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Nonlocal boundary value problems with two nonlinear boundary conditions. (English) Zbl 1198.34025
Existence of positive solutions of the second order differential equation
$u'' + g(t) f(t,u) =0$
under the nonlinear boundary conditions
$u'(0) +H_1(\alpha[u])=0, \, \sigma u'(1)+u(\eta)=H_2(\beta[u])$
is established by the fixed point index theory for compact maps. Here, $$H_1$$ and $$H_2:[0,\infty)\to [0,\infty)$$ are continuous functions between two linear functions and
$\alpha[u]= \int_0^1 u(s)\, DA(s),\qquad \beta[u] = \int_0^1 u(s)\, DB(s)$
are Lebesgue-Stieltjes integrals.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems