Multiple Positive Solutions of a Fourth-order Boundary Value Problem

We consider the nonlinear fourth-order semipositone boundary value problem u(4)=f(t,u(t),u′(t)),t∈(0,1),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u^{(4)}=f(t,u(t),u'(t)), \quad t \in (0,1), \end{aligned}$$\end{document}u(0)=u′(0)=u′′(1)=u′′′(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u(0)=u'(0)=u''(1)=u'''(1)=0, \end{aligned}$$\end{document}where f:[0,1]×[0,∞)×[0,∞)→(-∞,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: [0,1] \times [0,\infty ) \times [0, \infty ) \rightarrow (-\infty , \infty )$$\end{document} has the property f(t,x,y)≥-g(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(t,x,y) \ge -g(t)$$\end{document} for a nonnegative continuous function g(t). This paper improves the results of Ma (Hiroshima Math J 33:217–227, 2013) and Spraker (Differ Equ Appl 8:21–31, 2016).