Webb, J. R. L. Nonlocal conjugate type boundary value problems of higher order. (English) Zbl 1181.34025 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 5-6, 1933-1940 (2009). In this interesting paper, the author studies the nonlocal boundary value problem \[ \begin{gathered} u^{(n)}(t)+g(t)f(t,u(t))=0, \;t \in (0,1),\\ u^{(k)}(0) = 0, \; 0 \leq k \leq n-2, \;\; u(1) = \alpha[u], \end{gathered} \]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 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)]. Reviewer: Gennaro Infante (Arcavata di Rende) Cited in 57 Documents MSC: 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations 47N20 Applications 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 PDFBibTeX XMLCite \textit{J. R. L. Webb}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 5--6, 1933--1940 (2009; Zbl 1181.34025) Full Text: DOI References: [1] Eloe, P. W.; Henderson, J., Positive solutions for \((n - 1, 1)\) conjugate boundary value problems, Nonlinear Anal., 28, 1669-1680 (1997) · Zbl 0871.34015 [2] Eloe, P. W.; Ahmad, B., Positive solutions of a nonlinear \(n\)-th order boundary value problem with nonlocal conditions, Appl. Math. 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