Funkcialaj Ekvacioj, 37 (1994) 505-520
Solutions with Prescribed Numbers of Zeros for
Nonlinear Second Order Differential Equations
By
Manabu NAITO and Y?ki NAITO
(Hiroshima University, Japan)
1. Introduction
This paper is concerned with nonlinear second order differential equations
of the form
(1.1)
,
$a¥leq t¥leq b$
$¥mathrm{x}^{¥prime¥prime}+p(t)f(x)=0$
In equation (1.1), we assume that
(1.2)
$p¥in C^{1}[a, b]$
and that
$f$
(1.3)
$f¥in C(-¥infty, ¥infty)$
$p$
and
.
satisfies
$p(t)>0$
for
$a¥leq t¥leq b$
,
satisfies
for
$u¥in(0, ¥infty)$
, $f(-u)=-f(u)$ for
$u¥in(-¥infty, ¥infty)$
, and $f(u)>0$
.
In addition we impose either of the following conditions (F. 1) and (F.2) on :
$f$
(F. 1)
(F.2)
$¥left¥{¥begin{array}{l}f¥mathrm{i}¥mathrm{s}¥mathrm{l}¥mathrm{o}¥mathrm{c}¥mathrm{a}¥mathrm{l}1¥mathrm{y}¥mathrm{L}¥mathrm{i}¥mathrm{p}¥mathrm{s}¥mathrm{c}¥mathrm{h}¥mathrm{i}¥mathrm{t}¥mathrm{z}¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{t}¥mathrm{i}¥mathrm{n}¥mathrm{u}¥mathrm{o}¥mathrm{u}¥mathrm{s}¥mathrm{o}¥mathrm{n}[0,¥infty)¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{s}¥mathrm{a}¥mathrm{t}¥mathrm{i}¥mathrm{s}fi ¥mathrm{e}¥mathrm{s}¥¥¥lim_{u¥rightarrow+0}¥frac{f(u)}{u}=0¥mathrm{a}¥mathrm{n}¥mathrm{d}¥lim_{u¥rightarrow¥infty}¥frac{f(u)}{u}=¥infty,.¥end{array}¥right.$
$¥left¥{¥begin{array}{l}f¥mathrm{i}¥mathrm{s}1¥mathrm{o}¥mathrm{c}¥mathrm{a}¥mathrm{l}¥mathrm{l}¥mathrm{y}¥mathrm{L}¥mathrm{i}¥mathrm{p}¥mathrm{s}¥mathrm{c}¥mathrm{h}¥mathrm{i}¥mathrm{t}¥mathrm{z}¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{t}¥mathrm{i}¥mathrm{n}¥mathrm{u}¥mathrm{o}¥mathrm{u}¥mathrm{s}o¥mathrm{n}(0,¥infty)¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{s}¥mathrm{a}¥mathrm{t}¥mathrm{i}¥mathrm{s}fi ¥mathrm{e}¥mathrm{s}¥¥¥lim_{u¥rightarrow+0}¥frac{f(u)}{u}=¥infty ¥mathrm{a}¥mathrm{n}¥mathrm{d}¥lim_{u¥rightarrow¥infty}¥frac{f(u)}{u}=0.¥¥¥mathrm{M}¥mathrm{o}¥mathrm{r}¥mathrm{e}¥mathrm{o}¥mathrm{v}¥mathrm{e}¥mathrm{r},f(u)¥mathrm{i}¥mathrm{s}¥mathrm{n}¥mathrm{o}¥mathrm{n}¥mathrm{d}¥mathrm{e}¥mathrm{c}¥mathrm{r}¥mathrm{e}¥mathrm{a}¥mathrm{s}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥mathrm{a}¥mathrm{n}¥mathrm{d}f(u)/u¥mathrm{i}¥mathrm{s}¥mathrm{n}¥mathrm{o}¥mathrm{n}¥mathrm{i}¥mathrm{n}¥mathrm{c}¥mathrm{r}¥mathrm{e}¥mathrm{a}¥mathrm{s}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥¥¥mathrm{o}¥mathrm{n}¥mathrm{s}¥mathrm{o}¥mathrm{m}¥mathrm{e}¥mathrm{i}¥mathrm{n}¥mathrm{t}¥mathrm{e}¥mathrm{r}¥mathrm{v}¥mathrm{a}¥mathrm{l}¥mathrm{o}¥mathrm{f}¥mathrm{t}¥mathrm{h}¥mathrm{e}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{m}(0,u_{0}),u_{0}>0.¥end{array}¥right.$
, is an important special
The case where $f(u)=|u|^{¥gamma}sgnu$ with $¥gamma>0,$
case. If $¥gamma>1$ , then condition (F. 1) is satisfied; and if $0<¥gamma<1$ , then condition
(F.2) is satisfied. For this case, equation (1.1) becomes the Emden-Fowler
equation
$¥neq 1$
(1.1)
,
$x^{¥prime¥prime}+p(t)|x|^{¥gamma}¥mathrm{s}¥mathrm{g}¥mathrm{n}x=0$
$a¥leq r¥leq b$
.
506
Manabu NAITO and Y?ki NAITO
In this paper we consider the solution of (1.1) satisfying the initial condition
(1.5)
$¥mathrm{x}(a)=0$
,
$¥mathrm{x}^{¥prime}(a)=¥lambda$
,
where
is a real parameter. We denote by
the solution of the initial
In
2
value problem (1.1)?(1.5).
Section we show that
exists and is unique
on the whole interval $[a, b]$ . Our purpose is to study the numbers of zeros
of
in ( , . Since
for $a¥leq t¥leq b$ , it is enough to consider
the case where the parameter is positive. The main results are as follows:
$¥lambda$
$x_{¥lambda}(t)$
$x_{¥lambda}(t)$
$x_{¥lambda}(t)$
$a$
$x¥_¥lambda(t)=-¥mathrm{x}_{¥lambda}(t)$
$b]$
$¥lambda$
Theorem 1.1.
such that
$¥{¥lambda_{k}¥}_{k=1}^{¥infty}$
(i) if
(ii) if
(iii) if
satisfied.
Then there exists
, and
$ 0<¥lambda_{1}<¥lambda_{2}<¥cdots<¥lambda_{k}<¥lambda_{k+1}<¥cdots,¥lim_{k¥rightarrow¥infty}¥lambda_{k}=¥infty$
has no zeros in ( ,
;
$=1,2$
then
,
,
has
at
most
zeros in
];
[
$¥lambda=¥lambda_{k}(k=1,2, ¥cdots)$ , then
has exactly $k-1$ zeros in $(a, b)$ and
$¥lambda¥in(0, ¥lambda_{1})$
$¥lambda¥in$
$¥lambda_{k},$
, then
$x_{¥lambda}(t)$
$b]$
$a$
$¥lambda_{k+1})(k$
$¥cdots)$
$k$
$x_{¥lambda}(t)$
$(a,$
$b$
$x_{¥lambda}(t)$
satisfies
Theorem 1.2.
such that
$¥{¥lambda_{k}¥}_{k=1}^{¥infty}$
(i) if
(ii) if
(iii) if
Suppose that condition (F.I) is
$x_{¥lambda}(b)=0$
.
Suppose that condition (F.2) is
satisfied.
