Skip to main content
Article
Licensed
Unlicensed Requires Authentication

Attractivity for differential equations of fractional order and ψ-Hilfer type

  • , and EMAIL logo
Published/Copyright: September 11, 2020

Abstract

This paper investigates the overall solution attractivity of the fractional differential equation involving the ψ-Hilfer fractional derivative and using the Krasnoselskii’s fixed point theorem. We highlight some particular cases of the results presented here, especially involving the Riemann-Liouville, thus illustrating the broad class of fractional derivatives to which these results can be applied.

Acknowledgements

JVCS acknowledges the financial support of a PNPD-CAPES (88882.305834/2018-01) scholarship of the Postgraduate Program in Applied Mathematics of IMECC-Unicamp. The authors are grateful to the Editor in Chief, Prof. V. Kiryakova for her helpful remarks and suggestions.

References

[1] S. Abbas, M. Benchohra, J.R. Graef, Coupled Sytems of Hilfer fractional differential inclusions in Banach spaces. Commun. Pure & Appl. Anal. 17, No 6 (2018), 2479–2493; 10.3934/cpaa.2018118.Search in Google Scholar

[2] S. Abbas, M. Benchohra, N. Hamidi, G. N’Guérékata, Existence and attractivity results for coupled systems of nonlinear Volterra–Stieltjes multidelay fractional partial integral equations. Abstr. Appl. Anal. 2018 (2018), Article ID 8735614, 10 pages; 10.1155/2018/8735614.Search in Google Scholar

[3] S. Abbas, R.P. Agarwal, M. Benchohra, and F. Berhoun, Global attractivity for Volterra type Hadamard fractional integral equations in Fréchet spaces. Demonstr. Math. 51 (2018), 131–140; 10.1515/dema-2018-0009.Search in Google Scholar

[4] S. Abbas, M. Benchohra, N. Hamidi, J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1027–1045; 10.1515/fca-2018-0056; https://www.degruyter.com/view/journals/fca/21/4/fca.21.issue-4.xml.Search in Google Scholar

[5] S. Abbas, M. Benchohra, and J. Henderson, Existence and attractivity results for Hilfer fractional differential equations. J. Math. Sci. 243, No 3 (2019), 347–357; 10.1007/s10958-019-04544-y.Search in Google Scholar

[6] S. Abbas and M. Benchohra, Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses. Appl. Math. Comput. 257 (2015), 190–198; 10.1016/j.amc.2014.06.073.Search in Google Scholar

[7] S. Abbas, M. Benchohra, and J.J. Nieto, Global attractivity of solutions for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations. Electron. J. Qual. Theory Differ. Equ. 2012, No 81 (2012), 1–15; 10.14232/ejqtde.2012.1.81.Search in Google Scholar

[8] S. Abbas, M. Benchohra, A. Petrusel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory. Fract. Calc. Appl. Anal. 20, No 2 (2017), 384–398; 10.1515/fca-2017-0020; https://www.degruyter.com/view/journals/fca/20/2/fca.20.issue-2.xml.Search in Google Scholar

[9] R. Agarwal, S. Hristova, and D. O’Regan, Non-instantaneous impulses in Caputo fractional differential equations. Fract. Calc. Appl. Anal. 20, No 3 (2017), 595–622; 10.1515/fca-2017-0032; https://www.degruyter.com/view/journals/fca/20/3/fca.20.issue-3.xml.Search in Google Scholar

[10] J. Banaś, D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order. J. Math. Anal. Appl. 345 No 1 (2008), 573–582; 10.1016/j.jmaa.2008.04.050.Search in Google Scholar

[11] M. Benchohra, Z. Bouteffal, J. Henderson, and S. Litimein, Measure of noncompactness and fractional integro-differential equations with state-dependent nonlocal conditions in Fréchet spaces. AMS Math. 5, No 1 (2019), 15–25; 10.3934/math.2020002.Search in Google Scholar

[12] M. Benchohra, S. Litimein, and J.J. Nieto, Semilinear fractional differential equations with infinite delay and non-instantaneous impulses. J. Fixed Point Theory Appl. 2019 (2019), # 21; 10.1007/s11784-019-0660-8.Search in Google Scholar

[13] T.A. Burton, A fixed point theorem of Krasnoselskii. Appl. Math. Lett. 11 (1998), 85—88, 10.1016/S0893-9659(97)00138-9.Search in Google Scholar

[14] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional Calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269 (2002), 1–27; 10.1016/S0022-247X(02)00001-X.Search in Google Scholar

[15] F. Chen, J.J. Nieto and Y. Zhou, Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. 13 No 1 (2012), 287–298; 10.1016/j.nonrwa.2011.07.034.Search in Google Scholar

[16] F. Chen and Y. Zhou, Attractivity of fractional functional differential equations. Comput. Math. Appl. 62, No 3 (2011), 1359–1369; 10.1016/j.camwa.2011.03.062.Search in Google Scholar

[17] J. Deng and L. Ma, Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. Appl. Math. Lett. 23, No 6 (2010), 676–680; 10.1016/j.aml.2010.02.007.Search in Google Scholar

[18] Z. Fan, Existence and regularity of solutions for evolution equations with Riemann–Liouville fractional derivatives. Indagationes Math. 25, No 3 (2014), 516–524; 10.1016/j.indag.2014.01.002.Search in Google Scholar

[19] J.K. Hale, Theory of Function Differential Equations. Springer-Verlag, New York (1977).10.1007/978-1-4612-9892-2Search in Google Scholar

[20] R. Hilfer, Applications of Fractional Calculus in Physics. World Sci., N. Jersey (2000).10.1142/3779Search in Google Scholar

[21] T.D. Ke, N.N. Quan, Finite-time attractivity for semilinear tempered fractional wave equations. Fract. Calc. Appl. Anal. 21, No 6 (2018), 1471–1492; 10.1515/fca-2018-0077; https://www.degruyter.com/view/journals/fca/21/6/fca.21.issue-6.xml.Search in Google Scholar

[22] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[23] V. Kiryakova, Y. Luchko, Multiple Erdélyi-Kober integrals and derivatives as operators of generalized fractional calculus. In: Handbook of Fractional Calculus with Applications, Chap. 6, Vol. 1, De Gruyter, Berlin (2019), 127–158; 10.1515/9783110571622-006.Search in Google Scholar

[24] J. Losada, J.J. Nieto, and E. Pourhadi, On the attractivity of solutions for a class of multi-term fractional functional differential equations. J. Comput. Appl. Math. 312 (2017), 2–12; 10.1016/j.cam.2015.07.014.Search in Google Scholar

[25] E. de Oliveira, J. Vanterler da C. Sousa, Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. Results Math. 73, No 3 (2018), # 111; 10.1007/s00025-018-0872-z.Search in Google Scholar

[26] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach. Sci. Publ., London - N. York (1993).Search in Google Scholar

[27] J. Vanterler da C. Sousa, K.D. Kucche, E. de Oliveira, On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Meth. Appl. Sci. 42 (2019), 3021–3032; 10.1002/mma.5562.Search in Google Scholar

[28] J. Vanterler da C. Sousa, E. de Oliveira, On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72–91; 10.1016/j.cnsns.2018.01.005.Search in Google Scholar

[29] J. Vanterler da C. Sousa, E. de Oliveira, Leibniz type rule: ψ-Hilfer fractional operator. Commun. Nonlinear Sci. Numer. Simul. 77 (2019), 305–311; 10.1016/j.cnsns.2019.05.003.Search in Google Scholar

[30] J. Vanterler da C. Sousa, E. de Oliveira, Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81 (2018), 50–56; 10.1016/j.aml.2018.01.016.Search in Google Scholar

[31] J. Vanterler da C. Sousa, E. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of Hilfer operator. Diff. Equ. & Appl. 11, No 1 (2019), 87–106; 10.7153/dea-2019-11-02.Search in Google Scholar

[32] J. Vanterler da C. Sousa, K. D. Kucche, E. de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations. Appl. Math. Lett. 88 (2019), 73–80; 10.1016/j.aml.2018.08.013.Search in Google Scholar

[33] J. Vanterler da C. Sousa, E. de Oliveira, On the Ψ-fractional integral and applications, Comput. Appl. Math. 38 No 1 (2019) 4; https://doi.org/10.1007/s40314-019-0774-z.Search in Google Scholar

[34] J. Vanterler da C. Sousa, E. de Oliveira, On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ-Hilfer operator. J. Fixed Point Theory Appl. 20, No 3 (2018), # 96; 10.1007/s11784-018-0587-5.Search in Google Scholar

[35] J. Vanterler da C. Sousa, E. de Oliveira, Fractional order pseudo-parabolic partial differential equations: Ulam–Hyers Stability, Bull. Braz. Math. Soc. 50 (2019), 481–496; 10.1007/s00574-018-0112-x.Search in Google Scholar

[36] J. Vanterler da C. Sousa, E. de Oliveira, On the stability of a hyperbolic fractional partial differential equation. Diff. Equ. Dyn. Sys. 2019 (2019); 10.1007/s12591-019-00499-3.Search in Google Scholar

[37] J. Vanterler da C. Sousa, E. de Oliveira, Capelas, Fractional order pseudoparabolic partial differential equation: Ulam–Hyers stability. Bull. Braz. Math. Soc., New Series50, No 2 (2019), 481–496; 10.1007/s00574-018-0112-x.Search in Google Scholar

[38] H.M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45 (2017), 192–204; 10.1016/j.apm.2016.12.008.Search in Google Scholar

[39] Z. Zhang, B. Liu, Existence of mild solutions for fractional evolution equations. J. Fract. Calc. Appl. 2, No 20 (2012), 1–10.Search in Google Scholar

[40] Y. Zhou, J.W. He, B. Ahmad, and A. Alsaedi, Existence and attractivity for fractional evolution equations. Discrete Dyn. Nat. Soc. (2018), Art. ID 1070713, 9 pp.; 10.1155/2018/1070713.Search in Google Scholar

[41] Y. Zhou, Attractivity for fractional differential equations in Banach space. Appl. Math. Lett. 75 (2018), 1–6; 10.1016/j.aml.2017.06.008.Search in Google Scholar

[42] Y. Zhou, Attractivity for fractional evolution equations with almost sectorial operators. Fract. Calc. Appl. Anal. 21, No 3 (2018), 786–800; 10.1515/fca-2018-0041; https://www.degruyter.com/view/journals/fca/21/3/fca.21.issue-3.xml.Search in Google Scholar

Received: 2019-12-17
Revised: 2020-08-10
Published Online: 2020-09-11
Published in Print: 2020-08-26

© 2020 Diogenes Co., Sofia

Downloaded on 22.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/fca-2020-0060/html
Scroll to top button