\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 148, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/148\hfil Nonlocal boundary-value problems]
{k-dimensional nonlocal boundary-value problems at resonance}

\author[K. Szyma\'nska-D\c{e}bowska \hfil EJDE-2015/148\hfilneg]
{Katarzyna Szyma\'nska-D\c{e}bowska}

\address{Katarzyna Szyma\'nska-D\c{e}bowska \newline
Institute of Mathematics, \L\'od\'z University of Technology,
90-924 \L\'od\'z, ul. W\'olcza\'nska 215, Poland}
\email{katarzyna.szymanska-debowska@p.lodz.pl}


\thanks{Submitted February 2, 2015. Published June 6, 2015.}
\subjclass[2010]{34B10, 34B15}
\keywords{Nonlocal boundary value problem; perturbation method; 
\hfill\break\indent boundary value problem at resonance;  Neumann  BVP}

\begin{abstract}
 In this article we show the existence of at least one solution to
 the system of nonlocal resonant boundary-value problem
 $$
 x''=f(t,x), \quad x'(0)=0, \quad x'(1)=\int_{0 }^{1}x'(s)\,dg(s),
 $$
 where $f:[0,1]\times\mathbb{R}^k\to\mathbb{R}^k$, $g:[0,1]\to\mathbb{R}^k$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

In this article we study the system of ordinary differential equations
\begin{equation} \label{upn}
x''=f(t,x), \quad x'(0)=0, \quad x'(1)=\int_{0 }^{1}x'(s)\,dg(s),
\end{equation}
where $f=(f_1,\dots , f_k):[0,1]\times\mathbb{R}^k\to\mathbb{R}^k$
is continuous, and  $g=(g_1,\dots , g_k):[0,1]\to\mathbb{R}^k$ 
has bounded variation.
Observe that \eqref{upn} can be written down as the system of equations
\begin{gather*}
x''_i(t)=f_i(t,x(t)),  \\
x'_i(0)=0,\\
x'_i(1)=\int_{0 }^{1}x'_i(s)dg_i(s),
\end{gather*}
where $ t\in [0,1]$, $i=1, \dots , k$ and the integrals
$\int_{0 }^{1}x'_i(s)dg_i(s)$ are meant in the sense of Riemann-Stieltjes.

Our main goal is to show that the problem \eqref{upn} has at least one 
solution. We impose on the function $f$ a sign condition, which we called: 
the asymptotic integral sign condition. The idea comes from \cite{prz}, 
where the author shows that the first order equation
$x'=f(t,x)$ has periodic solutions. The method can be successfully applied 
to other BVPs (not necessarily only for differential equations of the first 
or second order but, for instance, involving p-Laplacians), for which 
the function  $f$ does not depend on $x '$.

As far as we are aware, \eqref{upn} has not been studied in this generality so far.
Note that a special case of \eqref{upn} is the Neumann  BVP
\[
x''=f(t,x), \quad x'(0)=0, \quad x'(1)=0.
\]
Under suitable monotonicity conditions or nonresonance conditions, 
some existence or uniqueness theorems or methods for Neumann BVPs have 
been presented (see, for instance, \cite{a, er, liu, yan, s, v, y, w, w1} 
and the references therein).

In \cite{h}, the authors study the Neumann boundary value problem 
$x''+\mu(t)x_{+}-\nu(t)x_{-}=p(t,x)$, $x'(0)=0=x'(\pi)$, where $\mu$, $\nu$ 
lie in $L^1(0,\pi)$, $p(t,x)$ is a Carath\'eodory function, 
$p\geq0$, $x_{+}(t)=\max(x(t),0)$, and $x_{-}(t)=\max(-x(t),0)$. 
They obtain several necessary and sufficient conditions on $p$ so that 
the Neumann problem has a positive solution or a solution with a simple 
zero in $(0,\pi)$.

