Classification of Non-Oscillatory Solutions of Nonlinear Neutral Delay Impulsive Differential Equations

Abstract

In this paper, a general class of second order nonlinear neutral delay impulsive differential equation of the form ( ) ( ) ( ) ( ( ( ) ( ))) ( ) ( ) ( ( ( ) ( ( )))) m n i i j jl jl 0 k i 1 j 1 m n k ik k i jk k jl k jl k k 0 k i 1 j 1 y t p t y t f t, y g t , , y g 0, t t R , t t y t p y t f t ,y g t , , y g t 0, t t R , t t + = = + = =   ″   − −τ  + = ≥ ∈ ≠      ′ Δ  − −τ  + = ≥ ∈ =    Σ Σ Σ Σ   is considered. We classifyits non-oscillatory solutions into four types of solution sets, namely Λ(0,0,0) ,Λ(b,a,0) ,Λ(∞,∞,0) and Λ(∞,∞,d) and establish necessary and sufficient conditions for the existence of these nonoscillatory solutions by means of Schauder-Tychonoff fixed point theorem and Lebesgue’s Monotone Convergence Theorem. Some examples are given to illustrate the obtained results.