The existence of multiple positive solutions for a?class of?semipositone Dirichlet boundary value problems

In this paper, we consider the existence of multiple positive solutions for the following singular semipositone Dirichlet boundary value problem: $$\left\{\begin{array}{l}-x''(t)=p(t)f(t, x) +q(t),\quad t\in(0,1),\\4pt]x(0) =0,\qquad x(1) = 0,\end{array}\right.$$ where p:(0,1)??0,+??) and f:0,1]×0,+??)??0,+??) are continuous, q:(0,1)??(???,+??) is Lebesgue integrable. Under certain local conditions and superlinear or sublinear conditions on f, by using the fixed point theorem, some sufficient conditions for the existence of multiple positive solutions are established for the case in which the nonlinearity is allowed to be sign-changing.