In \cite{h2}, the author uses phase plane and asymptotic techniques to 
discuss the number of solutions of the problems $-x''=f(t,x)$, 
$x'(0)=\sigma_1$, $x'(\pi)=\sigma_1$.
It is assumed that $f: [0, \pi] \times \mathbb{R} \to \mathbb{R}$ is a continuous jumping 
nonlinearity with nonnegative asymptotic limits: $x^{-1}f(t,x)\to \alpha$ as 
$x\to -\infty$ and $x^{-1}f(t,x)\to \beta$ as $x\to \infty$. 
The limit problem where $f(t,x)=\alpha x_{-}+\beta x_{+}$ plays a key role 
in his methods. The authors describe how the number of solutions of the
problem depends on the four parameters: $\alpha, \beta, \sigma_1, \sigma_2$. 
The results differ from those obtained by various authors
who were mainly concerned with forcing the equation with large positive 
functions and keeping the boundary conditions homogeneous.


The boundary-value problem
$$
x'' =f(t,x,x'), \quad x'(0)=0, \quad x'(1)=0,
$$
is considered in \cite{gr2}. The authors obtain some results of existence 
of solutions assuming that there is a constant $M> 0$ such that $yf(t,x,y)> 0$ 
for $| y |> M$ and the function $f$ satisfies the Bernstein growth condition 
(or the Bernstein-Nagumo growth condition).

In \cite{mawhin} the author shows the existence of a solution  to the 
Neumann problem for the equation
\[
(d/dt)[A(t)dx/dt]=f(t,x,x'),
\]
where $A:[0,1]\to L(\mathbb{R}^k,\mathbb{R}^k)$ and 
$f:[0,1]\times \mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k$, 
applying the coincidence degree theory.

The generalization of the Neumann problem \eqref{upn} is a nonlocal problem. 
BVPs with Riemann-Stieltjes integral
boundary conditions include as special cases multi-point and integral BVPs.

The multi-point and integral BCs are widely studied objects. 
The study of multi-point BCs was initiated in 1908 by Picone \cite{picone}.
 Reviews on differential equations with BCs involving
Stieltjes measures has been written in 1942 by Whyburn \cite{wh} and 
in 1967 by Conti \cite{conti}.

Since then, the existence of
solutions for nonlocal nonlinear BVPs has been
studied by many authors by using, for instance, the Leray-Schauder degree theory, 
the coincidence degree theory of Mawhin, the fixed point theorems for cones. 
For such problems and comments on their importance, we refer the reader 
to \cite{ge2, ge, kt3, webbzima, y1, y2} and the references therein.

\section{The perturbed problem}

First, we shall introduce notation and terminology. 
Throughout the paper $|\cdot|$ will denote the Euclidean
norm on $\mathbb{R}^k$, while  the scalar product in $\mathbb{R}^k$ 
corresponding to the Euclidean norm will be denoted by $(\cdot|\cdot)$.
Denote by $C^{1}([0,1],\mathbb{R}^k)$ the Banach space of all continuous 
functions $x:[0,1]\to\mathbb{R}^k$ which have continuous first derivatives $x'$  
with the norm
\begin{equation} \label{eq:norm}
\|x\|=\max \big\{\sup_{t\in[0,1]}|x(t)|, \sup_{t\in[0,1]}|x'(t)|\big\} .
\end{equation}
The Lemma below, which is a straightforward consequence of the classical
Arzel\`a-Ascoli theorem, gives a compactness criterion in $ C^1([0,1],\mathbb{R}^k)$.

\begin{lemma}\label{wz}
For a set $Z \subset   C^1([0,1],\mathbb{R}^k)$ to be relatively compact,
it is necessary and sufficient that:
\begin{itemize}
\item[(1)] there exists $M > 0$ such that for any $x\in Z$ and $t\in [0 , 1]$
we have
$|x(t)|\leq M$ and $|x'(t)|\leq M$;

\item[(2)] for every $t_{0} \in[0,1]$ the families 
$Z:= \{x: x\in Z\}$ and $Z':= \{x': x\in Z\}$
are equicontinuous at $t_{0}$.
\end{itemize}
\end{lemma}

Now, let us consider problem \eqref{upn} and observe that the homogeneous 
linear problem, i.e.,
\[
x''=0, \quad x'(0)=0, \quad x'(1)=\int_{0 }^{1}x'(s)\,dg(s),
\]
has always nontrivial solutions, hence we deal with a resonant situation.

