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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 $$L\sb n u(t)- (-1)\sp n f(t,u(g(t)))= 0,\tag1$$ where $n\ge 3$, $L\sb 0 u(t)= {u(t)\over r\sb 0(t)}$, $L\sb j u(t)={1\over r\sb j(t)} (L\sb{j-1} u(t))'$, $j=1,2,\dots,n$, $r\sb j$, $g: [t\sb 0,\infty)\to \bbfR$, $f: [t\sb 0,\infty)\times \bbfR\to\bbfR$ are continuous, $r\sb j(t)> 0$ $(j=0,1,\dots,n)$, $g(t)\ge t$, $\text{sign}(f(t,x))=\text{sign}(x)$ for $x\ne 0$ and $t\ge t\sb 0$ and $\int\sp \infty r\sb j(s)ds= \infty$ $(j=1,\dots, n-1)$. It is shown (Lemma 1): If $u$ is a nonoscillatory solution of (1) then there exist a $t\sb 1$ and an integer $l$, $0\le l\le n$ such that $l$ is even and $u(t)L\sb j u(t)>0$ on $[t\sb 1,\infty)$, $0\le j\le l$, $(-1)\sp{j-1} u(t) L\sb j u(t)>0$ on $[t\sb 1,\infty)$, $l\le j\le n$. The integer $l$ is called the degree of $u$. This result generalizes a well-known lemma of {\it I. T. Kiguradze} [On the oscillation of solutions of the equation $u\sp{(m)}+ a(t)\vert u\vert\sp n\text{sgn }u= 0$, Mat. Sb., n. Ser. 65(107), 172-187 (1964; Zbl 0135.143)]. Denote by ${\cal N}\sb 1$ the set of all nonoscillatory solutions of degree $l$ of (1) and by ${\cal N}\sp +$ (resp. ${\cal N}\sp -$) 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 ${\cal N}\sp += {\cal N}\sb 0$ and (1) has property (B) if $n$ is even and ${\cal N}\sp -= {\cal N}\sb 0\cup {\cal N}\sb n$. In the paper, it is next considered a “comparison” equation (2) $M\sb n u(t)- (-1)\sp n z(t) h(u(\tau(t)))= 0$, where $M\sb 0 u(t)= {u(t)\over q\sb 0(t)}$, $M\sb j u(t)= {1\over q\sb j(t)} (M\sb{j-1} u(t))'$, $j=1,2,\dots,n$. 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 $r\sb j$, $f$, $g$ and $q\sb j$, $z$, $h$, $\tau$ hold. Similar results for (1) with $g(t)\le t$ were proved by {\it S. R. Grace} and {\it B. S. Lalli} [Math. Nachr. 144, 65-79 (1989; Zbl 0714.34106)].

34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory