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Existence of positive solutions to a boundary value problem for a delayed nonlinear fractional differential system. (English) Zbl 1219.34103
Summary: We consider a boundary value problem of a general delayed nonlinear fractional system. By means of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions for the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated by some examples.

MSC:
34K37 Functional-differential equations with fractional derivatives
34K10 Boundary value problems for functional-differential equations
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[1] Debnath, L, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54, 3413-3442, (2003) · Zbl 1036.26004
[2] Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007. · Zbl 1116.00014
[3] Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science, Amsterdam, The Netherlands; 2006:xvi+523.
[4] Lakshmikantham, V, Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69, 3337-3343, (2008) · Zbl 1162.34344
[5] Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge, UK; 2009. · Zbl 1188.37002
[6] Lakshmikantham, V; Vatsala, AS, Basic theory of fractional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 69, 2677-2682, (2008) · Zbl 1161.34001
[7] Podlubny I: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340. · Zbl 0924.34008
[8] Bai, C, Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 24, 10, (2008) · Zbl 1183.34005
[9] Bai, Z, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis: Theory, Methods & Applications, 72, 916-924, (2010) · Zbl 1187.34026
[10] Bai, Z; Lü, H, Positive solutions for boundary value problem of nonlinear fractional differential equation, Journal of Mathematical Analysis and Applications, 311, 495-505, (2005) · Zbl 1079.34048
[11] Benchohra, M; Hamani, S; Ntouyas, SK, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications, 71, 2391-2396, (2009) · Zbl 1198.26007
[12] El-Shahed, M, Positive solutions for boundary value problem of nonlinear fractional differential equation, No. 2007, 8, (2007) · Zbl 1149.26012
[13] Jafari, H; Daftardar-Gejji, V, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Applied Mathematics and Computation, 180, 700-706, (2006) · Zbl 1102.65136
[14] Kaufmann, ER; Mboumi, E, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electronic Journal of Qualitative Theory of Differential Equations, 3, 11, (2008) · Zbl 1183.34007
[15] Kosmatov, N, A singular boundary value problem for nonlinear differential equations of fractional order, Journal of Applied Mathematics and Computing, 29, 125-135, (2009) · Zbl 1191.34006
[16] Li, CF; Luo, XN; Zhou, Y, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Computers & Mathematics with Applications, 59, 1363-1375, (2010) · Zbl 1189.34014
[17] Liang, S; Zhang, J, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, 71, 5545-5550, (2009) · Zbl 1185.26011
[18] Su, X, Boundary value problem for a coupled system of nonlinear fractional differential equations, Applied Mathematics Letters, 22, 64-69, (2009) · Zbl 1163.34321
[19] Wang, J; Xiang, H; Liu, Z, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations, No. 2010, 12, (2010) · Zbl 1207.34012
[20] Yang, A; Ge, W, Positive solutions for boundary value problems of \(N\)-dimension nonlinear fractional differential system, No. 2008, 15, (2008) · Zbl 1167.34314
[21] Zhang, S, Existence of solution for a boundary value problem of fractional order, Acta Mathematica Scientia B, 26, 220-228, (2006) · Zbl 1106.34010
[22] Zhang, S, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electronic Journal of Differential Equations, 36, 12, (2006)
[23] Zhao, Y; Sun, S; Han, Z; Li, Q, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 16, 2086-2097, (2011) · Zbl 1221.34068
[24] Babakhani, A, Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay, No. 2010, 16, (2010) · Zbl 1197.34155
[25] Babakhani, A; Enteghami, E, Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients, No. 2009, 12, (2009) · Zbl 1181.34080
[26] Benchohra, M; Henderson, J; Ntouyas, SK; Ouahab, A, Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications, 338, 1340-1350, (2008) · Zbl 1209.34096
[27] Deng, W; Li, C; Lü, J, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynamics, 48, 409-416, (2007) · Zbl 1185.34115
[28] Hu, L; Ren, Y; Sakthivel, R, Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays, Semigroup Forum, 79, 507-514, (2009) · Zbl 1184.45006
[29] Maraaba, TA; Jarad, F; Baleanu, D, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China. Series A, 51, 1775-1786, (2008) · Zbl 1179.26024
[30] Mophou, GM; N’Guérékata, GM, A note on a semilinear fractional differential equation of neutral type with infinite delay, No. 2010, 8, (2010) · Zbl 1194.34148
[31] Zhang, X, Some results of linear fractional order time-delay system, Applied Mathematics and Computation, 197, 407-411, (2008) · Zbl 1138.34328
[32] Podlubny I: Fractional Differential Equations. Academic Press, San Diego, Calif, USA; 1993.
[33] Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Applications, Cambridge Tracts in Mathematics. Volume 141. Cambridge University Press, Cambridge, UK; 2001:x+170.
[34] Granas, A; Guenther, RB; Lee, JW, Some general existence principles in the Carathéodory theory of nonlinear differential systems, Journal de Mathématiques Pures et Appliquées, 70, 153-196, (1991) · Zbl 0687.34009
[35] Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan, New York, NY, USA; 1964:xi + 395.
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