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Comparison theorems for functional differential equations with advanced argument. (English) Zbl 0803.34066

The author considers the $n$th order functional differential equation

${L}_{n}u\left(t\right)-{\left(-1\right)}^{n}f\left(t,u\left(g\left(t\right)\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $n\ge 3$, ${L}_{0}u\left(t\right)=\frac{u\left(t\right)}{{r}_{0}\left(t\right)}$, ${L}_{j}u\left(t\right)=\frac{1}{{r}_{j}\left(t\right)}{\left({L}_{j-1}u\left(t\right)\right)}^{\text{'}}$, $j=1,2,\cdots ,n$, ${r}_{j}$, $g:\left[{t}_{0},\infty \right)\to ℝ$, $f:\left[{t}_{0},\infty \right)×ℝ\to ℝ$ are continuous, ${r}_{j}\left(t\right)>0$ $\left(j=0,1,\cdots ,n\right)$, $g\left(t\right)\ge t$, $\text{sign}\left(f\left(t,x\right)\right)=\text{sign}\left(x\right)$ for $x\ne 0$ and $t\ge {t}_{0}$ and ${\int }^{\infty }{r}_{j}\left(s\right)ds=\infty$ $\left(j=1,\cdots ,n-1\right)$. It is shown (Lemma 1): If $u$ is a nonoscillatory solution of (1) then there exist a ${t}_{1}$ and an integer $l$, $0\le l\le n$ such that $l$ is even and $u\left(t\right){L}_{j}u\left(t\right)>0$ on $\left[{t}_{1},\infty \right)$, $0\le j\le l$, ${\left(-1\right)}^{j-1}u\left(t\right){L}_{j}u\left(t\right)>0$ on $\left[{t}_{1},\infty \right)$, $l\le j\le n$. The integer $l$ is called the degree of $u$. This result generalizes a well-known lemma of I. T. Kiguradze [On the oscillation of solutions of the equation ${u}^{\left(m\right)}+a\left(t\right){|u|}^{n}\text{sgn}\phantom{\rule{4.pt}{0ex}}u=0$, Mat. Sb., n. Ser. 65(107), 172-187 (1964; Zbl 0135.143)].

Denote by ${𝒩}_{1}$ the set of all nonoscillatory solutions of degree $l$ of (1) and by ${𝒩}^{+}$ (resp. ${𝒩}^{-}$) the set of all nonoscillatory solutions of (1) with odd $n$ (resp. even $n$). We say that (1) has property (A) if $n$ is odd and ${𝒩}^{+}={𝒩}_{0}$ and (1) has property (B) if $n$ is even and ${𝒩}^{-}={𝒩}_{0}\cup {𝒩}_{n}$. In the paper, it is next considered a “comparison” equation (2) ${M}_{n}u\left(t\right)-{\left(-1\right)}^{n}z\left(t\right)h\left(u\left(\tau \left(t\right)\right)\right)=0$, where ${M}_{0}u\left(t\right)=\frac{u\left(t\right)}{{q}_{0}\left(t\right)}$, ${M}_{j}u\left(t\right)=\frac{1}{{q}_{j}\left(t\right)}{\left({M}_{j-1}u\left(t\right)\right)}^{\text{'}}$, $j=1,2,\cdots ,n$. The author proves that (A) property (resp. (B) property) of (2) implies (A) property (resp. (B) property) of (1) provides that some sign conditions among the functions ${r}_{j}$, $f$, $g$ and ${q}_{j}$, $z$, $h$, $\tau$ hold. Similar results for (1) with $g\left(t\right)\le t$ were proved by S. R. Grace and B. S. Lalli [Math. Nachr. 144, 65-79 (1989; Zbl 0714.34106)].

##### MSC:
 34K99 Functional-differential equations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory