The Existence Of Uncountably Many Bounded Positive Solutions For A Third Order Nonlinear Neutral Delay Differential Equation

This work deals with the existence of uncountably many bounded positive solutions for the third order nonlinear neutral delay differential equation 3d³/dt³ [x(t)+p(t)x(t-τ)]+f(t, x (t-τ₁ ),... x (t-τ_k)) = 0, t≥t₀, whereτ> 0,τ_i∈R⁺ , i∈{1,2,..., k }, p∈C ([t₀, +∞), R⁺)and f∈C([t₀ +∞)×R^k, R).Under suitable conditions, by using Banach fixed point theorem and Krasnoselskii's fixed point theorem, the existence of bounded positive solutions of the third order nonlinear neutral delay differential equation is proved in the text. On this basis, by using Banach fixed point theorem, the number of solutions are expanded to uncountably.Based on the existence of uncountably many bounded positive solutions for the third order nonlinear neutral delay differential equation, some examples are given to illustrate main resuts at the end of the work.