Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: \begin{equation*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{equation*} with mixed boundary conditions \begin{equation*} \begin{split} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{split} \end{equation*} where $\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}$, $\psi(t)>0$ for all $t \in [\rho(c),\,\sigma(d)]_\mathbb{T}$; both $g$ and $h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}$ are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when $g$ and $h$ are both superlinear or sublinear or one is superlinear and the other is sublinear for $\lambda_i>0;\,i=1,\,2$ are sufficiently small.