Then there exists
, and
$0<¥cdots<¥lambda_{k+1}<¥lambda_{k}<¥cdots<¥lambda_{2}<¥lambda_{1},¥lim_{k¥rightarrow¥infty}¥lambda_{k}=0$
has no zeros in ( , ;
$(k
=1,2,
¥
cdots)$
, then
has at most zeros in ( ,
;
]
$¥lambda=¥lambda_{k}(k=1,2, ¥cdots)$ , then
has exactly $k-1$ zeros in $(a, b)$ and
$¥lambda¥in$
$(¥lambda_{1}, ¥infty)$
$¥lambda¥in(¥lambda_{k+1},$
satisfies
, then
$x_{¥lambda}(t)$
$a$
$k$
$x_{¥lambda}(t)$
$¥lambda_{k}$
$b]$
$a$
$b]$
$x_{¥lambda}(t)$
$x_{¥lambda}(b)=0$
.
By Theorems 1.1 and 1.2 we find that the Emden-Fowler equation (1.4)
has an explicit duality between the superlinear case $(¥gamma>1)$ and the sublinear
case
$(0<¥gamma<1)$ .
We now consider the boundary condition
(1.6)
$¥mathrm{x}(a)=¥mathrm{x}(b)=0$
.
Then Theorems 1.1 and 1.2 guarantee the existence of an infinite sequence of
solutions of the boundary value problem (1.1)?(1.6). More precisely, we have
the following corollary.
Corollary 1.1. Suppose that either (F.I) or (F.2) is satisfied. Then, for
any nonnegative integer , there exists a solution $x(t)$ of the boundary value
problem (1.1)?(1.6) which has exactly
zeros in $(a, b)$ .
$k$
$k$
For the case where (F.I) is satisfied, the existence of an infinite sequence
of solutions of (1.1)?(1.6) with prescribed numbers of zeros has been studied
by Hartman [8] and Hooker [9]. Actually, Corollary 1.1 is given in [8,
Theorem 5.1 and Corollary 5. 1] and [9, Theorem 4. 1] under a weaker condition
507
Nonlinear Second Order Dijfferentia[Equations
and /. For the superlinear Emden-Fowler equation (1.4) with $¥gamma>1$ , we
refer to Nehari [11] and Tal [12]. For the sublinear Emden-Fowler equation
(1.4) with $0<¥gamma<1$ , Corollary 1.1 is new.
Corollary 1.1 can be applied to nonlinear elliptic boundary value problems
in annular domains. Consider the Dirichlet problem
on
$p$
(1.7)
$¥Delta u+p(|x|)f(u)=0$
in
(1.8)
$u=0$
on
$¥Omega$
$¥partial¥Omega$
,
where $¥Omega=¥{x¥in R^{N} : 0<a<|x|<b¥}$ , $N¥geq 2$ . We assume that and satisfy
(1.2) and (1.3), respectively. Then, as a consequence of Corollary 1.1, we can
obtain the following corollary.
$p$
$f$
Corollary 1.2. Suppose that either (F.I) or (F.2) is satisfied. Then, for
any nonnegative integer , there exists a radial solution $u(r)$ , $r=|x|$ , of the
Dirichlet problem (1.7)?(1.8) such that $u(r)$ has exactly zeros in $(a, b)$ .
$k$
$k$
The existence of radial solutions with no zeros of (1.7)?(1.8) has recently
been studied by many authors (see, for example, [1?4, 6, 10]). The existence
of solutions with prescribed numbers of zeros is discussed by Coffman and
is essentially superlinear is
Marcus [4]. In these papers, the case where
considered.
In Section 2 we verify the global existence and uniqueness of solution
of the initial value problem (1.1)?(1.5), and give a few basic results on
which are crucial for the proofs of Theorems 1.1 and 1.2. The proofs
of Theorems 1.1 and 1.2 are given in Sections 3 and 4, respectively, by using
the Priufer transformation. Corollary 1.1 is a direct consequence of Theorems
1.1 and 1.2. Corollary 1.2 follows from Corollary 1.1. In fact, for a radial
solution $u(r)$ where $r=|x|$ , equation (1.7) is rewritten in the form
$f$
$x_{¥lambda}(t)$
$x_{¥lambda}(t)$
,
$¥frac{d^{2}u}{dr^{2}}+¥frac{N-1}{r}¥frac{du}{dr}+p(r)f(u)=0$
$a<r<b$ .
Let $v(t)=u(e^{t})$ for $N=2$ , and let $v(t)=u(t^{1/(2-N)})$ for $N¥geq 3$ . Then it is easy
to see that the problem (1.7)?(1.8) is transformed into the problem
(1.9)
$v^{¥prime¥prime}+q(t)f(v)=0$
(1.10)
$A<t<B$ ,
,
$v(A)=v(B)=0$ ,
for $N=2$ , and
$q(t)=(2-N)^{-2}t^{-2(1-N)/(2-N)}p(t^{1/(2-N)})$ ,
for $N¥geq 3$ .
and
Thus, applying Corollary 1.1 to (1.9)?(1.10), we have Corollary 1.2.
where
$=d/dt$ ,
and
$q(t)=e^{2t}p(e^{¥mathrm{t}})$
,
$ A=¥log$
$a$
and
$A=b^{2-N}$
$B=¥log b$
$B=a^{2-N}$
508
Manabu NAITO and Yuki NAITO
2. Preliminaries
In this section we show the existence and uniqueness of a global solution
on $[a, b]$ of the problem (1.1)?(1.5), and give a few basic results on
which are crucial for the proofs of Theorems 1.1 and 1.2. Throughout this
section, we assume that either (F.I) or (F.2) is satisfied.
First we consider equation (1.1) together with the general initial condition
$x_{¥lambda}(t)$
$x_{¥lambda}(t)$
(2. 1)
$¥mathrm{x}(t_{0})=¥alpha$
,
,
$ x^{¥prime}(t_{0})=¥beta$
where $t_{0}¥in[a, b]$ and ,
are arbitrarily given. It is well known that the
uniqueness problem and the continuability problem of solution of the general
initial value problem (1.1)?(2.1) have a very delicate aspect. For the
Emden-Fowler equation (1.4), the results concerning these problems may be
found in the survey paper of Wong [13]. For a more general equation than
(1.4), we refer to Coffman and Wong [5]. The technique used in this section
is adapted from [5].
Let us first discuss the uniqueness of a local solution of the problem
(1.1)?(2.1). Note that the existence of a local solution of (1.1)?(2.1) is
guaranteed by the Peano existence therorem. Under condition (F. ), the
uniqueness of local solution of (1.1)?(2.1) is clear since the function satisfies
¥
¥
a Lipschitz condition on the compact interval ¥ ¥ ¥ ¥ ¥
,
$
¥
delta>0$
where
is an arbitrary constant. Under condition (F.2), if
in (2.1),
then the uniqueness of local solution of (1.1)?(2.1) is also clear since satisfies
a Lipschitz condition on $[¥alpha-¥delta, ¥alpha+¥delta]¥subset(-¥infty, 0)¥cup(0, ¥infty)$, where $¥delta>0$ is
taken sufficiently small. Thus, under condition (F.2), the question is the case
of $¥alpha=0$ . In this case we make a distinction between
and $¥beta=0$ .
The case where $¥alpha=0$ and
in (2.1). We may suppose without loss
of generality that $¥beta>0$ . By (F.2), there exists a positive constant
such
that $f(u)$ is nondecreasing on
and $f(u)/u$ is nonincreasing on
. Let
$
¥
alpha=0$
and
be local solutions of (1.1) satisfying (2.1) with
and
$¥beta>0$ .
$t_{0}<t_{1}
¥
leq
b$
There exists a number
such that
and
$¥alpha$
$¥beta¥in R$
$¥mathrm{I}$
$f$
$[ alpha- delta,
alpha+ delta] subset(- infty,
$¥alpha¥neq 0$
$f$
$¥beta¥neq 0$
$¥beta¥neq 0$
$u_{0}$
$(0, u_{0})$
$(0, u_{0})$
$x_{2}$
$¥mathrm{x}_{1}$
$t_{1}$
$¥frac{¥beta}{2}$
Since
$x_{1}$
and
$(t -t_{0})¥leq x_{i}(t)¥leq u_{0}$
$x_{2}$
for
$t_{0}¥leq t¥leq t_{1}$
$i=1,2$ .
satisfy
$¥mathrm{x}_{i}(t)=¥beta(t-t_{0})-¥int_{t¥mathrm{o}}^{t}(t-s)p(s)f(x_{i}(s))ds$
we have
,
for
$t_{0}¥leq t¥leq t_{1}$
,
$i=1,2$ ,
infty)$
Differential
Nonlinear Second Order
509
Equations
for
$|x_{1}(t)-x_{2}(t)|¥leq(t-t_{0})¥int_{t_{0}}^{t}p(s)|f(x_{1}(s))-f(x_{2}(s))|ds$
By the monotone property of
$f(u)$
and
$f(u)/u$ ,
$t_{0}¥leq t¥leq t_{1}$
.
we easily find that
$|f(¥chi_{1}(t))-f(¥chi_{2}(t))|¥leq¥frac{f(¥beta(t-t_{0})/2)}{¥beta(t-t_{0})/2}|¥chi_{1}(t)-¥chi_{2}(t)|$
for
$t_{0}¥leq t¥leq t_{1}$
.
Then we obtain
$¥frac{|x_{1}(t)-¥chi_{2}(t)|}{t-t_{0}}¥leq¥frac{2}{¥beta}¥int_{t_{0}}^{t}p(s)f(¥frac{¥beta}{2}(s-t_{0}))¥frac{|x_{1}(s)-x_{2}(s)|}{s-t_{0}}ds$
. Notice here that the function $¥varphi(t)=|x_{1}(t)-¥chi_{2}(t)|/(t-t_{0})$ has
for
can be regarded as a continuous
, and hence
the finite limit 0 as
. Then by Gronwall’s inequality we see
function on the closed interval
¥
$
¥
varphi(t)=0$
.
for
for
, which implies that
that
has been proved. The
Thus the uniqueness in a right-neighborhood of
is similarly proved.
uniqueness in a left-neighborhood of
$¥alpha=0$
$
¥
beta=0$
in (2.1). In this case the next energy
and
The case where
important
role. Let
plays an
be a solution of (1.1), and let
function
$(¥subset[a, b])$ be a maximal interval of existence for
. Then we define the
energy function $V[x]$ as follows:
$t_{0}¥leq t¥leq t_{1}$
$t¥rightarrow t_{0}+0$
$¥varphi(t)$
$[t_{0}, t_{1}]$
$x_{1}(t) equiv x_{2}(t)$
$t_{0}¥leq t¥leq t_{1}$
$t_{0}¥leq t¥leq t_{1}$
$t_{0}$
$t_{0}$
$V[¥mathrm{x}]$
$x$
$I[¥mathrm{x}]$
(2.2)
$x$
$V[¥mathrm{x}](t)=¥frac{[x^{¥prime}(t)]^{2}}{2}+p(t)F(x(t))$
,
$t¥in I[x]$
,
where
(2.3)
$F(u)=¥int_{0}^{u}f(v)dv$
,
$ u¥in$
$(-¥infty, ¥infty)$
.
It is easy to see that $F(u)=F(|u|)>0$ for all
, $F(0)=0$ , and
$F(u)$ is strictly increasing in $u¥in(0, ¥infty)$ .
Thus $V[x](t)$ is always nonnegative
on $I[x]$ , and $V[x](t)=0$ on $I[x]$ if and only if $x(t)=0$ on $I[x]$ . Furthermore
we have
$ u¥in$ $(-¥infty, ¥infty)¥backslash ¥{0¥}$
(2.4)
$V[¥mathrm{x}](t)¥leq V[x](t_{0})¥exp(¥int_{t_{0}}^{t}¥frac{[p^{¥prime}(s)]_{+}}{p(s)}ds)$
for
$V[x](t)¥leq V[x](t_{0})¥exp(¥int_{t}^{t_{0}}¥frac{[p^{¥prime}(s)]_{-}}{p(s)}ds)$
for
$t_{0}$
,
$t¥in I[¥mathrm{x}]$
,
$t_{0}<t$
,
,
$t¥in I[x]$
,
$t<t_{0}$
,
and
(2.5)
where
$[u]_{+}=¥max¥{u, 0¥}$
and
$[u]¥_=¥max¥{-u, 0¥}$ .
$t_{0}$
In fact, we see that
510
Manabu NAITO and Yuki NAITO
$¥frac{d}{dt}V[x](t)=p^{¥prime}(t)F(x(t))¥leq¥frac{[p^{¥prime}(t)]_{+}}{p(t)}V[x](t)$
or equivalently
$¥frac{d}{dt}(V[x](t)¥exp(-¥int_{t¥mathrm{o}}^{t}¥frac{[p^{¥prime}(s)]_{+}}{p(s)}ds))¥leq 0$
for , $t¥in I[x]$ , $t_{0}<t$ . Then an integration of the above over
gives
inequality (2.4). Inequality (2.5) can be obtained in a similar fashion.
Now let us return to the uniqueness problem of solution of (1.1)?(2.1)
with $¥alpha=¥beta=0$. Suppose that
is a local solution of (1.1)?(2.1), $¥alpha=¥beta=0$,
and that it exists on an interval
. Then, making use of (2.4) and noting
$V[x](t)
¥equiv 0$ for
that $V[x](t_{0})=0$ , we see that
. Consequently
for
right-neighborhood
implies
which
the
uniqueness
on a
,
of . Similarly, the use of (2.5) yields the uniqueness on a left-neighborhood
of . Thus we conclude that, for each $t_{0}¥in[a, 6]$ and each ,
, the initial
value problem (1.1)?(2.1) has a unique local solution.
We can employ inequalities (2.4) and (2.5) to prove the global existence
of every solution of (1.1). Let
be a solution of (1.1). By (2.4) and (2.5)
we easily see that
$[t_{0}, t]$
$t_{0}$
$x$
$[t_{0}, t_{1}]$
$t_{0}¥leq t¥leq t_{1}$
$¥mathrm{x}(t)¥equiv 0$
$t_{0}¥leq t¥leq t_{1}$
$t_{0}$
$¥alpha$
$t_{0}$
$¥beta¥in R$
$x$
$V[x](t)¥leq V[x](t_{0})¥exp(¥int_{a}^{b}¥frac{|p^{¥prime}(s)|}{p(s)}ds)$
is the maximal interval of existence for . This
means that both $x(t)$ and
are bounded as far as the solution $x(t)$
exists. Thus, by a standard argument, we conclude that $x(t)$ is continuable
on $[a, b]$ , that is, $x(t)$ exists on $[a, b]$ .
We have proved in the above that, for each $t_{0}¥in[a, 6]$ and each ,
,
the initial value problem (1.1)?(2.1) has a unique solution which exists on all
of the interval $[a, b]$ . Let
be the solution of (1.1)?(2.1). Then,
by a general theory on the continuous dependence of solutions on initial
conditions (see, for example, [7, Chap. 1, Theorem 2.4]), it follows that
and
are continuous in
on the set
$[a, b]¥times[a, b]¥times R¥times R$ .
By uniqueness we easily see that if
has
$[a,
b]$
infinitely many zeros in the finite interval
, then
on
$[a, b]$ .
Thus, if
on $[a, b]$ , then the number of zeros of
in $[a, b]$ is finite.
Denote by
the solution of the initial value problem (1.1)?(1.5), where
¥
is a parameter. In the preceding notation we have ¥ ¥
.
Hereafter we restrict our attention to the solutions
. Then the results
for all
$t¥in I[x]$
, where
$I[x]$
$x$
$x^{¥prime}(t)$
$¥alpha$
$¥beta¥in R$
$x(t;t_{0}, ¥alpha, ¥beta)$
$x(t;t_{0}, ¥alpha, ¥beta)$
$x^{¥prime}(t;t_{0}, ¥alpha, ¥beta)$
$(t, t_{0}, ¥alpha, ¥beta)$
$x(t;t_{0}, ¥alpha, ¥beta)$
$x(t;t_{0}, ¥alpha, ¥beta)¥equiv 0$
$x(t;t_{0}, ¥alpha, ¥beta)¥not¥equiv 0$
$x(t;t_{0}, ¥alpha, ¥beta)$
$x_{¥lambda}(t)$
$¥lambda¥in R$
$ chi_{ lambda}(t)=x(t;a, 0,
$x_{¥lambda}(t)$
lambda)$
Nonlinear Second Order
Differential
511
Equations
mentioned above can be summarized as follows:
Theorem 2.1. Suppose that either (F.I) or (F.2) is satisfied. Then, for
, the solution
each
of the problem (1.1)?(1.5) exists on $[a, b]$ and is
unique.
$¥lambda¥in R$
$x_{¥lambda}(t)$
Theorem 2.2. Suppose that either (F.I) or (F.2) is satisfied. Then,
and
are continuous functions of $(¥lambda, t)¥in R¥times[a, b]$ ;
(i)
has a finite number of zeros in $[a, b]$ ;
with
,
(ii) for each
, we have
(iii) for each
$x_{¥lambda}(t)$
$x_{¥lambda}^{¥prime}(t)$
$¥lambda¥in R$
$¥lambda¥neq 0$
$x_{¥lambda}(t)$
$¥lambda¥in R$
(2.6)
$V[x_{¥lambda}](t)¥leq¥frac{¥lambda^{2}}{2}¥exp(¥int_{a}^{b}¥frac{[p^{¥prime}(s)]_{+}}{p(s)}ds)$
,
$a¥leq t¥leq b$
,
and
(2.7)
$V[x_{¥lambda}](t)¥geq¥frac{¥lambda^{2}}{2}¥exp(-¥int_{a}^{b}¥frac{[p^{¥prime}(s)]_{-}}{p(s)}ds)$
,
$a¥leq t¥leq b$
.
Theorem 2.1 forms the basis of Theorems 1.1 and 1.2, and Theorem 2.2
plays a crucial part in the proofs of Theorems 1.1 and 1.2.
3. Proof of Theorem 1.1
In this section we prove Theorem 1.1. Throughout this section we assume
that (F.I) is satisfied. We introduce the following notations:
(3. 1)
$p_{*}=¥min¥{p(t):a¥leq t¥leq b¥}$ ,
(F.2)
$P_{*}=¥exp(-¥int_{a}^{b}¥frac{[p^{¥prime}(s)]_{-}}{p(s)}ds)$
$p^{*}=¥max¥{p(t):a¥leq t¥leq b¥}$
,
$P^{*}=¥exp(¥int_{a}^{b}¥frac{[p^{¥prime}(s)]_{+}}{p(s)}ds)$
Lemma 3.1. There exists a positive constant
.
has no zeros in
,
$0<¥lambda¥leq¥lambda_{*}$
$y$
of
Proof.
$(a,$
$x_{¥lambda}(t)$
$¥lambda_{*}$
such that,
for
.
any
$¥lambda$
with
$b]$
We can choose a number
(3.3)
;
$¥mu>0$
$y^{¥prime¥prime}+¥mu p(t)y=0$
,
so small that a nontrivial solution
$a¥leq t¥leq b$
,
satisfying $y(a)=0$ has no zeros in ( , . Note that, under condition (F. ),
$f(u)/u¥rightarrow 0$ as
. Thus it is possible to take $Z>0$ so small that
$a$
$b]$
$¥mathrm{I}$
$u¥rightarrow 0$
(3.4)
$f(u)$
?
$u$
$¥leq¥mu$
for
$0<|u|¥leq Z$ .
512
Manabu NAITO and Y?ki NAITO
For this $Z>0$ , take
$¥lambda_{*}>0$
(3.5)
so that
$¥lambda_{*}^{2}=¥frac{2p_{*}F(Z)}{P^{*}}$
,
where
is defined by (2.3). We verify that, for any satisfying
,
has no zeros in ( , . Assume to the contrary that
has at least
one zero in ( , . By (2.6) in Theorem 2.2 and (3.5) we have
$F$
$0<¥lambda¥leq¥lambda_{*}$
$¥lambda$
$x_{¥lambda}(t)$
$a$
$a$
$b]$
$x_{¥lambda}(t)$
$b]$
$p_{*}F(¥chi_{¥lambda}(t))¥leq V[x_{¥lambda}](t)¥leq¥frac{1}{2}¥lambda^{2}P^{*}$
for
$¥leq¥frac{1}{2}¥lambda_{*}^{2}P^{*}=p_{*}F(Z)$
$a¥leq t¥leq b$
,
which implies that
(3.6)
for
$|x_{¥lambda}(t)|¥leq Z$
$a¥leq t¥leq b$
.
Then, from (3.4) and (3.6), we have
for
$p(t)¥frac{f(x_{¥lambda}(t))}{x_{¥lambda}(t)}¥leq¥mu p(t)$
We note that
$x_{¥lambda}(t)$
$a¥leq t¥leq b$
.
is a solution of the linear equation
,
$a¥leq t¥leq b$
$z^{¥prime¥prime}+p(t)¥frac{f(¥chi_{¥lambda}(t))}{x_{¥lambda}(t)}z=0$
.
It is to be remarked that
is continuous in on the interval
$[a, b]$ .
Indeed, this follows from the fact that $f(u)/u$ can be considered as a
continuous function of $u¥in(-¥infty, ¥infty)$ . (Recall that, under (F. ), $¥lim f(u)/u¥rightarrow 0$
as
) Then, using Sturm’s comparison theorem, we see that the solution
of (3.3) has at least one zero in ( , . This contradicts the assumption
that has no zeros in ( , . Thus, for any satisfying
has
,
no zeros in ( , . The proof of Lemma 3.1 is complete.
