Oscillations of first order linear retarded differential equations (Q807821)

The author considers the first order retarded differential equation \[ (E)\quad x'(t)+\sum^{n}_{k=1}p_ k(t)x(t-\tau_ k(t))=0, \] where \(p_ k\) and \(\tau_ k\) \((k=1,2,...,n)\) are non-negative continuous functions on an interval \([t_ 0,\infty)\), and \(\lim_{t\to \infty}(t- \tau_ k(t))=\infty\) \((k=1,2,...,n)\). Main results of the paper establish sufficient conditions for all solutions of (E) to be oscillatory and conditions, under which (E) has at least one positive solution x with \(\lim_{t\to \infty}x(t)=0\) and such that \(x(t)\leq \exp \{-\lambda \int^{t}_{t_ 0}[\sum^{n}_{j=1}p_ j(s)]ds]\) for all large t(\(\lambda\) is a positive number). In the paper we find several relations with earlier known results.