Abstract
By using the extension of the continuation theorem of Ge and Ren and constructing suitable Banach spaces and operators, we investigate the existence of solutions for a p-Laplacian boundary value problem with integral boundary condition at resonance on the half-line.
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1 Introduction
A boundary value problem is said to be at resonance one if the corresponding homogeneous boundary value problem has non-trivial solutions. Mawhin’s continuation theorem [1] is an effective tool to investigate the boundary value problems at resonance with linear or semilinear differential operators (see [2–7] and the references cited therein). Boundary value problems with p-Laplacian have been widely studied owing to their importance in theory and application of mathematics and physics (see [8–13]). But the p-Laplacian boundary value problems at resonance cannot be solved by Mawhin’s continuation theorem. In order to solve these problems, Ge and Ren extended Mawhin’s continuation theorem and used it to study boundary value problems with p-Laplacian [11]. In their new theorem, two projectors (Definition 2.2) P and Q must be constructed. But it is difficult to give the projector Q in many boundary value problems with p-Laplacian. In [14], the author extended the theorem in [11] and studied the problem
in finite interval, where Q is not a projector but satisfies suitable conditions, \(\varphi_{p}(s)=|s|^{p-2}s\), \(p>1\), \(\int_{0}^{1}g(t)\,dt=1\), \(\int_{0}^{1}h(t)\,dt=1\).
Boundary value problems on the half-line arise in various applications such as in the study of the unsteady flow of gas through a 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. [15]. In [16], using the continuation theorem of Ge and Ren [11], the author investigated the existence of solutions for the problem
on the half-line, where Q is a projector, \(\alpha_{i}>0\), \(i=1,2,\ldots,n\), \(\sum_{i=1}^{n}\alpha_{i}=1\).
In this paper, we study the boundary value problem
in infinite interval, where Q is not a projector, \(\varphi_{p}(s)=|s|^{p-2}s\), \(p>1\). To the best of our knowledge, this is the first paper to study the boundary value problems at resonance on the half-line where the operator Q is not a projector.
In this paper, we will always suppose that the following conditions hold.
- (H1):
-
\(\int_{0}^{+\infty}h(t)\,dt=1\), \(th(t)\in L^{1}[0,+\infty)\), \(\psi(t)\in L^{1}[0,+\infty)\cap C[0,+\infty)\), \(h(t)\geq0\), \(\psi(t)> 0\), \(t\in[0,+\infty)\).
- (H2):
-
\(f(t,u,v)\) is continuous in \([0,\infty)\times\mathbb{R}^{2}\). For any \(r>0\), there exists a constant \(M_{r}>0\) such that if \(\frac{|u|}{1+t}\leq r\), \(|v|\leq r\), \(t\in [0,+\infty)\) then \(|f(t,u,v)|\leq M_{r}\), and for any \(\varepsilon>0\) there exists \(\delta>0\) such that \(|f(t,u_{1},v_{1})-f(t,u_{2},v_{2})|<\varepsilon\) for \(t\in [0,+\infty)\), \(u_{i}, v_{i}\in\mathbb{R}\), \(i=1,2\), satisfying \(\frac{|u_{1}-u_{2}|}{1+t}<\delta\), \(|v_{1}-v_{2}|<\delta\) and \(\frac{|u_{i}|}{1+t}\leq r\), \(|v_{i}|\leq r\).
The paper is organized as follows. The first section provides a short overview of the problem. Section 2 recalls some preliminary facts. Section 3 contains the main result of the paper.
2 Preliminaries
Definition 2.1
([11])
Let X and Z be two Banach spaces with norms \(\|\cdot\|_{X}\), \(\|\cdot\|_{Z}\), respectively. An operator \(M: X\cap \operatorname{dom}M\rightarrow Z\) is said to be quasi-linear if
-
(i)
\(\operatorname{Im}M:=M(X \cap \operatorname{dom}M)\) is a closed subset of Z,
-
(ii)
\(\operatorname{Ker}M:=\{x\in X \cap \operatorname{dom}M: Mx=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\), \(n<\infty\),
where domM denotes the domain of the operator M.
In this paper, an operator \(T:X\rightarrow Z\) is said to be bounded if \(T(V)\subset Z\) is bounded for any bounded subset \(V\subset X\).
Definition 2.2
P is a projector if \(P:Y\rightarrow Y\) is linear and \(P^{2}x=Px\), where Y is a vector space.
Let \(X_{1}=\operatorname{Ker}M\), \(P:X\rightarrow X_{1}\) be a projector and \(X_{2}\) be the complement space of \(X_{1}\) in X with \(X=X_{1}\oplus X_{2}\). Let \(\Omega\subset X\) be an open and bounded set with the origin \(\theta\in\Omega\).
Definition 2.3
([14])
Suppose that \(N_{\lambda}:\overline{\Omega}\rightarrow Z\), \(\lambda\in[0,1]\) is a continuous and bounded operator. Denote \(N_{1}\) by N. Let \(\Sigma{_{\lambda}}=\{x\in\overline{\Omega}:Mx=N_{\lambda}x\}\). \(N_{\lambda}\) is said to be M-quasi-compact in Ω̅ if there exists a vector subspace \(Z_{1}\) of Z satisfying \(\operatorname{dim} Z_{1}=\operatorname{dim} X_{1}\) and two operators Q and R such that the following conditions hold:
-
(a)
\(\operatorname{Ker}Q=\operatorname{Im}M\),
-
(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)
\(M[P+R(\cdot,\lambda)]=(I-Q)N_{\lambda}\),
where \(Q:Z\rightarrow Z_{1}\) is continuous, bounded with \(Q(I-Q)=0\), \(QZ=Z_{1}\) and \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) is continuous and compact with \(Pu+R(u,\lambda)\in \operatorname{dom} M\), \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\).
Theorem 2.1
([14])
Let X and Z be two Banach spaces with the norms \(\|\cdot\|_{X}\), \(\|\cdot\|_{Z}\), respectively, and \(\Omega\subset X\) be an open and bounded nonempty set. Suppose that
is a quasi-linear operator and that \(N_{\lambda}:\overline{\Omega}\rightarrow Z\), \(\lambda\in[0,1]\) is M-quasi-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 \(J(\theta)=\theta\), \(deg\) is the Brouwer degree.
Remark
In the proof of Theorem 2.1 in [14], the continuity of the operator M is not needed. And domM may not be a linear space. But the operators P, R satisfy \(Pu+R(u,\lambda)\in \operatorname{dom}M\), \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\).
Lemma 2.1
([17])
Let \(\varphi_{p} : \mathbb{R}\rightarrow \mathbb{R}\) be a function given by the formula \(\varphi_{p}(s) = |s|^{p-2}s\), where \(p > 1\). Then, for any \(u, v \geq0\), we have
-
(1)
\(\varphi_{p}(u+v)\leq\varphi_{p}(u)+\varphi_{p}(v)\), \(1< p\leq2\).
-
(2)
\(\varphi_{p}(u+v)\leq2^{p-2}(\varphi_{p}(u)+\varphi_{p}(v))\), \(p\geq 2\).
3 Main results
In the following, we will always assume that q satisfies \(1/p+1/q=1\).
Let \(X= \{u\in C^{1}[0,+\infty):u'(+\infty)=0, \sup_{t\in [0,+\infty)}\frac{|u(t)|}{1+t}<+\infty \}\) be endowed with the following norm \(\|u\|_{X}=\max \{\|u'\|_{\infty}, \Vert \frac {u(t)}{1+t}\Vert _{\infty}\}\), \(Y=\{y\in C[0,+\infty):\sup_{t\in [0,+\infty)}|y(t)|<\infty\}\) be endowed with the following norm \(\|y\|_{Y}=\sup_{t\in[0,+\infty)}|y(t)|:=\|y\|_{\infty}\). Take \(Z=\{\psi y:y\in Y\}\times\mathbb{R}\), with norm \(\|(\psi y,c)\|_{Z}=\max\{\|y\|_{\infty},|c|\}\). We know that \((X,\|\cdot\|_{X})\) and \((Z,\|\cdot\|_{Z})\) are Banach spaces.
