Classifications and existence of positive solutions of higher order nonlinear iterative functional differential equations
The authors consider a class of higher-order nonlinear iterative functional-differential equations of the form \[ (r(t) x^{(m- 1)}(t))'+ f(t, x(t), x(\Delta(t, x(t))))= 0\tag{1} \] in the case when \(\int^\infty ds/r(s)= \infty\). It should be noted that the deviating arguments depend on both the state variable \(x\) and the time \(t\) while in most of the papers the deviating arguments depend only on time. The authors investigate the asymptotic behaviour of the nonoscillatory solutions to (1). More precisely, they give a classification scheme of the eventually positive solutions according to their asymptotic behaviour as \(t\to\infty\). Further, they obtain necessary or sufficient conditions (or both in some cases) for the existence of solutions to (1) of any of the types introduced in their classification scheme.
scientific article; zbMATH DE number 1743077
scientific article