Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System

Debnath L: Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003, (54):3413-3442. Sabatier J, Agrawal OP, Tenreiro Machado JA: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007. 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. Lakshmikantham V: Theory of fractional functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3337-3343. 10.1016/j.na.2007.09.025 Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific, Cambridge, UK; 2009. Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2677-2682. 10.1016/j.na.2007.08.042 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. Bai C: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation. Electronic Journal of Qualitative Theory of Differential Equations 2008, (24):-10. Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2010, 72(2):916-924. 10.1016/j.na.2009.07.033 Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495-505. 10.1016/j.jmaa.2005.02.052 Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7-8):2391-2396. 10.1016/j.na.2009.01.073 El-Shahed M: Positive solutions for boundary value problem of nonlinear fractional differential equation. Abstract and Applied Analysis 2007, 2007:-8. Jafari H, Daftardar-Gejji V: Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method. Applied Mathematics and Computation 2006, 180(2):700-706. 10.1016/j.amc.2006.01.007 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 2008, (3):-11. Kosmatov N: A singular boundary value problem for nonlinear differential equations of fractional order. Journal of Applied Mathematics and Computing 2009, 29(1-2):125-135. 10.1007/s12190-008-0104-x Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers & Mathematics with Applications 2010, 59(3):1363-1375. Liang S, Zhang J: Positive solutions for boundary value problems of nonlinear fractional differential equation. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(11):5545-5550. 10.1016/j.na.2009.04.045 Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009, 22(1):64-69. 10.1016/j.aml.2008.03.001 Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. International Journal of Differential Equations 2010, 2010:-12. Yang A, Ge W: Positive solutions for boundary value problems of N -dimension nonlinear fractional differential system. Boundary Value Problems 2008, 2008:-15. Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia B 2006, 26(2):220-228. 10.1016/S0252-9602(06)60044-1 Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, (36):-12. 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 2011, 16(4):2086-2097. 10.1016/j.cnsns.2010.08.017 Babakhani A: Positive solutions for system of nonlinear fractional differential equations in two dimensions with delay. Abstract and Applied Analysis 2010, 2010:-16. Babakhani A, Enteghami E: Existence of positive solutions for multiterm fractional differential equations of finite delay with polynomial coefficients. Abstract and Applied Analysis 2009, 2009:-12. 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 2008, 338(2):1340-1350. 10.1016/j.jmaa.2007.06.021 Deng W, Li C, Lü J: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics 2007, 48(4):409-416. 10.1007/s11071-006-9094-0 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 2009, 79(3):507-514. 10.1007/s00233-009-9164-y 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 2008, 51(10):1775-1786. 10.1007/s11425-008-0068-1 Mophou GM, N'Guérékata GM: A note on a semilinear fractional differential equation of neutral type with infinite delay. Advances in Difference Equations 2010, 2010:-8. Zhang X: Some results of linear fractional order time-delay system. Applied Mathematics and Computation 2008, 197(1):407-411. 10.1016/j.amc.2007.07.069 Podlubny I: Fractional Differential Equations. Academic Press, San Diego, Calif, USA; 1993. 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. Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. Journal de Mathématiques Pures et Appliquées 1991, 70(2):153-196. Krasnosel'skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan, New York, NY, USA; 1964:xi + 395.