The following assumptions will be needed throughout this article:
\begin{itemize}
\item[(i)] $f=(f_1,\dots , f_k):[0,1]\times\mathbb{R}^k\to\mathbb{R}^k$ 
 is a continuous function.
\item[(ii)] $g=(g_1,\dots , g_k):[0,1]\to\mathbb{R}^k$ 
 has bounded variation on the interval $[0,1]$.
\item[(iii)] There exists a uniform finite limit
\[
h(t,\xi):=\lim_{\lambda \to\infty}f(t,\lambda \:\xi)
\]
with respect to $t$ and $\xi\in \mathbb{R}^k$, $|\xi|=1$.

\item[(iv)] Set
\[
h_0(\xi):=\int_{0 }^{1}h(u,\xi)du -\int_{0}^{1}\int_{0}^{s}h(u,\xi)du\,dg(s).
\]
For every $\xi\in \mathbb{R}^k$, $|\xi|=1$, we have $(\xi : h_0(\xi))< 0$.
\end{itemize}



Problem \eqref{upn} is resonant. Hence, there is no equivalent
integral equation. The existence of a solution will be obtained by considering 
the  perturbed boundary-value problem
\begin{gather} \label{rr3}
x''=f(t,x), \quad t\in [0,1], \\
\label{wp3}
x'(0)=0, \\
\label{non3}
x'(1)=\int_{0 }^{1}x'(s)\,dg(s) +\alpha_n x(0), \quad \alpha_n \in(0,1),\quad \alpha_n\to 0.
\end{gather}
Notice that problem \eqref{rr3}, \eqref{wp3}, \eqref{non3} is always nonresonant.


Now, let us consider the equation \eqref{rr3} and integrate it from $0$ to $t$. 
By \eqref{wp3}, we obtain
\begin{equation} \label{ixprim}
x'(t)=\int_{0}^{t}f(u,x(u))du.
\end{equation}
By \eqref{non3} and \eqref{ixprim}, we obtain
\[
\int_{0}^{1}f(u,x(u))du=\int_{0}^{1}\int_{0}^{s}f(u,x(u))du\,dg(s)+\alpha_n x(0),
\]
so
\[
x(0)=\frac{1}{\alpha_n}\Big[\int_{0}^{1}f(u,x(u))du
-\int_{0}^{1}\int_{0}^{s}f(u,x(u))du\,dg(s)\Big],
\]
Moreover, by \eqref{ixprim}, we have
\[
x(t)=x(0) +\int_{0}^{t}\int_{0}^{s}f(u,x(u))\,du\,ds.
\]
Now, it is easily seen that the following Lemma holds.


\begin{lemma} \label{rownowaznosc}
A function $x\in C^{1}([0,1],\mathbb{R}^k)$ is a solution of
 \eqref{rr3}, \eqref{wp3}, \eqref{non3} if and only if  $x$ satisfies the 
 integral equation
\[ %\label{iks}
x(t)= \int_{0}^{t}\int_{0}^{s}f(u,x(u))\,du\,ds
+ \frac{1}{\alpha_n}\Big[\int_{0}^{1}f(u,x(u))du
 -\int_{0}^{1}\int_{0}^{s}f(u,x(u))du\,dg(s)\Big]. 
\]
\end{lemma}

