Oscillations of Nth-order Functional Differential Equations

-Some new oscillation criteria for the even order damped functional differential equation (a(t)z(n-1)(t)) ' ÷ p(t)lz(n-1)(Ol#x(n-a)(O ÷ q(t)f(xIaa(t)] . . . . . =Lq.~(t)]) = 0 are established, where ~ _> O. These criteria are an extension of some of the known results. 1. I N T R O D U C T I O N Recently, Grace and Lalli [1] discussed the oscillation of the nth-order functional differential equation (a ( t )x ("-a) ( t ) ) ' + p ( t ) l z (" -a ) ( t ) l%("-~) ( t ) + q( t ) f ( z [g( t ) ] ) = 0 where/~ > 0, n is even. In this paper, we establish some new oscillation criteria for the nth-order damped functional differential equation (aCt)zC"-l)Ct))' + pCt)lz("-l)(t)l~z("-x)(t) + q ( t ) f ( x [ g l ( t ) ] , . . . , z[gm(t)]) = 0 (1) where/~ >_ 0, n is even. In what follows, we restrict our discussion to nontrivial solutions of Eq.(1) which are indefinitely continuable to the right. The oscillatory property is considered in the usual sense, i.e. a solution of Eq.(1) is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. Eq. (1) is said to be oscillatory ff every solution of Eq.(1) is oscillatory. In the sequel we will assume the following conditions: (a) a, p, q: [to, ~ ) ---, [0, ~ ) are continuous, a(t) > O, q(t) is not eventually identically zero; (b) f : R m --* R is continuous, i fy l _< z i , i = 1,2, . . . . m, then f ( Y l , Y2 , . . . ,Ym) _< <_/(zl, z 2 , . . . , z~), and f(Yl,Y2, . . . ,Ym) > 0 if Yi > 0 for all i, f ( Y l , Y z , . . . ,Yra) > 0 if Yi < 0 for all i; (c) gi : [t0, oo) are continuous, i = 1 ,2 , . . . ,m , gi(t) ~ oo as t --+ oo, and there exist differentiable functions ~q : [to, c~) ---, [0, or) such that ~'i(t) = inf n~>n{s,gi(s)) , b , ( t ) > 0 and ai( t ) ~ oo as t ~ oo. (d) ~v, exist and ~ ~ a , > 0for y, ~ 0, i 1 ,2 , . . . ,m. AMS subject classification: 34015, 34010 Typeset by .AAdS-~X 95 CN4WA 21:2/3-G