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

The author considers the $$n$$th order functional differential equation $L_ n u(t)- (-1)^ n f(t,u(g(t)))= 0,\tag{1}$ where $$n\geq 3$$, $$L_ 0 u(t)= {u(t)\over r_ 0(t)}$$, $$L_ j u(t)={1\over r_ j(t)} (L_{j-1} u(t))'$$, $$j=1,2,\dots,n$$, $$r_ j$$, $$g: [t_ 0,\infty)\to \mathbb{R}$$, $$f: [t_ 0,\infty)\times \mathbb{R}\to\mathbb{R}$$ are continuous, $$r_ j(t)> 0$$ $$(j=0,1,\dots,n)$$, $$g(t)\geq t$$, $$\text{sign}(f(t,x))=\text{sign}(x)$$ for $$x\neq 0$$ and $$t\geq t_ 0$$ and $$\int^ \infty r_ j(s)ds= \infty$$ $$(j=1,\dots, n-1)$$. It is shown (Lemma 1): If $$u$$ is a nonoscillatory solution of (1) then there exist a $$t_ 1$$ and an integer $$l$$, $$0\leq l\leq n$$ such that $$l$$ is even and $$u(t)L_ j u(t)>0$$ on $$[t_ 1,\infty)$$, $$0\leq j\leq l$$, $$(-1)^{j-1} u(t) L_ j u(t)>0$$ on $$[t_ 1,\infty)$$, $$l\leq j\leq n$$. 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 $$u^{(m)}+ a(t)| u|^ n\text{sgn }u= 0$$, Mat. Sb., n. Ser. 65(107), 172-187 (1964; Zbl 0135.143)].
Denote by $${\mathcal N}_ 1$$ the set of all nonoscillatory solutions of degree $$l$$ of (1) and by $${\mathcal N}^ +$$ (resp. $${\mathcal N}^ -$$) 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 $${\mathcal N}^ += {\mathcal N}_ 0$$ and (1) has property (B) if $$n$$ is even and $${\mathcal N}^ -= {\mathcal N}_ 0\cup {\mathcal N}_ n$$. In the paper, it is next considered a “comparison” equation (2) $$M_ n u(t)- (-1)^ n z(t) h(u(\tau(t)))= 0$$, where $$M_ 0 u(t)= {u(t)\over q_ 0(t)}$$, $$M_ j u(t)= {1\over q_ j(t)} (M_{j-1} u(t))'$$, $$j=1,2,\dots,n$$. 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 $$r_ j$$, $$f$$, $$g$$ and $$q_ j$$, $$z$$, $$h$$, $$\tau$$ hold. Similar results for (1) with $$g(t)\leq t$$ were proved by S. R. Grace and B. S. Lalli [Math. Nachr. 144, 65-79 (1989; Zbl 0714.34106)].

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Citations:

Zbl 0135.143; Zbl 0714.34106