A higher order nonlinear neutral differential equation

This paper is concerned with the higher order nonlinear neutral differential equation\[[a(t)(x(t)+b(t)x(\tau(t))) ]^{(n1)}+f(t, x(g_1(t)),\ldots,x(g_k(t)))=c(t),\quad t\ge t_0.\] By dint of the LeraySchauder nonlinear alternative, Rothe fixed point theorem and some new techniques, we prove the existence of uncountably many bounded positive solutions for the equation. Several nontrivial examples are given to illustrate the applications and advantages of the results presented in this paper.