ON OSCILLATION OF SOLUTIONS OF HIGHER ORDER DIFFERENTIAL RETARDED INEQUALITIES

ABSTRACT. Some oscillation criteria for certain differential retarded inequalities with forcing terms of the kind introduced by Kartsatos are discussed. In a recent paper Lu-San Chen and Cheh-Chih Yeh [ 1] established an oscillation criterion for the n th order differential retarded inequalities: (1) x(t)(Lnx(t) + (-1)n+l,I;•lPi(t)fi(x[gi(t)])- i,, q, for n odd, and (2) x(t){Lnx(t) + (-1) n+l •;• = 1Pi(t)fi(x[gi(t)] )- h(t))> • O, for n even, where L0x(t) = x(t), LkX(t)-- 1.. (L k 1 x(t))', k = 1,2,...,n rk(t)-r0(t) = rn(t) = 1,( ' = d-•t' However there appears to be an error in the main result, in that the inequality (7) in that paper does not necessarily follow from (6). Moreover the inequalities (9), (10) and (11) in [ 1] hold only if h = 0. Therefore the result in [ 1] for h:P 0 is vacuous. The main purpose of this note is to establish some oscillation criteria for inequalities (1) (or (2)) which involve forcing terms of the kind introduced by Kartsatos [3,4]. We are concerned only with continuable solutions of (1) (or (2)) which exist on some half line [t0,oo), t0•0. A solution x(t) of (1) (or (2)) will be called nonoscillatory if there exists a t 1 • to such that x(t):P 0 for t • t 1, and x(t) will be called oscillatory if it has arbitrary large zeros. ASSUMPTIONS. (i) Pi,gi,rk 6 C[R+ =[0,oo),R+\{0)], gi(t) • t, tli_•m gi(t) =oo, gi oo is non-decreasing g(t) = max gi(t) and f rk(S)ds = 0 % k = 1,2,...,n-1 i = 1,2,...,m, 1-•i-•.rn &apos