Define operator \(T:Y\rightarrow\mathbb{R}\) by \(Ty=c\), for \(y\in Y\), where c satisfies
The following lemma shows that the operator T is well defined.
Lemma 3.1
For \(y\in Y\), there is only one constant \(c\in \mathbb{R}\) such that \(Ty=c\) with \(|c|\leq\|y\|_{\infty}\), \(T:Y\rightarrow\mathbb{R}\) is continuous and \(T(ky)=kT(y)\), \(k\in \mathbb{R}\).
Proof
For \(y\in Y\), let
Obviously, F is continuous and strictly decreasing in \(\mathbb{R}\). If y is a constant, the results hold, clearly. Assume y is not a constant. Take \(a=\inf_{t\in[0,+\infty)}y(t)\), \(b=\sup_{t\in[0,+\infty)}y(t)\). It is easy to see that \(F(a)> 0\), \(F(b)< 0\). So, there exists a unique constant \(c\in(a,b)\) such that \(F(c)=0\), i.e., there is only one constant \(c\in\mathbb{R}\) such that \(Ty=c\) with \(|c|\leq \|y\|_{\infty}\).
For \(y_{1}, y_{2}\in Y\), assume \(Ty_{1}=a\), \(Ty_{2}=b\). By \(h(t)\geq0\), \(\psi(t)> 0\), \(\int_{0}^{+\infty}h(t)\,dt=1\) and \(\varphi_{q}\) being strictly increasing, we obtain that if \(b-a> \sup_{t\in[0,+\infty)}(y_{2}(t)-y_{1}(t))\), then
a contradiction. On the other hand, if \(b-a< \inf_{t\in[0,+\infty)}(y_{2}(t)-y_{1}(t))\), then
a contradiction, too. So, we have \(\inf_{t\in[0,+\infty)}(y_{2}(t)-y_{1}(t))\leq b-a\leq \sup_{t\in[0,+\infty)}(y_{2}(t)-y_{1}(t))\), i.e., \(| b-a|\leq \|y_{2}-y_{1}\|_{\infty}\). So, \(T:Y\rightarrow\mathbb{R}\) is continuous. Obviously, \(T(ky)=kT(y)\), \(k\in\mathbb{R}\). The proof is completed. □
Define operators \(M:X\cap \operatorname{dom}M\rightarrow Z\), \(N_{\lambda}:X\rightarrow Z\) as follows:
where \(\operatorname{dom}M= \{u\in X |\frac{(\varphi_{p}(u'))'}{ \psi(t)}\in Y \}\).
Definition 3.1
u is a solution of (1.1) if \(u\in \operatorname{dom}M\) satisfies (1.1).
It is clear that \(u\in \operatorname{dom}M\) is a solution of (1.1) if and only if it satisfies \(Mu=Nu\), where \(N=N_{1}\).
Lemma 3.2
M is a quasi-linear operator.
Proof
It is easy to get that \(\operatorname{Ker}M=\{c | c\in \mathbb{R}\}:=X_{1}\).
For \(u\in X\cap \operatorname{dom}M\), if \(Mu=(\psi y,c)\), then c satisfies (3.1) with y. On the other hand, if \(y\in Y\), \(Ty=c\), take
By a simple calculation, we get \(u\in X\cap \operatorname{dom}M\) and \(Mu=(\psi y,c)\). Thus
By the continuity of T, we get that \(\operatorname{Im}M\subset Z\) is closed. So, M is quasi-linear. The proof is completed. □
Take a projector \(P: X\rightarrow X_{1}\) and an operator \(Q: Z\rightarrow Z_{1}\) as follows:
where \(Z_{1}=\{(0,c) | c\in\mathbb{R}\}\). Obviously, \(QZ=Z_{1}\) and \(\operatorname{dim} Z_{1}=\operatorname{dim} X_{1}\).
Define an operator R as
Lemma 3.3
([6])
\(V\subset X\) is relatively compact if \(\{\frac{u(t)}{1+t} | u\in V \}\) and \(\{u'(t)| u\in V\}\) are both bounded, equicontinuous on any compact intervals of \([0,+\infty)\) and equiconvergent at infinity.
