Comparison and oscillation theorems for functional differential equations with deviating arguments. (English) Zbl 0714.34106

The authors are interested in comparing the oscillatory behaviour of the two equations (I) \(L_ nx(t)+f(t,x(r(t)))=0,\) n even, \(L_ 0x(t)=x(t)\), \(L_ kx(t)=(1/a_ k(t))(L_{k-1}x(t))',\) \(k=1,...,n-1\) and (II) \(M_ nx(t)+q(t)h(x(s(t)))=0,\) n even \(M_ 0x(t)=x(t)\), \(M_ kx(t)=1/b_ k(t)(M_{k-1}x(t))',\) \(k=1,...,n-1\). This problem includes the earlier studied equations \(x^{(n)}(t)+P(t)F(x(r(t)))=0\) (Mahfoud) and also \(M_ nx(t)+q(t)x(t)=0\) (Kusano, Kitamura). The results connected with (I) and (II) are of high degree of generality and includes a lot of former theorems. Also a comparison technique between (II) and a set of first order delay differential equations is studied.
Reviewer: T.Dłotko


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
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