$p(t)f(x_{¥lambda}(t))/x_{¥lambda}(t)$
$t$
$¥mathrm{I}$
$u¥rightarrow 0.$
$a$
$y$
$a$
$x_{¥lambda}(t)$
Proof.
(3.7)
$¥lambda$
Choose
$k$
$k$
be any positive integer.
.
zeros in
$¥tau>0$
$(a,$
$b]$
so small that
$¥tau¥leq¥frac{b-a}{3k}$
$¥mu>0$
$0<¥lambda¥leq¥lambda_{*}$
$x_{¥lambda}(t)$
$b]$
Lemma 3.2. Let
that
has at least
Take
$b]$
$a$
$y$
$b]$
and $Z>0$ such that both
.
Then there exists
$¥lambda>0$
such
Nonlinear Second Order
(3.8)
Differential
and
$¥frac{¥pi}{(¥mu p_{*})^{1/2}}¥leq¥tau$
$f(u)$
(3.9)
?
for
$¥geq¥mu$
513
Equations
$|u|¥geq Z$
,
$u$
hold. (Recall that, under (F. ),
large that
$¥mathrm{I}$
(3. 10)
$ f(u)/u¥rightarrow¥infty$
as
$|u|¥rightarrow¥infty.$
$¥frac{1}{2}¥lambda^{2}P_{*}¥geq¥frac{1}{2}(¥frac{Z}{¥tau})^{2}+p^{*}F(Z)$
) Let
$¥lambda>0$
be so
.
We verify that, for this ¥
has at least zeros in ( , . We notice
,
here that if
on an interval I of the form $=[c, d]$ , $a¥leq c<d¥leq b$ ,
then
on this interval $=[c, d]$ . In fact, by (2.7) in Theorem
2.2 and (3.10), we have
$ lambda>0$
$k$
$x_{¥lambda}(t)$
$a$
$|¥mathrm{x}_{¥lambda}(t)|¥leq Z$
$b]$
$I$
$I$
$|x_{¥lambda}^{¥prime}(t)|¥geq Z/¥tau$
$¥frac{1}{2}[¥chi_{¥lambda}^{¥prime}(t)]^{2}+p(t)F(¥chi_{¥lambda}(t))¥geq¥frac{1}{2}¥lambda^{2}P_{*}¥geq¥frac{1}{2}(¥frac{Z}{¥tau})^{2}+p^{*}F(Z)$
for
$a¥leq t¥leq b$
Therefore, if
.
$|x_{¥lambda}(t)|¥leq Z$
for
then
$c¥leq t¥leq d(a ¥leq c<d¥leq b)$ ,
$¥frac{1}{2}[¥chi_{¥lambda}^{¥prime}(t)]^{2}¥geq¥frac{1}{2}(¥frac{Z}{¥tau})^{2}$
for
$c¥leq t¥leq d$
see that if
, and consequently
$|x_{¥lambda}(t)|¥leq Z$
on
$I$
$=[c,
for $c¥leq r¥leq d$ . By this fact we
d]$ $(¥subset[a, b])$ , then
is strictly monotone
$|¥mathrm{x}_{¥lambda}^{¥prime}(t)|¥geq Z/¥tau$
$x_{¥lambda}(t)$
and satisfies
on $=[c, d]$ .
¥
Let
, where
satisfies (3.10). We
) be an arbitrary zero of
suppose without loss of generality that
. In order to show that
$t_{0}+3
¥
tau]$
has at least one zero in ( ,
, we shall argue by dividing into the
following three steps:
for some
];
(i)
$0<x_{
¥
lambda}(t_{2})<Z$
and
for some
];
(ii)
$t_{2}+
¥
tau]$
has at least one zero in ( ,
; and hence,
has at
(iii)
$t_{0}+3¥tau]$ .
least one zero in
$t_{0}+¥tau]$ .
for some
(i) As the first step we assert that
$x_{
¥
lambda}(t)<Z$
Assume to the contrary that
for $ t_{0}<t¥leq t_{0}+¥tau$ . By the previous
$0<x_{
¥
lambda}(t)<Z$
notice we have
and
for $ t_{0}<t¥leq t_{0}+¥tau$ . Therefore
$I$
$|x_{¥lambda}^{¥prime}(t)|¥geq Z/¥tau$
$t_{0} in[a,$
$b$
$¥lambda$
$¥mathrm{x}_{¥lambda}(t)$
$x_{¥lambda}^{¥prime}(t_{0})>0$
$x_{¥lambda}(t)$
$t_{0}$
$x_{¥lambda}(t_{1})¥geq Z$
$t_{1}¥in(t_{0},$
$ t_{0}+¥tau$
$t_{2}¥in(t_{1},$
$x_{¥lambda}^{¥prime}(t_{2})<0$
$x_{¥lambda}(t)$
$ t_{1}+¥tau$
$x_{¥lambda}(t)$
$t_{2}$
$(t_{0},$
$t_{1}¥in(t_{0},$
$¥mathrm{x}_{¥lambda}(t_{1})¥geq Z$
$¥mathrm{x}_{¥lambda}^{¥prime}(t)¥geq Z/¥tau$
$¥chi_{¥lambda}(t_{0}+¥tau)=¥int_{t¥mathrm{o}}^{t_{¥mathrm{O}}+¥tau_{¥mathrm{X}_{¥lambda}^{¥prime}}}(t)dt¥geq Z$
,
$t_{0}+¥tau]$ ,
which is a contradiction. Thus, for some
$0<x_{
¥
lambda}(t_{2})<Z$
and
(ii) Next we claim that
$t_{1}¥in(t_{0},$
$x_{¥lambda}(t_{1})¥geq Z$
$¥chi_{¥lambda}^{¥prime}(t_{2})<0$
.
for some
Manabu NAITO and Y?ki NAITO
514
$t_{2}¥in(t_{1},$
for
$ t_{1}+¥tau$
is the number in the step (i). Assume that
], where
¥
. We consider the linear differential equation
$t_{1}$
$x_{¥lambda}(t)¥geq Z$
$ t_{1}¥leq t¥leq t_{1}+ tau$
(3.11)
$y^{¥prime¥prime}+¥mu p_{*}y=0$
.
Equation (3.11) has the solution $y(t)=¥sin[(¥mu p_{*})^{1/2}(t-t_{1})]$ , which vanishes at
It follows from (3.8) that $ t_{1}+¥pi/(¥mu p_{*})^{1/2}¥leq t_{1}+¥tau$ .
$t=t_{1}$ and $t_{1}+¥pi/(¥mu p_{*})^{1/2}$ .
By virtue of (3.9) we have
for
$¥mu p_{*}¥leq p(t)¥frac{f(¥chi_{¥lambda}(t))}{¥chi_{¥lambda}(t)}$
Since
$ t_{1}¥leq t¥leq t_{1}+¥tau$
.
is a solution of the linear equation
$x_{¥lambda}(t)$
$z^{¥prime¥prime}+p(t)¥frac{f(¥chi_{¥lambda}(t))}{x_{¥lambda}(t)}z=0$
,
$ t_{1}¥leq t¥leq t_{1}+¥tau$
,
has at least one zero in
using Sturm’s comparison theorem, we see that
$[t_{1}, t_{1}+¥pi/(¥mu p_{*})^{1/2}]$ .