Lemma 3.4
\(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\) is continuous and compact, \(Pu+R(u,\lambda)\in \operatorname{dom} M\), \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\), where \(X_{2}=\{u\in X:u(0)=0\}\), \(\Omega\subset X\) is an open bounded set.
Proof
Firstly, we prove that \(R:\overline{\Omega}\times [0,1]\rightarrow X_{2}\) and \(Pu+R(u,\lambda)\in \operatorname{dom} M\), \(u\in\overline{\Omega}\), \(\lambda\in[0,1]\).
Obviously, \(R(u,\lambda)(t)\in C^{1}[0,+\infty)\), \(R(u,\lambda)'(+\infty)=-\lim_{t\rightarrow+\infty}\varphi _{q}(\int_{t}^{+\infty}\lambda\psi(s) f(s,u(s), u'(s))\,ds)=0\). By (H2), we get \(\frac{|R(u,\lambda)(t)|}{1+t}\leq \varphi_{q}(\|\psi\|_{1}M_{\|u\|_{X}})<+\infty\), \(u\in X\). Therefore, \(R(u,\lambda)\in X\). It is clear that \(R(u,\lambda)(0)=0\). Thus \(R(u,\lambda)\in X_{2}\). Clearly, \(R(u,\lambda)+Pu\in X\). It follows from \((\varphi_{p}(R(u,\lambda)(t)+Pu(t))')'=\lambda\psi(t) f(t,u(t),u'(t))\) and (H2) that \(\frac{(\varphi_{p}(R(u,\lambda)(t)+Pu(t))')'}{\psi (t)}=\lambda f(t,u(t),u'(t))\in Y\). So, \(R(u,\lambda)+Pu\in \operatorname{dom}M\).
Secondly, we show that R is continuous.
Since Ω is bounded, there exists a constant \(r>0\) such that \(\|u\|_{X}\leq r\), \(u\in\overline{\Omega}\). By (H2), there exists a constant \(M_{r}>0\) such that \(|f(t,u(t),u'(t))|\leq M_{r}\), \(u\in \overline{\Omega}\), \(t\in[0,+\infty)\). So, we get
By the uniform continuity of \(\varphi_{q}(x)\) in \([-\|\psi\|_{1}M_{r}, \max\{1,\|\psi\|_{1}M_{r}\}]\), we obtain that for any \(\varepsilon>0\), there exists a constant \(\delta_{\varepsilon}>0\) such that
For \(\alpha= \frac{\delta_{\varepsilon}}{\|\psi\|_{1}}\), by (H2), there exists a constant \(\delta_{\alpha}>0\) such that if \(u,v\in\overline{\Omega}\), \(\|u-v\|_{X}<\delta_{\alpha}\), then \(|f(t,u(t),u'(t))-f(t,v(t),v'(t))|<\alpha\), \(t\in[0,\infty)\). So, we have
Take \(\delta=\min\{\delta_{\varepsilon}, \delta_{\alpha}\}\). For \(u, v\in\overline{\Omega}\), \(\lambda, \mu\in[0,1]\), if \(\|u-v\| _{X}<\delta\), \(|\lambda-\mu|<\delta\), then
This, together with
means that \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\cap \operatorname{dom}M\) is continuous.
We will prove that \(R:\overline{\Omega}\times[0,1]\rightarrow X_{2}\cap \operatorname{dom}M\) is compact.
It is easy to get that \(\{\frac{R(u,\lambda)(t)}{1+t}:u\in\overline{\Omega },\lambda\in [0,1] \}\) and \(\{R(u,\lambda)'(t):u\in\overline{\Omega},\lambda \in [0,1]\}\) are bounded.
For any \(T>0\), \(t_{1}, t_{2}\in[0,T]\), \(t_{1}>t_{2}\), \(u\in \overline{\Omega}\), \(\lambda\in[0,1]\), we have
Since t and \(\frac{1}{1+t}\) are uniformly continuous on \([0,T]\), we get that \(\{\frac{R(u,\lambda)(t)}{1+t}, u\in \overline{\Omega}, \lambda\in[0,1] \}\) is equicontinuous on \([0,T]\).
