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\markboth{\hfil Positive solutions for a nonlocal boundary-value problem
 \hfil EJDE--2002/46}
{EJDE--2002/46\hfil Andrzej Nowakowski \& Aleksandra Orpel \hfil}

\begin{document}

\title{\vspace{-1in}%
\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 46, pp. 1--15. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)} 
\vspace{\bigskipamount} \\
%
Positive solutions for a nonlocal boundary-value problem with vector-valued
response 
%
\thanks{\emph{Mathematics Subject Classifications:} 34B18. 
\hfil\break\indent 
\emph{Key words:} Nonlocal boundary-value problems, positive
solutions, duality method, \hfil\break\indent
 variational method. \hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted February 27, 2002. Published May 28, 2002.} }

\date{}

\author{Andrzej Nowakowski \&  Aleksandra Orpel}
\maketitle

\begin{abstract}
  Using variational methods, we study the existence of positive 
  solutions for a nonlocal boundary-value problem with vector-valued
  response. We develop duality and variational principles for this 
  problem and present a numerical version which enables the 
  approximation of solutions and gives a measure of a duality gap
  between primal and dual functional for approximate solutions
  for this problem.
\end{abstract}

\newtheorem{theorem}{Theorem}[section] 
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma} 
\newtheorem{corollary}[theorem]{Corollary}
\numberwithin{equation}{section}

\allowdisplaybreaks

\section{Introduction}

The aim of this paper is to establish the conditions for which the
differential equation
\begin{equation}
(k(t)| x'(t)| ^{q-2}x'(t))'+V_{x}(t,x(t))=0, \quad
\mbox{a.e. in }[0,T]  \label{1.1}
\end{equation}
possesses a positive solution $x:[ 0,T] \to \mathbb{R}^n$ such that $x(0)=0$, 
and satisfies the non-local boundary condition
\begin{equation}
| x'(T)| ^{q-2}x'(T)=\int_{t_{0}}^{T}| x'(s)|
^{q-2}x'(s)dg(s),  \label{boc}
\end{equation}
where $|z|=\sqrt{\sum_{i=1}^nz_{i}^{2}}$ and the integral is understood in
the Riemann-Stieltjes sense. The general assumptions for this article are as
follows:

\begin{description}
\item[(H)] The number $q$ is even and greater than zero, $T$ is an arbitrary
positive number, $t_{0}$ is a real number in the open interval $(0,T)$, 
$g=(g_{1},\dots ,g_{n}):[0,T]\rightarrow \mathbb{R}^{n}$ where $g_{i} $
increases and $g(t_{0})=0$, $V:[0,T]\times \mathbb{R}^{n}\rightarrow 
\mathbb{R}$ is Gateaux differentiable in the second variable and measurable 
in $t$, $k:[0,T]\rightarrow \mathbb{R}^{+}$, and $k(T)=1$.
\end{description}

Of course, when $k$ is an absolutely continuous, the solution of (\ref{1.1})
belongs to $C^{1,+}([0,T],\mathbb{R}^{n})$, the space of continuously
differentiable functions whose first derivative is absolutely continuous.
Note that we do not assume that $V_{x}$ is superlinear or sublinear.

In this paper we shall apply some variational methods and consider (\ref{1.1})
 as the Euler-Lagrange equation to the functional
\begin{equation}
J(x)=\int_{0}^{T}(-V(t,x(t))+\frac{k(t)}{q}|x^{\prime }(t)|^{q})dt
\label{1.2}
\end{equation}
defined on the space $A_{0}$ of absolutely continuous functions 
$x:[0,T]\rightarrow \mathbb{R}^{n}$, $x(0)=0$ with $x^{\prime }$ in 
$L^{q}([0,T],\mathbb{R}^{n})$ and the norm $\Vert x\Vert _{A_{0}}=\big(
\int_{0}^{T}|x^{\prime }(t)|^{q}dt\big)^{1/q}$. We shall denote by $A_{0b}$
the subset of $A_{0}$ consisting of functions satisfying (\ref{boc}) .

This problem appears in mathematical models of physical phenomena and it is
associated with the principle of minimal action, which holds true
universally in nature. Problems like (\ref{1.1}), (\ref{boc}) have been
studied by many authors, mainly in one-dimensional case ($n=1$) with $q=2$.
This problem is discussed also in \cite{K-T}, where a topological approach
is presented and the methods are based on the fixed point theorem in cones
by Krasnosielski \cite{dd}. In \cite{K-T}, it is assumed that $g(t_{0}+)>0$,
that $V$ has the special form $V_{x}(t,x)=q(t)f(x)$ for some continuous
functions $q:[0,1]\to \mathbb{R}$, $f:\mathbb{R}\to \mathbb{R}$, $f$ is
nonnegative for $x>0$, and $V$ quiet at infinity, and
\begin{equation}
\sup_{x\in [ 0,v]} f(x)\leq \theta v  \label{war}
\end{equation}
for some $v>0$ and $\theta >0$. It appears that weaker assumptions 
($n\geq 1$, $t$ and $x$ not separated in the left-hand side of the equation, 
$V_{x}(\cdot ,x)$ only measurable, and $V_{x}(t,\cdot )$ not necessarily
quiet at infinity) are still sufficient to conclude the existence of
solutions for (\ref{1.1}). We consider the general case when $V$ satisfies
hypothesis (\textbf{H}) which are not as strong as in \cite{K-T}. We are
also able to omit the condition $g(t_{0}+)>0$.

In this paper we study (\ref{1.1}), (\ref{boc}) by duality methods analogous
to the methods developed for (\ref{1.1}) in sublinear cases \cite{M-W,N}.
Functional (\ref{1.2}) is, in general, unbounded in $A_{0}$ (especially in
the superlinear case), so that we must look for critical points of 
(\ref{1.2}) of ''minmax'' type, or find subsets $X$ and $X^{d}$ on which 
the action
functional $J$ or its dual $J_{D}$ is bounded. We shall apply the second
approach; i.e., choose the special sets over which we calculate minimum of $J
$ and $J_{D}$ and then link this value with critical points of $J$. Of
course, we have the Morse theory and its generalization, the saddle points
theorems, and the mountain pass theorems \cite{M-W,R,RM}. However, because
of the boundary condition (\ref{boc}) they cannot be applied directly to
find critical points of $J$. Moreover, our assumptions are not strong enough
to use, for example, the Mountain Pass Theorem: $V$ is not sufficiently
smooth, $V_{x}$ and $V$ do not have growth conditions. In consequence, $J$
is not necessary of $C^{1}$class and it does not satisfy, in general, the
PS-condition. We shall develop duality, and because of this theory we are
able to omit in our proof of the existence of critical points, the
deformation lemmas, the Ekeland variational principle, and the PS type
conditions. Our approach also enables the numerical characterization of
solutions for this problem. It seems to us that this is the first
publication that applies variational methods to problem (\ref{1.1}), 
(\ref{boc}).