This contradicts our assumption. Thus it is impossible
$
, we easily
for t_{1}¥leq t¥leq t_{1}+¥tau$. Then, in view of
to have
.
see that there is
] satisfying $0<x_{¥lambda}(t_{2})<Z$ and
$t_{2}+
¥
tau]$
, where
has at least one zero in ( ,
(iii) Finally we show that
for
is the number in the step (ii). Assume to the contrary that
¥
¥
¥
so
and
,
¥
¥
¥
for
equation
. By
(1.1),
for $ t_{2}¥leq t¥leq t_{2}+¥tau$ . Hence we have $0<x_{¥lambda}(t)¥leq x_{¥lambda}(t_{2})<Z$
¥
¥
¥
, we obtain
for
for $ t_{2}¥leq t¥leq t_{2}+¥tau$ . Since
$¥mathrm{x}_{¥lambda}(t)$
$x_{¥lambda}(t_{1})¥geq Z$
$x_{¥lambda}(t)¥geq Z$
$t_{2}¥in(t_{1},$
$x_{¥lambda}^{¥prime}(t_{2})<0$
$ t_{1}+¥tau$
$¥mathrm{x}_{¥lambda}(t)$
$t_{2}$
$x_{¥lambda}(t)>0$
$t_{2}$
$ t_{2} leq t leq t_{2}+ tau$
$x_{¥lambda}^{¥prime¥prime}(t)<0$
$ t_{2} leq t leq t_{2}+ tau$
$¥chi_{¥lambda}^{¥prime}(t)¥leq¥chi_{¥lambda}^{¥prime}(t_{2})<0$
$¥chi_{¥lambda}^{¥prime}(t)¥leq-Z/¥tau$
$ t_{2} leq t leq t_{2}+ tau$
,
$x_{¥lambda}(t_{2}+¥tau)=¥mathrm{x}_{¥lambda}(t_{2})+¥int_{t_{2}}^{t_{2}+¥tau}x_{¥lambda}^{¥prime}(t)dt<Z-Z=0$
which contradicts our assumption.
$(t_{2}$
,
$t_{2}+¥tau]$
Thus
$x_{¥lambda}(t)$
has at least one zero in
.
¥
has a zero
), then it has another
We have proved that if
zero in ( , $t_{0}+3¥tau]$ . Then, by virtue of (3.7) and $x_{¥lambda}(a)=0$ , we conclude that
has at least zeros in ( , . The proof of Lemma 3.2 is complete.
$x_{¥lambda}(t)$
$t_{0} in[a,$
$b$
$t_{0}$
$k$
$x_{¥lambda}(t)$
$a$
$b]$
To prove Theorem 1.1 we employ the Priufer transformation. For the
by
and
, we define the functions
with ¥
solution
$x_{¥lambda}(t)$
$ lambda>0$
$r_{¥lambda}(t)$
,
(3. 12)
$¥mathrm{x}_{¥lambda}(t)=r_{¥lambda}(t)¥sin¥theta_{¥lambda}(f)$
(3. 13)
$¥chi_{¥lambda}^{¥prime}(t)=r_{¥lambda}(t)¥cos¥theta_{¥lambda}(t)$
and
Since
in the forms
$x_{¥lambda}(t)$
$¥chi_{¥lambda}^{¥prime}(t)$
cannot vanish simultaneously,
$¥theta_{¥lambda}(t)$
.
$r_{¥lambda}(t)$
and
$¥theta_{¥lambda}(t)$
are written
Nonlinear Second Order
Differential
and
$r_{¥lambda}(t)=([x_{¥lambda}^{¥prime}(t)]^{2}+[¥chi_{¥lambda}(t)]^{2})^{1/2}>0$
$¥theta_{¥lambda}(t)=¥arctan¥frac{x_{¥lambda}(t)}{¥chi_{¥lambda}^{¥prime}(t)},$
515
Equations
,
are determined as continuously differenand
respectively. Therefore
tiable functions with respect to $t¥in[a, b]$ . By a simple calculation we see that
$r_{¥lambda}(t)$
$¥theta_{¥lambda}(t)$
$¥theta_{¥lambda}^{¥prime}(t)=¥cos^{2}¥theta_{¥lambda}(t)+p(t)¥frac{¥sin¥theta_{¥lambda}(t)f(r_{¥lambda}(t)¥sin¥theta_{¥lambda}(t))}{r_{¥lambda}(t)}>0$
is strictly increasing in $t¥in[a, b]$ for each
for $a¥leq t¥leq b$ , which implies that
and
. From the initial condition (1.5) it follows that
fixed
. For simplicity we take
$¥theta_{¥lambda}(t)$
$¥lambda¥in(0, ¥infty)$
$ r_{¥lambda}(a)=¥lambda$
$¥theta_{¥lambda}(a)¥equiv 0(¥mathrm{m}¥mathrm{o}¥mathrm{d} 2¥pi)$
(3. 14)
$¥theta_{¥lambda}(a)=0$
.
is continuous in $(¥lambda, t)¥in(0, ¥infty)¥times[a, b]$ .
By (i) of Theorem 2.2,
has exactly zeros in $(a, b)$ if and only if
to see that
$¥theta_{¥lambda}(t)$
(3. 15)
by
$ k¥pi<¥theta_{¥lambda}(b)¥leq(k+1)¥pi$
Proof of
It is easy
$k$
$x_{¥lambda}(t)$
Theorem 1.1.
Let $k=1,2$ ,
$¥cdots$
.
.
Define the subsets
$¥Lambda_{k}=¥{¥lambda¥in(0, ¥infty):¥theta_{¥lambda}(b)¥geq k¥pi¥}$
$¥Lambda_{k}$
of
$(0, ¥infty)$
.
¥
such that ¥ ¥
for any ,
Lemma 3.1 admits the existence of
. For each integer , Lemma 3.2 guarantees the existence of $¥lambda>0$
is nonempty. By virtue of the relation
. Therefore,
satisfying
$ 0< theta_{ lambda}(b)< pi$
$¥lambda_{*}>0$
$¥lambda$
$k$
$0<¥lambda¥leq¥lambda_{*}$
$¥theta_{¥lambda}(b)¥geq k¥pi$
$[¥lambda_{*}$
$¥Lambda_{k}$
,
$¥infty)¥supset¥Lambda_{1}¥supset¥Lambda_{2}¥supset¥cdots¥supset¥Lambda_{k}¥supset¥Lambda_{k+1}¥supset¥cdots$
,
. Let
each
is bounded below by the positive constant
the continuity of
with respect to , we easily find that
$¥Lambda_{k}$
$¥lambda_{*}$
$¥lambda_{k}=¥inf¥Lambda_{k}$
.
By
$¥lambda$
$¥theta_{¥lambda}(b)$
(3.16)
for
$¥theta_{¥lambda}(b)=k¥pi$
.
