Existence of oscillatory solutions of second order delay differential equations

In this paper, we investigate existence of oscillatory solutions for a forced second order nonlinear delay differential equations [r(t)Φ(x′(t))]′+∑i=1mfi(t,x(gi(t)))=q(t), where fi∈C([t0,∞)×R,R),gi(t)≤t,limt→∞gi(t)=∞,i=1,2,⋯,m,Φ∈C1(R,R),Φ(u)fi∈C([t0,∞)×R,R),gi(t)≤t,limt→∞gi(t)=∞,i=1,2,⋯,m,Φ∈C1(R,R),Φ(u) is an increasing function for all u∈Ru∈R, Φ−1(u)Φ−1(u) satisfies the local LipischitzLipischitz condition. A new sufficient condition for global existence of oscillatory solution is obtained by the Schauder–Tychonoff theorem. When Φ(u)=uαΦ(u)=uα with α≥1α≥1 being the ratio of two positive odd integers has also been studied. We give examples to illustrate the applicability of our results.