Let the positive cone in $\mathbb{R}^n$ be denoted by
\begin{equation*}
P=\big\{ x=(x_{1},\dots ,x_{n})\in \mathbb{R}^n: x_{i}>0,\; i=1,\dots ,n \big\}
\end{equation*}
and let $\bar{P}=\{ x\in \mathbb{R}^n:x_{i}\geq 0,\;i=1,\dots ,n\}$. We say
that $x\geq y$ for $x$, $y\in \mathbb{R}^n$ if $x-y\in \bar{P}$. If $b,c\in
\mathbb{R}^n$ by $bc$ we always mean a vector $[b_{i}c_{i}]_{i=1,\dots ,n}$
and by $\sqrt[q-1]{b}=b^{\frac{1}{q-1}}$ a vector $[\sqrt[q-1]{b_{i}}%
]_{i=1,\dots ,n}=[b_{i}^{\frac{1}{q-1}}]_{i=1,\dots ,n}$. We denote 
$\mathbf{1}=(1,\dots 1)$ the vector in $\mathbb{R}^n$.

Let us investigate the operator $\widetilde{\mathbf{A}}$, for $x\in A_{0b}$,
$x(t)\in P$, $t\in \lbrack 0,T]$.
\begin{equation*}
\widetilde{\mathbf{A}}x(t)=\alpha \frac{1}{k(t)}\int_{t_{0}}^{T}\frac{1}{k(r)%
}\int_{r}^{T}V_{x}(s,x(s))dsdg(r)+\frac{1}{k(t)}\int_{t}^{T}V_{x}(s,x(s))ds.
\end{equation*}
where $\alpha :=[\alpha _{i}]_{i=1,\dots ,n}=[(1-a_{i})^{-1}]_{i=1,\dots ,n}$
and
\begin{equation*}
a:=\int_{t_{0}}^{T}\frac{1}{k(t)}dg(t)=\left[ \int_{t_{0}}^{T}\frac{1}{k(t)}%
dg_{i}(t)\right] _{i=1,\dots ,n}.
\end{equation*}
For the functions that appear in this paper we assume the following:

\begin{description}
\item[(H1)] The function $k$ is continuous and positive, $V(t,\cdot )$ is
convex in $\overline{P}$, $V_{x}(t,\cdot )$ is continuous and nonnegative in
$P$, $t\in \lbrack 0,T]$, $|\int_{0}^{T}V(t,0)dt|<\infty $, and $\alpha >0$.

\item[(H2)] There exist a function $u\in L^{\infty }([0,T],\mathbb{R}^{n})$,
$u(t)\in P$, for a.e. $t\in (0,T)$, and constants $c$, $e\in P$, such that
for
\begin{gather*}
b(t):=\frac{(\alpha a+\mathbf{1)}c+e}{k(t)},\quad v(t)=\int_{0}^{t}|b(s)|
^{-\frac{q-2}{q-1}}u(s)ds, \\
z(t)=\int_{0}^{t}|u(s)|^{-\frac{q-2}{q-1}}b(s)ds
\end{gather*}
we have $v(t)<z(t)$, $t\in (0,T)$ and
\begin{equation*}
\int_{t_{0}}^{T}V_{x}(t,z(t))dt\leq c,\quad \widetilde{\mathbf{A}}v(t)\geq
u(t),\;t\in (0,T).
\end{equation*}
\end{description}

We see that our hypotheses on $V$ concern only convexity of $V(t,\cdot )$ in
$\bar{P}$ and that this function is rather of general type.

To construct the set $X$ first we put
\begin{align*}
\overline{X}=\Big\{& x\in A_{0b}: p(t)=k(t)| x'(t)|
^{q-2}x'(t),\; t\in [0,T]\quad \text{ belongs to }A^{q'}, \\
&x(t)\geq v(t),\;t\in (0,T)\text{ and } \int_{t_{0}}^{T}V_{x}(t,x(t))dt\leq 
c\Big\}
\end{align*}
where $A^{q'}$ is the set of absolutely continuous functions 
$x:[0,T]\to \mathbb{R}^n$ with $x'$ in $L^{q'}([0,T],
\mathbb{R}^n) $ and $t_{0}$ is given above.

\begin{remark} \label{rmk1.1}  \rm
Assumptions \textbf{(H1)} and \textbf{(H2)} imply that $
\overline{X}\neq \emptyset $. Indeed; each element $x\in A_{0b}$ such that
$v(t)=\int_{0}^{t}| b(s)| ^{-\frac{q-2}{q-1}}u(s)ds<x(t)$
for all $t\in (0,T)$ and $x(t)<z(t)$ for all $t\in [ t_{0},T] $
belongs to $\overline{X}$.
\end{remark}

We reduce the space $\overline{X}$ to the set
\begin{equation*}
X=\{ x\in \overline{X}:x\in \bar{P},\;x'(t)\geq 0,\text{ }t\in [
0,T]\} .
\end{equation*}
It is clear that $X$ depends strongly on the type of nonlinearity of $V$. We
easily see that $X$ is not in general a closed set in $A$. Assume
additionally that

\begin{description}
\item[(H3)] For $e\in P$ given in (\textbf{H2})
\begin{equation*}
\int_{0}^{t_{0}}V_{x}(t,x(t))dt\leq e\text{ for all }x\in X
\end{equation*}
and there exists $d\in \mathbb{R}$ such that
\begin{equation*}
\Big|\int_{0}^{t_{0}}V(t,x(t))dt\Big|\leq d\text{ for all }x\in X.
\end{equation*}
\end{description}

Let an operator $\mathbf{A}:X\to \bar{P}$ be given by
\begin{equation*}
\mathbf{A}x(t):=\int_{0}^{t}\widetilde{\mathbf{A}}x(s)| 
\widetilde{\mathbf{A}}x(s)| ^{-\frac{q-2}{q-1}}ds.
\end{equation*}
Since $\widetilde{\mathbf{A}}x(t)>0$ for all $t\in [ 0,T] $, so the operator
$\mathbf{A}$ is well-defined. We easily see that solving the boundary
problem (\ref{1.1})-(\ref{boc}) is equivalent to solve the operator equation
$x=\mathbf{A}x$ in $A_{0b}$.

Now we give some auxiliary results

\begin{lemma} \label{lm1.1}
The set $X$ satisfies $\mathbf{A}X\subset X$;
i.e. for each $x\in X$ there exists $w\in X$ such that
$w=\mathbf{A}x$.
\end{lemma}

\paragraph{Proof.}

It is clear that if $x\in X$ then $\mathbf{A}x(t)\geq 0$ and
\begin{equation*}
(\mathbf{A}x(t))^{\prime }=\widetilde{\mathbf{A}}x(t)|\widetilde{\mathbf{A}}%
x(t)|^{-\frac{q-2}{q-1}}\geq 0.
\end{equation*}
First we will prove that $\widetilde{\mathbf{A}}x(t)\leq b(t)$, $t\in (0,T]$
for $x\in X$. Indeed; fix $x\in X$, then using definition of $X$ we have for
all $t\in (0,T]$
\begin{align*}
\widetilde{\mathbf{A}}x(t)=& \alpha \frac{1}{k(t)}\int_{t_{0}}^{T}\frac{1}{%
k(r)}\int_{r}^{T}V_{x}(s,x(s))dsdg(r)+\frac{1}{k(t)}%
\int_{t}^{T}V_{x}(s,x(s))ds \\
\leq & \alpha \frac{1}{k(t)}\int_{t_{0}}^{T}\frac{1}{k(r)}dg(r)\int_{\
t_{0}}^{T}V_{x}(s,x(s))ds+\frac{1}{k(t)}\int_{0}^{T}V_{x}(s,x(s))ds \\
\leq & \frac{(\alpha a+\mathbf{1})c+e}{k(t)}=b(t)
\end{align*}
and
\begin{equation*}
\big|\widetilde{\mathbf{A}}x(t)\big|^{-\frac{q-2}{q-1}}\leq \big|\widetilde{%
\mathbf{A}}v(t)\big|^{-\frac{q-2}{q-1}}\leq |u(t)|^{-\frac{q-2}{q-1}}.
\end{equation*}
Combining these results we obtain
\begin{equation*}
\mathbf{A}x(t)=\int_{0}^{t}\widetilde{\mathbf{A}}x(s)|\widetilde{\mathbf{A}}%
x(s)|^{-\frac{q-2}{q-1}}ds\leq \int_{0}^{t}b(s)|u(s)|^{-\frac{q-2}{q-1}%
}ds=z(t).
\end{equation*}
Thus, assumptions \textbf{(H1)} and \textbf{(H2) }imply
\begin{equation*}
\int_{t_{0}}^{T}V_{x}(s,\mathbf{A}x(s))ds\leq
\int_{t_{0}}^{T}V_{x}(s,z(s))ds\leq c.
\end{equation*}
On the other hand we have for all $t\in (0,T)$
\begin{align*}
\mathbf{A}x(t)=& \int_{0}^{t}\widetilde{\mathbf{A}}x(s)|\widetilde{\mathbf{A}%
}x(s)|^{-\frac{q-2}{q-1}}ds \\
\geq & \int_{0}^{t}|b(s)|^{-\frac{q-2}{q-1}}\widetilde{\mathbf{A}}x(s)ds\geq
\int_{0}^{t}|b(s)|^{-\frac{q-2}{q-1}}\widetilde{\mathbf{A}}v(s)ds \\
\geq & \int_{0}^{t}|b(s)|^{-\frac{q-2}{q-1}}u(s)ds=v(t).
\end{align*}
Summarizing: $\mathbf{A}x\in X$. As the dual set to $X$ we shall consider
the set
\begin{align*}
X^{d}=\Big\{& p\in A^{q^{\prime }}:\text{ there exists $x\in X$ such that} \\
p(t)& =k(t)|x^{\prime }(t)|^{q-2}x^{\prime }(t),\;t\in \lbrack 0,T]\;\Big\}.
\end{align*}

\begin{remark} \label{rmk1.2} \rm
From the definition of $\mathbf{A}$ and $X$, we derive
that for each $x\in X$ there exists $p\in X^{d}$ such that
$p'(\cdot )=-V_{x}(\cdot ,x(\cdot ))$ and therefore
\begin{equation*}
\int_{0}^{T}\langle -p'(t),x(t)\rangle dt-\int_{0}^{T}V^{\ast }(t,-p'(t))dt=
\int_{0}^{T}V(t,x(t))dt.
\end{equation*}
Indeed; fixing  $x\in X$, Lemma \ref{lm1.1} leads to the existence of
$\widetilde{x}\in X$ such that $\widetilde{x}=\mathbf{A}x$.
So that we have
\begin{align*}
| \widetilde{x}'(t)| ^{q-2}\widetilde{x}'(t)
=&[ | \widetilde{\mathbf{A}}x(t)| |
\widetilde{\mathbf{A}}x(t)| ^{-\frac{q-2}{q-1}}] ^{q-2}
\widetilde{\mathbf{A}}x(t)| \widetilde{\mathbf{A}}x(t)|
^{-\frac{q-2}{q-1}} \\
=&[ | \widetilde{\mathbf{A}}x(t)| ^{\frac{1}{q-1}}
] ^{q-2}\widetilde{\mathbf{A}}x(t)| \widetilde{\mathbf{A}}
x(t)| ^{-\frac{q-2}{q-1}}
=\widetilde{\mathbf{A}}x(t).
\end{align*}
Set $p(t)=k(t)| \widetilde{x}'(t)| ^{q-2}
\widetilde{x}'(t)$, $t\in [ 0,T]$. Since $\widetilde{x}\in X$
, $p\in X^{d}$ we get what follows
\begin{align*}
( p(t)) '=&( k(t)| \widetilde{x}'(t)| ^{q-2}\widetilde{x}'(t)) ' \\
=&( k(t)\widetilde{\mathbf{A}}x(t)) '\\
=&\Big( \alpha \int_{t_{0}}^{T}\frac{1}{k(r)}
\int_{r}^{T}V_{x}(s,x(s))dsdg(r)+\int_{t}^{T}V_{x}(s,x(s))ds\Big)' \\
=&-V_{x}(t,x(t)).
\end{align*}
Taking into account the properties of the subdifferential, we can infer that
the required relation is satisfied.
\end{remark}

Now we study the action functional
\begin{equation*}
J(x)=\int_{0}^{T}(-V(t,x(t))+\frac{1}{q}k(t)| x'(t)| ^{q})dt-\langle
x(T),| x'(T)| ^{q-2}x'(T)\rangle
\end{equation*}
on the set $X$. To show that $\bar{x}\in X$ realizing ``min'' is a critical
point of $J$ we develop a duality theory between $J$ and dual to it $J_{D}$.

