数理解析研究所講究録
第 1838 巻 2013 年 41-47

41

Blow-up and pointwise comparison principles forthe generalized Riccati differential equation and application tooscillation of some nonlinear ode’s
Mervan Pa\v{s}i\v{c}University of Zagreb, Faculty of Electrical Engeneenng and Computing
Department of Mathematics, 10000 Zagreb, Croatia
Abstract We give a short presentation on the blow-up and pointwise comparison principles for thegeneralized Riccati differential equation recently applied to a study of oscillation of a class of generalsecond-order differential equations with damping term.
part of this review article was presented on
the RIMS workshop: Global qualitative theory of ordinary differential equations and its applications,
Kyoto 2012. All details have been published in author’s paper [4].
$A$

1Introduction

We consider the following class of the second-order differential equation with damping:$(r(t)\Phi(x, x’))’+p(t)\Psi(x, x’)+q(t)f(x)=0, t\geq t_{0}$ ,

..

(1.1)

where we adopt the following:
solutions: $x=x(t),$$x\in C([t_{0}, \infty),\mathbb{R})\cap C^{2}((t_{0}, \infty),\mathbb{R})$nonlinear or linear functions:$\Phi=\Phi(u, v),$ $\Psi=\Psi(u, v),$ $f=f(u),$$eco$ efficients:$\Psi$:$\mathbb{R}\cross \mathbb{R}arrow \mathbb{R},$

$A$

:

$f$

$\mathbb{R}arrow \mathbb{R}$, are smooth enough;.

$r\in C^{1}([t_{0}, \infty), (0, \infty)),$ $p,$ $q\in C([t_{0}, \infty),\mathbb{R})$Definition 1.1
as$t_{n}arrow\infty$$\Phi,$;

function $x(t)$ is oscillatory if there is a sequence$narrow\infty.$$A$$t_{n}\geq t_{0}$such that $x(t_{n})=0$ anddifferential equation is oscillatory if all its solutions are oscillatory.

The main assumptions on the differential operators $(r(t)\Phi(x, x’))’$ and the damped term $p(t)\Psi(x,x’)$

are:$|u|^{\gamma-2}v\Phi(u,v)\geq g(|\Phi(u, v)|)$

and$\Psi(u, v)u\geq 0$

,

(1.2)

or
$\Phi(u, v)v\geq 0$

where$\gamma\geq 2$

and$g:\mathbb{R}+arrow \mathbb{R}_{+}$

and$u|u|^{\gamma-2}\Psi(u, v)\geq g(|\Phi(u,v)|)$

(1.3)

:

$\{\begin{array}{l}g(cs)\geq c^{\gamma}g_{0}(s) for all c>0 and s>0,g_{0} :For instance, if $g(s)=g_{0}(s)=s^{\gamma},$\mathbb{R}_{+}arrow \mathbb{R}+ is a locally Lipschitz function on \mathbb{R}+,\exists M_{0}>0 such that g_{0}(s)+M_{0}\geq s^{2}, \forall \mathcal{S}\in \mathbb{R}_{+}.\end{array}$$\gamma\geq 2$, then (1.2) is:$|u|^{\gamma-2}v\Phi(u,v)\geq|\Phi(u, v)|^{\gamma}$

and$\Psi(u, v)u\geq 0,$

(1.4)

42
Next we give the main examples for the second-order differential operators $(r(t)\Phi(x,x’))’$ thatsatisfy required assumption (1.2) or (1.3).
Example 1.1 (linear second-order differential operators in$x’$) We consider the equation:$(r(t)A(x)x’)’+p(t)B(x)\psi(x’)+q(t)f(x)=0, t\geq t_{0}.$Condition (1.2) for$\gamma=2$

is fulfilled provided functions $A(u),$ $B(u)$ and$0\leq A(u)\leq 1,$ $A(u)\neq 0,$ $uB(u)\geq 0$

For instance:$B(u)\psi(v),$$A(u)=|\sin(u)|$$\gamma=2$$\psi(v)$satisfy:

and $\psi(v)\geq 0.$

and $A(u)=|u|/(1+|u|)$ . Indeed, for $\Phi(u, v)=A(u)v,$ $\Psi(u,v)=$

we have:$|u|^{\gamma-2}v\Phi(u, v)=A(u)v^{2}\geq A^{2}(u)v^{2}=[A(u)v]^{2}=|\Phi(u, v)|^{\gamma},$$\Psi(u, v)u=uB(u)\psi(v)\geq 0.

\square$Example 1.2 (quasilinear prescribed mean-curvature differential operators in $x’$ ) We consider theequation:$(r(t)A(x) \frac{x’}{\sqrt{1+x^{2}}})’+p(t)B(x)\psi(x’)+q(t)f(x)=0, t\geq t_{0}.$Condition (1.2) for any$\gamma\geq 2$

is fulfilled provided $A(u),$ $B(u)$ and$0\leq A^{\gamma-1}(u)\leq|u|^{\gamma-2},$$A(u)\neq 0,$ $uB(u)\geq 0$

For instance: $A(u)=|\sin(u)|$ and $A(u)=|u|/(1+|u|)$ .
Indeed, forand$\Phi(u,v)=A(u)\frac{v}{(1+v^{2})^{1/2}}$$\psi(v)$satisfy:

and $\psi(v)\geq 0.$$\Psi(u,v)=B(u)\psi(v)$ ,we have:$|u|^{\gamma-2}v \Phi(u, v)=|u|^{\gamma-2}A(u)\frac{v^{2}}{(1+v^{2})^{1/2}}\geq A^{\gamma}(u)\frac{v^{2}}{(1+v^{2})^{1/2}}$$\geq A^{\gamma}(u)\frac{|v|^{\gamma}}{(1+v^{2})^{\gamma/2}}=|\Phi(u,v)|^{\gamma}.

\square$Example 1.3 (half-hnear second-order differential operators in $x’$ ) We consider the equation:$(r(t)A(x)|x’|^{\beta-1}x’)’+p(t)B(x)|x’|^{\beta\gamma}+q(t)f(x)=0, t\geq t_{0},$where$\beta\geq 1$. Condition (3) for any$\gamma\geq 2$

is fulfilled provided $A(u)$ and $B(u)$ satisfy:$0\leq A(u),$ $A(u)\neq 0$

and$u|u|^{\gamma-2}B(u)\geq A^{\gamma}(u)$.

Indeed, for $\Phi(u, v)=A(u)|v|^{\beta-1}v$ and $\Psi(u, v)=B(u)|v|^{\beta\gamma}$ , we have:$\Phi(u,v)v=A(u)|v|^{\beta-1}vv=A(u)|v|^{\beta+1}\geq 0$

and$|u|^{\gamma-1}\Psi(u, v)=|u|^{\gamma-1}B(u)|v|^{\beta\gamma}\geq A^{\gamma}(u)|v|^{\beta\gamma}$$=|A(u)|v|^{\beta-1}v|^{\gamma}=|\Phi(u, v)|^{\gamma}.

\square$43
2

Riccati transformation under assumption (1.2)

In this section we recall the well known Riccati transformation of the equation (1.1) by supposing themain assumption (1.2).

Lemma 2.1 Let $p(t)\geq 0$ and $q(t)>0$ . Let $\Phi(u,v)$ and $\Psi(u, v)$ satish (1.2), and $f(u)$ satisfy:$f(u)/u\geq K>0$ for all $u\neq 0$ . Let $x(t)$ be a nonoscillatory solution of equation (1.1). Then th$(t)$

defined by:$\overline{w}(t)=-\frac{r(t)\Phi(x(t),x’(t))}{x(t)}, t\geq T$

,

(2.1)

satisfies the inequality;$\vec{w}\geq(r(t))^{1-\gamma}g_{0}(|\overline{w}|)+Kq(t) t>T,$

