Existence of Positive Solutions for a Class of Higher Order Neutral Functional Differential Equations

The higher order neutral functional differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$(1) \frac {d^n}{dt^n} [x(t)+ h(t)x ( \tau(t))]+ \sigma f (t,x (g(t)))=0$$ \end{document} is considered under the following conditions: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$n \geqslant 2, \sigma = \pm 1, \tau (t)$$ \end{document} is strictly increasing in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$t \in \left[ {t_0 ,\infty } \right), \tau (t) < t {\text{for}} t \geqslant t_0 ,\mathop { \lim }\limits_{t \to \infty } \tau (t) = \infty ,\mathop { \lim }\limits_{t \to \infty } g(t) = \infty , {\text{and}} f(t,u)$$ \end{document} is nonnegative on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left[ {t_0 ,\infty } \right) \times \left( {0,\infty } \right)$$ \end{document} and nondecreasing in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u \in (0,\infty )$$ \end{document}. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).