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\usepackage{hyperref}
\begin{document}
\title{Focal points of half--linear second order differential equations}
\author{Robert Ma\v r\'\i k\\
        Masaryk University, Department of Mathematics,\\
        Jan\'a\v{c}kovo n\'am.~2A, CZ-662 95 Brno.\\
         E-mail: marik\symbol{64}math.muni.cz\\[2mm]
{\footnotesize
 Author supported by the  Grant 201/99/0295 of the
 Czech Grant Agency}
}


\date{\today}
\maketitle
\noindent\textbf{Keywords.} Half--linear differential equation, focal points,
$p-$degree functional, comparison theorems.

\noindent\textbf{AMS Class.} 34C10, 49-99.

\begin{abstract}
In the paper the solutions of the half-linear differential equation
\begin{equation*}
\Bigl(r(t)\Phi(x')\Bigr)'+c(t)\Phi(x)=0,\quad \Phi(x)=|x|^{p-1}\sgn x, \quad
p>1
\end{equation*}
with $x(a)\neq 0=x(b)$ are studied. The equation is compared with
another half--linear differential equation and with nonhomogenous
equation  having  nonpositive right hand side. The presented results
extend several theorems from the theory of linear differential equations
to the half--linear differential equations.
\end{abstract}

\section{Introduction}
The aim of this paper is to study focal points of the half--linear
differential equation
\begin{equation}
l[x]\equiv \Bigl(r(t)\Phi(x')\Bigr)'+c(t)\Phi(x)=0 \label{E}
\end{equation}
where $\Phi(x)=x |x|^{p-2}$, $p>1$, is a real constant and $r(t)$, $c(t)$
are real-valued continuous functions defined on the interval
$I=[a,\infty)$, $r(t)>0$ on~$I$.

The domain of the operator $l$ is defined to be the set of all
continuous real-valued functions $x$ defined on $I$ such that $x$
and $r\Phi(x')$ are continuously differentiable on $I$. The existence
and uniqueness of solution and basic properties of the
Eq. \eqref{E} were established in \cite{E-79,M-76}.

Remark that if $p=2$, then Eq. \eqref{E} is the well-known
linear Sturm--Liouville second order differential equation.

New comparison theorems for the half--linear equations based on the
existence of focal points are established in Section 2. These results
extend the comparison theorems of Leighton and Reid known in the
linear case. In the Section 3 the relationship between homogeneous
Eq.~\eqref{E} and  the  nonhomogenous equation
\begin{equation}
 l[y]\equiv \Bigl(r(t)\Phi(y')\Bigr)'+c(t)\Phi(y)=f(t),
 \quad f(t)\leq 0 \text{ on $[a,\infty)$}\label{N}
\end{equation}
is studied. These investigations are motivated by the paper Navarro and
Sarabia \cite{N-S}, where the linear equation is considered. The main
tool in the proofs are Riccati technique and comparison theorems from
Section 2, since the tool from \cite{N-S} has no extension in the
half--linear case.

First let us recall the main ideas of the variational technique for the
Eq. \eqref{E}, i.e. the relationship between Eq. \eqref{E} and
the corresponding $p-$degree functional
\begin{equation} \J(\eta;a,b)=\alpha |\eta(a)|^p + \beta
|\eta(b)|^p +\int_a^b \Bigl(r(t)|\eta'(t)|^p-c(t)|\eta(t)|^p\Bigr)\dt,
\label{J}
\end{equation}
which was studied in \cite{M-99}.

The equivalence between nonnegativity (positivity)
of functional \eqref{J} and nonexistence of conjugate (focal, coupled)
point is a powerful tool in the comparison and oscillation theory. See
\cite{J-K}, where the Sturm-Picone and Leighton type comparison
theorems for \eqref{E} are derived and \cite{D-98, D-99}, where the
variational technique is used to establish oscillation criteria for
\eqref{E}.

\begin{definition}
A point $d\in(a,b]$ is said to be the \textit{focal point} to the point
$a$ \textit{relative to the functional} \eqref{J} if there exists a
nontrivial solution $y(t)$ of the Eq. \eqref{E} satisfying
\begin{equation}
 \alpha \Phi(y(a))-r(a)\Phi(y'(a))=0         \label{*}
\end{equation}
and $y(d)=0$.

A point $b\in(a,\infty)$ is said to be the \textit{focal point} to the
point $a$ if there exists a nontrivial solution of the Eq. \eqref{E}
such that $x'(a)=0=x(b)$.

