International Journal of Nonlinear Analysis and Applications

Multi-point boundary value problems of higher-order nonlinear fractional differential equations

Document Type : Research Paper

Author

Department of Mathematics, Science and Art Faculty, Pamukkale University, Denizli, Turkey

Abstract

‎We investigate the existence and uniqueness of solutions for multi-point nonlocal boundary value problems of higher-order nonlinear fractional differential equations by using some well known fixed point theorems‎.

Keywords

[1] B. Ahmad and J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal. Art. ID 494720 (2009) 1–9.
[2] B. Ahmad, J.J. Nieto and A. Alsaedi, Existence and uniqueness of solutions for nonlinear fractional differential equations with non-separated type integral boundary conditions, Acta Math. Sci. 31 (2011) 2122–2130.
[3] B. Ahmad and S.K. Ntouyas, Fractional differential inclusions with fractional separated boundary conditions, Fract. Calc. Appl. Anal. 15 (2012) 362–382.
[4] B. Ahmad and S.K. Ntouyas, Initial value problems of fractional order Hadamard-type functional differential equations, Electron. J. Differ. Equ. 2015 (2015) 77.
[5] A.S. Berdyshev, A. Cabada and B.K. Turmetov, On solvability of a boundary value problem for a nonhomogeneous biharmonic equation with a boundary operator of fractional order, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014) 1695–1706.
[6] S. Das, Functional Fractional Calculus for System Identification and Controls, New York, Springer, 2008.
[7] M. El-Shahed and J.J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl. 59 (2010) 3438–3443.
[8] F.T. Fen, On the solutions of a multi-point BVP for fractional differential equations with impulses, Miskolc Math. Notes 19 (2018) 847–863.
[9] F.T. Fen, I.Y. Karaca and O.B. Ozen, Positive solutions of boundary value problems for p-Laplacian fractional differential equations, Filomat 31 (2017) 1265–1277.
[10] M. Gunendi and I. Yaslan, Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions, Fract. Calc. Appl. Anal. 19 (2016) 989–1009.
[11] D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Orlando, Academic Press, 1988.
[12] Y. Jia and X. Zhang, Positive solutions for a class of fractional differential equation multi-point boundary value problems with changing sign nonlinearity, J. Appl. Math. Comput. 47 (2015) 15–31.
[13] M. Jia, X. Zhang and X. Gu, Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions, Bound. Value Probl. 70 (2012) 1–16.
[14] J. Jiang and H. Wang, Existence and uniqueness of solutions for a fractional differential equation with multi-point boundary value problems, Journal of Applied Analysis and Computation 9 (2019) 2156–2168.
[15] A.A. Kilbas, HM. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006.
[16] Y. Li and W. Jiang, Existence and nonexistence of positive solutions for fractional three-point boundary value problems with a parameter, J. Funct. Space. 2019  (2019) 1–10.
[17] Y. Li and A. Qi, Positive solutions for multi-point boundary value problems of fractional differential equations with p- Laplacian, Math. Meth. Appl. Sci. 39 (2016) 1425–1434.
[18] X. Liu and M. Jia, Multiple solutions for fractional differential equations with nonlinear boundary conditions, Comput. Math. Appl. 59 (2010) 2880–2886.
[19] X. Liu, M. Jia and W. Ge, Multiple solutions of a p-Laplacian model involving a fractional derivative, Adv. Differ. Equ. 2013 (2013) No. 126 1–12.
[20] J. Podlubny, Fractional differential equations, New York, Academic Press, 1999.
[21] H. Qu and X. Liu, Existence of nonnegative solutions for a fractional m-point boundary value problem at resonance, Bound. Value Probl. 127 (2013) 1–10.
[22] H.A.H. Salem, On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies, J. Comput. Appl. Math. 224 (2009) 565–572.
[23] H.A.H. Salem and A.M.A. El-Sayed, Weak solution for fractional order integral equations in reflexive Banach spaces, J. Comput. Appl. Math. 224 (2009) 565–572.
[24] K. Shah, S. Zeb and R.A. Khan, Multiplicity results of multi-point boundary value problem of nonlinear fractional differential equations, Appl. Math. Inf. Sci. 12 (2018) 727–734.
[25] G.T. Wang, B. Ahmad and L.H. Zhang, Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions, Comput. Math. Appl. 62 (2011) 1389–1397.
[26] G.T. Wang, B. Ahmad and L.H. Zhang, On nonlocal integral boundary value problems for impulsive nonlinear differential equations of fractional order, Fixed Point Theory 15 (2014) 265–284.
[27] Y. Wang, Existence and nonexistence of positive solutions for mixed fractional boundary value problem with parameter and p-Laplacian operator, J. Funct. Space. 2018 (2018) 1–6.
[28] X. Xu, D. Jiang and C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Anal. 71 (2009) 4676–4688.
[29] X. Xu and H. Zhang, Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations, Bound. Value Probl. 34 (2018) 1–18.
[30] A. Yang, J. Henderson and C. Nelms Jr., Extremal points for a higher-order fractional boundary-value problem, Electron. J. Differential Equations 2015 (2015) No. 161 1–12.
[31] I. Yaslan and M. Gunendi, Higher order multi-point fractional boundary value problems with integral boundary conditions, Int. J. Nonlinear Anal. Appl. 9 (2018) 247–260.
[32] X. Zhang and Q. Zhong, Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables, Appl. Math. Lett. 80 (2018) 12–19.
[33] C. Zhou, Existence and uniqueness of positive solutions to a higher-order nonlinear fractional differential equation with integral boundary conditions, Electron. J. Differential Equations 2012 (2012)  234.
[34] Y. Zou and G. He, On the uniqueness of solutions for a class of fractional differential equations, Appl. Math. Lett. 74 (2017) 68–73.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 337-349
  • Receive Date: 28 March 2019
  • Revise Date: 08 January 2020
  • Accept Date: 25 January 2020