摘要
Consider a retarded differential equationx^(α-1)(t)x'(t)+P_0(t)x~α(t)+sum from i=1 to N P_i(t)x~α(g_i(t))=0, g_i(t)<t, (1)and an advanced differential equationx^(α-2)(t)x'(t)-P_0(t)x~α(t)-sum from i=1 to N P_i(t)x~α(g_i(t))=0, g_i(t)>t, (2)where a=m/n, m and n are odd natural numbers, P_0(t), P_i(t) and g_i(t) are continuous functions,and P_i(t) are positive-valued on [t_0, ∞), lim g_i(t)=∞. i=1,2.…, N. We prove the followingTheorem. Suppose that there is a constant T such thatinfμ>0,t≥T α:μ sum from i=1 to N P_i(t) exp[αB_i+μT_i(t)]>1. (3) Then all solutions of (1) and (2) are oscillatory.Here B_i=inf t≥T. P_0(s)ds>∞, D_i=[g_i(t), t], T_i(t)=t-g_i(t), for (1), and D_i=[t, g_i(t)]. T_i(t)=g_i(t)-t for (2), i=1,2,…,N.
Consider a retarded differential equationx^(α-1)(t)x'(t)+P_0(t)x~α(t)+sum from i=1 to N P_i(t)x~α(g_i(t))=0, g_i(t)<t, (1)and an advanced differential equationx^(α-2)(t)x'(t)-P_0(t)x~α(t)-sum from i=1 to N P_i(t)x~α(g_i(t))=0, g_i(t)>t, (2)where a=m/n, m and n are odd natural numbers, P_0(t), P_i(t) and g_i(t) are continuous functions,and P_i(t) are positive-valued on [t_0, ∞), lim g_i(t)=∞. i=1,2.…, N. We prove the followingTheorem. Suppose that there is a constant T such thatinfμ>0,t≥T α:μ sum from i=1 to N P_i(t) exp[αB_i+μT_i(t)]>1. (3) Then all solutions of (1) and (2) are oscillatory.Here B_i=inf t≥T. P_0(s)ds>∞, D_i=[g_i(t), t], T_i(t)=t-g_i(t), for (1), and D_i=[t, g_i(t)]. T_i(t)=g_i(t)-t for (2), i=1,2,…,N.
