Existence Of Nonoscillatory Solutions For A Higher Order Nonlinear Neutral Delay Differential Equation

The aim of this paper is to investigate the solvability of a higher order nonlinear neutral delay differential equation of the form where n,m,l∈N,τ>0, functions a,b∈C([t0,+∞),R\{0}), c,g,ff∈C([t0,+∞),R) and f∈C([t0,+∞)xRk,R) with limfj,(t)=+∞.Applying Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem, this paper, respectively, shows the existence of uncountably many bounded nonoscillatory solutions for the above differential equation under seven cases below:(1)-c2≤c(t)≤C1, t≥T0>t0, where min{C1,C2}≥0, C1+C2<1,0<N<(1-C1-C2)M;(2) O≤C2≤C(t)≤C1<1,t≥T0>t0, where0<(1-C2)N<(1C1)M;(3)-1<-C1≤C(t)≤-C2≤0,t≥T0>t0, where0<(1-C2)N<(1-C1)M;(4) C2≤C(t)≤C1,t≥T0≥t0+τ,(5)-∞<-C1<C(t)≤-C2<-1, t≥T0≥t0+τ, where0<(C1,-1)N<(C2-1)M;(6) C(t)=1, t≥t0;(7) C(t)=-1, t≥t0.The cases discussed here make this research more comprehensive, also extend and complement the results of many previous. When the nonlinear term turns to a linear function, n,m,I take the special values, functions a, b, care constant functions or g(t)=0,t≥t0, the above equation can contain all the equations enumerated in this article, which embodies the equation of this form is more general. Seven non-trivial examples are given in the last to illustrate the superiority of our results.