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On nonlinear boundary conditions satisfying certain asymptotic behavior. (English) Zbl 1264.34030
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.

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