The focal point $b\in (a,\infty)$ is said to be \textit{the first
focal point} to the point $a$ if the interval $(a,b)$ does not contain a
focal point to the point $a$.
\end{definition}

For the focal point criteria for the Eq. \eqref{E} see \cite{D-E}.
The connection between functional \eqref{J} and Eq. \eqref{E} is
expressed by the following lemma and presents the main tool used in
Section 2.
\begin{lemma}[\cite{M-99}]
\label{mojelemma}
The functional \eqref{J} is nonnegative over the
class of functions
$$U=\{\eta\in AC[a,b]:\eta'\in L^p[a,b], \eta(b)=0, \eta\not\equiv 0\}$$
if and only if the interval $(a,b)$ does not contain a focal point to the
point $a$ relative to the functional \eqref{E},
i.e. the solution which satisfies \eqref{*} has no zero on $(a,b)$.
\end{lemma}

\begin{remark}
A closer examination of the proof of Lemma \ref{mojelemma} shows that the
lemma holds \textit{verbatim et literatim} if the words nonnegative
and $(a,b)$ are replaced by positive and $(a,b]$, respectively.
\end{remark}


\section{Comparison theorems}\label{sec:Comp}


Let us consider together with Eq. \eqref{E} the equation
\begin{equation}
L[y]\equiv \Bigl(R(t)\Phi(y')\Bigr)'+C(t)\Phi(y)=0 ,\label{velka}
\end{equation}
where $R(t)$ and $C(t)$ are continuous on $[a,b]$, $R(t)$ is a positive
function and the inequality
\begin{equation}
R(t)\leq r(t) \text{ for } t\in [a,b] \label{Rr}
\end{equation}
is satisfied. If, in addition,
\begin{equation}
 c(t)\leq C(t) \text{ for } t\in [a,b] \label{Cc}
\end{equation}
is satisfied, then \eqref{velka} is called the \textit{Sturmian majorant} of
\eqref{E}. Elbert \cite{E-79} and Jaro\v s and Kusano \cite{J-K}
proved that under conditions \eqref{Rr} and \eqref{Cc} either between two
zeros of the solution of Eq. \eqref{E} there exists zero of the
solution of Eq. \eqref{velka} or both solutions are constant multiples.
This statement is a corollary of the following statement.

\begin{Theorem}[\cite{J-K}]\label{Jarosuv}
Let $x$ be a solution of \eqref{E} such that $x(a)=0=x(b)$, $x(t)\neq 0$
for every $t\in(a,b)$. If
$$
\int_a^b\biggl[\Bigl(r(t)-R(t)\Bigr)|x'(t)|^p+\Bigl(C(t)-c(t)\Bigr)
|x(t)|^p\biggr]\dt\geq 0,
$$
then every solution $y$ of \eqref{velka} has zero in $(a,b)$, or the
Eq. \eqref{E} and \eqref{velka} are identical and $y$ is a constant
multiple of $x$. If, in addition, strict inequality is satisfied, then
$y$ has a zero in $(a,b)$. If $r,R\in C^1((a,b);(0,\infty))$ and
$$
\int_a^b\biggl[\Bigl(C(t)-{R(t)\over r(t)}
c(t)\Bigr)|x(t)|^p+r(t)\biggl({R(t)\over r(t)}\biggr)'
x(t)\Phi\bigl(x'(t)\bigr)\biggr]\dt> 0,
$$
then every solution of \eqref{velka} has a zero on $(a,b)$.
\end{Theorem}

Remark that these results are usually referred as Leighton type, since
in the linear case they are due to W. Leighton. Theorem \ref{Jarosuv}
can be extended in a natural way to the case when $x(a)\neq 0=x(b)$.
This is the contents of the following theorem and its corollaries.


