A two-point problem for first-order systems

Let f be a continuous function from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ [a, b] \times \mathbb{R}^n $\end{document} into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \mathbb{R}^n $\end{document}. In this paper we prove that the problem¶¶\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \left \{ \begin{array}{ll}u^{\prime}= f(t,u)+ \lambda \\[3pt]u(a)=u(b)=0\end{array}\right.\ $\end{document}¶¶ has a (classical) solution for a wide class of functions f. Next we point out a particular case.