Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.455998 Mathematical Sciences Matematik Green's functions for boundary value problems of generalized fractional differential equations with p-Laplacian https://orcid.org/0000-0002-4219-8153 Seemab Arjumand National University of Sciences and Technology https://orcid.org/0000-0003-2511-8622 Rehman Mujeeb Ur National University of Sciences and Technology 08 06 2020 49 4 1355 1372 08 30 2018 09 28 2019 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

We utilize the recently presented generalized fractional derivatives, which are not the same as standard Caputo and Riemann-Liouville fractional derivatives, to reformulate some boundary value problems of fractional differential equations. For some classes of generalized fractional differential equations with boundary conditions build up, we find the corresponding Green's functions and establish their properties under suitable assumptions and we also demonstrate the applicability of these properties of the Green's functions to establish some existence results via fixed point theorems.

 Generalized fractional derivatives Positive solutions p-Laplician Green's function fixed-point theorem [1] B. Ahmad, M. Alghanmi, S.K. Ntouyas and A. Alsaedi, Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions, Appl. Math. Lett. 84, 111–117, 2018. [2] A. Ali, F. Rabiei and K. Shah, On Ulam’s type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions, J. Nonlinear. Sci. Appl. 10, 4760–4775, 2017. [3] A. Ali, K. Shah and D. Baleanu, Ulam stability results to a class of nonlinear implicit boundary value problems of impulsive fractional differential equations, Adv. Differ. Equ. 5, 1–21, 2019. [4] R. Almeida, A.B. Malinowska and M.T.T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications, Math. Method. Appl. Sci. 41, 336–352, 2018. [5] Asma, A. Ali, K. Shah and F. Jarad, Ulam-Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions, Adv. Differ. Equ. 7, 1–27, 2019. [6] Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal. 72, 916–924, 2010. [7] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311, 495–505, 2005. [8] A. Benlabbes, M. Benbachir and M. Lakrib, Boundary value problems for nonlinear fractional differential equations, Facta. Univ. Ser. Math. Inform. 30 (2), 157–168, 2015. [9] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389, 403–411, 2012. [10] T. Chen, W. Liu and Z. Hu, A boundary value problem for fractional differential equation with p-Laplacian operator at resonance, Nonlinear Anal. 75, 3210–3217, 2012. [11] W. Chen and Y. Zhao, Solvability of boundary value problems of nonlinear fractional differential equations, Adv. Differ. Equ., Art. No. 36, 2015. [12] Y. Cui, S. Kang and Z. Liu, Existence of positive solutions to boundary value problem of Caputo fractional differential equation, Discrete Dyn. Nat. Soc., Art. ID 708053, 6 pp., 2015. [13] X. Dong, Z. Bai and S. Zhang, Positive solutions to boundary value problems of p- Laplacian with fractional derivative, Bound. Value. Probl., Art. No. 5, 2017. [14] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, 1988. [15] F. Jarad, T. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl. 10, 2607–2619, 2017. [16] D. Jiang and C. Yuan, The positive properties of the Green function for Dirichlettype boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Anal. 72, 710–719, 2010. [17] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218, 860–865, 2011. [18] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. App. 6 (4), 1–15, 2014. [19] U.N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, arXiv:1411.5229v2[math.CA]. [20] Y. Li and G. Li, Positive solutions of p-Laplacian fractional differential equations with integral boundary value conditions, J. Nonlinear Sci. Appl. 9, 717–726, 2016. [21] Y. Li, K. Shah and R.A. Khan, Iterative technique for coupled integral boundary value problem of non-linear of non-integer order differential equations, Adv. Differ. Equ. 251, 2017. [22] S. Liang and J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal. 71, 5545–5550, 2009. [23] K. Shah, A. Ali and S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive condition, Math. Method. Appl. Sci. 41 (17), 8329–8343, 2018. [24] K. Shah, H. Khalil and R.A. Khan, Upper and lower solutions to a coupled system of nonlinear fractional differential equations, Progr. Fract. Differ. Appl. 2 (1), 1–10, 2016. [25] K. Shah, R.A. Khan, Study of solution to a toppled system of fractional differential equations with integral boundary conditions, Int. J. Appl. Comput. Math. 3 (3), 2369– 2388, 2017. [26] X. Su and S. Zhang, Solutions to boundary value problems for nonlinear differential equations of fractional order, Electron. J. Differ. Eq. 26, 1–15, 2009. [27] Y. Sun, Positive solutions for third-order three-point nonhomogeneous boundary value problems, Appl. Math. Lett. 22, 45–51, 2009. [28] J. Tan and M. Li, Solutions of fractional differential equations with p-Laplacian operator in Banach spaces, Bound. Value. Probl., Art. No. 15, 2018. [29] Y. Wang and L. Liu, Positive properties of the Green function for two-term fractional differential equations and its application, J. Nonlinear Sci. Appl. 10, 2094–2102, 2017. [30] L. Yang, X.P. Liu and M. Jia, Multiplicity results for second order m-point boundary value problem, J. Math. Anal. Appl. 324 (1), 532–542, 2006. [31] S. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equations, Electron. J. Differ. Eq. 36, 1–12, 2006. [32] S. Zhang, Positive solution of singular boundary value problem for nonlinear fractional differential equation with nonlinearity that changes sign, Positivity 16, 177–193, 2012.