Positive Solutions for Semipositone m-point Boundary-value Problems

Let ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < ··· < ξm−2 < 1, ai, bi ∈ [0,∞) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0 < {\sum\nolimits_{i = 1}^{m - 2} {a_{i} < 1} } $$\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}$$ {\sum\nolimits_{i = 1}^{m - 2} {b_{i} < 1} } $$\end{document}. We consider the m-point boundary-value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {u}\ifmmode{''}\else$''$\fi + \lambda f{\left( {t,u} \right)} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} t \in {\left( {0,1} \right)}, $$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {x}\ifmmode{'}\else$'$\fi{\left( 0 \right)} = {\sum\limits_{i = 1}^{m - 2} {b_{i} {x}\ifmmode{'}\else$'$\fi{\left( {\xi _{i} } \right)},{\kern 1pt} {\kern 1pt} {\kern 1pt} x{\left( 1 \right)} = {\sum\limits_{i = 1}^{m - 2} {a_{i} x{\left( {\xi _{i} } \right)},} }} } $$\end{document} where f(x, y) ≥ −M, and M is a positive constant. We show the existence and multiplicity of positive solutions by applying the fixed point theorem in cones.