Oscillation Results for Nonlinear Functional Differential Equations (1

Funkcialaj Ekvacioj, 27, (1984), 255-260

Oscillation Results for Nonlinear Functional Differential Equations By

John R. GRAEF*, Samuel M. RANKIN, III and Paul W. SPIKES* (Mississippi State University, and West Virginia University, U.S.A.)

§1.

Introduction.

In this paper we obtain three oscillation results for forced second order nonlinear functional differential equations. One of these results is obtained by comparing the functional equation to a second order unforced ordinary linear equation while the other two are obtained by comparing the functional equation to a second order unforced nonlinear ordinary equation. What makes our theorems different from other comparison type results is that the explicit knowledge of the distance between consecutive zeros of the forcing term is utilized. The types of forcing terms allowed do not have to be small, or periodic, or strongly bounded as was needed in [3-6]. Neither do we impose any sign conditions on the functions and $F$ below, nor do we use any type of integral conditions. In order to prove our theorems we use, in addition to the well-known Picone identity for linear equations, a nonlinear version of that identity. This approach was successfully used in [2] to obtain nonoscillation results for nonlinear ordinary equations. $b$

§2.

Linear comparison. We consider the equations

(1)

$(a_{1}(t)x^{¥prime})^{¥prime}+b(t, x(t), x(g(t)))=f(t)$

and

(2)

$(a_{2}(t)y^{¥prime})^{¥prime}+k(t)y=0$

and $b:[t_{0},$ ) are continuous, $a_{1}(t)>0$ , $a_{2}(t)>0$ , and as . A solut.ion $u(t)$ of (1) or (2) is said to be oscillatory if its set of zeros is unbounded and it is called nonoscillatory otherwise. The points and , , are called consecutive sign change points of a continuous function

where

$a_{1}$

,

, ,

$a_{2},f$

$g$

$k:[t_{0},$

$ g(t)¥rightarrow¥infty$

$t_{1}$

$t_{2}$

.

$¥infty$

)

$¥rightarrow R$

$¥infty$

$¥times R^{2}¥rightarrow R$

$ t¥rightarrow¥infty$

$t_{1}0$

$(t_{1}-¥epsilon, t_{1})$

and

$(t_{2}, t_{2}+¥epsilon)$

and

$[t_{1}, t_{2}]$

(3)

$a_{2}(t)¥geq a_{1}(t)$

and

(4)

$u[b(t, u, v)-k(t)u]¥geq 0$

if $uv>0$ .

Moreover, we assume throughout this section that either (3) or (4) does not become an identity on any open interval where $f(t)¥equiv 0$ . Theorem 1. Suppose that $(3)-(4)$ hold, equation (2) is oscillatory, and the distance between any two consecutive sign change points of $f$ is greater than the distance between consecutive zeros of any solution of (2). Then every solution of (1) is oscillatory.

. be a nonoscillatory solution of (1), say $x(t)>0$ for ¥ $t_{2}>t_{1}>T_{1}$ $x(g(t))>0$ . Let be consecusuch that Then there exists for , and let $y(t)$ be a tive sign change points of $f(t)$ such that $f(t)¥leq 0$ for . There exists such that and solution of (2) with $y(t_{1})=0$ and $y(t_{3})=0$ . We then have

Proof. Let $x(t)$

$t¥geq T¥geq t_{0}$

$t¥geq T_{1}$

$T_{1} geq T$

$t_{1}¥leq t¥leq t_{2}$

$y^{¥prime}(t_{1})>0$

$t_{3}$

$t_{1}t_{0}$ such that for every pair of consecutive zeros $T$ and of $u(t)$ in [ , we have . The following corollary is then an immediate consequence of Theorem 1. $¥epsilon>0$

$t_{2}$

$t_{1}$

$¥infty)$

$|t_{2}-t_{1}|0$ , and every solution of (2) is quickly oscillatory, then equation (1) is oscillatory. Corollary 2 generalizes Theorem 5 in [7], We conclude this section with a simple example of Theorem 1.

(5)

The equation

$x^{¥prime¥prime}+4x=1/2+¥sin t$

can be compared to the unforced linear equation $x^{¥prime¥prime}+4x=0$

.

It is easy to see that the hypotheses of Theorem 1 are satisfied and so equation (5) is oscillatory. Observe that the function $R(t)=t^{2}/4-¥sin t$ has the property that $R^{¥prime¥prime}(t)=1/2+¥sin t$ . None of the results in [3?6] can be applied to equation (5). §3.

Nonlinear comparision.

In this section we consider the equations

(6)

$(a_{1}(t)x^{¥prime})^{¥prime}+H(t)f(x(t))F(x(g(t)))=r(t)$

and

(7)

$(a_{2}(t)y^{¥prime})^{¥prime}+G(t)h(y(t))=0$

where , and are as before, ous, and $H(t)¥geq 0$ . We will also assume that $a_{1}$

$a_{2}$

$g$

(8) (9)

(11)

$G$

,

$r:[t_{0},$

and

$uh(u)>0$ $k_{1}$

and

$k_{2}$

$¥infty$

)

$¥rightarrow R$

, and $f$, $F$,

$h:R¥rightarrow R$

are continu-

if $u¥neq 0$ ,

$f^{¥prime}(u)>0$

and there exist constants

(10)

,

$H$

if

$uf(u)>0$

$u¥neq 0$

,

such that if $u¥neq 0$ ,

$0¥leq h^{¥prime}(v)/f^{¥prime}(u)¥leq k_{1}$

$F(u)¥geq k_{2}$

if

$u¥neq 0$

(12)

$a_{2}(t)¥geq k_{1}a_{1}(t)$

(13)

$k_{2}H(t)¥geq G(t)$

,

, ,

and either (12) or (13) does not become an identity on any open interval where .

$r(t)¥equiv 0$

258

J. R. GRAEF, S. M. RANKIN NI and P. W. SPIKES

Theorem 3. If $(8)-(13)$ hold, equation (7) is oscillatory, and the distance between any two consecutive sign change points of is greater than the distance between consecutive zeros of any solution of (7), then every solution of (6) is oscillatory. $r$

Suppose that (6) has a nonoscillatory solution $x(t)$ . $lfx(t)$ is eventually $x(t)>0$ and $x(g(t))>0$ for say positive, , let be consecutive sign change points of such that $r(t)¥leq 0$ on . Let $y(t)$ be a solution of (7) satisfying $y(t_{1})=0$ and . Then there exists such that , $y(t_{3})=0$ , and $y(t)>0$ . on It is easy to see that

Proof.

$T¥geq t_{0}$

$t_{2}>¥mathrm{r}_{1}>T$

$[t_{1}, t_{2}]$

$r$

$y^{¥prime}(t_{1})>0$

$t_{1}0$ and $x(g(t))>0$ for $t¥geq T$. eventually positive. Then there exists Now let , and and a solution $y(t)$ of (7) be chosen as in the proof of Theorem 3; then

Proof. Assume that

$x(t)$

$T¥geq t_{0}$

$t_{1}$

$t_{2}$

$t_{3}$

$¥{h(y(t))[k_{1}^{2}f(x(t))a_{2}(t)y^{¥prime}(t)-k_{1}h(y(t))a_{1}(t)x^{¥prime}(t)]/f(x(t))¥}^{¥prime}$

$=k_{1}[H(t)F(x(g(t)))-k_{1}G(t)]h^{2}(y(t))-k_{1}r(t)h^{2}(y(t))/f(x(t))$ $+k_{1}h^{¥prime}(y(t))[k_{1}a_{2}(t)-h^{¥prime}(y(t))a_{1}(t)/f^{¥prime}(x(t))][y^{¥prime}(t)]^{2}$

$+k_{1}a_{1}(t)f^{¥prime}(x(t))[h^{¥prime}(y(t))y^{¥prime}(t)/f^{¥prime}(x(t))-h(y(t))x^{¥prime}(t)/f(x(t))]^{2}$

$¥geq k_{1}[k_{2}H(t)-k_{1}G(t)]h^{2}(y(t))+k_{1}^{2}h^{¥prime}(y(t))[a_{2}(t)-a_{1}(t)][y^{¥prime}(t)]^{2}$ $+k_{1}a_{1}(t)f^{¥prime}(x(t))[h^{¥prime}(y(t))y^{¥prime}(t)/f^{¥prime}(x(t))-h(y(t))x^{¥prime}(t)/f(x(t))]^{2}$

$-k_{1}r(t)f¥iota^{2}(y(t))/f(x(t))$

.

Integrating the last inequality from to leads to a contradiction. A similar argument also leads to a contradiction if it is assumed that (6) has a solution which is eventually negative. $t_{1}$

$t_{3}$

The same reasoning indicated after the proof of Theorem 3 leads to following corollary to Theorem 5. Corollary 6. If $(8)-(11)$ and (14)?(15) hold, equation (7) has an oscillatory solution, and $r(t)¥equiv 0$ , then every solution of (6) is oscillatory.

Remark. $(14^{¥prime})$

If $F(u)¥equiv 1$ so that (6) is an ordinary equation, then (14) becomes $H(t)¥geq k_{1}G(t)$

and once again the requirement that

$H(t)¥geq 0$

is not needed.

It is interesting to notice that if $k_{1}