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Nonlocal conjugate type boundary value problems of higher order. (English) Zbl 1181.34025

In this interesting paper, the author studies the nonlocal boundary value problem

$\begin{array}{c}{u}^{\left(n\right)}\left(t\right)+g\left(t\right)f\left(t,u\left(t\right)\right)=0,\phantom{\rule{0.277778em}{0ex}}t\in \left(0,1\right),\\ {u}^{\left(k\right)}\left(0\right)=0,\phantom{\rule{0.277778em}{0ex}}0\le k\le n-2,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u\left(1\right)=\alpha \left[u\right],\end{array}$

where $\alpha \left[·\right]$ is a linear functional on $C\left[0,1\right]$ given by a Riemann-Stieltjes integral, namely

$\alpha \left[u\right]={\int }_{0}^{1}u\left(s\right)dA\left(s\right),$

with $dA$ a signed measure. This formulation is quite general and covers classical $m$-point boundary conditions and integral conditions as special cases. The author proves, under suitable growth conditions on the nonlinearity $f$, existence of multiple positive solutions.

Interesting features of this paper are that the theory is illustrated with explicit examples, including a 4-point problem with coefficients with both signs, and that all the constants that appear in the theoretical results are explicitly determined.

The methodology involves classical fixed point index theory and makes extensive use of the results in J. R. L. Webb and G. Infante [NoDEA, Nonlinear Differ. Equ. Appl. 15, No. 1–2, 45–67 (2008; Zbl 1148.34021)], J. R. L. Webb and K. Q. Lan [Topol. Methods Nonlinear Anal. 27, 91–115 (2006; Zbl 1146.34020)], J. R. L. Webb and G. Infante [J. Lond. Math. Soc., II. Ser. 74, No. 3, 673–693 (2006; Zbl 1115.34028)].

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B27 Green functions 47N20 Applications of operator theory to differential and integral equations
##### Keywords:
positive solutions; fixed point index; cone