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Zbl 1181.34025
Webb, J.R.L.
Nonlocal conjugate type boundary value problems of higher order.
(English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 5-6, A, 1933-1940 (2009). ISSN 0362-546X

In this interesting paper, the author studies the nonlocal boundary value problem $$\gathered u^{(n)}(t)+g(t)f(t,u(t))=0, \;t \in (0,1),\\ u^{(k)}(0) = 0, \; 0 \leq k \leq n-2, \;\; u(1) = \alpha[u], \endgathered$$ where $\alpha[\cdot]$ is a linear functional on $C[0,1]$ given by a Riemann-Stieltjes integral, namely $$\alpha[u]=\int_0^1 u(s) dA(s),$$ with $dA$ a {\it 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 {\it J. R. L. Webb} and {\it G. Infante} [NoDEA, Nonlinear Differ. Equ. Appl. 15, No.~1--2, 45--67 (2008; Zbl 1148.34021)], {\it J. R. L. Webb} and {\it K. Q. Lan} [Topol. Methods Nonlinear Anal. 27, 91--115 (2006; Zbl 1146.34020)], {\it J. R. L. Webb} and {\it G. Infante} [J. Lond. Math. Soc., II. Ser. 74, No.~3, 673--693 (2006; Zbl 1115.34028)].
[Gennaro Infante (Arcavata di Rende)]
MSC 2000:
*34B10 Multipoint boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
34B27 Green functions
47N20 Appl. of operator theory to differential and integral equations

Keywords: positive solutions; fixed point index; cone

Citations: Zbl 1148.34021; Zbl 1146.34020; Zbl 1115.34028

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