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

This paper deals with the following higher order nonlinear neutral differential equation[a(t)(b(t)(x(t)+c(t)x(t-τ))’)’](n-2)+g(t,x(g1(t))),…,xgk(t))=h(t),t≥t0.First section of this paper states the meaning of the above equation in its entirety. Massive literature just discuss linear functions, but not to discuss nonlinear function in depth. This paper discusses and studies the multivariate nonlinear function and obtains five results. For these situations of c(t)=±1, those literature don’t provide discussions in [1],[2],[4],[5]-[14],[16]-[22]. However, this paper gives methods studies for the above two kinds of situations.Second section of this paper appoints some related notations, definitions and lemmas which be applied in this paper.Third section of this paper deals with five different situations of c(t)=±1,-1<c(t)<1,1<c(t)<+∞and-∞<c(t)<-1for the higher order nonlinear neutral differential equation. Using the Krasnoselskii fixed point theorem, the Schauder fixed point theorem and some interrelated analysis technics, this paper establishes the existence of uncountably many bounded positive solutions for the above five kinds of situations of differential equation.Fourth section of this paper enumerates five examples which illustrate the main results of the third section.