$¥lambda=¥lambda_{k}$
Then we have
$ 0<¥lambda_{1}<¥lambda_{2}<¥cdots<¥lambda_{k}<¥lambda_{k+1}<¥cdots$
From the definition of
$¥lambda_{k}$
it follows that
$¥theta_{¥lambda}(b)<k¥pi$
Thus
that if
$¥mathrm{x}_{¥lambda}(t)$
for
$0<¥lambda<¥lambda_{k}$
for
has at most $k-1$ zeros in ( ,
has exactly $k-1$ zeros in
, then
$¥lambda=¥lambda_{k}$
$a$
$x_{¥lambda}(t)$
.
$b]$
.
. We see by (3.16)
and satisfies $x_{¥lambda}(b)=0$ .
$0<¥lambda<¥lambda_{k}$
$(a, b)$
516
Manabu NAITO and Y?ki NAITO
Finally we claim that
$¥lim_{k¥rightarrow¥infty}¥lambda_{k}=¥infty$
.
Assume to the contrary that
$¥{¥lambda_{k}¥}$
Then, by the continuity of
tends to the finite number
with
with respect to ,
tends to
with
. On the other hand, by (3.16),
. This is a contradiction. Therefore we conclude that
The proof of Theorem 1.1 is complete.
a finite limit as
$ k¥rightarrow¥infty:¥lim_{k¥rightarrow¥infty}¥lambda_{k}=¥lambda^{*}<¥infty$
$¥lambda$
.
$¥lambda=¥lambda_{k}$
$¥theta_{¥lambda}(b)$
$¥theta_{¥lambda}(b)$
$¥theta_{¥lambda^{*}}(b)$
$ k¥rightarrow¥infty$
$¥lambda=¥lambda_{k}$
$¥theta_{¥lambda}(b)$
$ k¥rightarrow¥infty$
4.
has
$¥infty$
as
as
$¥lim_{k¥rightarrow¥infty}¥lambda_{k}=¥infty$
.
Proof of Theorem 1.2
In this section we prove Theorem 1.2. We assume (F.2) throughout this
section. Notations (3.1) and (3.2) are still used.
Lemma 4.1. There exists a positive constant
.
has no zeros in
,
satisfying
$¥lambda¥geq¥lambda^{*}$
Proof.
$(a,$
$x_{¥lambda}(t)$
We choose
$¥mu>0$
(4. 1)
$¥lambda^{*}$
such that,
for any
$¥lambda$
$b]$
so small that
$¥frac{¥pi}{3(¥mu p^{*})^{1/2}}¥geq b-a$
.
Take $Z>0$ so large that
(4.2)
for
$¥frac{f(u)}{u}<¥mu$
This is possible since $f(u)/u¥rightarrow 0$ as
such that
take
$|u|¥geq¥frac{Z}{2}$
$ u¥rightarrow¥pm¥infty$
.
(see condition (F.2)). Further,
$¥lambda^{*}>0$
(4.3)
$¥lambda^{*2}>¥frac{2p^{*}F(Z)}{P_{*}}$
,
has no
,
is defined by (2.3). We claim that, for any
in
one
zero
has
at
least
contrary
that
to
the
.
Assume
zeros in ( ,
, there
. By virtue of
] be the first zero of
( , . Let
. By (2.7) in Theorem 2.2 and (4.3) we have
satisfying
exists
where
$¥lambda¥geq¥lambda^{*}$
$F$
$a$
$a$
$b]$
$b]$
$x_{¥lambda}(t)$
$x_{¥lambda}(t)$
$t_{1}¥in(a,$
$t_{2}¥in(a, t_{1})$
$6$
$¥mathrm{x}_{¥lambda}(a)=0$
$¥mathrm{x}_{¥lambda}(t)$
$¥chi_{¥lambda}^{¥prime}(t_{2})=0$
$p^{*}F(x_{¥lambda}(t_{2}))¥geq V[x_{¥lambda}](t_{2})$
$¥geq¥frac{1}{2}¥lambda^{2}P_{*}¥geq¥frac{1}{2}¥lambda^{*2}P_{*}>p^{*}F(Z)$
which implies that
(4.4)
$¥mathrm{x}_{¥lambda}(t_{2})>Z$
$x_{¥lambda}(t)>¥frac{Z}{2}$
.
Then there exists
for
$t_{2}¥leq t<t_{3}$
and
,
$t_{3}¥in(t_{2}, t_{1})$
such that
$¥mathrm{x}_{¥lambda}(t_{3})=¥frac{Z}{2}$
.
Nonlinear Second Order
Let
Differential
517
Equations
be a solution of the initial value problem
$y$
(4.5)
$y^{¥prime¥prime}+¥mu p^{*}y=0$
$y(t_{2})=Z$ ,
(4.6)
We easily see that
(4.7)
is given by
$y(t)$
$y^{¥prime}(t_{2})=0$
.
$y(t)=Z¥cos[(¥mu p^{*})^{1/2}(t-t_{2})]$
for
$y(t)¥geq¥frac{1}{2}Z$
,
$t_{2}¥leq t¥leq t_{2}+¥frac{¥pi}{3(¥mu p^{*})^{1/2}}$
; and so
.
We assert that
(4.8)
.
$t_{3}>t_{2}+¥frac{¥pi}{3(¥mu p^{*})^{1/2}}$
To see this, assume to the contrary that
$t_{3}¥leq t_{2}+¥frac{¥pi}{3(¥mu p^{*})^{1/2}}$
Then there exists some
(4.9)
$t^{*}¥in(t_{2},$
$t_{3}$
for
$x_{¥lambda}(t)>y(t)$
.
] such that
$t_{2}¥leq t<t^{*}$
and
$x_{¥lambda}(t^{*})=y(t^{*})$
.
and (4.5) by
by
, subtract and integrate
We multiply (1.1) with
rearranging
the
Then,
by
terms, we find
.
over
$x=x_{¥lambda}$
$y$
$x_{¥lambda}$
$[t_{2}, t^{*}]$
(4. 10)
$y(t^{*})x_{¥lambda}^{¥prime}(t^{*})-x_{¥lambda}(t^{*})y^{¥prime}(t^{*})$
.
$=¥int_{t_{2}}^{t^{*}}y(t)¥chi_{¥lambda}(t)[¥mu p^{*}-p(t)¥frac{f(¥mathrm{x}_{¥lambda}(t))}{¥chi_{¥lambda}(t)}]dt$
Since (4.9) is satisfied, the left-hand side of (4.10) is nonpositive. On the other
hand, from (4.2), (4.4) and (4.7) we see that the right-hand side of (4.10) is
positive, which yields a contradiction. Hence (4.8) holds. Then, it follows that
$t_{1}-a>t_{3}-t_{2}>¥frac{¥pi}{3(¥mu p^{*})^{1/2}}$
Thus, by (4.1) we have
proof of Lemma 4.1.
Lemma 4.2. Let
that
has at least
$x_{¥lambda}(t)$
Proof. Consider
(4.11)
$k$
$k$
$t_{1}>b$
.
, which is a contradiction.
be any positive integer.
