International Journal of Nonlinear Analysis and Applications

Positive solutions for fractional-order nonlinear boundary value problems on infinite interval

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Faculty of Science‎, ‎Ege University‎, ‎35100 Bornova‎, ‎Izmir‎, ‎Turkey‎

Abstract

‎In this paper‎, ‎Avery-Henderson (Double) fixed point theorem and Ren fixed point theorem are used to investigate the existence of positive solutions for fractional-order nonlinear boundary value problems on infinite interval‎. ‎As applications‎, ‎some examples are given to illustrate the main results‎.

Keywords

[1] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, 2006.
[2] G. Wang, Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval, Appl. Math. Lett. 47 (2015) 1–7.
[3] I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
[4] I. Yaslan and M. Gunendi, Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions, Fract. Calc. Appl. Anal. 19(4) (2016) 989–1009.
[5] J. Graef, L. Kong, Q. Kong and M. Wang, Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions, Fract. Calc. Appl. Anal. 15(3) (2012) 509–528.
[6] J. Henderson and R.I. Avery, Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8(1) (2001) 27–36.
[7] J.L. Ren, W. Ge and B.X. Ren, Existence of three positive solutions for quasi-linear boundary value problem, Acta Math. Appl. Sin. Engl. Ser. 21(3) (2005) 353–358.
[8] K.B. Oldham and J. Spanier, Fractional Calculus, Dover, 2006.
[9] K. Zhang and J. Xu, Unique positive solution for a fractional boundary value problem, Fract. Calc. Appl. Anal. 16(4 (2013) 937–948.
[10] M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci. (Ruse) 4 (2010) 1021–1032.
[11] M. Rehman and R.A. Khan, Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations, Appl. Math. Lett. 23(9) (2010) 1038–1044.
[12] N. Abel, Solution de quelques probl`emes `a I’aide d’integrales definies, Mag. Naturv 1(2) (1823) 1–27.
[13] R.P. Agarwal, D. O’Regan and M. Meehan, Fixed Point Theory and Applications, Cambridge University Press, 2004.
[14] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3(1) (1922) 133–181.
[15] S. Liang and J. Zhang, Existence of Three Positive Solutions of m-point Boundary Value Problems for Some Nonlinear Fractional Differential Equations on an Infinite Interval, Comput. Math. Appl. 61(11) (2011) 3343– 3354.
[16] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22(1) (2009) 64–69.
[17] W. Ge and X. Zhao, Unbounded solutions for fractional boundary value problems on the infinite interval, Acta Appl. Math. 109 (2008) 495–505.
[18] Y. Gholami, Existence of an unbounded solution for multi-point boundary value problems of fractional differential equations on an infinite domain, Fract. Differ. Calc. 4(2) (2014) 125–136.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 317-335
  • Receive Date: 26 June 2018
  • Revise Date: 28 October 2019
  • Accept Date: 16 October 2020