On nonlinear boundary conditions satisfying certain asymptotic behavior (Q715678)

In this interesting paper, the author studies the existence of a positive solution to a boundary value problem (BVP), where one of the boundary conditions is allowed to be nonlinear, namely \[ \begin{aligned} u''(t)&+\lambda a(t) g(u(t))=0,\;t \in (0,1),\\ &u(0) =\varphi (u) ,\;u(1) =0.\\ \end{aligned} \] Here, \(\varphi\) is a nonlinear functional and \(\lambda\) a parameter. The functional \(\varphi\) satisfies some asymptotic conditions. The author proves the existence of at least one positive solution for the BVP, improving some earlier results in the literature. A number of examples is given to illustrate the theory.