denote the domain of the operator M.

Let \(X_{1}=\operatorname{Ker} M\) and \(X_{2}\) be the complement space of \(X_{1}\) in X, then \(X=X_{1}\oplus X_{2}\). On the other hand, suppose \(Y_{1}\) is a subspace of Y and that \(Y_{2}\) is the complement of \(Y_{1}\) in Y, i.e. \(Y=Y_{1}\oplus Y_{2}\). Let \(P:X\rightarrow X_{1}\) and \(Q:Y\rightarrow Y_{1}\) be two projectors and \(\Omega\subset X\) an open and bounded set with the origin \(\theta\in\Omega\).

Definition 2.2

[23]

Suppose that \(N_{\lambda}:\overline{\Omega}\rightarrow Y\), \(\lambda\in[0,1]\) is a continuous operator. Denote \(N_{1}\) by N. Let \(\Sigma{_{\lambda}}=\{x\in\overline{\Omega}:Mx=N_{\lambda}x\}\). \(N_{\lambda}\) is said to be M-compact in \(\overline{\Omega}\) if there exist a vector subspace \(Y_{1}\) of Y satisfying \(\operatorname{dim} Y_{1}=\operatorname{dim} X_{1}\) and an operator \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) being continuous and compact such that for \(\lambda\in[0,1]\),
  1. (a)

    \((I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname{Im} M\subset(I-Q)Y\),

     
  2. (b)

    \(QN_{\lambda} x=\theta, \lambda\in(0,1)\Leftrightarrow QNx=\theta\),

     
  3. (c)

    \(R(\cdot,0)\) is the zero operator and \(R(\cdot,\lambda)|_{\Sigma{_{\lambda}}}=(I-P)|_{\Sigma{_{\lambda}}}\),

     
  4. (d)

    \(M[P+R(\cdot,\lambda)]=(I-Q)N_{\lambda}\).

     

Theorem 2.1

[23]

Let X and Y be two Banach spaces with the norms \(\|\cdot\|_{X}\), \(\|\cdot\|_{Y}\), respectively, and \(\Omega\subset X\) an open and bounded nonempty set. Suppose that
$$M:X\cap \operatorname{dom}M\rightarrow Y $$
is a quasi-linear operator and \(N_{\lambda}:\overline{\Omega}\rightarrow Y\), \(\lambda\in[0,1]\) is M-compact. In addition, if the following conditions hold:
(C1): 

\(Mx\neq N_{\lambda}x\), \(\forall x\in\partial\Omega\cap \operatorname{dom} M\), \(\lambda\in(0,1)\),

(C2): 

\(\operatorname{deg}\{JQN, \Omega\cap \operatorname{Ker}M, 0\}\neq0\),

then the abstract equation \(Mx=Nx\) has at least one solution in \(\operatorname{dom} M\cap\overline{\Omega}\), where \(N=N_{1}\), \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}M\) is a homeomorphism with \(J(\theta)=\theta\).

3 Main results

In the following, we will always suppose that q satisfies \(1/p+1/q=1\).

Let \(\mathbb{R}^{+}=[0,+\infty)\), \(J'=\mathbb{R}^{+}{\backslash} \{t_{1},t_{2},\ldots,t_{k}\}\), \(Y=L(\mathbb{R}^{+})\) with norm \(\|y\|_{1}=\int_{0}^{+\infty}|y(t)|\,dt\),
$$\begin{aligned}& \begin{aligned}[b] PC^{1}\bigl(\mathbb{R}^{+}\bigr)={}& \bigl\{ u: u\in C^{1} \bigl(J'\bigr), u'(t_{i}-0), u'(t_{i}+0) \mbox{ exist and}\\ &{} u'(t_{i}-0)=u'(t_{i}), i=1,2,\ldots,k \bigr\} , \end{aligned} \\& X= \biggl\{ u: u(0)=0, u\in C\bigl(\mathbb{R}^{+}\bigr)\cap PC^{1}\bigl( \mathbb{R}^{+}\bigr), \sup_{t\in\mathbb{R}^{+}}\frac{|u(t)|}{1+t}< +\infty, \lim _{t\rightarrow+\infty}u'(t) \mbox{ exists} \biggr\} \end{aligned}$$
with norm \(\|u\|=\max \{\Vert \frac{u}{1+t}\Vert _{\infty}, \| u'\|_{\infty}\}\), where \(\|u\|_{\infty}=\sup_{t\in\mathbb{R}^{+}}|u(t)|\).

Let \(Z=Y\times\mathbb{R}^{k}\), with norm \(\|(y,c_{1},c_{2},\ldots,c_{k})\|=\max\{\|y\|_{1},|c_{1}|,|c_{2}|,\ldots,|c_{k}|\}\). Then \((X,\|\cdot\|)\) and \((Z,\|\cdot\|)\) are Banach spaces.