To search for solutions of \eqref{rr3}, \eqref{wp3}, \eqref{non3}, 
we first reformulate the problem as an operator equation.
Given $x\in C^{1}([0,1],\mathbb{R}^k)$ and fixed $n\in\mathbb{N}$ let
\begin{align*} %\label{opa}
(A_{n}x)(t)
&=\int_{0}^{t}\int_{0}^{s}f(u,x(u))\,du\,ds \\
&\quad + \frac{1}{\alpha_n}\Big[\int_{0}^{1}f(u,x(u))du
 -\int_{0}^{1}\int_{0}^{s}f(u,x(u))du\,dg(s)\Big]. 
\end{align*}
Then
\begin{equation} \label{opaprim}
(A_{n}x)'(t)=\int_{0}^{t}f(u,x(u))du.
\end{equation}
It is clear that $A_{n}x, (A_{n}x)':[0,1]\to\mathbb{R}^k$ are continuous.
It follows that the operator
\[
A_{n}: C^1([0,1],\mathbb{R}^k) \to  C^1([0,1],\mathbb{R}^k)
\]
is well defined.

By assumption (iii), function $f$ is bounded and we put
\[
M:=\sup_{t\in[0,1],x\in\mathbb{R}^k}|f(t,x)|.
\]
By \eqref{opaprim}, we have
\begin{equation} \label{ogr}
\sup_{t\in [0,1]}|(A_{n}x)'(t)|\leq M .
\end{equation}
Moreover, we obtain
\begin{equation}
\sup_{t\in [0,1]}|(A_{n}x)(t)|\leq M+\frac{1}{\alpha_n}\left(M+M\operatorname{Var}(g) \right),
\end{equation}
where $\operatorname{Var}(g) $ means the variation of $g$ on the interval $[0,1]$.


From (ii), $L: = \operatorname{Var}(g) < \infty$. Put $M_{n}:=M+\frac{1}{\alpha_n}\left(M+M\:L \right)$, 
then $\|A_{n}x\|\leq M_{n}$ for every $n\in\mathbb{N}$.
Moreover, $(A_{n}x)''(t)$ and $(A_{n}x)'(t)$, $t\in[0,1]$, are bounded, hence 
the families $(A_{n}x)'$ and $(A_{n}x)$ are equicontinuous. 
Now, by Lemma \ref{wz}, the following Lemma holds.

\begin{lemma} \label{pelopA}
The operator $A_{n}$ is completely continuous.
\end{lemma}


Let $B_{n}:=\{x\in  C^1([0,1],\mathbb{R}^k) : \|x\|\leq M_{n}\}$. 
Now, considering operator
\[
A_{n}:B_{n}\to B_{n},
\]
by Schauder's fixed point Theorem, we get that the operator $A_{n}$ has 
a fixed point in $B_{n}$ for every $n$.
We have proved the following result.

\begin{lemma} \label{lem2.4}
Under assumptions {\rm (i)--(iii)}, problem \eqref{rr3}, \eqref{wp3}, \eqref{non3} 
has at least one solution for every $n\in\mathbb{N}$.
\end{lemma}


\section{Main results}

Let $ \varphi _{n}$ be a solution of the problem \eqref{rr3}, \eqref{wp3}, \
eqref{non3}, where $n$ is fixed.

\begin{lemma} \label{fogr}
The sequence $(\varphi_n)$ is bounded in $ C^1([0,1],\mathbb{R}^k)$.
\end{lemma}

\begin{proof}
Assume that the sequence $(\varphi_n)$ is unbounded. Then, passing to a 
subsequence if necessary, we have $\|\varphi_n \|\to\infty$. We can proceed
analogously as in \eqref{ogr} to show that
\[
\sup_{t\in [0,1]}|(\varphi_n)'(t)|\leq M,
\]
for every $n$. Hence, $\sup_{t\in[0,1]}|\varphi_n(t)|\to\infty$, when $n\to\infty$.