\section{Duality results}

In this section we shall develop the duality, which describes the
relationship between the critical value of $J$ and the infimum of the dual
functional $J_{D}:X^{d}\to \mathbf{R}$,
\begin{equation}
J_{D}(p)=-\int_{0}^{T}\frac{1}{q'}[ k(t)] ^{1-q'}| p(t)|
^{q'}+\int_{0}^{T}V^{\ast }(t,-p'(t))dt,  \label{funkdual}
\end{equation}
where $q':=q/(q-1)$.

Now we need a kind of perturbation of $J$ and the convexity of a function
considered on a whole space. Therefore, for each $x\in X$ the perturbation $%
J_{x}:L^{q}([0,T],\mathbb{R}^{n})\times \mathbb{R}^{n}\rightarrow \mathbb{R}$
for the functional $J$ is defined as
\begin{align*}
J_{x}(y,a)=& \int_{0}^{T}(\breve{V}(t,x(t)+y(t))-\frac{k(t)}{q}|x^{\prime
}(t)|^{q})dt+\langle x(T)-a,|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle \\
=& \int_{0}^{T}(\breve{V}(t,x(t)+y(t))-\frac{k(t)}{q}|x^{\prime }(t)|^{q})dt
\\
& +\langle x(T),|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle -\langle
a,|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle ,
\end{align*}
with
\begin{equation*}
\breve{V}(t,x)=
\begin{cases}
V(t,x) & \text{if }x\in \bar{P},\;t\in \lbrack 0,T] \\
\infty & \text{if }x\notin \bar{P},\;t\in \lbrack 0,T].%
\end{cases}%
\end{equation*}
We use this notation only for the purpose of duality and we will not change
a notation for the functional $J$ containing $V$ or $\breve{V}$. It is
associated with the fact that our all investigation reduce to the set $X$ on
which $\breve{V}(t,x)=V(t,x)$ for all $t\in \lbrack 0,T]$. Let $%
J_{x}^{\#}:X^{d}\rightarrow \mathbb{R}$, where $x\in X$, be defined as a
type of conjugate of $J_{x}$:
\begin{equation}
\begin{aligned} J_{x}^{\#}(p) =&\sup_{y\in L^{q}([0,T],\mathbb{R}^n),a\in
\mathbb{R}^n} \Big\{\int_{0}^{T}\langle y(t),p'(t)\rangle dt +\langle
p(T),a\rangle -J_{x}(y,a)\Big\} \\ =&\sup_{y\in
L^{q}([0,T],\mathbb{R}^n)}\Big\{\int_{0}^{T} \langle y(t),p'(t)\rangle
dt-\int_{0}^{T}\breve{V}(t,x(t)+y(t))dt\Big\} \\ &+\sup_{a\in
\mathbb{R}^n}\big\{\langle a,p(T)\rangle +\langle a,| x'(T)|
^{q-2}x'(T)\rangle \big\} \\ &+\int_{0}^{T}\frac{k(t)}{q}| x'(t)|
^{q}dt-\langle x(T),| x'(T)| ^{q-2}x'(T)\rangle . \end{aligned}  \label{2.1}
\end{equation}
Put $l:\mathbb{R}^{n}\rightarrow \{0,+\infty \}$, $l(b)=
\begin{cases}
0, & \text{for }b=0 \\
+\infty , & \text{for }b\neq 0.%
\end{cases}
$ We see at once that
\begin{equation}
\begin{aligned} J_{x}^{\#}(p)=&-\int_{0}^{T}\langle x(t),p'(t)\rangle
dt+\frac{1}{q} \int_{0}^{T}k(t)| x'(t)| ^{q}dt +\int_{0}^{T}V^{\ast
}(t,p'(t))dt\\ &-\langle x(T),| x'(T)| ^{q-2}x'(T)\rangle +l(|
x'(T)|^{q-2}x'(T)+p(T)) \\ =&\int_{0}^{T}\langle x'(t),p(t)\rangle
dt-\langle x(T),p(T)\rangle\\ &+\frac{1}{q}\int_{0}^{T}k(t)|
x'(t)|^{q}dt+\int_{0}^{T}V^{\ast }(t,p'(t))dt\\ &-\langle x(T),| x'(T)|
^{q-2}x'(T)\rangle +l(| x'(T)| ^{q-2}x'(T)+p(T)). \end{aligned}  \label{2.2}
\end{equation}

\begin{theorem} \label{thm2.1}
For functionals $J$\textit and $J_{D}$,
 we have the duality relation
\begin{equation*}
\inf_{x\in X}J(x)\leq \inf_{p\in X^{d}}J_{D}(p).
\end{equation*}
\end{theorem}