Take \(G(t)=\int_{t}^{+\infty}\lambda\psi(r) f(r,u(r),u'(r))\,dr\). We have
It follows from the absolute continuity of integral and the uniform continuity of \(\varphi_{q}(t)\) in \([-M_{r}\|\psi\|_{1},M_{r}\|\psi\|_{1}]\) that \(\{R(u,\lambda)'(t), u\in \overline{\Omega}, \lambda\in[0,1]\}\) is equicontinuous on \([0,T]\).
For \(u\in \overline{\Omega}\), since
and
for any \(\varepsilon>0\), there exists a constant \(T_{1}>0\) such that
Obviously, there exists a constant \(T>T_{1}\) such that, for any \(t>T\),
Thus, for any \(t_{1}, t_{2}>T\), we have
and
By Lemma 3.3, we get that \(\{R(u,\lambda)| u\in \overline{\Omega}, \lambda\in[0,1]\}\) is relatively compact. The proof is completed. □
Lemma 3.5
Assume that \(\Omega\subset X\) is an open bounded set. Then \(N_{\lambda}\) is M-quasi-compact in Ω̅.
Proof
It is clear that \(\operatorname{Im}P=\operatorname{Ker}M\), \(\operatorname{Ker} Q=\operatorname{Im} M\) and \(QN_{\lambda} x=\theta\), \(\lambda\in(0,1)\Leftrightarrow QNx=\theta\), i.e., Definition 2.3(a) and (b) are satisfied.
For \(u\in\Sigma{_{\lambda}}=\{\omega\in\overline{\Omega}\cap \operatorname{dom}M:M\omega=N_{\lambda}\omega\}\), by (H1) and (H2), we get
Clearly, \(R(\cdot,0)=0\). Thus, Definition 2.3(c) is satisfied. For \(u\in\overline{\Omega}\), we have
So, Definition 2.3(d) is satisfied.
Considering (H2),
and
we can obtain that \(N_{\lambda}\) is continuous and bounded in Ω̅.
It follows from the continuity and boundedness of T that Q is continuous and bounded in Z. By a simple calculation, we can obtain that \(Q(I-Q)(\psi y,c)=(0,0)\), \((\psi y,c)\in Z\).
These, together with Lemma 3.4, mean that \(N_{\lambda}\) is M-quasi-compact in Ω̅. The proof is completed. □
Theorem 3.1
Suppose that (H1), (H2) and the following conditions hold:
- (H3):
-
There exist constants \(c_{0}>0\) and \(l>0\) such that
$$\int_{0}^{+\infty}h(t)\int_{0}^{t} \varphi_{q} \biggl(\int_{s}^{+\infty}\psi (r)f \bigl(r,u(r),u'(r)\bigr)\,dr \biggr)\,ds\,dt\neq 0,\quad \bigl|u(t)\bigr|>c_{0}, t\in[0,l], u\in X. $$ - (H4):
-
There exist nonnegative functions \(a(t)\), \(b(t)\), \(c(t)\) with \((1+t)^{p-1}a(t)\psi(t), b(t)\psi(t), c(t)\psi(t)\in L^{1}[0,+\infty)\) such that
$$\bigl|f(t,x,y)\bigr|\leq a(t)\bigl|\varphi_{p}(x)\bigr|+b(t)\bigl|\varphi_{p}(y)\bigr|+c(t), \quad\textit{a.e. } t\in[0,+\infty), $$where \(\|(1+t)^{p-1}a(t)\psi(t)\|_{1}l_{0}^{p-1}+\|b(t)\psi(t)\|_{1}<1\), if \(1< p\leq2\); \(2^{p-2}\|(1+t)^{p-1}a(t) \psi(t)\|_{1}l_{0}^{p-1}+\|b(t)\psi(t)\|_{1}<1\), if \(p\geq2\), where \(l_{0}=\max\{1,l\}\).
