Nonoscillatory solutions for higher-order nonlinear neutral delay differential equations

Abstract

Abstract This paper deals with the solvability of the higher-order nonlinear neutral delay differential equation d n d t n [ x ( t ) + p ( t ) x ( t − τ ) ] + ( − 1 ) n + 1 ∑ i = 1 m q i ( t ) x ( α i ( t ) ) + ( − 1 ) n + 1 f ( t , x ( β 1 ( t ) ) , … , x ( β l ( t ) ) ) = r ( t ) , t ≥ t 0 , where τ > 0 , n , m , l ∈ N , p , r , q i , α i , β j ∈ C ( [ t 0 , + ∞ ) , R ) , and f ∈ C ( [ t 0 , + ∞ ) × R l , R ) satisfying lim t → + ∞ α i ( t ) = lim t → + ∞ β j ( t ) = + ∞ , i ∈ { 1 , 2 , … , m } , j ∈ { 1 , 2 , … , l } . With respect to various ranges of the function p, we investigate the existence of uncountably many bounded nonoscillatory solutions for the equation. The main tools used in this paper are the Krasnoselskii and Schauder fixed point theorems together with some new techniques. Six nontrivial examples are given to illustrate the superiority of the results presented in this paper. MSC:39A10, 39A20, 39A22.