Interval criteria for oscillation of second-order functional differential equations

Abstract

By using averaging functions, new interval oscillation criteria are established for the second-order functional differential equation, (r(t)l~'(t)l~-l~'(t))'+t,~(t,~(t),~(~-(t)),~'(t),~'(,-(t)))=o, t>to, that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [to, oo), rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional differential equations, our criteria implies that the T(t) < t delay and v(t) >__ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results. (~) 2005 Elsevier Ltd. All rights reserved.

Oscillation of second order differential equations with mixed nonlinearities

“…(1.1)-(1.2). Our results generalize and improve some known results in [4,13,15,16,20,22]. Finally, several examples are also given to illustrate the importance of our results.…”

“…(1.2), Theorems 3.4-3.7 reduce to corresponding all theorems given in [20]. However, our conditions do not require…”

“…Therefore, the problem is to find oscillation criteria that use only the information about the involved functions on these intervals; outside of these intervals the behavior of the functions is irrelevant. Such type of oscillation criteria are referred to as interval oscillation criteria, see, e.g., [4,20,22].…”

New interval oscillation criteria for second-order functional differential equations with nonlinear damping

“…Since the functional term contained in (9) has a very general form, oscillation of the Eq. (9) cannot be demonstrated by other oscillation criteria given in the papers cited in this study, except [15,23]. In example 2.8, we have considered a very simple case k 1 .u; v/ D v, but if we consider Eq.…”

“…where the functions w, w 1 and w 2 are defined with (15) and (25). Now, by using Lemma 3.2 with (30), (31), (32) and taking (27) into account, we obtain…”