Jiang, Daqing; Yuan, Chengjun The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. (English) Zbl 1192.34008 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 710-719 (2010). The authors discuss some new positive properties of the Green function for boundary value problems of nonlinear Dirichlet-type fractional differential equation \[ \begin{aligned} &D_{0^+}^{\alpha}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ &u(0)=u(1)=0 \end{aligned} \]Applications are also given. Reviewer: Minghe Pei (Jilin) Cited in 110 Documents MSC: 34A08 Fractional ordinary differential equations 34B27 Green’s functions for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:Riemann-Liouville’s fractional derivative; boundary value problem; positive solution; fixed point theorem in cones PDFBibTeX XMLCite \textit{D. Jiang} and \textit{C. Yuan}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 710--719 (2010; Zbl 1192.34008) Full Text: DOI References: [1] Agrawal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272, 368-379 (2002) · Zbl 1070.49013 [2] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Appl., 204, 609-625 (1996) · Zbl 0881.34005 [3] Leggett, R. W.; Williams, L. R., Multiple positive solutions of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033 [4] Podlubny, I., (Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198 (1999), Academic Press: Academic Press New Tork, London, Toronto) · Zbl 0918.34010 [5] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integral And Derivatives (Theory and Applications) (1993), Gordon and Breach: Gordon and Breach Switzerland · Zbl 0818.26003 [6] Bai, Zhanbing; Lü, Haishen, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048 [7] Zhang, Shu-qin, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252, 804-812 (2000) · Zbl 0972.34004 [8] Zhang, Shu-qin, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 278, 1, 136-148 (2003) · Zbl 1026.34008 [9] Wei, Zhongli, Positive solution of singular Dirichlet boundary value problems for second order differential equation system, J. Math. Anal. Appl., 328, 1255-1267 (2007) · Zbl 1115.34025 [10] Zhang, Xinguang; Liu, Lishan; Wu, Yonghong, Positive solutions of nonresonance semipositone singular Dirichlet boundary value problems, Nonlinear Anal. TMA, 68, 1, 97-108 (2008) · Zbl 1135.34016 [11] Agarwal, Ravi P.; O’Regan, Donal, Twin solutions to singular Dirichlet problems, J. Math. Anal. Appl., 240, 2, 433-445 (1999) · Zbl 0946.34022 [12] Tersenov, Alkis S.; Tersenov, Aris S., The problem of Dirichlet for evolution one-dimensional p-Laplacian with nonlinear source, J. Math. Anal. Appl., 340, 1109-1119 (2008) · Zbl 1137.35386 [13] Lin, Xiaoning; Jiang, Daqing, Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl., 321, 2, 501-514 (2006) · Zbl 1103.34015 [14] O’Regan, Donal, Singular Dirichlet boundary value problems. Superlinear and nonresonant case, Nonlinear Anal., 29, 2, 221-245 (1997) · Zbl 0884.34028 [15] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff Gronigen: Noordhoff Gronigen Netherland · Zbl 0121.10604 [16] Lakshmikantham, V.; Vatsala, A. S., Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11, 3-4, 395-402 (2007) · Zbl 1159.34006 [17] Lakshmikantham, V.; Vatsala, A. S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21, 828-834 (2008) · Zbl 1161.34031 [18] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal. TMA, 69, 8, 2677-2682 (2008) · Zbl 1161.34001 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.