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A comparison theorem for certain functional differential equations
| Content Provider | Semantic Scholar |
|---|---|
| Author | True, Ernest DeCarteteret |
| Copyright Year | 1975 |
| Abstract | The oscillatory character of solutions to the functional differential equation x(n (t) + a(t)f(x(g(t)) = Q(t) is investigated, by comparison with the oscillatory character of solutions to x(n) (t) + s(t)f(x(t))=O where s(t)> jya(t), 0 t c for large t but is otherwise arbitrary. Consider the functional differential equation (1) x(n )(t) + a(t)/(x(g(t))) = Q(t), whose solutions may be compared with those of the differential equation (2) x(n)(t) + s(t)/(x(t)) = 0, where s(t) > ya(t) for all sufficiently large t, 0 < y < 1. The questions of existence and uniqueness of solutions to initial value problems with functional arguments have been considered recently in [41, [51 and [6]. The oscillatory behaviour of solutions to functional differential equations has been examined by Waltman [81 for second order equations of the form x(t) + a(t) / (x(t), x(g(t))) = 0. Some generalizations of these results can be found in [1]. Recently, Henry [21 has submitted some numerical techniques for approximating solutions to functional differential equations. The choice of algorithm employed in approximating such solutions is often determined by prior knowledge of the behaviour of these solutions. The purpose of this work is to determine the oscillatory behaviour of solutions to (1), by comparing the solutions of (1) to the solutions of the well-known equation (2), whose solutions have been examined considerably in the literature; see for example [7]. Presented to the Society, January 26, 1973; received by the editors April 2, 1973 and, in revised form, November 23, 1973. AMS (MOS) subject classifications (1970). Primary 34J05, 34K05; Secondary 34C10. |
| Starting Page | 127 |
| Ending Page | 132 |
| Page Count | 6 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1975-0352657-7 |
| Volume Number | 47 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1975-047-01/S0002-9939-1975-0352657-7/S0002-9939-1975-0352657-7.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1975-0352657-7 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