\begin{theorem} \label{theo:V[x]}
Let $x$ be solution of \eqref{E} such that $x(b)= 0\neq x(a)$, $y$ be a
solution of \eqref{velka} such that $y(a)\neq 0$. Denote
$$
\alpha=r(a)\Phi\Bigl({x'(a)\over x(a)}\Bigr), \qquad
A=R(a) \Phi\Bigl({y'(a)\over y(a)}\Bigr).
$$
If
$$
V[x]\equiv (\alpha-A)|x(a)|^{p}+
\int_a^b\biggl[\Bigl(r(t)-R(t)\Bigr)|x'(t)|^p+\Bigl(C(t)-c(t)\Bigr)
|x(t)|^p\biggr]\dt\geq 0,
$$
then $y$ has a zero on $(a,b]$. If, in addition, the sharp inequality is
satisfied, then $y$ has a zero on $(a,b)$.
\end{theorem}

\begin{proof}
Denote
\begin{gather*}
\J_\alpha(\eta;a,b)=\alpha |\eta(a)|^p+\int_a^b (r|\eta'|^p-c|\eta|^p)\dt\\
\J_A(\eta;a,b)=A|\eta(a)|^p+\int_a^b (R|\eta'|^p-C|\eta|^p)\dt.
\end{gather*}
The integration by parts yields that $\J_\alpha(x;a,b)=0$ and hence
\begin{equation*}
\J_A(x;a,b)=\J_A(x;a,b)-\J_\alpha(x;a,b)=-V[x]\leq 0 \quad (<0).
\end{equation*}
Now the conclusion follows from Lemma \ref{mojelemma} and the remark
below this lemma.
\end{proof}

\begin{corollary}[\cite{M-98}] \label{Srovnavaci}
Suppose \eqref{Rr} and \eqref{Cc}. Let $x$, $y$ be solutions of \eqref{E},
\eqref{velka} from Theorem \ref{theo:V[x]}. If
\begin{equation}
%R(a)\Phi\Bigl({y'(a)\over y(a)}\Bigr)\leq
%r(a)\Phi\Bigl({x'(a)\over x(a)}\Bigr),\label{nerovnost}
A\leq \alpha, \label{nerovnost}
\end{equation}
then  $y$ has a zero on $(a,b]$. Moreover, $y$ has a zero
on $(a,b)$ in the case when any of the following conditions is satisfied:
\bod (i) the sharp inequality in \eqref{nerovnost} is satisfied, or

\bod (ii) $c(\cdot)\not\equiv C(\cdot)$ on $(a,b)$, or

\bod (iii) $R(t_0)<r(t_0)$ and $c(t_0)\neq 0$ in some
$t_0\in(a,b)$.

\end{corollary}
\begin{proof}Follows immediately from Theorem \ref{theo:V[x]}.
\end{proof}

Under additional condition on nonnegativity of the function $c(x)$ the
inequalities \eqref{Cc} and \eqref{nerovnost} can be replaced by more general
integral inequality, as shows the following corollary. In the linear case
is this corollary due to Leighton \cite[Theorem 1]{L-83}.

\begin{corollary} \label{Srovnavaci2}
Suppose \eqref{Rr}, $\alpha\leq 0$ and $c(t)\geq 0$ for $t\in [a,b]$.
Let $x$, $y$ be solutions of \eqref{E}, \eqref{velka} from Theorem
\ref{theo:V[x]}. If
\begin{equation}
%r(a)\Phi\Bigl({x'(a)\over x(a)}\Bigr) -
%R(a)\Phi\Bigl({y'(a)\over y(a)}\Bigr) +
\alpha-A+
\int_a^t \Bigl[C(s)-c(s)\Bigr]\ds\geq 0
\quad\text{for every $t\in[a,b]$}\label{nerovnost2}
\end{equation}
then the function $y(t)$ has a zero for some $t\in(a,b]$. Moreover, $y(t)$
has a zero on $(a,b)$ if the sharp inequality in \eqref{nerovnost2} is
satisfied, or if the condition \textup{(iii)} of Corollary \ref{Srovnavaci}
holds.
\end{corollary}
\begin{proof}
Without loos of generality we can consider $x(t)>0$ for $t\in[a,b)$.
%Let $\alpha$, $A$ be the numbers from Theorem \ref{theo:V[x]}.
The following integral can be written as a double integral
$$
\int_a^b \Bigl[C(t)-c(t)\Bigr]|x(t)|^p\dt=
\int_a^b\int_0^{|x(t)|^p} \Bigl[C(t)-c(t)\Bigr]\ds\dt.
$$
From \eqref{E} it follows
$$
r(t)\Phi(x'(t))=\alpha\Phi(x(a))-\int_a^t c(s)\Phi(x(s))\ds
\qquad t\in [a,b)
$$
and with regard to $\alpha\leq 0$ and $c(t)\geq 0$ the function $x(t)$
is decreasing or it is constant in some interval $[a,d]\subset [a,b)$
and decreasing on $[d,b]$. The same is true also for
$|x(t)|^p$. Then we can interchange the order of the integration and we
obtain
\begin{multline*}
\int_a^b \Bigl[C(t)-c(t)\Bigr]|x(t)|^p\dt=
\int_0^{|x(a)|^p} \int_0^{\phi(s)} \Bigl[C(t)-c(t)\Bigr]\dt\ds\\
\geq\int_0^{|x(a)|^p}[A-\alpha]\ds=[A-\alpha]|x(a)|^p.
\end{multline*}
Note that $\phi(s)$ is well-defined and at least continuous for
$0\leq s\leq|x(a)|^p$. By this computation we get
$$V[x]\geq\int_a^b\Bigl[r(t)-R(t)\Bigr]|x'(t)|^p\dt\geq0,$$
and by Theorem \ref{theo:V[x]} $y$ has a zero on $(a,b]$. If the strict
inequality is satisfied, then $V[x]>0$ and $y$ has a zero on $(a,b)$.
\end{proof}


The next comparison theorem is an alternative to Theorem
\ref{theo:V[x]}. It can be considered as a supplement to the second
inequality in Theorem \ref{Jarosuv}.

\begin{theorem}\label{theo:c1}
Let $r,R\in C^1([a,b];(0,\infty))$, $\alpha\leq 0$,
$x$, $y$ be solutions of \eqref{E}, \eqref{velka} from Theorem
\ref{theo:V[x]}.
If
%$$
\begin{multline*}
{\cal V}[x]\equiv \Bigl({R(a)\over r(a)} \alpha-A\Bigr) |x(a)|^p +\\
+ \int_a^b
\biggl[\Bigl(C(t)-{R(t)\over r(t)} c(t)\Bigr)|x(t)|^p + r(t)
\biggl({R(t)\over r(t)}\biggr)' x(t) \Phi\bigl(x'(t)\bigr)\biggr]\dt\geq 0,
\end{multline*}
%$$
then $y$ has a zero on $(a,b]$. If in addition the sharp inequality
holds, then $y$ has zero on $(a,b)$.
\end{theorem}

\begin{proof}
Without loos of generality suppose that $x(t)>0$ for $t\in[a,b)$.
Then
\begin{multline}
L[x]=\Bigl({R\over r} r \Phi(x')\Bigr)' +C\Phi(x)=
\Bigl({R\over r}\Bigr)'r\Phi(x') + {R\over r} \Bigl(r\Phi(x')\Bigr)'
+C\Phi(x) \\=
\biggl[C-{R\over r} c\biggr]\Phi(x)+
\Bigl( {R\over r}\Bigr)' r \Phi(x') \label{xL[x]}
\end{multline}
and since integration by parts shows that
\begin{multline*}
\int_a^b \Bigl({R\over r}\Bigr)' r \Phi(x') x\dt=\bigl[R \Phi(x')x \bigr]_a^b
- \int_a^b\biggl[{R\over r}x \Bigl(r\Phi(x')\Bigr)'+{R\over r} r \Phi(x') x'
\biggr]\dt \\=
-R(a) \Phi\bigl(x'(a)\bigr)x(a)-\int_a^b\Bigl[R|x'|^p-{R\over r}c|x|^p
\Bigr]\dt,
\end{multline*}
it holds
\begin{equation*}
\int_a^b xL[x] \dt=
-{R(a) \over r(a)} \alpha |x(a)|^p-\int_a^b\Bigl[R|x'|^p-C|x|^p\Bigr]\dt.
\end{equation*}
Hence
\begin{multline}
\J_A(x;a,b)=\Bigl(A-{R(a)\over r(a)}\alpha\Bigr) |x(a)|^p+ \int_a^b
\Bigl[R|x'|^p-C|x|^p\Bigr]\dt+{R(a) \over r(a)} \alpha |x(a)|^p\\=
\Bigl(A-{R(a)\over r(a)}\alpha\Bigr) |x(a)|^p-\int_a^b xL[x]\dt
\end{multline}
and by \eqref{xL[x]}
$$
\J_A(x;a,b)=\Bigl(A-{R(a)\over r(a)}\alpha\Bigr) |x(a)|^p-
\int_a^b \biggl[\bigl(C-{R\over r} c\bigr)\Phi(x)+
\Bigl( {R\over r}\Bigr)' r \Phi(x')\biggr] \dt=-{\cal V}[x]
$$
Now the statement follows from Lemma \ref{mojelemma}.
\end{proof}

