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 ) | x ′ ( t ) | α − 1 x ′ ( t ) ) ′ + F ( t , x ( t ) , x ( τ ( t ) ) , x ′ ( t ) , x ′ ( τ ( t ) ) ) = 0 , t ≥ t 0 that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of [ t 0, ∞], 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 τ(t) ≤ t delay and Gt( t) ≥ t advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.