Let us consider the following sequence $(\frac{\varphi_n}{\|\varphi_n\|})\subset  C^1([0,1],\mathbb{R}^k)$ 
and notice that the norm of the sequence equals 1. Hence, the sequence is bounded. 
Moreover, the family $(\frac{\varphi_n}{\|\varphi_n\|})$ (and simultaneously 
$(\frac{\varphi_n'}{\|\varphi_n\|})$) is equicontinuous, since $\frac{\varphi_n'(t)}{\|\varphi_n\|}$ 
(or $\frac{\varphi_n''(t)}{\|\varphi_n\|}$) is bounded. By Lemma \ref{wz}, there exists 
a convergent subsequence of $(\frac{\varphi_n}{\|\varphi_n\|})$. To simplify the notation, 
let us denote this subsequence as $(\frac{\varphi_n}{\|\varphi_n\|})$.

First, observe that $\frac{\varphi_n' (t)}{\|\varphi_n \|}\to 0\in\mathbb{R}^k$. 
Now, we shall show that
\begin{equation} \label{granicat}
\frac{\varphi_n (t)}{\|\varphi_n \|}\to \xi,
\end{equation}
where $\xi=(\xi_1, \dots , \xi_k)$ does not depend on t and $|\xi|=1$.

Indeed, notice that $\frac{\varphi_n (t)}{\|\varphi_n \|}$ is given by
\begin{equation} \label{de}
\begin{aligned}
\frac{\varphi_n (t)}{\|\varphi_n \|}
&= \frac{\int_{0}^{t}\int_{0}^{s}f(u,\varphi_n(u))\,du\,ds}{ \|\varphi_n \| }  \\
&\quad +\frac{\int_{0}^{1}f(u,\varphi_n(u))du
 -\int_{0}^{1}\int_{0}^{s}f(u,\varphi_n(u))du\,dg(s)}{\alpha_n \|\varphi_n \| }.
\end{aligned}
\end{equation}
Since $f$ is bounded, we obtain
\begin{equation} \label{lim1}
\lim_{n\to\infty}\frac{\int_{0}^{t}\int_{0}^{s}f(u,\varphi_n(u))\,du\,ds}{ \|\varphi_n \| }=0
\in\mathbb{R}^k.
\end{equation}
Now, by \eqref{de} and \eqref{lim1}, we can easily observe that the limit
\eqref{granicat} does not depend on $t$.
The norm of the sequence $(\frac{\varphi_n}{\|\varphi_n\|})$ equals 1.
Hence $\frac{\varphi_n (t)}{\|\varphi_n \|}\to \xi$, where $|\xi|=1$.

On the other hand, 
\begin{equation}\label{rownanie}
\begin{aligned}
\xi &= \lim_{n\to\infty}\frac{\varphi_n (t)}{\|\varphi_n \|}\\
&= \frac{\int_{0}^{t}\int_{0}^{s}f(u,\varphi_n(u))\,du\,ds}{ \|\varphi_n \| }  \\
&\quad +\frac{\int_{0}^{1}f(u,\varphi_n(u))du-\int_{0}^{1}
 \int_{0}^{s}f(u,\varphi_n(u))du\,dg(s)}{\alpha_n \|\varphi_n \| }   \\
&= \lim_{n\to\infty}\Big( \frac{\int_{0 }^{1}f(u,\|\varphi_n \|\frac{\varphi_n(u)}{\|\varphi_n \|})du }
{\alpha_n \|\varphi_n \| }  
- \frac{\int_{0}^{1}\int_{0}^{s}f(u,\|\varphi_n \|\frac{\varphi_n(u)}{\|\varphi_n \|})du\,dg(s)}
{\alpha_n \|\varphi_n \| } \Big) .
\end{aligned}
\end{equation}
Now, observe, that there exist a uniform limits of
\[
\int_{0 }^{1}f(u,\|\varphi_n \|\frac{\varphi_n(u)}{\|\varphi_n \|})du
\]
and
\[
\int_{0}^{1}\int_{0}^{s}f(u,\|\varphi_n \|\frac{\varphi_n(u)}{\|\varphi_n \|})du\,dg(s)
\]
Moreover, by (iv), the sum of the limits is different from zero.
Hence, since \eqref{granicat} holds, there exists $\gamma \in (0,\infty)$
such that $\gamma:=\lim_{n\to\infty}1/(\alpha_n \|\varphi_n \|)$.

