Extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions

Document Type : Research Paper

Authors

Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.

Abstract

This paper is devoted to prove the existence of extremal solutions for multi-term nonlinear fractional differential equations with nonlinear boundary conditions. The fractional derivative is of Caputo type and the inhomogeneous term depends on the fractional derivatives of lower orders. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions. The method is a constructive method that yields monotone sequences that converge to the extremal solutions. As an application, some examples are presented to illustrate the main results.

Keywords

References

  • [1] P. Agarwal, R. P. Agarwal, and M. Ruzhansky, (Eds.), Special Functions and Analysis of Differential Equations (1st ed.), Chapman and Hall/CRC, 2020.
  • [2] R. P. Agarwal, M. Benchohra, and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.
  • [3] P. Agarwal, A. Berdyshev, and E. Karimov, Solvability of a Non-local Problem with Integral Transmitting Condi- tion for Mixed Type Equation with Caputo Fractional Derivative, Results Math., 71 (2017), 1235-1257.
  • [4] P. Agarwal, S. Jain, and T. Mansour, Further extended Caputo fractional derivative operator and its applications, Russ. J. Math. Phys., 24 (2017), 415-425.
  • [5] A. Aghajani, Y. Jalilian, and J. J. Trujillo, On the existence of solutions of fractional integro-differential equations, Fract. Calc. Appl. Anal., 15 (2012), 44-69.
  • [6] F. Bahrami, H. Fazli, and A. Jodayree Akbarfam, A new approach on fractional variational problems and Euler- Lagrange equations, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 39-50.
  • [7] B. Bonilla, M. Rivero, L. Rodriguez-Germa, and J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 187 (2007), 79-88.
  • [8] L. Byszewski, Theorems about the existence and uniqueness of solution of semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.
  • [9] J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 23 (2010), 676-680
  • [10] J. Deng and Z. Deng, Existence of solutions of initial value problems for nonlinear fractional differential equations, Appl. Math. Lett., 32 (2014), 6-12
  • [11] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
  • [12] K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Comput. Math., 154 (2004), 621-640.
  • [13] Y. Ding, Z. Wei, J. Xu, and D. O´Regan, Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian, J. Comput. Appl. Math., 288 (2015), 151-158.
  • [14] H. Fazli and F. Bahrami, On the steady solutions of fractional reaction-diffusion equations, Filomat, 31 (2017), 1655-1664.
  • [15] H. Fazli, J. J. Nieto, and F. Bahrami, On the existence and uniqueness results for nonlinear sequential fractional differential equations, Appl. Comput. Math., 17 (2018), 36-47.
  • [16] H. Fazli and J. J. Nieto, An investigation of fractional Bagley-Torvikequation, Open Math., 17 (2019), 499-512.
  • [17] H. Fazli and J. J. Nieto, Fractional Langevin Equation with Anti-Periodic Boundary Conditions, Chaos, Solitons and Fractals, 114 (2018), 332-337.
  • [18] H. Fazli and J. J. Nieto, Nonlinear sequential fractional differential equations in partially ordered spaces, Filomat, 32 (2018), 4577–4586.
  • [19] H. Fazli, H. Sun, and S.Agchi, Existence of extremal solutions of fractional Langevin equation involving nonlinear boundary conditions, Int. J. Comput. Math., (2020), 1720662
  • [20] H. Fazli, H. Sun, and J. J. Nieto, New existence and stability results for fractional Langevin equation with three- point boundary conditions, Comp. Appl. Math., 40 (2021), 48.
  • [21] H. Fazli, H. Sun, and J. J. Nieto, Fr´echet-Kolmogorov compactness of Prabhakar integral operator, RACSAM, 115 (2021), 165.
  • [22] H. A Hammad, P. Agarwal, S. Momani, and F. Alsharari, Solving a Fractional-Order Differential Equation Using Rational Symmetric Contraction Mappings. Fractal Fract., 5 (2021), 159.
  • [23] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  • [24] N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order, Nonlinear Anal., 70 (2009), 2521-2529.
  • [25] M. P. Lazarevic and A. M. Spasic, Finite-time stability analysis of fractional order time delay systems: Gronwall’s approach, Math. Comput. Modelling, 49 (2009), 475-481.
  • [26] Q. Li, C. Hou, L. Sun, and Z. Han, Existence and uniqueness for a class of multi-term fractional differential equations, J. Appl. Math. Comput., 53 (2017), 383-395.
  • [27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [28] 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 (2010), 1038-1044
  • [29] M. Ruzhansky, Y. J. Cho, P. Agarwal, I. Area, (Eds.), Advances in Real and Complex Analysis with Applications (Trends in Mathematics), Birkhuser, 2017.
  • [30] L. Schwan, O. Umnova, C. Boutin, and J. Groby, Nonlocal boundary conditions for corrugated acoustic metasurface with strong near-field interactions , Journal of Applied Physics, 123 (2018), 091712, doi.org/10.1063/1.5011385
  • [31] X. Su, Boundary value problem for a coupled systerm of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64-69.
  • [32] A. Sunarto, P. Agarwal, J. Sulaiman, and J. Vui, Computational Approach via Half-Sweep and Preconditioned AOR for Fractional Diffusion. Intelligent Automation & Soft Computing, 31(2) (2022), 1173-1184.
  • [33] G. T. Wang, R. P. Agarwal, and A. Cabada, Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations, Appl. Math. Lett., 25 (2012), 1019-1024.
Computational Methods for Differential Equations
Volume 11, Issue 1
January 2023
Pages 32-41

Files

History

  • Receive Date: 07 October 2021
  • Revise Date: 31 December 2021
  • Accept Date: 07 January 2022

Share

How to cite

Statistics