\paragraph{Proof.}

Our proof starts with the observation that for all $p\in X^{d}$
\begin{equation}
\inf_{x\in X}J_{x}^{\#}(-p)=J_{D}(p)  \label{2.4}
\end{equation}%
and for all $x\in X$
\begin{equation}
\underset{p\in X^{d}}{\inf }J_{x}^{\#}(-p)\geq J(x).  \label{2.5}
\end{equation}%
Because $X$ is not a linear space we need some trick to avoid calculation of
the conjugate with respect to a nonlinear space. To this effect we use the
special structure of the sets $X^{d}$ and $X$. Indeed; fix $p\in X^{d}$. The
definition of $X^{d}$ implies that there exists $x_{p}\in X$ satisfying the
equality
\begin{equation*}
p(t)=k(t)|x_{p}^{\prime }(t)|^{q-2}x_{p}^{\prime }(t)\quad \text{for all }%
t\in \lbrack 0,T]
\end{equation*}%
and, in consequence
\begin{equation*}
\int_{0}^{T}\langle x_{p}^{\prime }(t),p(t)\rangle dt-\int_{0}^{T}\frac{k(t)%
}{q}|x_{p}^{\prime }(t)|^{q}dt=\int_{0}^{T}\frac{1}{q^{\prime }}%
[k(t)]^{1-q^{\prime }}|p(t)|^{q^{\prime }}dt.
\end{equation*}%
An easy calculation yields
\begin{align*}
\int_{0}^{T}& \langle x_{p}^{\prime }(t),p(t)\rangle dt-\int_{0}^{T}\frac{%
k(t)}{q}|x_{p}^{\prime }(t)|^{q}dt \\
\leq & \sup_{x\in \{z\in X,p(T)=|z^{\prime }(T)|^{q-2}z^{\prime }(T)\}}\Big\{%
\int_{0}^{T}\langle x^{\prime }(t),p(t)\rangle dt-\int_{0}^{T}\frac{k(t)}{q}%
|x(t)^{\prime }|^{q}dt\Big\} \\
\leq & \sup_{x\in X}\Big\{\int_{0}^{T}\langle x^{\prime }(t),p(t)\rangle
dt-\int_{0}^{T}\frac{k(t)}{q}|x(t)^{\prime }|^{q}dt\Big\} \\
\leq & \sup_{x^{\prime }\in L^{q}([0,T],\mathbb{R}^{n})}\Big\{%
\int_{0}^{T}\langle x^{\prime }(t),p(t)\rangle dt-\int_{0}^{T}\frac{k(t)}{q}%
|x^{\prime }(t)|^{q}dt\Big\} \\
=& \int_{0}^{T}\frac{1}{q^{\prime }}[k(t)]^{1-q^{\prime }}|p(t)|^{q^{\prime
}}dt\,.
\end{align*}%
Actually all inequalities above are equalities. Finally for $p\in X^{d}$, we
obtain
\begin{align*}
-\inf_{x\in X}J_{x}^{\#}(-p)=& \sup_{x\in X}(-J_{x}^{\#}(-p)) \\
=& \sup_{x\in X}\Big\{\int_{0}^{T}\langle x^{\prime }(t),p(t)\rangle dt-%
\frac{1}{q}\int_{0}^{T}k(t)|x^{\prime }(t)|^{q}dt \\
& -\int_{0}^{T}V^{\ast }(t,-p^{\prime }(t))dt-\langle x(T),p(T)\rangle
+\langle x(T),|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle  \\
& -l(|x^{\prime }(T)|^{q-2}x^{\prime }(T)-p(T))\Big\} \\
=& -\int_{0}^{T}V^{\ast }(t,-p^{\prime }(t))dt+\int_{0}^{T}\frac{1}{%
q^{\prime }}[k(t)]^{1-q^{\prime }}|p(t)|^{q^{\prime }}dt=-J_{D}(p),
\end{align*}%
which proves our claim. To show the other assertion, fix $x\in X$. Remark %
\ref{rmk1.2} leads to the existence of $\bar{p}\in X^{d}$ such that
\begin{equation*}
\int_{0}^{T}\langle -\bar{p}^{\prime }(t),x(t)\rangle dt-\int_{0}^{T}V^{\ast
}(t,-\bar{p}^{\prime }(t))dt=\int_{0}^{T}\breve{V}(t,x(t))dt.
\end{equation*}%
Moreover,
\begin{align*}
\int_{0}^{T}& \langle -\bar{p}^{\prime }(t),x(t)\rangle
dt-\int_{0}^{T}V^{\ast }(t,-\bar{p}^{\prime }(t))dt \\
& \leq \sup_{p\in X^{d}}\{\int_{0}^{T}\langle -p^{\prime }(t),x(t)\rangle
dt-\int_{0}^{T}V^{\ast }(t,-p^{\prime }(t))dt\} \\
& \leq \sup_{p^{\prime }\in L^{q^{\prime }}([0,T],\mathbb{R}%
^{n})}\{\int_{0}^{T}\langle -p^{\prime }(t),x(t)\rangle
dt-\int_{0}^{T}V^{\ast }(t,-p^{\prime }(t))dt\} \\
& =\int_{0}^{T}\breve{V}(t,x(t))dt.
\end{align*}%
Combining both results we infer
\begin{align*}
\sup_{p\in X^{d}}(-J_{x}^{\#}(-p))=& \sup_{p\in X^{d}}\Big\{%
\int_{0}^{T}\langle x(t),-p^{\prime }(t)\rangle dt-\int_{0}^{T}V^{\ast
}(t,-p^{\prime }(t))dt \\
& -l(|x^{\prime }(T)|^{q-2}x^{\prime }(T)-p(T))\Big\} \\
& +\langle x(T),|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle -\frac{1}{q}%
\int_{0}^{T}k(t)|x^{\prime }(t)|^{q}dt \\
\leq & \sup_{p\in X^{d}}\big\{-l(|x^{\prime }(T)|^{q-2}x^{\prime }(T)-p(T))%
\big\} \\
& +\sup_{p\in X^{d}}\Big\{\int_{0}^{T}\langle x(t),-p^{\prime }(t)\rangle
dt-\int_{0}^{T}V^{\ast }(t,-p^{\prime }(t))dt\Big\} \\
& +\langle x(T),|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle -\frac{1}{q}%
\int_{0}^{T}k(t)|x^{\prime }(t)|^{q}dt \\
=& -\int_{0}^{T}(-\breve{V}(t,x(t))+\frac{k(t)}{q}|x^{\prime
}(t)|^{q})dt+\langle x(T),|x^{\prime }(T)|^{q-2}x^{\prime }(T)\rangle  \\
=& -J(x);
\end{align*}%
so that $\inf_{p\in X^{d}}J_{x}^{\#}(-p)\geq J(x)$. Combining (\ref{2.4})
and (\ref{2.5}) we obtain the chain of relations
\begin{equation*}
\inf_{x\in X}J(x)\leq \inf_{x\in X}\underset{p\in X^{d}}{\inf }%
J_{x}^{\#}(-p)=\underset{p\in X^{d}}{\inf }\inf_{x\in
X}J_{x}^{\#}(-p)=\inf_{p\in X^{d}}J_{D}(p).
\end{equation*}%
Denote by $\partial J_{x}(y)$ the subdifferential of $J_{x}$. Calculating $%
\partial J_{x}$ at $0$ we obtain
\begin{equation}
\begin{aligned} \partial J_{x}(0)=\Big\{ &p'\in L^{q'}( [ 0,T]
,\mathbb{R}^n) : \int_{0}^{T}V^{\ast }(t,p'(t))dt\\
&+\int_{0}^{T}\breve{V}(t,x(t))dt=\int_{0}^{T}\langle p'(t),x(t)\rangle
dt\Big\} \end{aligned}  \label{subr}
\end{equation}%
Our task is now to prove a variational principle for ``min'' arguments.

