Guo, Yanping; Ge, Weigao Positive solutions for three-point boundary value problems with dependence on the first order derivative. (English) Zbl 1054.34025 J. Math. Anal. Appl. 290, No. 1, 291-301 (2004). The existence of positive solutions for the second-order three-point boundary value problem \(x''+f(t,x,x')=0\), \(0<t<1\), \(x(0)=0\), \(x(1)=\alpha x(\eta)\), is considered. It is supposed that \(\alpha>0\), \(0<\eta<1\), \(1-\alpha\eta>0\), and \(f:[0,1]\times [0,\infty)\times \mathbb R\to[0,\infty]\) is continuous. Under some growth conditions, the authors prove that the problem has at least one positive solution. As a tool for the proof a new fixed-point theorem in a cone is used. Reviewer: Josef Diblík (Brno) Cited in 75 Documents MSC: 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H11 Degree theory for nonlinear operators 47N20 Applications of operator theory to differential and integral equations Keywords:three-point boundary value problem; fixed-point theorem in a cone; Green’s function; positive solution PDFBibTeX XMLCite \textit{Y. Guo} and \textit{W. Ge}, J. Math. Anal. Appl. 290, No. 1, 291--301 (2004; Zbl 1054.34025) Full Text: DOI References: [1] Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 34, 1-8 (1999) [2] Ma, R., Positive solutions for second order three-point boundary value problems, Appl. Math. Lett., 14, 1-5 (2001) · Zbl 0989.34009 [3] Ma, R., Existence theorems for a second order \(m\)-point boundary value problem, J. Math. Anal. Appl., 211, 545-555 (1997) · Zbl 0884.34024 [4] Ma, R., Existence of solutions of nonlinear \(m\)-point boundary value problems, J. Math. Anal. Appl., 256, 556-567 (2001) · Zbl 0988.34009 [5] Ma, R., Positive solutions of a nonlinear \(m\)-point boundary value problem, Comput. Math. Appl., 42, 755-765 (2001) · Zbl 0987.34018 [6] Anderson, D., Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling, 27, 49-57 (1998) · Zbl 0906.34014 [7] He, X.; Ge, W., Triple solutions for second order three-point boundary value problems, J. Math. Anal. Appl., 268, 256-265 (2002) · Zbl 1043.34015 [8] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045 [9] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033 [10] Dugundji, J., An extension of Tietze theorem, Pacific J. Math., 1, 353-367 (1951) · Zbl 0043.38105 [11] Avery, R. I.; Anderson, D. R., Fixed point theorem of cone expansion and compression of functional type, J. Difference Equations Appl., 8, 1073-1083 (2002) · Zbl 1013.47019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.