Now, by assumption (iii), we obtain
\begin{equation} \label{rownanie1}
\xi = \lim_{n\to\infty}\frac{\varphi_n (t)}{\|\varphi_n \|}   
= \gamma \Big[ \int_{0 }^{1}h(u,\xi)du
-\int_{0}^{1}\int_{0}^{s}h(u,\xi)du\,dg(s)\Big].
\end{equation}
Finally, by \eqref{rownanie1} and (iv), we obtain
\begin{align*}
1=(\xi \mid \xi)
&=\gamma\Big(\xi\mid  \int_{0 }^{1}h(u,\xi)du
-\int_{0}^{1}\int_{0}^{s}h(u,\xi)du\,dg(s) \Big) \\
&= \gamma(\xi \mid h_0(\xi))< 0
\end{align*}
a contradiction. Hence, the sequence $(\varphi_n)$ is bounded.
\end{proof}

Now, it is easy to see that the following lemma holds.


\begin{lemma}\label{fwz}
The set $Z=\{\varphi_n: n\in\mathbb{N}\}$ is relatively compact in $ C^1([0,1],\mathbb{R}^k)$.
\end{lemma}

By the above Lemmas, we get the proof of the following result.

\begin{theorem} \label{thm3.3}
Under assumptions {\rm (i)--(iv)} problem \eqref{upn} has at least one solution.
\end{theorem}

\begin{proof}
Lemma \ref{fwz} implies that $(\varphi_n)$ has a convergent subsequence
 $(\varphi_{n_l})$, $\varphi_{n_l}\to \varphi $. We know that $\varphi_{n_l} $ $(\varphi_{n_l} ')$ converges
 uniformly to $\varphi$ $(\varphi ')$ on $[0,1]$. Since $(\varphi_{n_l})$ is equibounded 
and $f$ is uniformly continuous on compact sets, one can see that $f(t,\varphi_{n_l} )$ 
is uniformly convergent to $f(t,\varphi )$. Since
\[
\varphi_{n_l} ''(t)=f(t,\varphi_{n_l} (t)),
\]
the sequence $\varphi_{n_l} ''(t)$ is also uniformly convergent. Moreover,
 $\varphi_{n_l} ''(t)$ converges uniformly to $\varphi ''(t)$.

Note that we have actually proved that function $\varphi \in  C^1([0,1],\mathbb{R}^k)$ is a 
solution of the equation of problem \eqref{upn}.
By \eqref{wp3} and \eqref{non3}, it is easy to see that $\varphi$ 
satisfies boundary conditions of problem \eqref{upn}. This completes the proof.
\end{proof}

\section{Applications}

To illustrate our results we shall present some examples.

\begin{example}\rm
Let us consider the Neumann  BVP
\[
x''=f(t,x), \quad x'(0)=0, \quad x'(1)=0.
\]
In this case $g_i(t)=\text{constant}$, $i=1, \dots, k$, $t\in [0,1]$ 
and condition (ii) always holds. Moreover, we have
\[
h_0(\xi)=\int_{0}^{1}h(s,\xi)ds   .
\]
Hence for any $f$ which satisfies conditions (i), (iii) and (iv) the Neumann  
BVP has at least one solution.
\end{example}


\begin{example} \rm
Let $k=1$, $g(t)=t$ and $f(t,x)=\frac{t-|x|x}{x^2+1}$. We have
\[
h(t,\xi)=\lim_{\lambda\to\infty}f(t,\lambda \:\xi)=
\begin{cases}
-1, &\xi=1 \\
 1, &\xi=-1\,.
\end{cases} 
\]
Then $h_0(1)=-1/2$ and $h_0(-1)=1/2$ and we get $(\xi|h_0(\xi))<0$.
Hence, problem \eqref{upn} has at least one nontrivial solution.
\end{example}