Proof. Taking the first derivative in (2.1), we obtain:$\overline{w}(t)=\frac{r(t)\Phi(x(t),x’(t))}{x^{2}(t)}x’(t)-\frac{(r(t)\Phi(x(t),x’(t)))’}{x(t)}$and using equation (1.1) we get:$\overline{w}(t)=\frac{r(t)\Phi(x(t),x’(t))}{x^{2}(t)}x’(t)+\frac{p(t)\Psi(x,x’)}{x(t)}+q(t)\frac{f(x(t))}{x(t)}$that is:$\overline{w}(t)=\frac{r(t)\Phi(x(t),x’(t))|x(t)|^{\gamma-2}x’(t)}{|x(t)|\gamma}+\frac{p(t)\Psi(x,x’)x(t)}{x^{2}(t)}+q(t)\frac{f(x(t))}{x(t)}.$Using assumptions (1.2) and $f(u)/u\geq K>0$ for all $u\neq 0$ , whereget:$arrow w(t)\geq\frac{r(t)g(|\Phi(x(t),x’(t))|)}{|x(t)|\gamma}+Kq(t)$$\gamma\geq 2$

and $g(s)$ satisfies (1.4), wefor $t>T.$

From(2.1) we get:$| \Phi(x(t), x’(t))|=\frac{|\overline{w}(t)||x(t)|}{r(t)}, t\geq T,$

and putting it into previous inequality we obtain:
for $t>T.$$\vec{w}(t)\geq\frac{r(t)}{|x(t)|\gamma}g(\frac{|\overline{w}(t)||x(t)|}{r(t)})+Kq(t)$Now we can use assumption (1.4) in previous inequality, and hence, we obtain:$\overline{w}(t)\geq\frac{r(t)}{|x(t)|\gamma}\frac{|x(t)|^{\gamma}}{r(t)^{\gamma}}g_{0}(|\varpi(t)|)+Kq(t)$$=(r(t))^{1-\gamma}g_{0}(|\overline{w}(t)|)+Kq(t)$that proves this lemma.

$\square$for $t>T,$44
3

Riccati transformation under assumption (1.3)

In this section, we repeat the(1.3) instead of (1.2).

$\infty$

nsideration from previous section but supposing the main assumption

Lemma 3.1 Let $p(t)\geq 0$ and $q(t)>0$ . Let $\Phi(u, v)$ and $\Psi(u, v)$ satisfy (1.3), and $f(u)$ satisfy:$f(u)/u\geq K>0$ for all $u\neq 0$ . Let $x(t)$ be a nonoscillatory solution of equation (1.1). Thendefined by;$\overline{w}(t)$$\overline{w}(t)=-\frac{r(t)\Phi(x(t),x’(t))}{x(t)}, t\geq T,$

satisfies the inequality:$\vec{w}\geq\frac{p(t)}{r(t)^{\gamma}}g_{0}(|\overline{w}|)+Kq(t) , t>T.$

Proof. We start as before. Taking the first derivative in (2.1), we obtain:$\vec{w}(t)=\frac{r(t)\Phi(x(t),x’(t))}{x^{2}(t)}x’(t)-\frac{(r(t)\Phi(x(t),x’(t)))’}{x(t)},$that is,$\overline{w}(t)=\frac{r(t)\Phi(x(t),x’(t))x’(t)}{|x(t)|^{2}}+\frac{p(t)\Psi(x,x’)x(t)|x(t)|^{\gamma-2}}{|x(t)|\gamma}+q(t)\frac{f(x(t))}{x(t)}.$Using assumptions $f(u)/u\geq K>0$ for all $u\neq 0$ , and (1.3), where$arrow w(t)\geq\frac{p(t)g(|\Phi(x(t),x’(t))|)}{|x(t)|^{\gamma}}+Kq(t)$$\gamma\geq 2$

and $g(s)$ satisfy (1.4), we get:for $t>T.$

From (2.1) we have in particular that:$| \Phi(x(t), x’(t))|=\frac{|\overline{w}(t)||x(t)|}{r(t)}, t\geq T,$

and puting it into
$arrow w(t)\geq\frac{p(t)g(|\Phi(x(t),x’(t))|)}{|x(t)|\gamma}+Kq(t)$for $t>T,$$\overline{w}’(t)\geq\frac{p(t)}{|x(t)|\gamma}g(\frac{|\overline{w}(t)||x(t)|}{r(t)})+Kq(t)$for $t>T.$

we obtain:
Now we use assumption (1.4) in previous inequality and so, we conclude that:$arrow w(t)\geq\frac{p(t)}{|x(t)|\gamma}\frac{|x(t)|^{\gamma}}{r(t)^{\gamma}}90(|\overline{w}(t)|)+Kq(t)$$= \frac{p(t)}{r(t)^{\gamma}}g_{0}(|\overline{w}(t)|)+Kq(t)$that shows this lemma.

$\square$for $t>T,$45
4pointwise comparison principle and a blow-up argument$A$

We consider the generalized Riccati differential equation:$w’(t)=a(t)g_{0}(|w(t)|)+Kq(t) , t>T$,

(4.1)

where
$a(t)=\{\begin{array}{ll}(r(t))^{1-\gamma} under condition (1.2),p(t)/r(t)^{\gamma} under condition (1.3).\end{array}$For
tions:$T_{0}$

and$T^{*},$, we associate to equation (4.1) the corresponding sub- and supersoludefined respectively by:$T\leq T_{0}<T^{*}$$\underline{w},\overline{w}\in C^{1}([T_{0}, T^{*}), \mathbb{R})$$\underline{w}’(t)\leq a(t)g_{0}(|\underline{w}(t)|)+Kq(t)$and$\overline{w}’(t)\geq a(t)g_{0}(|\overline{w}(t)|)+Kq(t)$in$[T_{0}, T^{*})$.

Definition 4.1 We says that the comparison principle holds for equation (4.1) with arbitrary
and$T\leq T_{0}<T^{*}$ , if the following statement holds forall sub- and supersolutionsof equation (4.1):$T_{0}$$T^{*},$$\underline{w},\varpi$$\underline{w}(T_{0})\leq\varpi(T_{0})$..

For a supersolution
two real numbers
anda subsolution

implies$\overline{w}\in C^{1}([T_{0}, \infty), \mathbb{R})$$T_{0}$$T^{*},$$\underline{w}(t)\leq\overline{w}(t)$for all $t\in[T_{0}, T^{*})$ .

of the Riccati differential equation (4.1), let find:$T\leq T_{0}<\tau*,$$\underline{w}\in C^{1}([T_{0}, T^{*}), \mathbb{R})$of equation (4.1),

such that the following initial and blow-up arguments are satisfied at the sam time:$\underline{w}(T_{0})\leq\overline{w}(T_{0})$and$\lim_{tarrow T}.\underline{w}(t)=\infty.$

By a combination of the preceding comparison principle and the initial and blow-up arguments wecan conclude:$\lim_{tarrow T^{*}}\underline{w}(t)\leq\lim_{tarrow T^{*}}\varpi(t)$and hence, every supersoluton$\varpi(t)$,satisfies:$\lim_{tarrow T^{*}}\overline{w}(t)=\infty.$

It shows the nonexistence of a global supersolution of the Riccati differential equation (4.1).
In conclusion:th step: if there is a nonoscillatory solution $x(t)$ of the main equation (1.1):$1^{o}$$(r(t)\Phi(x, x’))’+p(t)\Psi(x, x’)+q(t)f(x)=0, t\geq t_{0},$then the hmction$\overline{w}(t)$defined by:$\overline{w}(t)=-\frac{r(t)\Phi(x(t),x’(t))}{x(t)}, t\geq T,$

is a GLOBAL supersolution of the Riccati differential equation (4.1);
th and th steps: for some sufficient “ oscillation” conditions, the comparison and blow-up principles
for equation (4.1) hod and it implies:, that is,
LOCAL supersolution of(4.1), which implies that there is $NO$ any nonoscillatory solution of the main equation (1.1). By thiscontradiction, we conclude that equation (1.1) is oscillatory.
$2^{o}$$3^{o}$$\lim_{tarrow T^{*}}\overline{w}(t)=\infty$

Now we present the main results of this section.

