The existence of solutions for impulsive p-Laplacian boundary value problems at resonance on the half-line
- Weihua Jiang1Email author
https://doi.org/10.1186/s13661-015-0299-3
© Jiang; licensee Springer. 2015
Received: 1 October 2014
Accepted: 28 January 2015
Published: 24 February 2015
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
MSC
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]
- (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]
- (i)
\(\operatorname{Im}M:=M(X \cap \operatorname{dom}M)\) is a closed subset of Y,
- (ii)
\(\operatorname{Ker}M:=\{x\in X \cap \operatorname{dom} M: Mx=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\), \(n<\infty\),
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]
- (a)
\((I-Q)N_{\lambda}(\overline{\Omega})\subset \operatorname{Im} M\subset(I-Q)Y\),
- (b)
\(QN_{\lambda} x=\theta, \lambda\in(0,1)\Leftrightarrow QNx=\theta\),
- (c)
\(R(\cdot,0)\) is the zero operator and \(R(\cdot,\lambda)|_{\Sigma{_{\lambda}}}=(I-P)|_{\Sigma{_{\lambda}}}\),
- (d)
\(M[P+R(\cdot,\lambda)]=(I-Q)N_{\lambda}\).
Theorem 2.1
[23]
- (C1):
-
\(Mx\neq N_{\lambda}x\), \(\forall x\in\partial\Omega\cap \operatorname{dom} M\), \(\lambda\in(0,1)\),
- (C2):
-
\(\operatorname{deg}\{JQN, \Omega\cap \operatorname{Ker}M, 0\}\neq0\),
3 Main results
In the following, we will always suppose that q satisfies \(1/p+1/q=1\).
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.
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 \(\operatorname{Ker}M=\{at \mid a\in \mathbb{R}\}:=X_{1}\).
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.
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.
Theorem 3.1
- (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:where \(t\in[0,+\infty)\). Then boundary value problem (1.1) has at least one solution.$$\begin{aligned}& (1)\quad u' (t) \int_{0}^{+\infty}h(t) \biggl(\int_{t}^{+\infty} f\bigl(s,u(s),u'(s) \bigr)\,ds-\sum_{t_{i}\geq t}I_{i} \bigl(u(t_{i}),u'(t_{i})\bigr) \biggr)\,dt>0; \\& (2)\quad u' (t) \int_{0}^{+\infty}h(t) \biggl(\int_{t}^{+\infty} f\bigl(s,u(s),u'(s) \bigr)\,ds-\sum_{t_{i}\geq t}I_{i} \bigl(u(t_{i}),u'(t_{i})\bigr) \biggr)\,dt< 0, \end{aligned}$$
In order to prove Theorem 3.1, we show two lemmas.
Lemma 3.5
Proof
Lemma 3.6
Proof
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 \(J(ae^{-t},0,\ldots,0)^{T}=at\), \(\rho= \Bigl\{ \scriptsize{\begin{array}{@{}l@{\quad}l} -1, &\mbox{if } (\mathrm{H}_{4})(1) \mbox{ holds},\\ 1, &\mbox{if } (\mathrm{H}_{4})(2) \mbox{ holds}. \end{array} }\)
By Theorem 2.1, we can find that \(Mu=Nu\) has at least one solution in \(\overline{\Omega}\). The proof is completed. □
4 Example
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
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