The following corollary is of Reid's type, see \cite[p. 29]{S-66} and
follows immediately from Theorem \ref{theo:c1}.


\begin{corollary}
Let $r,R\in C^1([a,b];(0,\infty))$, $c(t)\geq 0$ for $t\in[a,b]$. Further let
$x$ and $y$ be solutions of \eqref{E} and \eqref{velka}, respectively,
such that $x'(a)\leq 0=x(b)$, $x$ is positive on $[a,b)$, $y(a)\neq 0$
and ${x'(a)\over x(a)}\geq {y'(a)\over y(a)}$. If
$$
\Bigl({R(t)\over r(t)}\Bigr)'\leq 0 \quad\text{and}\quad
{C(t)\over R(t)}\geq {c(t)\over r(t)} \quad \text{for $t\in[a,b]$,}
$$
then $y$ has a zero on $(a,b]$.
\end{corollary}

If we apply the method from the proof of Corollary \ref{Srovnavaci2}, we get
the following

\begin{corollary}
Let $r,R\in C^1([a,b];(0,\infty))$, $c(t)\geq 0$ for $t\in[a,b]$. Further let
$x$ and $y$ be solutions of \eqref{E} and \eqref{velka}, respectively,
such that
$x'(a)\leq 0=x(b)$, $x$ is positive on $[a,b)$, $y(a)\neq 0$.
If $\Bigl({R(t)\over r(t)}\Bigr)'\leq 0$ and
$$
{R(a)\over r(a)}\alpha - A + \int_a^t\Bigl[C(s)-{R(s)\over r(s)} c(s)\Bigr]
\ds\geq 0 \quad \text{for every $t\in[a,b]$,}
$$
then $y$ has a zero on $(a,b]$.
\end{corollary}

%\begin{remark}
%In many papers, especially when the equation \eqref{E} is considered without
%the functional \eqref{J}, the focal point is defined with $\alpha=0$ in
%Definition \ref{Def:focal}, i.e. a point $x_1\in (x_0,b)$ is said to be focal
%point to the point $x_0$ if there exists a nontrivial solution of
%\eqref{E} such that $y'(x_0)=0=y(x_1)$. If there exists in every
%neighborhood of $+\infty$ such a point $x_0$ and it's focal point $x_1$, then
%the \eqref{E} is called \textit{focal oscillatory}. This is motivated by
%the property which follows from Sturmian type theorem of separation of zeros
%which states, that \eqref{E} is oscillatory iff there exists a pair of
%conjugate points in every neighborhood of $+\infty$. Due the separation of
%zeros of two solutions of \eqref{E}, oscillation implies focal
%oscillation. The reverse statement does not hold, which can be showed on the
%example of Euler equation, which is nonoscillatory, but it is focal
%oscillatory, for more details see \cite{N}, where the linear systems are
%studied. Corollary \ref{Srovnavaci} then states, that if the equation
%\eqref{E} is focal oscillatory, then it's Sturmian majorant is focal
%oscillatory.
%\end{remark}

The same method as in Theorem \ref{theo:V[x]} can be applied to the case of
the functional \eqref{J} defined on the class of functions with
$x(a)=0\neq x(b)$. This is sketched in the following theorem.

\begin{theorem}\label{theo:V_b[x]}
Let $x$ and $y$ be solutions of \eqref{E} and \eqref{velka},
respectively, such that $x(a)= 0\neq x(b)$, $y(b)\neq 0$. Denote
$$
\beta=-r(b)\Phi\Bigl({x'(b)\over x(b)}\Bigr), \qquad
B=-R(b) \Phi\Bigl({y'(b)\over y(b)}\Bigr).
$$
If
$$
V_b[x]\equiv (\beta-B)|x(b)|^{p}+
\int_a^b\biggl[\Bigl(r(t)-R(t)\Bigr)|x'(t)|^p+\Bigl(C(t)-c(t)\Bigr)
|x(t)|^p\biggr]\dt\geq 0,
$$
then $y$ has a zero on $[a,b)$. If in addition the sharp inequality is
satisfied, then $y$ has a zero on $(a,b)$.
\end{theorem}

\begin{proof}
Theorem follows from the fact that
$$
\J_B(x;a,b)\equiv B|x(b)|^p + \int_a^b \Bigl[ R(t)|x'(t)|^p - C(t)|x(t)|^p
\Bigr]\dt = -V_b[x]\leq 0
$$
and from \cite[Theorem 1]{M-99}
\end{proof}

