The existence of positive solutions to neutral differential equations
From authors' abstract: We consider a class of neutral differential equations of the form \[ \Biggl[x(t)- {r(t)\over r(t-\tau)} x(t- \tau)\Biggr]'+ Q(t) x(t-\sigma)= 0,\quad t\geq t_0,\tag{\(*\)} \] with \(\tau\in (0,\infty)\), \(\sigma\in [0,\infty)\), \(Q(t)\in C([t_0, \infty),\mathbb{R}^+)\), \(r(t)\in C([t_0, \infty),(0, \infty))\) with \(r(t)\) nondecreasing on \([t_0- \tau,\infty)\). We show that all positive solutions to \((*)\) can be classified into four types, \(A\), \(B\), \(C\), and \(D\). Sufficient and necessary conditions are obtained for the existence of \(A\)-type, \(B\)-type, and \(D\)-type positive solutions to \((*)\), respectively. A sufficient condition for the existence of \(C\)-type positive solutions to \((*)\) is also given. Finally, a sharp oscillation result is offered for all solutions to \((*)\). These results generalize and improve those established in \textit{B. Yang} and \textit{B. G. Zhang} [Funkcial. Ekvacioj, Ser. Int. 39, No. 2, 347-362 (1996; Zbl 0865.34054)].
scientific article; zbMATH DE number 1417575
scientific article