- (H5):
-
There exists a constant \(d_{0}>0\) such that if \(|d|>d_{0}\), then one of the following inequalities holds:
-
(1)
\(d f(t,d,0)>0\), \(t\in[0,l)\);
-
(2)
\(d f(t,d,0)<0\), \(t\in[0,l)\).
Then the boundary value problem (1.1) has at least one solution.
-
(1)
In order to prove Theorem 3.1, we show two lemmas.
Lemma 3.6
Assume (H1)-(H4) hold. Then the set
is bounded in X.
Proof
For \(u\in\Omega_{1}\), we have \(QN_{\lambda}u=0\), i.e., \(T(f(t,u(t),u'(t)))=0\). By (H3), there exists a constant \(t_{0}\in[0,l]\) such that \(|u(t_{0})|\leq c_{0}\). Since \(u(t)=u(t_{0})+\int_{t_{0}}^{t}u'(s)\,ds\), then
Thus
It follows from \(Mu=N_{\lambda}u\), (H4) and (3.2) that
Whenever \(1< p\leq2\), by Lemma 2.1, we get
Whenever \(p>2\), by Lemma 2.1, we get
These, together with (3.2), mean that \(\Omega_{1}\) is bounded in X. □
Lemma 3.7
Assume (H1)-(H3) and (H5) hold. Then
is bounded in X, where \(N=N_{1}\).
Proof
For \(u\in\Omega_{2}\), we have \(u=a\), \(a\in\mathbb{R}\) and \(Q(N u)=0\), i.e.,
By (H5), we get that \(|a|\leq d_{0}\). So, \(\Omega_{2}\) is bounded. The proof is completed. □
Proof of Theorem 3.1
Let \(\Omega=\{u\in X | \|u\|< r\}\), where \(r>d_{0}\) is large enough such that \(\Omega\supset \overline{\Omega}_{1}\cup\overline{\Omega}_{2}\).
By Lemmas 3.6 and 3.7, we know \(Mu\neq N_{\lambda}u\), \(u\in \operatorname{dom}M\cap\partial\Omega\) 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 \(J(0,a)=a\),
Define a function
For \(u\in \operatorname{Ker}M\cap\partial\Omega\), we have \(u=a\neq0\). Thus
If \(\delta=1\), \(H(u,1)=\rho a\neq0\). If \(\delta=0\), by \(QNu\neq0\), we get \(H(u,0)=JQN(a)\neq0\). For \(0<\delta<1\), we now prove that \(H(u,\delta )\neq0\). Otherwise, if \(H(u,\delta)=0\), then
Since \(\|u\|_{X}=|a|=r>d_{0}\), for \(t\in[0,l)\), by (H5) and Lemma 3.1, we have
a contradiction with 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 that
By Theorem 2.1, we can get that \(Mu=Nu\) has at least one solution in Ω̅. The proof is completed. □
4 Example
Example 4.1
Let us consider the following boundary value problem at resonance:
where \(p= \frac{4}{3}\), \(h(t)=e^{-t}\),
\(\psi(t)= \frac{1}{3}(1+t)^{-\frac{4}{3}}(1+l_{0})^{-\frac {4}{3}}e^{-t}\), \(l_{0}=\max\{1,l\}\). Take \(a(t)=b(t)=(l-t)\), \(c(t)=0\), \(d_{0}=c_{0}=27\). By a simple calculation, we can obtain \(\|(1+t)^{p-1}a(t)\psi(t)\|_{1}l_{0}^{p-1}+\|b(t)\psi(t)\|_{1}<1\). It is easy to get that conditions (H1)-(H5) are satisfied. It follows from Theorem 3.1 that problem (4.1) has at least one solution.
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Acknowledgements
The author is grateful to anonymous referees for their constructive comments and suggestions which led to improvement of the original manuscript. This work is supported by the Natural Science Foundation of China (11171088) and the Natural Science Foundation of Hebei Province (A2013208108).
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Jiang, W., Zhang, Y. & Qiu, J. The existence of solutions for p-Laplacian boundary value problems at resonance on the half-line. Bound Value Probl 2015, 179 (2015). https://doi.org/10.1186/s13661-015-0439-9
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Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-015-0439-9