Goodrich, Christopher S. Existence of a positive solution to a class of fractional differential equations. (English) Zbl 1204.34007 Appl. Math. Lett. 23, No. 9, 1050-1055 (2010). The author considers a class of fractional boundary value problem involving the Riemann-Liouville derivative. The main contribution of the author is to improve certain recent results by showing that the Green function associated to the mentioned problem satisfies, among other properties, a Harnack-like inequality. Also, the author shows that the mentioned boundary problem has a positive solution under standard assumptions on the nonlinearity part of the fractional differential equation. Reviewer: Juan J. Trujillo (La Laguna) Cited in 1 ReviewCited in 162 Documents MSC: 34A08 Fractional ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations Keywords:fractional calculus; fractional boundary value problem; Green’s function; fixed point theorem in cones; Riemann-Liouville derivative PDFBibTeX XMLCite \textit{C. S. Goodrich}, Appl. Math. Lett. 23, No. 9, 1050--1055 (2010; Zbl 1204.34007) Full Text: DOI References: [1] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048 [2] Xu, X.; Jiang, D.; Yuan, C., Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. TMA, 71, 4676-4688 (2009) · Zbl 1178.34006 [3] (Schuster, H. G., Reviews of Nonlinear Dynamics and Complexity (2008), Wiley-VCH: Wiley-VCH Weinheim) · Zbl 1141.37001 [4] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal. TMA, 69, 2677-2682 (2008) · Zbl 1161.34001 [5] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl., 59, 1300-1309 (2010) · Zbl 1189.34050 [6] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 3, 743-748 (1994) · Zbl 0802.34018 [7] Davis, J. H.; Henderson, J., Triple positive solutions for \((k, n - k)\) conjugate boundary value problems, Math. Slovaca, 51, 3, 313-320 (2001) · Zbl 0996.34017 [8] Graef, J. R.; Yang, B., Positive solutions of a nonlinear fourth order boundary value problem, Comm. Appl. Nonlinear Anal., 14, 1, 61-73 (2007) · Zbl 1136.34024 [9] Ma, R.; Xu, L., Existence of positive solutions of a nonlinear fourth-order boundary value problem, Appl. Math. Lett., 23, 5, 537-543 (2010) · Zbl 1195.34037 [10] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010 [11] Agarwal, R.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027 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.