A uniqueness result for a class of infinite semipositone problems with nonlinear boundary conditions

We study positive solutions to the two-point boundary value problem: $$\begin{aligned} \begin{matrix} -u''=\lambda h(t) f(u)~;~(0,1) \\ u(0)=0\\ u'(1)+c(u(1))u(1)=0,\end{matrix} \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$h \in C^1((0,1],(0,\infty ))$$ is a decreasing function, $$f \in C^1((0,\infty ),\mathbb {R}) $$ is an increasing concave function such that $$\lim \limits _{s \rightarrow \infty }f(s)=\infty $$ , $$\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0$$ , $$\lim \limits _{s \rightarrow 0^+}f(s)=-\infty $$ (infinite semipositone) and $$c \in C([0,\infty ),(0,\infty ))$$ is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for $$\lambda \gg 1$$ .