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Positive solutions of nonlocal boundary value problems involving integral conditions. (English) Zbl 1148.34021

The paper deals with the second-order nonlinear differential equation

$-{u}^{\text{'}\text{'}}\left(t\right)=g\left(t\right)f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,1\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $g\in {L}_{1}\left[0,1\right]$ and $f\in Car\left(\left[0,1\right]×{R}_{+}\right)$ are nonnegative functions. Equation (1) is subject to various nonlocal boundary conditions including the following

$u\left(0\right)=0,\phantom{\rule{4pt}{0ex}}u\left(1\right)=\alpha \left[u\right]\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}{u}^{\text{'}}\left(0\right)=0,u\left(1\right)=\alpha \left[u\right]\phantom{\rule{4.pt}{0ex}}\text{or}\phantom{\rule{4.pt}{0ex}}u\left(0\right)=0,\phantom{\rule{4pt}{0ex}}{u}^{\text{'}}\left(1\right)=\alpha \left[u\right]·$

Here, $\alpha \left[u\right]$ is a linear functional on $C\left[0,1\right]$ given by $\alpha \left[u\right]={\int }_{0}^{1}u\left(s\right)dA\left(s\right)$ involving a Stieltjes integral with a signed measure. This includes the special cases of so called $m$-point problems when $\alpha \left[u\right]={\sum }_{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_{i}\right)$ and ${\alpha }_{i}$ need not have the same sign. The authors prove the existence of multiple positive solutions of the above boundary value problems. The common feature is that each problem can be written as an integral equation, in the space $C\left[0,1\right]$, of the form

$u\left(t\right)=\gamma \left(t\right)\alpha \left[u\right]+{\int }_{0}^{1}k\left(t,s\right)g\left(s\right)f\left(s,u\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds,$

where $\gamma$ depends on boundary conditions. The assumption $\alpha \left[u\right]\ge 0$ for all $u\ge 0$ is not used here. The proofs are based on fixed point theory on a suitable cone. Illustrative examples are given as well.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H30 Particular nonlinear operators