Define the operators \(M:X\cap \operatorname{dom}M\rightarrow Z\), \(N_{\lambda}:X\rightarrow Z\) as follows:
$$Mu=\left [ \begin{array}{@{}c@{\quad}c@{}} (\varphi_{p}(u') )'(t)\\ \triangle\varphi_{p}(u'(t_{1}))\\ \cdots\\ \triangle\varphi_{p}(u'(t_{k})) \end{array} \right ], \qquad N_{\lambda}u=\left [ \begin{array}{@{}c@{\quad}c@{}} -\lambda f (t,u(t),u'(t) )\\ \lambda I_{1}(u(t_{1}),u'(t_{1}))\\ \cdots\\ \lambda I_{k}(u(t_{k}),u'(t_{k})) \end{array} \right ], $$
where \(\operatorname{dom}M= \{u\in X: (\varphi_{p}(u'))'\in Y, \varphi_{p}(u'(+\infty))=\int_{0}^{+\infty}h(t)\varphi_{p}(u'(t))\,dt \}\).

It is clear that \(u\in \operatorname{dom}M\) is a solution of the problem (1.1) if it satisfies \(Mu=Nu\), where \(N=N_{1}\). For convenience, let \((a,b)^{T}:=\bigl [ {\scriptsize\begin{matrix}a\cr b \end{matrix}} \bigr ]\), denote \(J_{0}=[0,t_{1}]\), \(J_{i}=(t_{i},t_{i+1}]\), \(i=1,2,\ldots,k-1\), \(J_{k}=(t_{k},+\infty)\).

Lemma 3.1

M is a quasi-linear operator.

Proof

It is easy to get \(\operatorname{Ker}M=\{at \mid a\in \mathbb{R}\}:=X_{1}\).

For \(u\in X\cap \operatorname{dom}M\), if \(Mu=(y,c_{1},c_{2},\ldots,c_{k})^{T}\), then
$$\bigl(\varphi_{p}\bigl(u'\bigr)\bigr)'(t)=y(t), \qquad \triangle\varphi_{p}\bigl(u'(t_{i}) \bigr)=c_{i},\quad i=1,2,\ldots,k. $$
For \(t\in J_{0}\), we get
$$\varphi_{p}\bigl(u'(t)\bigr)=\int_{0}^{t}y(s) \,ds+a . $$
For \(t\in J_{1}\), considering \(\triangle\varphi_{p}(u'(t_{1}))=c_{1}\), we get
$$\varphi_{p}\bigl(u'(t)\bigr)=\int_{0}^{t}y(s) \,ds+a+c_{1} . $$
For \(t\in J_{i}\), \(i=2,3,\ldots,k\), considering \(\triangle\varphi_{p}(u'(t_{i}))=c_{i}\), we get
$$\varphi_{p}\bigl(u'(t)\bigr)=\int_{0}^{t}y(s) \,ds+a+\sum_{t_{i}< t}c_{i} . $$
By \(\varphi_{p}(u'(+\infty))=\int_{0}^{+\infty}h(t)\varphi_{p}(u'(t))\,dt\) and \(\int_{0}^{+\infty}h(t)\,dt=1\), we find that \((y,c_{1},c_{2},\ldots,c_{k})^{T}\) satisfies
$$ \int_{0}^{+\infty}h(t)\int_{t}^{+\infty}y(s) \,ds+\int_{0}^{t_{k}}\sum _{t_{i}\geq t}c_{i}h(t)\,dt=0. $$
(3.1)
On the other hand, if \((y,c_{1},c_{2},\ldots,c_{k})^{T}\) satisfies (3.1), take
$$u(t)=\int_{0}^{t}\varphi_{q} \biggl( \int_{0}^{s}y(r)\,dr+\sum _{t_{i}< s}c_{i} \biggr)\,ds. $$
By a simple calculation, we get \(u\in X\cap \operatorname{dom}M\) and \(Mu=(y,c_{1},c_{2},\ldots,c_{k})^{T}\). Thus
$$\operatorname{Im}M= \bigl\{ (y,c_{1},c_{2}, \ldots,c_{k})^{T} \mid y\in Y, c_{1},c_{2}, \ldots,c_{k} \mbox{ satisfies }(3.1) \bigr\} . $$
Obviously, \(\operatorname{Im}M\subset Z\) is closed. So, M is quasi-linear. The proof is completed. □
Take projectors \(P: X\rightarrow X_{1}\) and \(Q: Z\rightarrow Z_{1}\) as follows:
$$\begin{aligned}& (Pu) (t)=u'(+\infty)t, \\& Q(y,c_{1},c_{2},\ldots,c_{k})^{T}= \biggl(\frac{\int_{0}^{+\infty}h(t)\int_{t}^{+\infty}y(s)\,ds\,dt+\int_{0}^{t_{k}}\sum_{t_{i}\geq t}c_{i}h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-t},0,\ldots,0 \biggr)^{T}, \end{aligned}$$
where \(Z_{1}=\{(ce^{-t},0,\ldots,0)^{T} \mid c\in\mathbb{R}\}\). Obviously, \(QZ=Z_{1}\) and \(\operatorname{dim} Z_{1}=\operatorname{dim} X_{1}\).
Define an operator \(R:X\times[0,1]\rightarrow X_{2}\) as
$$\begin{aligned} R(u,\lambda) (t) ={}& \int_{0}^{t} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &{}\! - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r}\! \biggr]\,dr \\ &{}\!+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda\sum _{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds-u'(+\infty)t,\quad t\in J_{i}, i=0,1, \ldots,k, \end{aligned}$$
where \(X_{1}\oplus X_{2}=X\).

By [1, 24], we get the following lemma.

Lemma 3.2

Assume that \(V\subset X\) is bounded. V is compact if \(\{ \frac{u(t)}{1+t}:u\in V \}\) and \(\{u'(t):u\in V\}\) are both equicontinuous on \(J_{i}\), \(i=0,1,\ldots,k-1\), and \(J_{T}=(t_{k},T]\), for any given \(T>t_{k}\), respectively, and equiconvergent at infinity.

Lemma 3.3

\(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) is continuous and compact, where \(\Omega\subset X\) is an open bounded set.

Proof

By (H1), (H2), the continuity of \(\varphi_{q}\) and Lebesgue’s dominated convergence theorem, we find that R is continuous and \(\{R(u,\lambda)\mid u\in \overline{\Omega}, \lambda\in[0,1]\}\) is bounded. We will prove that \(R(\overline{\Omega}\times[0,1])\) is compact.

Since \(\Omega\subset X\) is bounded, there exists a constant \(r>0\) such that \(\|u\|\leq r\), \(u\in\overline{\Omega}\). It follows from (H2) that there exist a function \(h_{r}\in L(\mathbb{R}^{+})\) and a constant \(M_{r}>0\) such that \(|f(t,u(t),u'(t))|\leq h_{r}(t)\), \(|I_{i}(u(t_{i}),u'(t_{i}))|\leq M_{r}\), \(i=1,2,\ldots,k\), \(t\in \mathbb{R}^{+}\), \(u\in\overline{\Omega}\). For any given \(T>t_{k}\), \(x_{1}, x_{2}\in J_{i}\), \(i=0,1,\ldots,k-1,T\), \(x_{1}< x_{2}\), we have
$$\begin{aligned} &\biggl\vert \frac{R(u,\lambda)(x_{2})}{1+x_{2}}-\frac{R(u,\lambda )(x_{1})}{1+x_{1}}\biggr\vert \\ &\quad\leq \biggl| \frac{1}{1+x_{2}} \int_{0}^{x_{2}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(x,u(x),u'(x)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-x} \biggr]\,dx \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds \\ &\qquad{}- \frac{1}{1+x_{1}} \int_{0}^{x_{1}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(x,u(x),u'(x)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-x} \biggr]\,dx \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds \biggr| + \biggl\vert \frac{x_{2}}{1+x_{2}}-\frac{x_{1}}{1+x_{1}}\biggr\vert \bigl|u'(+\infty)\bigr| \\ &\quad\leq \biggl\vert \frac{1}{1+x_{2}}-\frac{1}{1+x_{1}}\biggr\vert T \varphi_{q} \biggl(\int_{0}^{+\infty} \biggl[h_{r}(x) +\frac{\int_{0}^{+\infty}h_{r}(t)\,dt+\sum_{i=1}^{k}|I_{i}(u(t_{i}),u'(t_{i}))|}{\int_{0}^{+\infty }h(t)e^{-t}\,dt}e^{-x} \biggr]\,dx\\ &\qquad{}+ \varphi_{p}(r)+kM_{r} \biggr) \\ &\qquad{}+ \frac{x_{2}-x_{1}}{1+x_{2}}\varphi_{q} \biggl(\int _{0}^{+\infty} \biggl[h_{r}(x) + \frac{\int_{0}^{+\infty}h_{r}(t)\,dt+\sum_{i=1}^{k}|I_{i}(u(t_{i}),u'(t_{i}))|}{\int_{0}^{+\infty }h(t)e^{-t}\,dt}e^{-x} \biggr]\,dx\\ &\qquad{}+\varphi_{p}(r)+kM_{r} \biggr) + \biggl\vert \frac{x_{2}}{1+x_{2}}-\frac{x_{1}}{1+x_{1}}\biggr\vert r \\ &\quad\leq \biggl[\biggl\vert \frac{1}{1+x_{2}}-\frac{1}{1+x_{1}}\biggr\vert T+(x_{2}-x_{1}) \biggr]\varphi_{q} \biggl( \|h_{r}\|_{1} +\frac{\|h_{r}\|_{1}+kM_{r}}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}+\varphi _{p}(r)+kM_{r} \biggr)\\ &\qquad{}+\biggl\vert \frac{x_{2}}{1+x_{2}}- \frac{x_{1}}{1+x_{1}}\biggr\vert r. \end{aligned}$$
Since t, \(\frac{1}{1+t}\), and \(\frac{t}{1+t}\) are equicontinuous on \(J_{i}\), \(i=1,2,\ldots,k-1,T\), we find that \(\{\frac{R(u,\lambda)(t)}{1+t}, u\in \overline{\Omega}, \lambda\in[0,1] \}\) are equicontinuous on \(J_{i}\), \(i=1,2,\ldots,k-1,T\). We have
$$\begin{aligned} &\bigl|R(u,\lambda)'(x_{1})-R(u,\lambda)'(x_{2})\bigr| \\ &\quad= \biggl|\varphi_{q} \biggl(\int_{x_{1}}^{+\infty} \lambda \biggl[f\bigl(s,u(s),u'(s)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds \\ &\qquad{} + \varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq x_{1}}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr) \\ &\qquad{} - \varphi_{q} \biggl(\int_{x_{2}}^{+\infty} \lambda \biggl[f\bigl(s,u(s),u'(s)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds \\ &\qquad{} + \varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq x_{2}}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr) \biggr|. \end{aligned}$$
For \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\), define
$$\begin{aligned} F(u,\lambda) (t)={}& \int_{t}^{+\infty}\lambda \biggl[f\bigl(s,u(s),u'(s)\bigr) \\ &{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s}\! \biggr]\,ds \\ &{} +\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda\sum _{t_{j}\geq t}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr). \end{aligned}$$
Obviously,
$$\begin{aligned}& \bigl|F(u,\lambda) (t)\bigr|\leq \|h_{r}\|_{1} +\frac{\|h_{r}\|_{1}+kM_{r}}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}+ \varphi _{p}(r)+kM_{r}:=K, \quad u\in \overline{\Omega}, \lambda\in[0,1], t\in\mathbb{R}^{+}, \\& \bigl|F(u,\lambda) (x_{1})-F(u,\lambda) (x_{2})\bigr| \\& \quad= \biggl|\int_{x_{1}}^{x_{2}}\lambda \biggl[f \bigl(s,u(s),u'(s)\bigr) \\& \qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds \biggr| \\& \quad\leq \int_{x_{1}}^{x_{2}}h_{r}(t)\,dt+ \frac{\|h_{r}\|_{1}+kM_{r}}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}\bigl(e^{-x_{1}}-e^{-x_{2}}\bigr),\quad u\in \overline{ \Omega}, \lambda\in[0,1]. \end{aligned}$$
It follows from the absolute continuity of integral and the equicontinuity of \(e^{-t}\) that \(\{F(u,\lambda)(t), u\in\overline{\Omega}, \lambda\in[0,1]\}\) are equicontinuous on \(J_{i}\), \(i=1,2,\ldots,k-1,T\). By the uniform continuity of \(\varphi_{q}(t)\) in \([-K,K]\), we find that \(\{R(u,\lambda)'(t), u\in \overline{\Omega}, \lambda\in[0,1]\}\) are equicontinuous on \(J_{i}\), \(i=1,2,\ldots,k-1,T\).
For any \(u\in \overline{\Omega}\), \(\lambda\in[0,1]\), since
$$\begin{aligned} & \biggl| \int_{t}^{+\infty}\lambda \biggl[f \bigl(s,u(s),u'(s)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds \biggr| \\ &\quad\leq \int_{t}^{+\infty}h_{r}(s)+ \frac{\|h_{r}\|_{1}+kM_{r}}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s}\,ds\rightarrow 0\quad (t\rightarrow\infty) \end{aligned}$$
and \(\varphi_{q}(u)\) is uniform continuous on \([-K-\varphi_{p}(r),K+\varphi_{p}(r)]\), for any \(\varepsilon>0\), there exists a constant \(T_{1}>t_{k}\) such that
$$\begin{aligned} & \biggl| \varphi_{q} \biggl(\int_{s}^{+\infty} \lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr\\ &\qquad{}+\varphi_{p} \bigl(u'(+\infty)\bigr) \biggr) -u'(+\infty) \biggr|\leq \frac{\varepsilon}{4},\quad s>T_{1}, u\in \overline{\Omega}, \lambda\in[0,1]. \end{aligned}$$
Obviously, there exists a constant \(T>T_{1}\) such that, for any \(t>T\),
$$\frac{1}{1+t}\bigl(\varphi_{q}(K)+r\bigr)T_{1}< \frac{\varepsilon}{4}. $$
Thus, for any \(x_{1}, x_{2}>T\), we have
$$\begin{aligned} &\biggl\vert \frac{R(u,\lambda)(x_{1})}{1+x_{1}}-\frac{R(u,\lambda )(x_{2})}{1+x_{2}}\biggr\vert \\ &\quad= \biggl| \frac{1}{1+x_{1}} \biggl\{ \int_{0}^{x_{1}} \varphi _{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds-u'(+\infty)x_{1} \biggr\} \\ &\qquad{}- \frac{1}{1+x_{2}} \biggl\{ \int_{0}^{x_{2}}\varphi _{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds-u'(+\infty)x_{2} \biggr\} \biggr| \\ &\quad\leq \biggl| \frac{1}{1+x_{1}} \biggl\{ \int_{0}^{T_{1}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds-u'(+\infty)T_{1} \biggr\} \biggr| \\ &\qquad{}+ \biggl| \frac{1}{1+x_{1}} \biggl\{ \int_{T_{1}}^{x_{1}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{}+\varphi_{p}\bigl(u'(+\infty)\bigr) \biggr) \,ds-u'(+\infty ) (x_{1}-T_{1}) \biggr\} \biggr| \\ &\qquad{}+ \biggl| \frac{1}{1+x_{2}} \biggl\{ \int_{0}^{T_{1}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{} +\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq s}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr) \biggr)\,ds-u'(+\infty)T_{1} \biggr\} \biggr| \\ &\qquad{}+ \biggl| \frac{1}{1+x_{2}} \biggl\{ \int_{T_{1}}^{x_{2}} \varphi_{q} \biggl(\int_{s}^{+\infty}\lambda \biggl[f\bigl(r,u(r),u'(r)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{} +\varphi_{p}\bigl(u'(+\infty)\bigr) \biggr) \,ds-u'(+\infty ) (x_{2}-T_{1}) \biggr\} \biggr| \\ &\quad\leq \frac{1}{1+x_{1}}\bigl(\varphi_{q}(K)+r\bigr)T_{1}+ \frac {x_{1}-T_{1}}{1+x_{1}}\frac{\varepsilon}{4} +\frac{1}{1+x_{2}}\bigl(\varphi_{q}(K)+r \bigr)T_{1}+\frac{x_{2}-T_{1}}{1+x_{2}}\frac {\varepsilon}{4}< \varepsilon, \\ &\bigl|R(u,\lambda)'(x_{1})-R(u,\lambda)'(x_{2})\bigr| \\ &\quad\leq \biggl|\varphi_{q} \biggl(\int_{x_{1}}^{+\infty} \lambda \biggl[f\bigl(s,u(s),u'(s)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds\\ &\qquad{}+\varphi_{p} \bigl(u'(+\infty)\bigr) \biggr) -u'(+\infty) \biggr| + \biggl|\varphi_{q} \biggl(\int_{x_{2}}^{+\infty} \lambda \biggl[f\bigl(s,u(s),u'(s)\bigr) \\ &\qquad{} - \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-s} \biggr]\,ds\\ &\qquad{}+\varphi_{p} \bigl(u'(+\infty)\bigr) \biggr)-u'(+\infty) \biggr| < \frac{\varepsilon}{4}+ \frac{\varepsilon }{4}< \varepsilon. \end{aligned}$$

By Lemma 3.2, we find that \(\{R(u,\lambda)\mid u\in \overline{\Omega}, \lambda\in[0,1]\}\) is compact. The proof is completed. □

Lemma 3.4

Assume that \(\Omega\subset X\) is an open bounded set. Then \(N_{\lambda}\) is M-compact in \(\overline{\Omega}\).

Proof

By (H1), we get \(N_{\lambda}:\overline{\Omega}\rightarrow Y\), \(\lambda\in[0,1]\) is continuous. It is clear that \(\operatorname{Im}P=\operatorname{Ker}M\), \(QN_{\lambda} x=\theta, \lambda\in(0,1)\Leftrightarrow QNx=\theta\), i.e. Definition 2.2(b) holds.

For \(u\in\overline{\Omega}\), it follows from \(Q(I-Q)N_{\lambda}u=\theta\) that \((I-Q)N_{\lambda}u\) satisfies (3.1). So, \((I-Q)N_{\lambda}u\in \operatorname{Im}M\), i.e. \((I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname{Im}M\). Furthermore, by \(\operatorname{Im}M= \operatorname{Ker}Q\) and \(z=Qz+(I-Q)z\) we find that \(z\in \operatorname{Im}M\) implies \(z=(I-Q)z\in(I-Q)Z\), i.e. \(\operatorname{Im}M\subset(I-Q)Z\). Thus, \((I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname{Im}M\subset(I-Q)Z\), i.e. Definition 2.2(a) holds.

Obviously, \(R(\cdot,0)=0\). For \(u\in \Sigma{_{\lambda}}=\{u\in\overline{\Omega}\cap \operatorname{dom}M:Mu=N_{\lambda}u\}\), we get \(QN_{\lambda}u=\theta\) and
$$\varphi_{p}\bigl(u'(t)\bigr)= \int_{t}^{+\infty} \lambda f\bigl(s,u(s),u'(s)\bigr)\,ds+\varphi_{p} \bigl(u'(+\infty)\bigr)-\lambda\sum_{t_{i}\geq t}I_{i} \bigl(u(t_{i}),u'(t_{i})\bigr). $$
So, we have
$$R(u,\lambda)= \int_{0}^{t}\varphi_{q} \bigl(\varphi_{p}\bigl(u'(s)\bigr) \bigr)\,ds -u'(+\infty)t =(I-P)u, $$
i.e. Definition 2.2(c) holds.
For \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\), \(t\in J_{i}\), \(i=0,1,2,\ldots,k\), we have
$$\begin{aligned} &\bigl(\varphi_{p}\bigl(Pu+R(u,\lambda)\bigr)' \bigr)'(t) \\ &\quad=-\lambda f\bigl(t,u(t),u'(t)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} -\lambda f(s,u(s),u'(s))\,ds\,dt+\lambda\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-t} \end{aligned}$$
and
$$\begin{aligned} &\varphi_{p}\bigl(\bigl(Pu+R(u,\lambda)\bigr)'(t)\bigr) \\ &\quad=\int_{t}^{+\infty}\lambda \biggl[f \bigl(r,u(r),u'(r)\bigr) \\ &\qquad{}- \frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,u(s),u'(s))\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(u(t_{i}),u'(t_{i}))h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}e^{-r} \biggr]\,dr \\ &\qquad{} +\varphi_{p}\bigl(u'(+\infty)\bigr)-\lambda \sum_{t_{j}\geq t}I_{j}\bigl(u(t_{j}),u'(t_{j}) \bigr). \end{aligned}$$
By a simple calculation, we can get
$$M\bigl[Pu+R(u,\lambda)\bigr]=(I-Q)N_{\lambda}u. $$
So, Definition 2.2(d) holds. These, together with Lemma 3.3, mean that \(N_{\lambda}\) is M-compact in \(\overline{\Omega}\). The proof is completed. □

Theorem 3.1

Assume that (H1), (H2), and the following conditions hold:
(H3): 
There exist nonnegative functions \(a(t)\), \(b(t)\), \(c(t)\), and nonnegative constants \(d_{i}\), \(g_{i}\), \(e_{i}\), \(i=1,2,\ldots,k\) with \((1+t)^{p-1}a(t), b(t), c(t)\in Y\), and \(\|a(t)(1+t)^{p-1}\|_{1}+\| b\|_{1}+\sum_{i=1}^{k} [d_{i}(1+t_{i})^{p-1}+g_{i} ]<1\) such that
$$\begin{aligned}[b] &\bigl|f(t,x,y)\bigr|\leq a(t)\bigl|\varphi_{p}(x)\bigr|+b(t)\bigl|\varphi_{p}(y)\bigr|+c(t), \quad\textit{a.e. } t\in[0,+\infty), x,y\in\mathbb{R}, \\ &\bigl|I_{i}(x,y)\bigr|\leq d_{i}\bigl|\varphi_{p}(x)\bigr|+g_{i}\bigl| \varphi_{p}(y)\bigr|+e_{i},\quad i=1,2,\ldots ,k, x,y\in\mathbb{R}. \end{aligned} $$
(H4): 
There exists a constant \(e_{0}>0\) such that if \(\inf_{t\in\mathbb{R}^{+}}|u' (t)|>e_{0}\), then one of the following inequalities holds:
$$\begin{aligned}& (1)\quad u' (t) \int_{0}^{+\infty}h(t) \biggl(\int_{t}^{+\infty} f\bigl(s,u(s),u'(s) \bigr)\,ds-\sum_{t_{i}\geq t}I_{i} \bigl(u(t_{i}),u'(t_{i})\bigr) \biggr)\,dt>0; \\& (2)\quad u' (t) \int_{0}^{+\infty}h(t) \biggl(\int_{t}^{+\infty} f\bigl(s,u(s),u'(s) \bigr)\,ds-\sum_{t_{i}\geq t}I_{i} \bigl(u(t_{i}),u'(t_{i})\bigr) \biggr)\,dt< 0, \end{aligned}$$
where \(t\in[0,+\infty)\). Then boundary value problem (1.1) has at least one solution.

In order to prove Theorem 3.1, we show two lemmas.

Lemma 3.5

Suppose that (H1)-(H4) hold. Then the set
$$\Omega_{1}=\bigl\{ u\in \operatorname{dom}M \mid Mu=N_{\lambda}u, \lambda\in(0,1)\bigr\} $$
is bounded in X.

Proof

For \(u\in\Omega_{1}\), we have \(QN_{\lambda}u=0\), i.e.
$$\begin{aligned} &\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f\bigl(s,u(s),u'(s)\bigr)\,ds\,dt-\int_{0}^{t_{k}} \sum_{t_{i}\geq t}I_{i}\bigl(u(t_{i}),u'(t_{i}) \bigr)h(t)\,dt \\ &\quad= \int_{0}^{+\infty}h(t) \biggl[ \int _{t}^{+\infty} f\bigl(s,u(s),u'(s)\bigr) \,ds-\sum_{t_{i}\geq t}I_{i}\bigl(u(t_{i}),u'(t_{i}) \bigr) \biggr]\,dt=0. \end{aligned}$$
By (H4), there exists a constant \(t_{0}\in\mathbb{R}^{+}\) such that \(|u'(t_{0})|\leq e_{0}\). Assume \(t_{0}\in J_{m}\), \(m=0,1,\ldots,k\). It follows from \(Mu=N_{\lambda}u\) that
$$ \varphi_{p}\bigl(u'(t)\bigr)=\left \{ \begin{array}{@{}l} \int_{t}^{t_{0}}\lambda f(s,u(s),u'(s))\,ds+ \varphi_{p}(u'(t_{0}))-\lambda\sum_{j=i+1}^{m}I_{j}(u(t_{j}),u'(t_{j})),\\ \quad t\in J_{i}, i=0,1,\ldots,m-1,\\ \int_{t}^{t_{0}}\lambda f(s,u(s),u'(s))\,ds+ \varphi_{p}(u'(t_{0})),\quad t\in J_{m},\\ \int_{t}^{t_{0}}\lambda f(s,u(s),u'(s))\,ds+ \varphi_{p}(u'(t_{0}))+\lambda\sum_{j=m+1}^{i}I_{j}(u(t_{j}),u'(t_{j})),\\ \quad t\in J_{i}, i=m+1,m+2,\ldots,k. \end{array} \right . $$
(3.2)
Since \(u(t)=\int_{0}^{t}u'(s)\,ds\),
$$ \frac{|u(t)|}{1+t}\leq\bigl\| u'\bigr\| _{\infty},\quad t\in[0,+ \infty). $$
(3.3)
By (3.2), (H3), and (3.3), we obtain
$$\begin{aligned} \bigl|\varphi_{p}\bigl(u'(t)\bigr)\bigr| \leq{}& \int _{0}^{+\infty} \bigl[a(t)\bigl|\varphi_{p} \bigl(u(t)\bigr)\bigr|+b(t)\bigl|\varphi _{p}\bigl(u'(t) \bigr)\bigr|+c(t) \bigr]\,dt+\varphi_{p}(e_{0}) \\ &{} +\sum_{i=1}^{k} \bigl(d_{i}\bigl| \varphi _{p}\bigl(u(t_{i})\bigr)\bigr|+g_{i}\bigl| \varphi_{p}\bigl(u'(t_{i})\bigr)\bigr|+e_{i} \bigr) \\ \leq{}& \Biggl(\bigl\| a(t) (1+t)^{p-1}\bigr\| _{1}+\sum _{i=1}^{k}d_{i}(1+t_{i})^{p-1} \Biggr)\varphi_{p} \biggl(\biggl\Vert \frac{u}{1+t}\biggr\Vert _{\infty} \biggr) \\ &{} + \Biggl(\| b\|_{1}+\sum_{i=1}^{k}g_{i} \Biggr)\varphi_{p}\bigl(\bigl\| u'\bigr\| _{\infty}\bigr)+\| c \| _{1}+\varphi_{p}(e_{0})+\sum _{i=1}^{k}e_{i} \\ \leq{}& \Biggl(\bigl\| a(t) (1+t)^{p-1}\bigr\| _{1}+\| b\|_{1}+ \sum_{i=1}^{k} \bigl[d_{i}(1+t_{i})^{p-1}+g_{i} \bigr] \Biggr)\varphi_{p} \bigl( \bigl\| u'\bigr\| _{\infty}\bigr) \\ &{} +\| c\|_{1}+\varphi_{p}(e_{0})+\sum _{i=1}^{k}e_{i}. \end{aligned}$$
Thus
$$\bigl\| u'\bigr\| _{\infty}\leq \varphi_{q} \biggl( \frac{\| c\|_{1}+ \varphi_{p}(e_{0})+\sum_{i=1}^{k}e_{i}}{ 1- (\|a(t)(1+t)^{p-1}\|_{1}+\| b\|_{1}+\sum_{i=1}^{k} [d_{i}(1+t_{i})^{p-1}+g_{i} ] )} \biggr). $$
This, together with (3.3), means that \(\Omega_{1}\) is bounded in X. □

Lemma 3.6

Assume that (H1), (H2), and (H4) hold. Then
$$\Omega_{2}=\{u\in \operatorname{Ker}M \mid QN u=0\} $$
is bounded in X, where \(N=N_{1}\).

Proof

For \(u\in\Omega_{2}\), we have \(u=at\), \(a\in\mathbb{R}\), and \(Q(N u)=0\), i.e.
$$\begin{aligned} &\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,as,a)\,ds\,dt-\int_{0}^{t_{k}}\sum _{t_{i}\geq t}I_{i}(at_{i},a)h(t)\,dt \\ &\quad=\int_{0}^{+\infty}h(t) \biggl[\int _{t}^{+\infty} f(s,as,a)\,ds-\sum _{t_{i}\geq t}I_{i}(at_{i},a) \biggr]\,dt=0. \end{aligned}$$
By (H4), we get \(\|u\|=|a|=|u'(t)|\leq e_{0}\). So, \(\Omega_{2}\) is bounded. The proof is completed. □

Proof of Theorem 3.1

Let \(\Omega=\{u\in X \mid \|u\|< r\}\), where \(r>e_{0}\) is large enough such that \(\Omega\supset \overline{\Omega}_{1}\cup\overline{\Omega}_{2}\).

By Lemmas 3.5 and 3.6, we have \(Mu\neq N_{\lambda}u\), \(u\in \operatorname{dom}M\cap\partial\Omega\), and \(QN u\neq0\), \(u\in \operatorname{Ker}M\cap\partial\Omega\).

Let \(H(u,\delta)=\rho\delta u+(1-\delta)JQNu\), \(\delta\in[0,1]\), \(u\in \operatorname{Ker}M\cap\overline{\Omega}\), where \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}M\) is a homeomorphism with \(J(ae^{-t},0,\ldots,0)^{T}=at\), \(\rho= \Bigl\{ \scriptsize{\begin{array}{@{}l@{\quad}l} -1, &\mbox{if } (\mathrm{H}_{4})(1) \mbox{ holds},\\ 1, &\mbox{if } (\mathrm{H}_{4})(2) \mbox{ holds}. \end{array} }\)

For \(u\in \operatorname{Ker}M\cap\partial\Omega\), we have \(u=at\neq0\). Thus
$$H(u,\delta)=\rho\delta at-(1-\delta)\frac{\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,as,a)\,ds\,dt-\int_{0}^{t_{k}}\sum_{t_{i}\geq t}I_{i}(at_{i},a)h(t)\,dt}{\int_{0}^{+\infty}h(t)e^{-t}\,dt}t . $$
If \(\delta=1\), \(H(u,1)=\rho at\neq0\). If \(\delta=0\), by \(QNu\neq0\), we get \(H(u,0)=JQN(at)\neq0\). For \(0<\delta<1\), we now prove that \(H(u,\delta )\neq0\). Otherwise, if \(H(u,\delta)=0\), then
$$\int_{0}^{+\infty}h(t) \int_{t}^{+\infty} f(s,as,a)\,ds\,dt-\int_{0}^{t_{k}}\sum _{t_{i}\geq t}I_{i}(at_{i},a)h(t)\,dt= \frac{\rho\delta a}{1-\delta}\int_{0}^{+\infty}h(t)e^{-t} \,dt. $$
Thus
$$a\int_{0}^{+\infty} h(t) \biggl[\int _{t}^{+\infty} f(s,as,a)\,ds-\sum _{t_{i}\geq t}I_{i}(at_{i},a) \biggr]\,dt= \frac{\rho\delta a^{2}}{1-\delta}\int_{0}^{+\infty}h(t)e^{-t} \,dt. $$
Since \(|u'(t)|=|a|=\|u\|=r>e_{0}\), this is a contradiction with (H4) and the definition of ρ. So, \(H(u,\delta)\neq0\), \(u\in \operatorname{Ker}M\cap\partial\Omega\), \(\delta\in[0,1]\).
By the homotopy of degree, we get
$$\begin{aligned} \operatorname{deg}\Bigl(JQN, \Omega\cap \operatorname{Ker}M,0\Bigr) &=\operatorname{deg}\Bigl(H(\cdot,0),\Omega\cap \operatorname{Ker}M,0 \Bigr) \\ &=\operatorname{deg}\Bigl( H(\cdot,1),\Omega\cap \operatorname{Ker}M,0\Bigr)\\ &=\operatorname{deg}\Bigl(\rho I, \Omega\cap \operatorname{Ker}M,0\Bigr)\neq0. \end{aligned}$$

By Theorem 2.1, we can find that \(Mu=Nu\) has at least one solution in \(\overline{\Omega}\). The proof is completed. □

4 Example

Let us consider the following impulsive p-Laplacian boundary value problems at resonance on the half-line
$$ \left \{ \begin{array}{@{}l} (\varphi_{p}(u'(t)))'+f(t,u(t),u'(t))=0,\quad t\in[0,\infty){\backslash} \{t_{1},t_{2},\ldots,t_{k}\},\\ \triangle\varphi_{p}(u'(t_{i}))=c_{i},\quad i=1,2,\ldots,k,\\ u(0)=0, \qquad\varphi_{p}(u'(+\infty))= \int_{0}^{+\infty }e^{-t}\varphi_{p}(u'(t))\,dt, \end{array} \right . $$
(4.1)
where \(0< t_{1}<t_{2}<\cdots<t_{k}<+\infty\), \(p=\frac{4}{3}\), \(f(t,x,y)=\frac {e^{-4t}}{(1+t)^{\frac{1}{3}}}\sqrt[3]{\sin x}+e^{-4t}\sqrt[3]{y}+e^{-4t}\).

Corresponding to the problem (1.1), we have \(h(t)=e^{-t}\), \(I_{i}(u,v)=c_{i}\), \(i=1,2,\ldots,k\). Take \(h_{r}(t)= ((1+t)^{-\frac{1}{3}}+r^{\frac{1}{3}}+1 )e^{-4t}\), \(a(t)=\frac{e^{-4t}}{(1+t)^{\frac {1}{3}}}\), \(b(t)=c(t)=e^{-4t}\), \(d_{i}=g_{i}=0\), \(e_{i}=c_{i}\), \(i=1,2,\ldots,k\), \(e_{0}=e^{12(1+t_{k})}(1+20\sum_{i=1}^{k}|c_{i}|)^{3}\), \(M_{r}=\max_{1\leq i\leq k}\{|c_{i}|\}\).

By a simple calculation, we find that (H1)-(H3) and (H4)(1) hold.

By Theorem 3.1, we find that the problem (4.1) has at least one solution.

Declarations

Acknowledgements

This work is supported by the Natural Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108). The author is grateful to anonymous referees for their constructive comments and suggestions, which led to improvement of the original manuscript.

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.

Authors’ Affiliations

(1)
College of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei, 050018, P.R. China

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