Existence of one-signed solutions of nonlinear four-point boundary value problems

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u'' + Mu = rg(t)f(u),u(0) = u(\varepsilon ),u(1) = u(1 - \varepsilon ) $$\end{document}, where ε ∈ (0, 1/2), M ∈ (0,∞) is a constant and r > 0 is a parameter, g ∈ C([0, 1], (0,+∞)), f ∈ C(ℝ,ℝ) with sf(s) > 0 for s ≠ 0. The proof of the main results is based upon bifurcation techniques.