\begin{example}\rm
Let $k=3$, $g(t)=(t,t,t)$ and
\begin{gather*}
f_1(t,x_1,x_2,x_3)=\frac{-x_1}{\sqrt{x_1^2 + x_2^2+x_3^2}+\sin ^2 t +1}, \\
f_2(t,x_1,x_2,x_3)=\frac{-x_2-t}{\sqrt{x_1^2 + x_2^2+x_3^2}+1}, \\
f_3(t,x_1,x_2,x_3)=\frac{-x_3+\arctan (x_2-t)}{\sqrt{x_1^2 + x_2^2+x_3^2} +1}.
\end{gather*}
For every $\xi=(\xi_1,\xi_2,\xi_3)$ with $|\xi|=1$, we obtain
\begin{gather*}
h(t,\xi)=\lim_{\lambda\to\infty}f(t,\lambda \xi)
=\Big(-\frac{\xi_1}{|\xi|},-\frac{\xi_2}{|\xi|},-\frac{\xi_3}{|\xi|}\Big),\\
h_0(\xi)=\Big(-\frac{\xi_1}{2|\xi|},-\frac{\xi_2}{2|\xi|},
-\frac{\xi_3}{2|\xi|}\Big).
\end{gather*}
Then
\[
(\xi|h_0 (\xi))=-\frac{1}{2} 
\Big(\frac{\xi_1^2}{|\xi|}+\frac{\xi_2^2}{|\xi|}+\frac{\xi_3^2}{|\xi|}\Big)
=-\frac{1}{2} |\xi| < 0.
\]
Hence, problem \eqref{upn} has at least one nontrivial solution.
\end{example}


\begin{thebibliography}{00}

\bibitem{a} G. Anichini, G. Conti; 
\emph{Existence of solutions of a boundary value problem through the solution 
map of a linearized type problem}, 
Rend. Sem. Mat. Univ. Politec. Torino  (1990), 149--159.

\bibitem{conti} R. Conti;
\emph{Recent trends in the theory of boundary value problems for ordinary 
differential equations}, Boll. Un. Mat. Ital. (3)  22  (1967), 135–178.

\bibitem{ge2} Z. Du, W. Ge, X. Lin;
\emph{Nonlocal boundary value problem of higher
order ordinary differential equations at resonance}, 
Rocky Mountain J. Math. 36  no. 5 (2006), 1471--1486.

\bibitem{er} L. H. Erbe;
\emph{Existence of solutions to boundary value problems for ordinary 
differential equations}, Nonlinear Anal., 6 (1982), 1155--1162.

\bibitem{ge} W. Ge, B. Sun, A. Yang;
\emph{Existence of positive solutions for self-adjoint
boundary-value problems with integral boundary conditions at resonance},
 Electron. J. Differential Equations, no. 11 (2011),  1--8.

\bibitem{gr2} A. Granas, R. Guenther, J. Lee;
\emph{Nonlinear boundary value problems for ordinary differential equations}, 
Dissertationes Math. (Rozprawy Mat.) 244 (1985) 128 pp.

\bibitem{gupta} C. P. Gupta;
\emph{Solvability of a three-point nonlinear boundary value problem for a 
second order ordinary differential equations}, 
J. Math. Anal. Appl. 168 (1998), 540--551.

\bibitem{h} P. Habets, M. Ramos, L. Sanchez;
\emph{Jumping nonlinearities for Neumann BVPs with positive forcing},
 Nonlinear Anal. 20  (1993), 533--549.


\bibitem{h2} G. A. Harris;
\emph{On multiple solutions of a nonlinear Neumann problem}, 
J. Differential Equations  95  (1992), 75--104.

\bibitem{kt3} G. L. Karakostas, P. Ch. Tsamatos;
\emph{Sufficient Conditions for the Existence of Nonnegative Solutions of 
a Nonlocal Boundary Value Problem}, Appl. Math. Letters 15 (2002) 401--407.

