Research article

Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

  • Received: 08 October 2021 Revised: 15 December 2021 Accepted: 20 December 2021 Published: 30 December 2021
  • Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.

    Citation: J. Vanterler da C. Sousa, Kishor D. Kucche, E. Capelas de Oliveira. Stability of mild solutions of the fractional nonlinear abstract Cauchy problem[J]. Electronic Research Archive, 2022, 30(1): 272-288. doi: 10.3934/era.2022015

    Related Papers:

  • Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.



    加载中


    [1] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. http://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [2] S. M. Ulam, Problems in Modern Mathematics, science editions, John-Wiley & Sons Inc., New York, 1964.
    [3] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
    [4] M. Akkouchi, A. Bounabat, M. H. L. Rhali, Fixed point approach to the stability of integral equation in the sense of Ulam-Hyers-Rassias, Ann. Math. Silesianae, 5 (2011), 27–44.
    [5] S. M. Ulam, A Collection of Mathematical Problems, 1960.
    [6] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Jpn., 2, (1950), 64–66. http://doi.org/10.2969/jmsj/00210064 doi: 10.2969/jmsj/00210064
    [7] C. Park, T. M. Rassias, Homomorphisms and derivations in proper JCQ-triples, J. Math. Anal. Appl., 337 (2008), 1404–1414. https://doi.org/10.1016/j.jmaa.2007.04.063 doi: 10.1016/j.jmaa.2007.04.063
    [8] J. V. da Sousa, E. C. de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simulat., 60 (2018), 72–91. https://doi.org/10.1016/j.cnsns.2018.01.005 doi: 10.1016/j.cnsns.2018.01.005
    [9] J. V. da Sousa, E. C. de Oliveira, On the $\psi$-fractional integral and applications, Comp. Appl. Math., 38 (2019). https://doi.org/10.1007/s40314-019-0774-z doi: 10.1007/s40314-019-0774-z
    [10] J. V. da Sousa, E. C. de Oliveira, Leibniz type rule: $\psi$-Hilfer fractional operator, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 305–311. https://doi.org/10.1016/j.cnsns.2019.05.003 doi: 10.1016/j.cnsns.2019.05.003
    [11] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063–1077. https://doi.org/10.1016/j.camwa.2009.06.026 doi: 10.1016/j.camwa.2009.06.026
    [12] M. Yang, Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions, Frac. Calc. Appl. Anal., 20 (2017), 679–705. https://doi.org/10.1515/fca-2017-0036 doi: 10.1515/fca-2017-0036
    [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, 2006.
    [14] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Switzerland, 1993.
    [15] J. Wang, Y. Zhou, Mittag-Leffler-Ulam stabilities of fractional evolution equations, Appl. Math. Lett., 25 (2012), 723–728. https://doi.org/10.1016/j.aml.2011.10.009 doi: 10.1016/j.aml.2011.10.009
    [16] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530–2538. https://doi.org/10.1016/j.cnsns.2011.09.030 doi: 10.1016/j.cnsns.2011.09.030
    [17] J. Wang, Y. Zhang, A class of nonlinear differential equations with fractional integrable impulses, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3001–3010. https://doi.org/10.1016/j.cnsns.2014.01.016 doi: 10.1016/j.cnsns.2014.01.016
    [18] J. Wang, Y. Zhou, M. Fec, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comput. Math. Appl., 64 (2012), 3389–3405. https://doi.org/10.1016/j.camwa.2012.02.021 doi: 10.1016/j.camwa.2012.02.021
    [19] M. Inc, M. Partohaghighi, M. A. Akinlar, P. Agarwale, Y. M. Chu, New solutions of fractional-order Burger-Huxley equation. Results Phys., 18 (2020), 103290. https://doi.org/10.1016/j.rinp.2020.103290 doi: 10.1016/j.rinp.2020.103290
    [20] H. Ahmad, T. A. Khan, I. Ahmad, P. S. Stanimirovic, Y. M. Chu, A new analyzing technique for nonlinear time fractional Cauchy reaction-diffusion model equations. Results Phys., 19 (2020), 103462. https://doi.org/10.1016/j.rinp.2020.103462 doi: 10.1016/j.rinp.2020.103462
    [21] Y. M. Chu, N. A. Shah, H. Ahmad, J. D. Chung, S. M. Khaled, A comparative study of semi-analytical methods for solving fctional-order cauchy reaction-fiffusion equation, Fractals, 29 (2021). https://doi.org/10.1142/S0218348X21501437 doi: 10.1142/S0218348X21501437
    [22] Y. M. Chu, N. A. Shah, P. Agarwal, J. D. Chung, Analysis of fractional multi-dimensional Navier–Stokes equation, Adv. Differ. Equations, 91 (2021), 1–18. https://doi.org/10.1186/s13662-021-03250-x doi: 10.1186/s13662-021-03250-x
    [23] M. Inc, M. Parto-Haghighi, M. A. Akinlar, Y. M. Chu, New numerical solutions of fractional-order Korteweg-de Vries equation, Results Phys., 19 (2020), 103326. https://doi.org/10.1016/j.rinp.2020.103326 doi: 10.1016/j.rinp.2020.103326
    [24] L. V. C. Hoan, Z. Korpinar, M. Inc, Y. M. Chu, B. Almohsen, On convergence analysis and numerical solutions of local fractional Helmholtz equation, Alex. Eng. J., 59 (2020), 4335–4341. https://doi.org/10.1016/j.aej.2020.07.038 doi: 10.1016/j.aej.2020.07.038
