An application of mixed monotone operator on a fractional differntial equation on an unbounded domain

Document Type : Research Paper


School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran



This paper provides sufficient conditions that guarantee the existence of positive solutions to a boundary value problem of a nonlinear fractional differential equation on the half line. Our analysis takes advantage of a theory on cones and mixed monotone operators combined with the diagonalization method. The paper also contains some examples that are numerically solved by the Adomian Decomposition Method.


[1] R.P. Agarwal and D. O’Regan, Non-linear boundary value problems on the semi-infinite interval: an upper and lower solution approach, Mathematika 49 (2002), no. 1-2, 129–140.
[2] R.P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Springer Science and Business Media, 2012.
[3] B. Ahmad and S. Ntouyas, A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips-type integral boundary conditions, J. Math. Sci. 226 (2017), no. 3.
[4] A. Arara, M. Benchohra, N. Hamidi, and J.J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal.: Theory Meth. Appl. 72 (2010), no. 2, 580–586.
[5] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, arXiv preprint arXiv:1602.03408 (2016).
[6] C. Bai and C. Li, Unbounded upper and lower solution method for third-order boundary-value problems on the half-line, Electronic J. Differ. Equ. 2009 (2009), no. 119, 1--12.
[7] Y. Bao, L. Wang, and M. Pei, Existence of positive solutions for a singular third-order two-point boundary value problem on the half-line, Boundary Value Prob.2022 (2022), no. 1, 1–11.
[8] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012), no. 1, 403–411.
[9] M. Caputo, Linear models of dissipation whose q is almost frequency independent, Ann. Geophys.19 (1966), no. 4, 383–393.
[10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fractional Differ. Appl. 1 (2015), no. 2, 73–85.
[11] K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), no. 2, 229–248.
[12] K. Diethelm and A.D. Freed, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, Scientific computing in chemical engineering II: computational fluid dynamics, reaction engineering, and molecular properties, Springer, 1999, pp. 217–224.
[13] W.G. Glockle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68 (1995), no. 1, 46–53.
[14] J.R. Graef, L. Kong, and Q. Kong, Application of the mixed monotone operator method to fractional boundary value problems, Fract. Calc. Differ. Calc. 2 (2011), 554–567.
[15] J.R. Graef, L. Kong, Q. Kong, and M. Wang, Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions, Fractional Calc. Appl. Anal. 15 (2012), 509–528.
[16] A. Guezane-Lakoud and A. Ashyralyev, Positive solutions for a system of fractional differential equations with nonlocal integral boundary conditions, Differ. Equ. Dyn. Syst. 25 (2017), 519–526.
[17] D. Guo, Fixed points of mixed monotone operators with applications, Appl. Anal. 31 (1988), no. 3, 215–224.
[18] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal.: Theory Meth. Appl. 11 (1987), no. 5, 623–632.
[19] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, 2014.
[20] J. Hadamard, Essai sur l’etude des fonctions donnees par leur developpement de taylor, J. Math. Pures Appl. 8 (1892), 101–186.
[21] J. Henderson and R. Luca, Existence of positive solutions for a system of semipositone coupled discrete boundary value problems, J. Differ. Equ. Appl. 25 (2019), no. 4, 516–541.
[22] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
[23] A.A. Kilbas, O.I. Marichev, and S.G. Samko, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993.
[24] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier, 2006.
[25] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, CSP, 2009.
[26] H. Lian, P. Wang, and W. Ge, Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals, Nonlinear Anal.: Theory Meth. Appl. 70 (2009), no. 7, 2627–2633.
[27] F. Mainardi, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, Springer, 1997.
[28] H.M. Malaikah, The Adomian decomposition method for solving Volterra-Fredholm integral equation using maple, Appl. Math. 11 (2020), 779–787.
[29] R. Metzler, W. Schick, H.-G. Kilian, and T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), no. 16, 7180–7186.
[30] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.
[31] E. Pourhadi, R. Saadati, and S.K. Ntouyas, Application of fixed-point theory for a nonlinear fractional three-point boundary-value problem, Mathematics 7 (2019), no. 6, 526.
[32] M. Sangi, S. Saiedinezhad, and M.B. Ghaemi, A system of high-order fractional differential equations with integral boundary conditions, J. Nonlinear Math. Phys.30 (2023), no. 2, 699–718.
[33] V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science & Business Media, 2011.
[34] W. Wang and X. Liu, Properties and unique positive solution for fractional boundary value problem with two parameters on the half-line, J. Appl. Anal. Comput 11 (2021), no. 5, 2491–2507.
[35] B. Yan, D. O’Regan, and R.P. Agarwal, Unbounded solutions for singular boundary value problems on the semiinfinite interval: upper and lower solutions and multiplicity, J. Comput. Appl. Math. 197 (2006), no. 2, 365–386.
[36] C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal.: Theory Methods Appl. 75 (2012), no. 4, 2542–2551.

Articles in Press, Corrected Proof
Available Online from 08 May 2024
  • Receive Date: 12 December 2023
  • Revise Date: 02 April 2024
  • Accept Date: 20 February 2024