Citation: Md. Asaduzzaman, Md. Zulfikar Ali. Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations[J]. AIMS Mathematics, 2019, 4(3): 880-895. doi: 10.3934/math.2019.3.880
[1] | K. Diethelm, The Analysis of Fractional Differential Equations, Springer, 2010. |
[2] | J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007. |
[3] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204 of North-Holland Mathematics Studies, Elsevier Science Limited, 2006. |
[4] | I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. |
[5] | K. S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993. |
[6] | N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45 (2006), 765-771. doi: 10.1007/s00397-005-0043-5 |
[7] | Q. Sun, H. Ji and Y. Cui, Positive Solutions for Boundary Value Problems of Fractional Differential Equation with Integral Boundary Conditions, J. Funct. Space. Appl., 2018 (2018), 1-6. |
[8] | W. Ma, S. Meng and Y. Cui, Resonant Integral Boundary Value Problems for Caputo Fractional Differential Equations, Math. Probl. Eng., 2018 (2018), 1-8. |
[9] | Y. Cu, W. Ma, Q. Sun, et al. New uniqueness results for boundary value problem of fractional differential equation, Nonlinear Anal-Model, 23 (2018), 31-39. |
[10] | X. Han and X. Yang, Existence and multiplicity of positive solutions for a system of fractional differential equation with parameters, Bound. Value Probl., 2017 (2017), 78. |
[11] | Y. Cui, Q. Sun and X. Su, Monotone iterative technique for nonlinear boundary value problems of fractional order p∈ (2 ,3], Adv. Differ. Equ-NY, 2017 (2017), 248. |
[12] | T. Qi, Y. Liu and Y. Cui, Existence of Solutions for a Class of Coupled Fractional Differential Systems with Nonlocal Boundary Conditions, J. Funct. Space. Appl., 2017 (2017), 1-9. |
[13] | T. Qi, Y. Liu and Y. Zou, Existence result for a class of coupled fractional differential systems with integral boundary value conditions, J. Nonlinear Sci. Appl., 10 (2017), 4034-4045. doi: 10.22436/jnsa.010.07.52 |
[14] | T. Bashiri, S. M. Vaezpour and C. Park, A coupled fixed point theorem and application to fractional hybrid differential problems, Fixed Point Theory and Applications, 2016 (2016), 23. |
[15] | B. Zhu, L. Liu, and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79. doi: 10.1016/j.aml.2016.05.010 |
[16] | Y. Wang, L. Liu, X. Zhang, et al. Positive solutions of an abstract fractional semi-positone differential system model for bioprocesses of HIV infection, Appl. Math. Comput., 258 (2015), 312-324. |
[17] | D. Luo and Z. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Differ. Equ-NY, 2019 (2019), 155. |
[18] | D. Luo, and Z. Luo, Uniqueness and Novel Finite-Time Stability of Solutions for a Class of Nonlinear Fractional Delay Difference Systems, Discrete Dyn. Nat. Soc., 2018 (2018), 1-7. |
[19] | P. Agarwal, M. Chand, D. Baleanu, et al. On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function, Adv. Differ. Equ-NY, 2018 (2018), 249. |
[20] | P. Agarwal, M. Chand, J. Choi, et al. Certain fractional integrals and image formulas of generalized k-Bessel function, Communications of the Korean Mathematical Society, 33 (2018), 423-436. |
[21] | P. Agarwal, A.A. El-Sayed, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Physica A, 500 (2018), 40-49. doi: 10.1016/j.physa.2018.02.014 |
[22] | K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti-periodic boundary conditions, Differ. Equ. Appl., 7 (2015), 245-262. |
[23] | M. Hao and C. Zhai, Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order, J. Nonlinear Sci. Appl., 7 (2014), 131-137. doi: 10.22436/jnsa.007.02.07 |
[24] | Y. Cui, Y. Zou, Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point boundary value problems, Abstr. Appl. Anal., 2014 (2014), 1-6. |
[25] | M. J. Li, Y. L. Liu, Existence and uniqueness of positive solutions for a coupled system of nonlinear fractional differential equations, Open Journal of Applied Sciences, 3 (2013), 53-61. |
[26] | C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62 (2011), 1251-1268. doi: 10.1016/j.camwa.2011.02.039 |
[27] | C. S. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055. doi: 10.1016/j.aml.2010.04.035 |
[28] | X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69. doi: 10.1016/j.aml.2008.03.001 |
[29] | D. R. Dunninger and H. Y. Wang, Existence and multiplicity of positive solutions for elliptic systems, Nonlinear Anal-Theor, 29 (1997), 1051-1060. doi: 10.1016/S0362-546X(96)00092-2 |
[30] | J. Leray, J. Schauder, Topologie et equations fonctionels, Ann. Sci. École Norm. Sup., 51 (1934), 45-78. doi: 10.24033/asens.836 |
[31] | M. Fréchet, Sur quelques points du calculfonctionnel, Rend. Circ. Mat. Palermo, 22 (1906), 1-74. doi: 10.1007/BF03018603 |