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On nonlocal BVPs with nonlinear boundary conditions with asymptotically sublinear or superlinear growth. (English) Zbl 1252.34032
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 $\begin{gathered} u''+\lambda f(t,u)=0,\;t \in (0,1),\\ u(0) =H(\varphi (u)) ,\;u(1) =0,\\ \end{gathered}$ 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.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B09 Boundary eigenvalue problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H10 Fixed-point theorems
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