\begin{theorem} \label{thm2.2}
Let $\bar{x}\in X$ be a minimizer of $J:X\to R$,
$J(\bar{x})=\inf_{x\in X}J(x)$.
Then there exists $\bar{p}\in X^{d}$ with
$-\bar{p}'\in \partial J_{\bar{x}}(0)$, such that $\bar{p}$
satisfies
$$
J_{D}(\bar{p})=\inf_{p\in X^{d}}J_{D}(p).
$$
Furthermore,
\begin{gather}
J_{\overline{x}}(0)+J_{\bar{x}}^{\#}(-\bar{p})=0   \label{d.8}\\
J_{D}(\bar{p})-J_{\bar{x}}^{\#}(-\bar{p})=0.  \label{2.9}
\end{gather}
\end{theorem}

\paragraph{Proof.}

From Theorem \ref{thm2.1}, $J(\bar{x})\leq \inf_{p\in X^{d}}J_{D}(p)$, so to
prove the first assertion we need to show only the existence of $\bar{p}\in
X^{d}$ such that $J(\bar{x})$ $\geq $ $J_{D}(\bar{p})$. To this effect we
use Remark \ref{rmk1.2} which implies the existence of $\bar{p}\in X^{d}$
such that
\begin{equation*}
\bar{p}^{\prime }(t)=-V_{x}(t,\overline{x}(t))\text{ a.e. on }[0,T]
\end{equation*}
and further
\begin{equation}
\int_{0}^{T}\langle -\bar{p}^{\prime }(t),\overline{x}(t)\rangle
dt-\int_{0}^{T}V^{\ast }(t,-\bar{p}^{\prime }(t))dt=\int_{0}^{T}\breve{V}(t,%
\overline{x}(t))dt.  \label{pom}
\end{equation}
Combining (\ref{pom}) and (\ref{subr}) we get the inclusion $-\bar{p}%
^{\prime }\in $ $\partial J_{\bar{x}}(0)$. On the other hand an easy
computation gives the equality $J_{\bar{x}}^{\ast }(-\bar{p}^{\prime },-%
\overline{p}(T))=J_{\bar{x}}^{\#}(-\bar{p})$ (where $J_{\bar{x}}^{\ast }(-%
\bar{p}^{\prime },-\overline{p}(T))$ denotes the Fenchel transform of $J_{%
\bar{x}}$ at $(-\bar{p}^{\prime },-\overline{p}(T))$ $)$. Indeed; from the
definitions of $J_{\bar{x}}^{\ast }$ and of $J_{\bar{x}}^{\#}$ we have
\begin{align*}
J_{\bar{x}}^{\ast }& (-\bar{p}^{\prime },-\overline{p}(T)) \\
=& \sup_{y\in L^{q}([0,T],\mathbb{R}^{n}),a\in \mathbb{R}^{n}}\Big\{%
\int_{0}^{T}\langle y(t),-\overline{p}^{\prime }(t)\rangle dt+\langle a,-%
\overline{p}(T)\rangle -J_{\bar{x}}(y,a)\Big\} \\
=& \sup_{y\in L^{q}([0,T],\mathbb{R}^{n})}\Big\{\int_{0}^{T}\langle y(t),-%
\overline{p}^{\prime }(t)\rangle dt-\int_{0}^{T}\breve{V}(t,\bar{x}%
(t)+y(t))dt\Big\} \\
& +\sup_{a\in \mathbb{R}^{n}}\big\{\langle a,-\overline{p}(T)\rangle
+\langle a,|\bar{x}^{\prime }(T)|^{q-2}\bar{x}^{\prime }(T)\rangle \big\} \\
& +\int_{0}^{T}\frac{k(t)}{q}|\bar{x}^{\prime }(t)|^{q}dt-\langle \bar{x}%
(T),|\bar{x}^{\prime }(T)|^{q-2}\bar{x}^{\prime }(T)\rangle \\
=& -\int_{0}^{T}\langle \bar{x}(t),-\overline{p}^{\prime }(t)\rangle dt+%
\frac{1}{q}\int_{0}^{T}k(t)|\bar{x}^{\prime }(t)|^{q}dt \\
& +\int_{0}^{T}V^{\ast }(t,-\overline{p}^{\prime }(t))dt-\langle \bar{x}(T),|%
\bar{x}^{\prime }(T)|^{q-2}\bar{x}^{\prime }(T)\rangle +l(|\bar{x}^{\prime
}(T)|^{q-2}\bar{x}^{\prime }(T)-\overline{p}(T)) \\
=& J_{\bar{x}}^{\#}(-\overline{p}).
\end{align*}
Therefore, we obtain (\ref{d.8}). Finally
\begin{equation*}
J(\bar{x})=J_{\bar{x}}^{\#}(-\bar{p})\geq \inf_{x\in X}J_{x}^{\#}(-\bar{p}%
)=J_{D}(\bar{p}),
\end{equation*}
where the last equality is due to (\ref{2.4}). Hence $J(\bar{x})\geq J_{D}(%
\bar{p})$. Now Theorem \ref{thm2.1} leads to $J(\bar{x})=J_{D}(\bar{p}%
)=\inf_{p\in X^{d}}J_{D}(p)$. (\ref{2.9}) follows from (\ref{d.8}) and the
chain of equalities $J_{\bar{x}}(0)=-J(\bar{x})=-J_{D}(\overline{p})$.

\begin{corollary} \label{coro2.1}
Let $\bar{x}\in X$ be such that $J(\bar{x})=\inf_{x\in X}J(x)$.
Then there exists $\bar{p}\in X^{d}$ such that the pair
$(\bar{x},\bar{p})$ satisfies the relations
\begin{equation}
J_{D}(\bar{p})=\inf_{p\in X^{d}}J_{D}(p)=\inf_{x\in X}J(x)=J(
\bar{x})    \label{2.12}
\end{equation}
and
\begin{equation}
-( k(t)| \bar{x}'(t)| ^{q-2}\bar{x}'(t)) '=V_{x}(t,\bar{x}(t)),   \label{2.13}
\end{equation}
\end{corollary}

\paragraph{Proof.}

From Theorems \ref{thm2.1} and \ref{thm2.2}, we obtain immediately (\ref%
{2.12}). To show (\ref{2.13}) we use (\ref{d.8}) and (\ref{2.9}) obtaining
the two equalities:
\begin{gather*}
\int_{0}^{T}V(t,\bar{x}(t))dt+\int_{0}^{T}V^{\ast }(t,-\bar{p}%
'(t))dt -\int_{0}^{T}\langle \bar{x}(t),-\bar{p}'(t)\rangle
dt=0, \\
\int_{0}^{T}\frac{1}{q'}[ k(t)] ^{1-q'}| \bar{p}(t)|
^{q'}dt+\int_{0}^{T}\frac{k(t)}{q}| \bar{x} '(t)|
^{q}dt-\int_{0}^{T}\langle \bar{x}'(t),\bar{p}(t)\rangle dt=0
\end{gather*}
and further
\begin{equation*}
-\bar{p}'(t)=V_{x}(t,\bar{x}(t))\text{ and }\bar{p}(t)=k(t)| \bar{x}%
'(t)| ^{q-2}\bar{x}'(t)
\end{equation*}
which gives (\ref{2.13}).

\section{Variational principles and a duality gap for minimizing sequences}

In this section we prove that a statement analogous to Theorem \ref{thm2.2}
is true for a minimizing sequence of $J$. It is worth noting that as a
consequence of our duality we obtain for the first time in the superlinear
case a measure of a duality gap between primal and dual functional for
approximate solutions to (\ref{1.1}) (for the sublinear case see \cite{N}).

\begin{theorem} \label{thm3.1}
Let $\{ x_{j}\} $, $x_{j}\in X$, $j=1,2,\dots $, be a minimizing sequence for
$J$. Then there exist $p_{j}\in X^{d}$ with $-p_{j}'\in \partial J_{x_{j}}(0)$
such that  $\{ p_{j}\}$ is a minimizing sequence for $J_{D}$ i.e.,
\begin{equation}
\inf_{x\in X}J(x)=\inf_{j\in N}J(x_{j})=\inf_{j\in N}J_{D}(p_{j})
=\inf_{p\in X^{d}}J_{D}(p).   \label{rr}
\end{equation}
Furthermore,
\begin{equation*}
J_{x_{j}}(0)+J_{x_{j}}^{\#}(-p_{j})=0,
\end{equation*}
and for all $\varepsilon >0$ there exists $j_{0}\in N$
 such that for all $j\geq j_{0}$,
\begin{gather}
J_{D}(p_{j})-J_{x_{j}}^{\#}(-p_{j})\leq \varepsilon ,   \label{rr1} \\
0\leq J(x_{j})-J_{D}(p_{j})\leq \varepsilon .   \label{rr2}
\end{gather}
\end{theorem}

\paragraph{Proof.}
We first show the boundedness of $J$ on $X$. By the assumptions made on $V$
and the definition of $X$ we get for all $x\in X$, $0\leq
\int_{0}^{T}V_{x}(r,x(r))dr\leq c+e$ and, in consequence,
\begin{equation*}
\Vert x\Vert \int_{0}^{T}V_{x}(r,x(r))dr\leq \Vert x\Vert (c+e),
\end{equation*}
where $\Vert x\Vert =\max \{|x(t)|:t\in \lbrack 0,T]\}$. Therefore,
\begin{equation*}
\int_{0}^{T}V_{x}(r,x(r))x(r)dr\leq \Vert x\Vert (c+e).
\end{equation*}
Hence, by the property of the subdifferential
\begin{equation*}
\int_{0}^{T}V(r,0)dr-\int_{0}^{T}V(r,x(r))dr\geq
\int_{0}^{T}V_{x}(r,x(r))[0-x(r)]dr\geq -\Vert x\Vert (c+e)
\end{equation*}
and further
\begin{equation}
\begin{aligned} J(x)\geq &\int_{0}^{T}\frac{k(t)}{q}| x'(t)|^{q}dt
-\int_{0}^{T}V(t,0)dt \\ &-\| x\| ( c+e)-\langle x(T),| x'(T)|
^{q-2}x'(T)\rangle \\ \geq &\int_{0}^{T}\frac{k(t)}{q}|
x'(t)|^{q}dt-\int_{0}^{T}V(t,0)dt\\ &-T\| x'\|_{L^{q}([0,T],\mathbb{R}^n)}(
c+e) -|x(T)||x'(T)|^{q-1}. \end{aligned}  \label{ogr}
\end{equation}
It is clear that the definition of $J$ and (\ref{ogr}) give the estimate
\begin{equation*}
\infty >\inf_{j\in N}J(x_{j})=a>-\infty .
\end{equation*}
Fix $\varepsilon >0$. From the above there exists $j_{0}$ such that $%
J(x_{j})-a<\varepsilon $, for all $j\geq j_{0}$. We can now proceed
analogously to the proof of Theorem \ref{thm2.2}. First we observe that for
each $j\in N$ there exists $p_{j}\in X^{d}$ such that $p_{j}^{\prime
}(t)=-V_{x}(t,x_{j}(t))$ a.e. on $[0,T]$. This implies
\begin{equation*}
-p_{j}^{\prime }\in \partial J_{x_{j}}(0)\quad \text{for all }j\in \mathbb{N}%
,
\end{equation*}
which gives $J_{x_{j}}(0)+J_{x_{j}}^{\#}(-p_{j})=0$ for all $j\in \mathbb{N}$%
. As in the proof of Theorem \ref{thm2.2}, taking into account (\ref{2.4}),
this assertion yields
\begin{equation*}
J(x_{j})=J_{x_{j}}^{\#}(-p_{j})\geq \inf_{x\in
X}J_{x}^{\#}(-p_{j})=J_{D}(p_{j}).
\end{equation*}
Thus, by Theorem \ref{thm2.1}, we deduce that (\ref{rr}) holds. Then (\ref%
{rr1}) and (\ref{rr2}) follow from (\ref{rr}) and two facts:
\begin{gather*}
J_{D}(p_{j})+\varepsilon \geq J(x_{j})\text{ \ for}\quad j\geq j_{0}, \\
-J(x_{j})=J_{x_{j}}(0)=-J_{x_{j}}^{\#}(-p_{j})\text{ for each }j\in N.
\end{gather*}

Theorem \ref{thm3.1} gives the below corollary

\begin{corollary} \label{coro3.1}
Let $\{ x_{j}\}$, $x_{j}\in X$, $j=1,2,\dots $, be a minimizing sequence
for $J$. If $-p_{j}'(t)=V_{x}(t,x_{j}(t))$, then
$$
p_{j}(t)=p_{j}(T)-\int_{t}^{T}p_{j}'(s)ds,p_{j}(T)=| x_{j}'(T)|^{q-2}x_{j}'(T)
$$
which belongs to $X^{d}$ and $\{ p_{j}\}$ is a
minimizing sequence for $J_{D}$; i.e.,
\begin{equation*}
\inf_{x\in X}J(x)=\inf_{j\in N}J(x_{j})=\inf_{j\in
N}J_{D}(p_{j})=\inf_{p\in X^{d}}J_{D}(p).
\end{equation*}
Furthermore for a given $\varepsilon >0$ and sufficiently large $j$,
\begin{gather*}
J_{D}(p_{j})-J_{x_{j}}^{\#}(-p_{j})\leq \varepsilon , \
0\leq J(x_{j})-J_{D}(p_{j})\leq \varepsilon .
\end{gather*}
\end{corollary}

\section{The existence result}

In this section we prove the existence of a minimizer of $\bar{x}\in X$ of $%
J $ on $X$; i.e., $J(\bar{x})=\min_{x\in X}J(x)$ being a solution of (\ref%
{1.1}) with the non-local boundary condition (\ref{boc}).

\begin{theorem} \label{thm4.1}
Under hypotheses (\textbf{H}) and (\textbf{H1})-(\textbf{H3}) there
 exists $\bar{x}\in X$ such that $J(\bar{x})=\min_{x\in X}J(x)$
and $\bar{x}$ satisfies (\ref{1.1})-(\ref{boc}).
\end{theorem}

\paragraph{Proof.}

Let $S_{z}=\{x\in X,J(x)\leq z\}$, where $z\in \mathbb{R}$. It follows from (%
\ref{ogr}), the assumptions made on $V$, and the definition of $X$ that $%
S_{z}$ is nonempty for sufficiently large $z$ and bounded with respect to
the norm $\Vert x^{\prime }\Vert _{L^{q}([0,T],\mathbb{R}^{n})}$. This
statement implies that $S_{z}$, $z\in \mathbb{R}$, are relatively weakly
compact in $A_{0b}$. It is a well known fact that the functional $J$ is
weakly lower semicontinuous in $A_{0}$ and thus also in $X$. Therefore,
there exists a sequence $\{x_{n}\}$, $x_{n}\in X$, such that $%
x_{n}\rightharpoonup \bar{x}$ weakly in $A_{0b}$ with $\bar{x}\in A_{0b}$
(we use the fact that $\{x_{n}\}$ is uniformly convergent to $\bar{x}$) and $%
\liminf_{n\rightarrow \infty }J(x_{n})\geq J(\bar{x})$. Moreover, the
uniform convergence of $\{x_{n}\}$ to $\bar{x}$, implies that $\bar{x}\in
\overline{X}.$

Our task is now to show that $\bar{x}\in X$. For this purpose, we recall
from Corollary \ref{coro3.1} that for
\begin{equation}
p_{n}^{\prime }(t)=-V_{x}(t,x_{n}(t)),\quad t\in \lbrack 0,T]  \label{4.2}
\end{equation}
$p_{n}(t)=p_{n}(T)$ $-\int_{t}^{T}p_{n}^{\prime }(s)ds$, where $%
p_{n}(T)=|x_{n}^{\prime }(T)|^{q-2}x_{n}^{\prime }(T)$, belongs to $X^{d}$.
Then $\{p_{n}\}$ is a minimizing sequence for $J_{D}$. We easily check that $%
\{p_{n}(T)\}$ is a bounded sequence and therefore we may assume (up to a
subsequence) that it is convergent. From (\ref{4.2}) we infer that $%
\{p_{n}^{\prime }\}$ is a bounded sequence in $L^{1}$ norm and that it is
pointwise convergent to
\begin{equation}
\bar{p}^{\prime }(t)=-V_{x}(t,\bar{x}(t)).  \label{4.3}
\end{equation}
Thus $\{p_{n}\}$ is uniformly convergent to $\bar{p}$ where $\bar{p}(t)=\bar{%
p}(T)-$ $\int_{t}^{T}\bar{p}^{\prime }(s)ds$. By Corollary 3.1 we also have
that for $\varepsilon _{n}\rightarrow 0$ ($n\rightarrow \infty )$
\begin{equation*}
0\leq \int_{0}^{T}(\frac{1}{q^{\prime }}[k(t)]^{1-q^{\prime
}}|p_{n}(t)|^{q^{\prime }}+\frac{k(t)}{q}|x_{n}^{\prime
}(t)|^{q})dt-\int_{0}^{T}\langle x_{n}^{\prime }(t),p_{n}(t)\rangle dt\quad
\leq \varepsilon _{n}
\end{equation*}
where $1/q+1/q^{\prime }=1$, and so, taking the limit
\begin{equation*}
0=\int_{0}^{T}\frac{1}{q^{\prime }}[k(t)]^{1-q^{\prime }}|\bar{p}%
(t)|^{q^{\prime }}dt+\lim_{n\rightarrow \infty }\int_{0}^{T}\frac{k(t)}{q}%
|x_{n}^{\prime }(t)|^{q}dt-\int_{0}^{T}\langle \bar{x}^{\prime }(t),\bar{p}%
(t)\rangle dt.
\end{equation*}
Therefore, in view of the property of Fenchel inequality,
\begin{equation*}
0=\int_{0}^{T}\frac{1}{q^{\prime }}[k(t)]^{1-q^{\prime }}|\bar{p}%
(t)|^{q^{\prime }}dt+\int_{0}^{T}\frac{k(t)}{q}|\bar{x}^{\prime
}(t)|^{q}dt-\int_{0}^{T}\langle \bar{x}^{\prime }(t),\bar{p}(t)\rangle dt
\end{equation*}
and further
\begin{equation}
\bar{p}(t)=k(t)|\bar{x}^{\prime }(t)|^{q-2}\bar{x}^{\prime }(t).  \label{4.4}
\end{equation}
Thus $\bar{x}\in X$. Substituting (\ref{4.4}) into (\ref{4.3}) we can assert
that the minimizer $\bar{x}\in X$ is the solution of problem (\ref{1.1})-(%
\ref{boc}).

\paragraph{Acknowledgments}

The authors are in debt to anonymous referee whose suggestions led to the
improvement of the presentation and the corrections of a few mistakes in
this paper.

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\smallskip

\noindent\textsc{Andrzej Nowakowski} (e-mail: annowako@math.uni.lodz.pl)%
\newline
\textsc{Aleksandra Orpel} (e-mail: orpela@math.uni.lodz.pl)\\[3pt]
Faculty of Mathematics, University of Lodz, \newline
Banacha 22, 90-238 Lodz, Poland

\end{document}