\bibitem{kosmatov} N. Kosmatov;
\emph{Multi-point boundary value problems on an unbounded domain at resonance},
 Nonlinear Anal. 68 (2008) 2158--2171.

\bibitem{liu} X. Liu, J. Qiu, Y. Guo;
\emph{Three positive solutions for second-order $m$-point boundary value problems}, 
Appl. Math. Comput.  156  no. 3 (2004) 733--742

\bibitem{ma} R. Ma;
\emph{Multiplicity of positive solutions for second-order three-point boundary 
value problems}, Comput. Math. Appl. 40 (2000) 193--204.

\bibitem{mawhin} J. Mawhin;
\emph{Probl\`emes aux limites du type de Neumann pour certaines \'equations 
diff\'erentielles ou aux d\'eriv\'ees partielles non lin\'eaires}. (in French) 
 \'Equations diff\'erentielles et fonctionnelles non lin\'eaires 
(Actes Conf. Internat. "Equa-Diff 73'', Brussels/Louvain-la-Neuve, 1973),  
 Hermann, Paris (1973) 123--134.

\bibitem{picone} M. Picone;
\emph{Su un problema al contorno nelle equazioni differenziali lineari 
ordinarie del secondo ordine}, (in Italian)
 Ann. Scuola Norm. Sup. Pisa Cl. Sci. 10 (1908), 1--95.


\bibitem{prz} B. Przeradzki;
\emph{Teoria i praktyka r\'owna\'n r\'o\.zniczkowych zwyczajnych},
U\L,     \L\'od\'z 2003 (in Polish).

\bibitem{s} J. Saranen, S. Seikkala;
\emph{Solution of a nonlinear two-point boundary value problem with
 Neumann-type boundary data}, J. Math. Anal. Appl.  135  no. 2 (1988), 691--701.

\bibitem{yan} Y. Sun, Y. J. Cho, D. O'Regan;
\emph{Positive solutions for singular second order Neumann boundary 
value problems via a cone fixed point theorem},
 Appl. Math. Comput.  210  no. 1 (2009) 80--86.


\bibitem{v} G. Vidossich;
\emph{A general existence theorem for boundary value problems for ordinary
 differential equations}, Nonlinear Anal.  15  no. 10 (1990), 897--914.

\bibitem{y} H. Z. Wang, Y. Li;
\emph{Neumann boundary value problems for second-order ordinary differential
equations across resonance}, SIAM J. Control Optim. 33 (1995), 1312--1325.

\bibitem{w} F. Wang, Y. Cui, F. Zhang;
\emph{Existence and nonexistence results for second-order Neumann boundary 
value problem}, Surv. Math. Appl. 4 (2009) 1--14.

\bibitem{w1} F. Wang, F. Zhang;
\emph{Existence of positive solutions of Neumann boundary value problem 
via a cone compression-expansion fixed point theorem of functional type}, 
J. Appl. Math. Comput.  35  no. 1-2 (2011) 341--349.

\bibitem{webbzima} J.R.L. Webb, M. Zima;
\emph{Multiple positive solutions of resonant and non-resonant nonlocal 
boundary value problems}, Nonlinear Anal. 71 (2009) 1369--1378.

\bibitem{wh} W .M. Whyburn;
\emph{Differential equations with general boundary conditions},
 Bull. Amer. Math. Soc.  48,  (1942), 692--704.

\bibitem{y1} Z. Yang;
\emph{Positive solutions to a system of second-order nonlocal boundary 
value problems}, Nonlinear Anal., 62 (2005) 1251--1265.

\bibitem{y2} Z. Yang and L. Kong;
\emph{Positive solutions of a system of second order boundary value problems
 involving first order derivatives via $R^n_+$-monotone matrices}, 
Nonlinear Anal., 75 (2012) 2037--2046.

\end{thebibliography}

\end{document}