$\overline{w}(t)is\cdot a$

46
Lemma 4.1 (a pointwise comparison principle) Let $a(t)=(r(t))^{1-\gamma}$ in the case of condition (1.2)
or $a(t)=p(t)/r(t)^{\gamma}$ in the case of condition (1.3) and let $a(t)$ be a locally integrable function on$T\leq T_{0}<\tau*$ , andandThen for every two points
for every sub- and supersolution
implies(4.1) we have:equation
Riccatigenemlizedthedifferentialof. That is:forallt,$\mathbb{R}+\cdot$$\underline{w}(t),\varpi(t)\in$$T^{*},$$T_{0}$$\underline{w}(T_{0})\leq\varpi(T_{0})$$C^{1}([T_{0}, T^{*}), \mathbb{R})$$\in[T_{0}, T^{*})$$\underline{w}(t)\leq\varpi(t)$$\underline{w}(T_{0})\leq\varpi(T_{0})$$\underline{w}’(t)\leq a(t)g_{0}(|\underline{w}(t)|)+Kq(t),$$t>T,$$arrow w(t)\geq a(t)g_{0}(|\varpi(t)|)+Kq(t),$$t>T,$

gives:$\underline{w}(t)\leq\varpi(t)$for all$t\in[T_{0},T^{*})$.

Lemma 4.2 (a blow-up principle) Let the coefficients $a(t)$ and $Kq(t)$ of the generalized Riccati differential equation (4.1) satisfy the following ‘’oscillation” condition: there is a $continuo’Lrs$ function $C(t)$such that:and a point$T_{1}\geq t_{0}$$C(t) \leq\min\{a(t), Kq(t)\},$

Then there are two points$T_{0}$

and$\tau*,$

and$t\geq T_{1}$$T_{0}<T^{n}$$\lim\sup_{tarrow\infty}\int_{T}^{t}C(\tau)d\tau=\infty$

.

(4.2)

, and a subsolution $w(t)$ of equation (4.1) such that:and$\underline{w}(T_{0})\leq\varpi(T_{0})$$\lim_{tarrow T}.\underline{w}(t)=\infty.$

Main results and examples

5

In this section we present the main results and their consequences. Also, a few examples are given toillustrate the importance of our main results.
Theorem 5.1 Let $\Phi(u, v)$ and $\Psi(u, v)$ satisfy condition (1.2) or (1.3). Let $f(u)/u\geq K>0$ for $u\neq 0,$
and let coefficients $r(t)$ and $q(t)$ satisfy “oscillatory condition” (4.2). Then equation (1.1) is oscillatory.

The main consequence is the following.
Corollary 5.1 Let $\Phi(u,v)$ and $\Psi(u,v)$ satisfy condition (1.2) or (1.3), and let $f(u)/u\geq K>0$ for, whereThen equation:. Let $\mu\leq 1/(\gamma-1)$ or $\nu\geq\gamma\mu-1$ and$u\neq 0$$\sigma\leq 1$$\gamma\geq 2$$(t^{\mu}\Phi(x,x’))’+t^{\nu}\Psi(x,x’)+t^{-\sigma}f(x)=0, t\geq t_{0}$

,

(5.1)

is oscillatory.