\begin{corollary}
Suppose that \eqref{Rr} and $c(t)\geq0$ for $t\in(a,b)$ holds. Let $x$ and
$y$ be solutions of \eqref{E} and \eqref{velka}, respectively,
$x'(b)\geq0=x(a)$, $x$ positive  on $[a,b)$, $y(b)\neq0$. Let $\beta$,
$B$ be the numbers from the preceding theorem. If
\begin{equation}
\beta-B+\int_t^b \Bigl[C(s)-c(s)\Bigr]\ds\geq0
\quad\text{for every $t\in[a,b]$,}\label{nerovnost22}
\end{equation}
then the function $y(t)$ has a zero for some $t\in[a,b)$. Moreover, $y(t)$
has a zero on $(a,b)$ if the sharp inequality in \eqref{nerovnost22} is
satisfied, or if the condition \textup{(iii)} of Theorem \ref{Srovnavaci}
holds.
\end{corollary}

\section{Nonhomogenous equation}
In this section we will study the solution of the half-linear equation \eqref{E}
\begin{equation}
l[x]\equiv \Bigl(r(t)\Phi(x')\Bigr)'+c(t)\Phi(x)=0, 
\end{equation}
and nonhomogenous differential equation \eqref{N}
\begin{equation}
l[y]\equiv \Bigl(r(t)\Phi(y')\Bigr)'+c(t)\Phi(y)=f(t),
\quad f(t)\leq 0 \text{ on $[a,b]$.} 
\end{equation}
%and the differential inequality
%\begin{equation}
%l[y]\leq 0 \label{I}
%\end{equation}
%with the initial conditions $x(a)\neq0$, $x'(a)=0$.

In the following we present an useful tool in investigating of the operator
$l[\cdot]$.


\textbf{Riccati technique.} If $x$ is a function which has no
zero on the interval $(a,b)$, then the function
$w(t)=r(t)\Phi\bigl({x'(t)\over x(t)}\bigr)$
is well defined on $(a,b)$ and satisfies
\begin{equation}
R[w]:=w'+c(t)+(p-1)r^{1-q}(t)|w|^{q}={l[x](t)\over \Phi(x(t))}, \label{R}
\end{equation}
where $q={p\over p-1}$ is the conjugate number to $p$. If $x$ is a
solution of the Eq. \eqref{E} then $w$ solves the Riccati equation
$R[w]=0$. The point $a$ ($b$) is a zero of the function $x$ if and only if
$w(a+)=\infty$ ($w(b-)=-\infty$). For every solution of Riccati equation
the corresponding solution of Eq. \eqref{E} is unique up to the constant
multiple.

\begin{theorem}\label{th31}
 Let $\alpha, \alpha'\in\R$, $\alpha \neq 0$, $x_{1}$ and $x_{2}$ be
 solutions of
 \begin{gather*}
   l[x_{1}]=0 \qquad x_{1}(a)=\alpha \qquad x'_{1}(a)=\alpha',\\
   l[x_{2}]\leq0 \qquad x_{2}(a)=\alpha \qquad x'_{2}(a)=\alpha'
 \end{gather*}
respectively. Suppose that the functions $x_{1}$, $x_2$ have zero on
$(a,\infty)$ and denote by $b_1$ and $b_2$ the first zero rigths of $a$
of the function $x_1$, $x_2$, respectively. Define $d=\min(b_1,b_2)$.
Then
$$x_2 (t) \leq x_1(t) \quad \text{for $t\in[a,d]$,}$$
i.e. $x_2(t)\leq x_1(t)$ on the interval, where both functions are
nonnegative (nonpositive).
\end{theorem}
\begin{proof}
 Let $\eps>0$ and denote $I_\eps=[a,d-\eps]$.
 Then $w_1(t)=r(t)\Phi\Bigl({x'_1(t)\over x_1(t)}\Bigr)$,
 $w_2(t)=r(t)\Phi\Bigl({x'_2(t)\over x_2(t)}\Bigr)$
 are solutions of the  Riccati equation $R[w_1]=0$,
 $R[w_2]={l[x_2](t)\over \Phi\bigl(x_2(t)\bigr)}$ on
 $I_\eps$, respectively and $w_1(a)=w_2(a)$.