    [25] S. Sahoo, S. Sahoo, S. S. Ray, M. A. M. Abdou, M. Inc, New soliton solutions of fractional Jaulent-Miodek system with symmetry analysis, Symmetry, 12 (2020), 1001. https://doi.org/10.3390/sym12061001 doi: 10.3390/sym12061001
    [26] P. O. Mohammed, T. Abdeljawad, F. Jarad, Y. M. Chu, Existence and uniqueness of uncertain fractional backward difference equations of Riemann–Liouville type, Math. Probl. Eng., 2020 (2020), 6598682. https://doi.org/10.1155/2020/6598682 doi: 10.1155/2020/6598682
    [27] M. I. Abbas, Existence and Uniqueness of Mittag-Leffler-Ulam stable solution for fractional integrodifferential equations with nonlocal initial conditions, Eur. J. Pure Appl. Math., 8 (2015), 478–498.
    [28] R. Saadati, E. Pourhadi, B. Samet, On the $\mathcal{PC}$-mild solutions of abstract fractional evolution equations with non-instantaneous impulses via the measure of noncompactness, Bound. Value Probl., 19 (2019). https://doi.org/10.1186/s13661-019-1137-9 doi: 10.1186/s13661-019-1137-9
    [29] J. V. da C. Sousa, D. S. Oliveira, E. C. de Oliveira, A note on the mild solutions of Hilfer impulsive fractional differential equations, Chaos Soliton. Fract., 147 (2021), 110944. https://doi.org/10.1016/j.chaos.2021.110944 doi: 10.1016/j.chaos.2021.110944
    [30] J. Dabas, A. Chauhan, M. Kumar, Existence of the mild solutions for impulsive fractional equations with infinite delay, Inter. J. Diff. Equations, 2011 (2011), 793023. https://doi.org/10.1155/2011/793023 doi: 10.1155/2011/793023
    [31] A. Jawahdou, Mild solutions of functional semilinear evolution Volterra integrodifferential equations on an unbounded interval, Nonlinear Anal. Theor. Methods Appl., 74 (2011), 7325–7332. https://doi.org/10.1016/j.na.2011.07.050 doi: 10.1016/j.na.2011.07.050
    [32] K. Balachandran, N. Annapoorani, Existence results for impulsive neutral evolution integrodifferential equations with infinite delay, Nonlinear Anal. Hybrid Syst., 3 (2009), 674–684. https://doi.org/10.1016/j.nahs.2009.06.004 doi: 10.1016/j.nahs.2009.06.004
    [33] L. Olszowy, Existence of mild solutions for the semilinear nonlocal problem in Banach spaces, Nonlinear Anal. Theory, Mett. Appl., 81 (2013), 211–223. https://doi.org/10.1016/j.na.2012.11.001 doi: 10.1016/j.na.2012.11.001
    [34] C. Chen, M. Li, On fractional resolvent operator functions, Semigroup Forum, 80 (2010), 121–142. https://doi.org/10.1007/s00233-009-9184-7 doi: 10.1007/s00233-009-9184-7
    [35] Y. Zhou, J. W. He, New results on controllability of fractional evolution systems with order $\alpha\in (1, 2)$, Evol. Equ. Control Theory, 10 (2021), 491–509. http://doi.org/10.3934/eect.2020077 doi: 10.3934/eect.2020077
    [36] Y. Zhou, B. Ahmad, A. Alsaedi. Existence of nonoscillatory solutions for fractional neutral differential equations, Appl. Math. Lett., 72 (2017), 70–74. https://doi.org/10.1016/j.aml.2017.04.016 doi: 10.1016/j.aml.2017.04.016
    [37] J. Wang, M. Feckan, Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bull. Sci. Math., 141 (2017), 727–746. https://doi.org/10.1016/j.bulsci.2017.07.007 doi: 10.1016/j.bulsci.2017.07.007
    [38] J. V. da C. Sousa, K. D. Kucche, E. C. de Oliveira, Stability of $\psi$-Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88 (2019), 73–80. https://doi.org/10.1016/j.aml.2018.08.013 doi: 10.1016/j.aml.2018.08.013
    [39] J. V. da C. Sousa, F. Jarad, T. Abdeljawad, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, Ann. Funct. Anal., 12 (2021), 12. https://doi.org/10.1007/s43034-020-00095-5 doi: 10.1007/s43034-020-00095-5
    [40] H. Gu, J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2015), 344–354. https://doi.org/10.1016/j.amc.2014.10.083 doi: 10.1016/j.amc.2014.10.083
    [41] J. V. da C. Sousa, E. C. de Oliveira, Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50–56. https://doi.org/10.1016/j.aml.2018.01.016 doi: 10.1016/j.aml.2018.01.016
    [42] P. Chen, Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Zeitschrift für Angewandte Mathematik und Physik, 65 (2014), 711–728. https://doi.org/10.1007/s00033-013-0351-z doi: 10.1007/s00033-013-0351-z
    [43] P. M. de Carvalho-Neto, G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^{N}$, J. Diff. Equ., 259 (2015), 2948–2980. https://doi.org/https://doi.org/10.1016/j.jde.2015.04.008 doi: 10.1016/j.jde.2015.04.008
    [44] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Disc. Conti. Dyn. Sys., 2019, 1775–1786.
    [45] J. V. da C. Sousa, S. Gala, E. C. de Oliveira, On the uniqueness of mild solutions to the time-fractional Navier-Stokes equations in $L^{N}(\mathbb{R}^{N})^{N}$, preprint, arXiv: 1907.06587.
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1397) PDF downloads(208) Cited by(4)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog