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Licensed Unlicensed Requires Authentication Published by De Gruyter October 28, 2021

Existence of solutions for the semilinear abstract Cauchy problem of fractional order

  • Hernán R. Henríquez EMAIL logo , Verónica Poblete and Juan C. Pozo


In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α-resolvent family for the homogeneous linear problem. By using this α-resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.


The authors are very grateful to the editor and the anonymous reviewers for their careful reading of the manuscript, comments and suggestions, which allowed to significantly improve the original version of the text.

H. R. Henríquez was partially supported by Vicerrectoría de Investigación, Desarrollo e Innovación de la Universidad de Santiago under Grant DICYT-USACH 041733HM; V. Poblete was partially supported by project Fondecyt 1191137, and J. C. Pozo was partially supported by project Fondecyt 1181084.


[1] S. Abbas, M. Benchohra, G.M. N'Guérékata, Topics in Fractional Differential Equations. Springer Science, New York (2012).10.1007/978-1-4614-4036-9Search in Google Scholar

[2] J.B. Baillon, Caractére borné de certains générateurs de semi-groupes linéaires dans les espaces de Banach. C. R. Acad. Sci. Paris 290 (1980), 757–760.Search in Google Scholar

[3] D. Baleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control. Springer, New York (2012).10.1007/978-1-4614-0457-6Search in Google Scholar

[4] E.G. Bazhlekova, Fractional Evolution Equations in Banach Spaces. Dissertation, Eindhoven University of Technology, Eindhoven (2001).Search in Google Scholar

[5] E. Bazhlekova, Existence and uniqueness results for fractional evolution equation in Hilbert space. Fract. Calc. Appl. Anal. 15, No 2 (2012), 232–243; 10.2478/s13540-012-0017-0; in Google Scholar

[6] S. Belarbi, Z. Dahmani, New controllability results for fractional evolution equations in Banach spaces. Acta Universitatis Apulensis 34 (2013), 17–33.Search in Google Scholar

[7] P. Chen, X. Zhang, Y. Li, Study on fractional non-autonomous evolution equations with delay. Comput. Math. Appl. 73, No 5 (2017), 794–803.10.1016/j.camwa.2017.01.009Search in Google Scholar

[8] D. Chyan, S. Shaw, S. Piskarev, On maximal regularity and semivariation of cosine operator functions. J. London Math. Soc. 59, No 3 (1999), 1023–1032.10.1112/S0024610799007073Search in Google Scholar

[9] S. Das, Functional Fractional Calculus. Springer-Verlag, Berlin (2011).10.1007/978-3-642-20545-3Search in Google Scholar

[10] A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems. Comput. Math. Appl. 62 (2011), 1442–1450.10.1016/j.camwa.2011.03.075Search in Google Scholar

[11] K. Diethelm, The Analysis of Fractional Differential Equations. Springer-Verlag, Berlin (2010).10.1007/978-3-642-14574-2Search in Google Scholar

[12] N. Dunford, J.T. Schwartz, Linear Operators, Part I. John Wiley & Sons, New York (1988).Search in Google Scholar

[13] R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014), 2nd ed. (2020).10.1007/978-3-662-43930-2Search in Google Scholar

[14] A. Granas, J. Dugundji, Fixed Point Theory. Springer-Verlag, New York (2003).10.1007/978-0-387-21593-8Search in Google Scholar

[15] H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011 (2011), Art. ID 298628, 51 pp.10.1155/2011/298628Search in Google Scholar

[16] H.R. Henríquez, Introducción a la Integración Vectorial. Editorial Académica Española, ISBN: 978-3-659-04096-2, Saarbrücken (2012).Search in Google Scholar

[17] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000).10.1142/3779Search in Google Scholar

[18] C.S. Hönig, The Green function of a linear differential equation with a lateral condition. Bull. Amer. Math. Soc. 79 (1973), 587–593.10.1007/BFb0057546Search in Google Scholar

[19] C.S. Hönig, The Abstract Riemann-Stieltjes Integral and Its Applications to Linear Differential Equations With Generalized Boundary Conditions. Universidade de São Paulo, São Paulo (1973).Search in Google Scholar

[20] C.S. Hönig, Semigroups and semivariation. In: Proc. 14 Seminário Brasileiro de Análise (1981), 185–193.Search in Google Scholar

[21] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006).10.1016/S0304-0208(06)80001-0Search in Google Scholar

[22] F. Li, M. Li, On maximal regularity and semivariation of α-times resolvent families. Advances in Pure Math. 3 (2013), 680–684.10.4236/apm.2013.38091Search in Google Scholar

[23] A. Lopushansky, O. Lopushansky, A. Szpila, Fractional abstract Cauchy problem on complex interpolation scales. Fract. Calc. Appl. Anal. 23, No 4 (2020), 1125-1140; 10.1515/fca-2020-0057; in Google Scholar

[24] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models. Imperial College Press, London (2010).10.1142/p614Search in Google Scholar

[25] F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118, No 1-2 (2000), 283–299.10.1016/S0377-0427(00)00294-6Search in Google Scholar

[26] C.-M. Marle, Mesures et Probabilités. Hermann, Paris (1974).Search in Google Scholar

[27] Z.-D. Mei, J.-G. Peng, Y. Zhang, A characteristic of fractional resolvents. Fract. Calc. Appl. Anal. 16, No 4 (2013), 777–790; 10.2478/s13540-013-0048-1; in Google Scholar

[28] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).10.1007/978-1-4612-5561-1Search in Google Scholar

[29] M. Pierri, D. O'Regan, On non-autonomous abstract nonlinear fractional differential equations. Appl. Anal. 94, No 8 (2015), 879–890.10.1080/00036811.2014.905679Search in Google Scholar

[30] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering. Vol. 198. Academic Press, Inc., San Diego (1999).Search in Google Scholar

[31] J. Prüss, Evolutionary Integral Equations and Applications. Monographs Math. Vol. 87, Birkhäuser Verlag, Basel (1993).10.1007/978-3-0348-8570-6Search in Google Scholar

[32] H. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem. J. Evol. Equ. 8, No 2 (2008), 283–305.10.1007/s00028-007-0355-2Search in Google Scholar

[33] C.C. Travis, Differentiability of weak solutions to an abstract inhomogeneous differential equation. Proc. Amer. Math. Soc. 82, No 3 (1981), 425–430.10.1090/S0002-9939-1981-0612734-2Search in Google Scholar

[34] K. Zhang, Existence results for a generalization of the time-fractional diffusion equation with variable coefficients. Bound. Value Probl. 2019, No 10 (2019), 1–11.10.1186/s13661-019-1125-0Search in Google Scholar

[35] 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; in Google Scholar

[36] Y. Zhou, J.W. He, New results on controllability of fractional evolution systems with order α ∈ (1, 2). Evol. Equ. Control Theory. 10, No 3 (2021), 491–509.10.3934/eect.2020077Search in Google Scholar

Received: 2020-10-29
Revised: 2021-08-17
Published Online: 2021-10-28
Published in Print: 2021-10-26

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