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 ...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.展开更多
The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),w...The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.展开更多
The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t...The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t)((r1(t)(y′(t))α)′)β)′+q(t)yγ(σ(t))=0,t≥t0>0,where∫∞r1-α/1(s)ds<∞and∫∞r2-1/β(s)ds<∞.The criteria in this paper improve and complement some existing ones.The results are illustrated by two Euler-type differential equations.展开更多
A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.
文摘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.
基金This research is supported by the Shandong Provincial Natural Science Foundation of China(ZR2017MA043).
文摘The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))')^(α))'+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.
基金Youth Program of National Natural Science Foundation of China under Grant 61304008Youth Program of Natural Science Foundation of Shandong Province under Grant ZR2013FQ033
文摘The theme of this article is to provide some sufficient conditions for the asymptotic property and oscillation of all solutions of third-order half-linear differential equations with advanced argument of the form(r2(t)((r1(t)(y′(t))α)′)β)′+q(t)yγ(σ(t))=0,t≥t0>0,where∫∞r1-α/1(s)ds<∞and∫∞r2-1/β(s)ds<∞.The criteria in this paper improve and complement some existing ones.The results are illustrated by two Euler-type differential equations.
基金supported by FONDECYT 1080034APIS 29-11 DIUMCEDI 0052-10 UNAP
文摘A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.
