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)].


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