Analysis of a System of Nonlinear Hadamard Type Fractional Boundary Value Problems

The aim of this work is to analyze the existence of positive solutions for a coupled system of Hadamard type fractional boundary value problems. By using the five functional fixed point theorem, the conditions for the existence of positive solutions are derived. Finally, to show the applicability of the main result, an illustrative example is also involved.

Keywords:

Fixed-point theorem, Hadamard fractional differential systems Positive solutions,

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[1] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
[2] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, In: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V, Amsterdam, 2006.
[5] V. Lakshmikantham, S. Leela, J.V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
[6] J. R. Graef, L. Kong, Q. Kong, M. Wang, Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary condition, Fract. Calc. Appl. Anal., 15(3) (2012), 509-528.
[7] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411.
[8] J. Tariboon, S. Ntouyas, W. Sudsutad, Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions, J. Nonlinear Sci. Appl., 9 (2016), 295-308.
[9] J. He, M. Jia, X. Liu, H. Chen, Existence of positive solutions for a high order fractional differential equation integral boundary value problem with changing sign nonlinearity, Adv. Dier. Equ., 49 (2018).
[10] K. R. Prasad, M. Khuddush, P. Veeraiah, Countably many positive solutions for singular R-L fractional order bvp with R-S integral boundary conditions, Nonlinear Stud., 27(4) (2020), 1075-1089.
[11] Z. B. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72 (2010), 916-924.
[12] T. S. Cerdik, F. Y. Deren, N. A. Hamal, Unbounded solutions for boundary value problems of Riemann Liouville fractional differential equations on the half-line, Fixed Point Theory, 19 (2018), 93-106.
[13] F. Y. Deren, T. S. Cerdik, R. P. Agarwal, Existence criteria of positive solutions for fractional p-Laplacian boundary value problems, Filomat, 34(11) (2020).
[14] N. Nyamoradi, Multiple positive solutions for fractional differential systems, Ann. Univ. Ferrara, 58 (2012), 359-369.
[15] S. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Math. Comput. Model., 57(5-6) (2013), 1053-1059.
[16] W. G. Yang, Y. P. Qin, Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions, Scienceasia, Electron., 43(3) (2017), 201-206.
[17] P. Thiramanus, S. K. Ntouyas, J. Tariboon, Positive solutions for Hadamard fractional differential equations on infinite domain, Adv. Dif. Equ., 83 (2016), https://doi.org/10.1186/s13662-016-0813-7.
[18] W. Zhang, W. Liu, Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval, Bound. Value Probl., 134 (2018), 1-27.
[19] K. Pei, G. Wang, Y. Sun, Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain, Appl. Math. Comput.,312 (2017), 158-168.
[20] G. Wang, K. Pei, R. P. Agarwal, L. Zhang, B. Ahmad, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math., 343 (2018), 230-239.
[21] W. Zhang, W. Liu, Existence, uniqueness, and multiplicity results on positive solutions for a class of Hadamard-type fractional boundary value problem on an infinite interval, Math. Meth. Appl. Sci., 43 (2020), 2251-2275, https://doi.org/10.1002/mma.6038.
[22] G. Wang, T. Wang, On a nonlinear Hadamard type fractional dierential equation with p-Laplacian operator and strip condition, J. Nonlinear Sci. Appl., 9 (2016), 5073-5081.
[23] C. Zhai, W. Wang, Solutions for a system of Hadamard fractional differential equations with integral conditions, Numer. Funct. Anal. Optim., 41(2) (2020), 209-229, https://doi.org/10.1080/01630563.2019.1620771.
[24] J. Tariboon, S. K. Ntouyas, S. Asawasamrit, C. Promsalon, Positive solutions for Hadamard differential systems with fractional integral conditions on unbounded domain, Open Math., 15(1) (2017), 645-666.
[25] B. Ahmad, S. K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 348-360.
[26] B. Ahmad, S. K. Ntouyas, A. Alsaedi, New results for boundary value problems of Hadamard-type fractional differential inclusion and integral boundary conditions, Bound. Value Probl., 275 (2013), https://doi.org/10.1186/1687-2770-2013-275.
[27] B. Ahmad, S. K. Ntouyas, On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput., 47(1-2), 119-131, https://doi.org/10.1007s12190-014-0765-6.
[28] G. Wang, K. Pei, D. Baleanu, Explicit iteration to Hadamard fractional integro-differential equations on infinite domain, Adv. Difference Equ., 299 (2016), https://doi.org/10.1186/s13662-016-1023-z.
[29] S. N. Rao, M. Singh, M. Z. Meetei, Multiplicity of positive solutions for Hadamard fractional differential equations with p-Laplacian operator, Bound. Value Probl., 43 (2020).
[30] H. Huang, W. Liu, Positive solutions for a class of nonlinear Hadamard fractional differential equations with a parameter, Adv. Dier. Equ., 96 (2018).
[31] W. Yang, Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions, J. Appl. Math. Comput., 49 (2015), 357–381.
[32] J. Jiang, D. O’Regan, J. Xu, Z. Fu, Positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions, J. Inequal. Appl., 204 (2019), https://doi.org/10.1186/s13660-019-2156-x.
[33] C. Zhai, W. Wang, H. Li, A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions, J. Inequal. Appl., 207 (2018).
[34] R. I. Avery, A generalization of the Leggett-Williams fixed point theorem, Math. Sci. Res. Hot-Line, 3(7) (1999), 9-14.