On nonlocal BVPs with nonlinear boundary conditions with asymptotically sublinear or superlinear growth (Q2909657)

The author studies the existence of one positive solution of a boundary value problem where one of the boundary conditions is allowed to be nonlinear, namely NEWLINE\[NEWLINE \begin{gathered} u''+\lambda f(t,u)=0,\;t \in (0,1),\\ u(0) =H(\varphi (u)) ,\;u(1) =0,\\ \end{gathered} NEWLINE\]NEWLINE where \(H\) is a continuous, possibly nonlinear, function and \(\varphi\) is a functional given by a (signed) Stieltjes measure. This is quite a general form and the author focuses on the asymptotically linear and sublinear growth in the nonlinear boundary conditions. The proofs use the fixed point theorem of Guo-Krasnoselskii on cone compressions and cone expansions. The author also provides some numerical examples to illustrate his results.