.
zeros in
$(a,$
Then, there exists
$b]$
the linear equation
$y^{¥prime¥prime}+¥mu p(t)y=0$
,
This completes the
$a¥leq t¥leq b$
,
$¥lambda>0$
such
518
Manabu NAITO and Y?ki NAITO
where
is a positive constant. We choose $¥mu>0$ so large that a nontrivial
solution of (4.11) satisfying $y(a)=0$ has at least zeros in ( , . Remember
that, under (F.2), $¥lim f(u)/u=¥infty$ as
. Then it is possible to take $Z>0$
such that
$¥mu$
$k$
$y$
$a$
$b]$
$u¥rightarrow 0$
$f(u)$
(4.12)
Let
for
$¥overline{u}>¥mu$
$¥lambda>0$
$0<|u|¥leq Z$ .
be a number satisfying
(4. 13)
$¥lambda^{2}¥leq¥frac{2p_{*}F(Z)}{P^{*}}$
We verify that, for this ¥
,
Theorem 2.2 and (4.13) we have
$ lambda>0$
$x_{¥lambda}(t)$
has at least
.
zeros in ( ,
$k$
$a$
for
$¥prime p_{*}F(¥mathrm{x}_{¥lambda}(t))¥leq V[x_{¥lambda}](t)¥leq¥frac{1}{2}¥lambda^{2}P^{*}¥leq p_{*}F(Z)$
$b]$
.
By (2.6) in
$a¥leq t¥leq b$
,
which implies
(4.14)
for
$|x_{¥lambda}(t)|¥leq Z$
$a¥leq t¥leq b$ .
and , $a¥leq t_{0}<t_{1}¥leq b$ , be any successive zeros of the solution
of
has at least one zero in
(4.11) with $y(a)=0$ . We claim that
. Assume to the contrary that
has no zeros in
. We may
suppose that $y(t)>0$ and ¥
for $t_{0}<t<t_{1}$ . From (4.12) and (4.14) we
have
Let
$t_{0}$
$t_{1}$
$y$
$x_{¥lambda}(t)$
$(t_{0}, t_{1})$
$(t_{0}, t_{1})$
$x_{¥lambda}(t)$
$x_{ lambda}(t)>0$
for
$¥mu<¥frac{f(¥mathrm{x}_{¥lambda}(t))}{¥mathrm{x}_{¥lambda}(t)}$
Multiplying (1.1) where
by
integrating over
, we have
$x=x_{¥lambda}$
$y$
$t_{0}<t<t_{1}$
and (4.11) by
.
$x_{¥lambda}(t)$
, subtracting, and
$[t_{0}, t_{1}]$
(4. 15)
$¥chi_{¥lambda}(t_{1})y^{¥prime}(t_{1})-¥chi_{¥lambda}(t_{0})y^{¥prime}(t_{0})$
$=¥int_{t_{0}}^{¥mathrm{r}_{1}}p(t)y(t)[f(¥chi_{¥lambda}(t))-¥mu ¥mathrm{x}_{¥lambda}(t)]dt$
.
The left-hand side of (4.15) is nonpositive, while the right-hand side of (4.15)
is positive, giving a contradiction. Thus
has at least one zero between
the successive zeros ,
of . Since
has at least $k+1$ zeros in $[a, b]$ ,
we can conclude that
has at least zeros in ( , . The proof is complete.
$¥chi_{¥lambda}(t)$
$t_{0}$
$t_{1}$
$y$
$y$
$k$
$x_{¥lambda}(t)$
$a$
$b]$
To prove Theorem 1.2 we employ the same Priufer transformation as in
Section 3. For the solution
with ¥
, we define the functions
$x_{¥lambda}(t)$
$ lambda>0$
$r_{¥lambda}(t)$
Nonlinear Second Order
Differential
519
Equations
are well defined
and
by (3.12) and (3.13). As in Section 3,
is
satisfies the following properties:
with
on $[a, b]$ , and
and is continuous in
strictly increasing in $t¥in[a, b]$ for each fixed
$(a,
b)$
$(¥lambda, t)¥in(0, ¥infty)¥times[a, b];x_{¥lambda}(t)$ has exactly
if and only if
zeros in
and
$r_{¥lambda}(t)$
$¥theta_{¥lambda}(t)$
$¥theta_{¥lambda}(t)$
$¥theta_{¥lambda}(a)=0$
$¥theta_{¥lambda}(t)$
$¥theta_{¥hat{¥lambda}}(t)$
$¥lambda¥in(0, ¥infty)$
$k¥pi<¥theta_{¥lambda}(b)$
$k$
$¥leq(k+1)¥pi$ .
Proof of
by
Theorem 1.2.
Let
$ k=1,2,¥cdots$
Define the subsets
.
$¥Lambda_{k}=¥{¥lambda¥in(0, ¥infty):¥theta_{¥lambda}(b)¥geq k¥pi¥}$
$¥lambda^{*}>0$
$ 0<¥theta_{¥lambda}(b)<¥pi$
$ lambda>0$
$¥lambda¥geq¥lambda^{*}$
of
$(0, ¥infty)$
.
such that
Lemma 4.1 admits the existence of
. Lemma 4.2 guarantees the existence of ¥
,
. From the relation
Therefore,
$¥lambda$
$¥Lambda_{k}$
satisfying
for any
$¥theta_{¥lambda}(b)¥geq k¥pi$
.
$¥Lambda_{k}¥neq¥phi$
(0,
$¥lambda^{*}$
]
,
$¥supset¥Lambda_{1}¥supset¥Lambda_{2}¥supset¥cdots¥supset¥Lambda_{k}¥supset¥Lambda_{k+1}¥supset¥cdots$
is bounded above.
it follows that each
with respect to , we find that
of
Let
(4.16)
for
$¥Lambda_{k}$
$¥lambda_{k}=¥sup¥Lambda_{k}$
.
By the continuity
$¥lambda$
$¥theta_{¥lambda}(b)$
$¥theta_{¥lambda}(b)=k¥pi$
$¥lambda=¥lambda_{k}$
.
Then we have
$0<¥cdots<¥lambda_{k+1}<¥lambda_{k}<¥cdots<¥lambda_{2}<¥lambda_{1}$
.
It is easy to see that
$¥theta_{¥lambda}(b)<k¥pi$
for
$¥lambda>¥lambda_{k}$
.
and has at most $k-1$ zeros in
for
has no zeros in ( ,
Then
has exactly $k-1$
,
. By (4.16) we see that, for
for
( ,
zeros in $(a, b)$ and satisfies $x_{¥lambda}(b)=0$ .
. Assume to the contrary that
Finally we claim that
$a$
$x_{¥lambda}(t)$
$a$
$b]$
$b]$
$¥lambda>¥lambda_{1}$
$¥lambda=¥lambda_{k}$
$¥lambda>¥lambda_{k}$
$x_{¥lambda}(t)$
$¥lim_{k¥rightarrow¥infty}¥lambda_{k}=0$
as
has the finite limit
with
. On the other hand, it follows from (4.16) that
, which is a contradiction. This completes the proof
as
diverges to
of Theorem 1.2.
$¥lim_{k¥rightarrow¥infty}¥lambda_{k}=¥lambda_{*}>0$
.
Then
$¥theta_{¥lambda}(b)$
with
$¥theta_{¥lambda^{*}}(b)$
$¥lambda=¥lambda_{k}$
$ k¥rightarrow¥infty$
$¥theta_{¥lambda}(b)$
$¥infty$
$¥lambda=¥lambda_{k}$
$ k¥rightarrow¥infty$
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nuna adreso .
Department of Mathematics
Faculty of Science
Hiroshima University
Higashi-Hiroshima 724
Japan
(Ricevita la 31-an de augusto, 1992)