Proof. The hypotheses on$\Psi(u, v)$$\Phi(u, v),$, and $f(u)$ are the same as in Theorem 5.1. Hence, weneed only to show that the coefficients:$r(t)=t^{\mu},$$p(t)=t^{\nu}$and$q(t)=t^{-\sigma},$$t\geq t_{0},$, satisfy the required oscillatory conditon (4.2). Indeed,
where $\mu\leq 1/(\gamma-1)$ or $\nu\geq\gamma\mu-1$ andin both cases (1.2) and (1.3), if $C(t)=c/t$ for some $c>0$ and all $t\geq t_{0}>0$ , then:$\sigma\leq 1$$\frac{c}{t}\leq(\frac{1}{t})^{\mu(\gamma-1)}$$\frac{c}{t}\leq(\frac{1}{t})^{\mu\gamma-\nu}$and$\frac{c}{t}\leq(\frac{1}{t})^{\sigma}$

and$\lim\sup_{tarrow\infty}\int_{T}^{t}C(\tau)d\tau=\lim_{tarrow\infty}\int_{T}^{t}\frac{c}{\tau}d\tau=\infty,$

which proves this corollary.
Finally, we present some concrete examples which can be shown by previous corollary.
$\square$47
Example 5.1 Let $K>0,$$\mu\leq 1$

or $\nu\geq 2\mu-1$ and$\sigma\leq 1$. Then the equation:$(t^{\mu} \frac{x^{2}}{1+x^{2}}x’)’+t^{\nu}x^{3}x^{\prime 2}+Kt^{-\sigma}x=0, t\geq t_{0}>0,$

is oscillatory.
Example 5.2 Let $K>0,$$\mu\leq 1$

or $\nu\geq 2\mu-1$ and$\sigma\leq 1$. Then the equation:$(t^{\mu}(\sin x)^{2}x’)’+t^{\nu}x^{3}x^{;2}+Kt^{-\sigma}x=0, t\geq t_{0}>0,$

is oscillatory.

Example 5.3 Let$\alpha\geq 1,$$n\in \mathbb{N},$$K>0,$ $\mu\leq 1$ or $\nu\geq 2\mu-1$ and$\sigma\leq 1$. Then the equation:$(t^{\mu} \frac{x^{2}}{1+x^{2}}\frac{x’}{(1+x^{\prime 2})\S})’+t^{\nu}x(\frac{xx’}{(1+x^{2})(1+x^{\prime 2})\yen})^{2n}+Kt^{-\sigma_{X}}=0,$

is oscillatory.
Example 5.4 Let$\beta\geq 1,$$K>0,$ $\nu\geq 2\mu-1$ and$\sigma\leq 1$. Then the equation:$(t^{\mu}(\sin x)^{2}x^{J\beta})’+t^{\nu}x^{3}x^{;2\beta}+Kt^{-\sigma}x=0, t\geqt_{0}>0,$

is oscillatory.
Example 5.5 Let $K>0,$$\mu\leq 1,$$\nu\geq 0,$$\lambda\geq 0$, and$\sigma\leq 1$. Then the equation:$(t^{\mu} \frac{x^{2}}{1+x^{2}}x’)’+t^{\nu}|x|^{\lambda}xsh(x’)x’+Kt^{-\sigma}x=0, t\geq t_{0}>0,$

is oscillatory.
The oscillation criterion presented in Theorem 5.1 can be called the Fite-Wintner-Leighton type
criterion. The reason for that can be found in papers by Fite [1], Wintner [2], Leighton [3], and Pasic[4].

References[1] W. B. Fite, Conceming the zeros of the solutions of certain differential equations, Trans. Amer.
Math. Soc. 19 (1918), 341-352.
[2] A. Wintner, A criterion of oscillatory stability, Quart. J. Appl. Math. 7 (1949), 115-117.
[3] W. Leighton, The detection of the oscillation of solutions of a second order linear differential
equation, Duke Math. J. 17 (1950), 57-62.
[4] M. Pa\v{s}i\v{c}, Fite-Wintner-Leighton type oscillation criteria for second-order differential equations

with nonlinear damping, Abstract and Applied Analysis, in press.