Abstract
The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [r(t)(x(t)+P(t)x(t-τ))(n-1)]′+∑i=1mQi(t)fi(x(t-σi))=0, t≥t0, where m≥1,n≥2 are integers, τ>0, σi≥0, r,P,Qi∈C([t0,∞),R), fi∈C(R,R) (i=1,2,…,m), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Qi(t) (i=1,2,…,m) which means that we allow oscillatory Qi(t) (i=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.
Highlights
Consider the higher-order nonlinear neutral differential equation m rtxtPtxt − τ n−1Qi t fi x t − σi 0, t ≥ t0. i1With respect to 1.1, throughout, we shall assume the following:i m ≥ 1, n ≥ 2 are integers, τ > 0, σi ≥ 0, ii r, P, Qi ∈ C t0, ∞, R, r t > 0, fi ∈ C R, R, i 1, 2, . . . , m.Let ρ max1≤i≤m{τ, σi}
The existence of nonoscillatory solutions of higher-order nonlinear neutral differential equations received much less attention, which is due mainly to the technical difficulties arising in its analysis
In 1998, Kulenovic and Hadziomerspahic 1 investigated the existence of nonoscillatory solutions of second-order nonlinear neutral differential equation x t cx t − τ
Summary
Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations. The existence of nonoscillatory solutions of the higher-order nonlinear differential equation rtxtPtxt − τ n−1 m i1. 0, t ≥ t0, where m ≥ 1, n ≥ 2 are integers, τ > 0, σi ≥ 0, r, P, Qi ∈ C t0, ∞ , R , f i ∈ C R, R i 1, 2, . Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general. Our results improve essentially and extend some known results in the recent references
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