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Positive solutions of a second-order integral boundary value problem. (English) Zbl 1106.34014

The author uses a well-known fixed-point theorem to study the existence of positive solutions of the nonlinear integral boundary value problem

$-{\left(a{u}^{\text{'}}\right)}^{\text{'}}+bu=f\left(t,u\right),$
$\left(cos{\gamma }_{0}\right)u\left(0\right)-\left(sin{\gamma }_{0}\right){u}^{\text{'}}\left(0\right)={H}_{1}\left({\int }_{0}^{1}\phantom{\rule{-0.166667em}{0ex}}u\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}d\alpha \left(\tau \right)\right),$
$\left(cos{\gamma }_{1}\right)u\left(1\right)+\left(sin{\gamma }_{1}\right){u}^{\text{'}}\left(1\right)={H}_{2}\left({\int }_{0}^{1}\phantom{\rule{-0.166667em}{0ex}}u\left(\tau \right)\phantom{\rule{0.166667em}{0ex}}d\beta \left(\tau \right)\right)·$

The method used is an interesting variant of the traditional means to show the existence of positive solutions via cone theoretic techniques. The author considers the above boundary value problem as a perturbation of the boundary value problem $-{\left(a{u}^{\text{'}}\right)}^{\text{'}}+bu=f\left(t,u\right),\left(cos{\gamma }_{0}\right)u\left(0\right)-\left(sin{\gamma }_{0}\right){u}^{\text{'}}\left(0\right)=0,\left(cos{\gamma }_{1}\right)u\left(1\right)+\left(sin{\gamma }_{1}\right){u}^{\text{'}}\left(1\right)=0$. Using the Green function associated with the boundary value problem with homogeneous boundary conditions, the author defines a cone preserving operator and obtains many fixed-point theorems.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 47H10 Fixed point theorems for nonlinear operators on topological linear spaces