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Second-order boundary value problems with integral boundary conditions. (English) Zbl 1169.34310

The author studies the existence of positive solutions of the solutions of the second-order boundary value problem

${y}^{\text{'}\text{'}}=f\left(t,y\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0
$y\left(0\right)-a{y}^{\text{'}}\left(0\right)={\int }_{0}^{1}\phantom{\rule{-0.166667em}{0ex}}{g}_{0}\left(s\right)y\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,$
$y\left(1\right)-b{y}^{\text{'}}\left(1\right)={\int }_{0}^{1}\phantom{\rule{-0.166667em}{0ex}}{g}_{1}\left(s\right)y\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds,$

where $f:\left[0,1\right]×ℝ\to ℝ$ and ${g}_{0},{g}_{1}:\left[0,1\right]\to \left[0,+\infty \right)$ are continuous and, $a$ and $b$ are nonnegative real numbers. The results are based on the classical cone expansion and contraction technique. The author provides and example at the end of the paper to illustrate his technique.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations