On some nonlinear ordinary differential equations with advanced arguments

• Published in: Nonlinear analysis Vol. 53; no. 3; pp. 495 - 505
• Main Authors: Augustynowicz, Antoni, Leszczyński, Henryk, Walter, Wolfgang
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.2003; Elsevier
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Ordinary differential equations; Sciences and techniques of general use; Non linear equation; Solution uniqueness; Differential equation; Analytical solution; Initial value problem; Classification
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/S0362-546X(02)00314-0

The paper deals with ODEs with an advanced argument, in particular, with such equations as y′( t)=( y( βt)) 1/ β , the simplest example is y′(t)= y(2t) . The corresponding initial-value problem with y(0)= y 0>0 has three types of solutions: (a) a unique analytic solution, (b) solutions of subexponential growth, and (c) for every λ>0 a solution that behaves like e λt as t→∞. In the case y 0=0 there is (besides y≡0) a positive solution, for some β>1 it can be analytic, but the above example β=2 admits infinitely many analytic solutions.