Existence for nonoscillatory solutions of second-order nonlinear differential equations

In this paper, the existence of nonoscillatory solutions of the second-order nonlinear neutral differential equation[r(t)(x(t)+P(t)x(t−τ))′]′+∑i=1mQi(t)fi(x(t−σi))=0,t⩾t0, where m⩾1m⩾1 is an integer, τ>0τ>0, σi⩾0σi⩾0, r,P,Qi∈C([t0,∞),R)r,P,Qi∈C([t0,∞),R), fi∈C(R,R)fi∈C(R,R) (i=1,2,…,mi=1,2,…,m), are studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general P(t)P(t) and Qi(t)Qi(t) (i=1,2,…,mi=1,2,…,m) which means that we allow oscillatory P(t)P(t) and Qi(t)Qi(t) (i=1,2,…,mi=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.