 Suppose $\alpha>0$. Then
 ${l[x_2](t)\over \Phi\bigl(x_2(t)\bigr)}\leq 0$  for $t\in I_\eps$ and
 by standard argumentation $w_2(t)\leq w_1(t)$ for $t\in I_\eps$.
 Hence,
 \begin{equation}
  x_2(t)=\alpha e^{\int_a^t \Phi^{-1}\bigl({w_2(s)\over
  r(s)}\bigr)\text{d}s}\leq
  \alpha e^{\int_a^t \Phi^{-1}\bigl({w_1(s)\over
  r(s)}\bigr)\text{d}s}=x_1(t)
  \label{1234}
 \end{equation}
for $t\in I_\eps$.

If $\alpha<0$, then ${l[x_2](t)\over \Phi\bigl(x_2(t)\bigr)}\geq 0$
and $w_2(t)\geq w_1(t)$ for $t\in I_\eps$, which implies \eqref{1234}.
The limit process $\eps\to0+$  completes the proof.
\end{proof}

The following corollary is an extension of Theorem 1 and 3 from \cite{N-S}.

\begin{corollary}
Let $b$ be the first right focal point to the point $a$. Then every solution
$y$ of Eq. \eqref{N} which satisfies $y(a)>0$ ($y(a)<0$) and $y'(a)=0$ has
a zero (has no zero) on the interval $(a,b]$ ($(a,b)$).
\end{corollary}


\begin{proof}
We choose $\alpha'=0$ and $\alpha=y(a)$ in the previous theorem. Then
$b_1$ is the first focal point to the point $a$ and the conclusion follows
from Theorem \ref{th31}.

An easy alternative proof of the part concerning the positive solutions
is the following:
on the interval where $y$ is positive is the function $y(t)$
solution of homogeneous equation
$$
\Bigl(r(t)\Phi(x')\Bigr)'+\Bigl[c(t)-{f(t)\over \Phi(y(t))}\Bigr]\Phi(x)=0.
$$
Now the conclusion follows from the fact that this equation is Sturmian
majorant to \eqref{E} and from the Corollary \ref{Srovnavaci}.
\end{proof}

The following theorem compares solutions of two nonhomogenous equations.

\begin{theorem}\label{Cor01}
Let $x_i$ $(i=1,2)$ be the solutions of the equation
$$l[x_i]=\Bigl(r(t)\Phi (x'_i)\Bigr)+c(t)\Phi(x_i)=f_i(t),$$
where $f_2(t)\leq f_1(t)\leq 0$ on $[a,b]$. If
$$x_1(a)>x_2(a)>0\quad \text{and} \quad {x'_1(a)\over x_1(a)}\geq
{x'_2(a) \over x_2(a)},$$ then $x_1(t)>x_2(t)$ on the interval,
where both $x_{1,2}$ are positive.
\end{theorem}
\begin{proof}
 Suppose, by contradiction, that there exists a point $d$, $d\geq a$ such that
 $x_1(t)> x_2(t)>0$ for $t\in [a,d)$ and $x_1(d)=x_2(d)>0$. For
 $i=1,2$ the Riccati variable  $w_i(t)=r(t)\Phi\Bigl({x'_i(t)\over x_i(t)}
 \Bigr)$
 solves the Riccati equation
 $$
 R[w_i]=w_i'+(p-1)r^{1-q}(t)|w_i|^{q}+c(t)-{f_i(t)\over
 \Phi\bigl(x_i(t)\bigr)}=0
 $$
 on $I$ and $w_1(a)\geq w_2(a)$. From the assumptions it follows
 $$c(t)-{f_2(t)\over \Phi\bigl(x_2(t)\bigr)}\geq
 c(t)-{f_2(t)\over \Phi\bigl(x_1(t)\bigr)} \geq
 c(t)-{f_1(t)\over \Phi\bigl(x_1(t)\bigr)} \quad \text{for $t\in [a,d]$.}
 $$
 Hence $w_2(t)\leq w_1(t)$ and
\begin{equation}
 x_2(t)=x_2(a) e^{\int_a^t \Phi^{-1}\bigl({w_2(s)\over
 r(s)}\bigr)\text{d}s}
 <x_1(a) e^{\int_a^t \Phi^{-1}\bigl({w_1(s)\over r(s)}\bigr)\text{d}s}=x_1(t)
\end{equation}
holds for $t\in [a,d]$. Particularly, for $t=d$ we have $x_2(d)<x_1(d)$, a
contradiction.
\end{proof}


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