Asymptotic behavior of solutions of fractional differential equations with Hadamard fractional derivatives

References

[1] D. Babusci, G. Dattoli, D. Sacchetti, The Lamb–Bateman integral equation and the fractional derivatives. Fract. Calc. Appl. Anal. 14, No 2 (2011), 317–320; 10.2478/s13540-011-0019-3; https://www.degruyter.com/journal/key/FCA/14/2/html.Suche in Google Scholar

[2] D. Babusci, G. Dattoli, D. Sacchetti, Integral equations, fractional calculus and shift operators. arXiv:1007.5211v1 (2010).Suche in Google Scholar

[3] D. Băleanu, O.G. Mustafa, R.P. Agarwal, On the solution set for a class of sequential fractional differential equations. J. Phys. A Math. Theor. 43, No 38 (2010), 1–10.10.1088/1751-8113/43/38/385209Suche in Google Scholar

[4] D. Băleanu, O.G. Mustafa, R.P. Agarwal, Asymptotically linear solutions for some linear fractional differential equations. Abstr. Appl. Anal. 2010, No 10 (2010), 1–8.10.1155/2010/865139Suche in Google Scholar

[5] D. Băleanu, O.G. Mustafa, R.P. Agarwal, Asymptotic integration of (1 + α)-order fractional differential equations. Comput. Math. with Appl. 62, No 3 (2011), 1492–1500.10.1016/j.camwa.2011.03.021Suche in Google Scholar

[6] D. Băleanu, R.P. Agarwal, O.G. Mustafa, M. Coşulschi, Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A Math. Theor. 44, No 5 (2011), 1–9.10.1088/1751-8113/44/5/055203Suche in Google Scholar

[7] D.S. Cohen, The asymptotic behavior of a class of nonlinear differential equations. Proc. Amer. Math. Soc. 18, No 4 (1967), 607–609.10.1090/S0002-9939-1967-0212289-3Suche in Google Scholar

[8] A. Constantin, On the asymptotic behavior of second order nonlinear differential equations. Rend. Mat. Appl. 13, No 7 (1993), 627–634.Suche in Google Scholar

[9] A. Constantin, On the existence of positive solutions of second order differential equations. Ann. di Mat. Pura ed Appl. 184, No 2 (2005), 131–138.10.1007/s10231-004-0100-1Suche in Google Scholar

[10] F.M. Dannan, Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations. J. Math. Anal. Appl. 108, No 1 (1985), 151–164.10.1016/0022-247X(85)90014-9Suche in Google Scholar

[11] G. de Barra, Measure Theory and Integration. Woodhead Publishing Limited, Cambridge (2003).10.1533/9780857099525Suche in Google Scholar

[12] R. Garra, F. Polito, On some operators involving Hadamard derivatives. Integr. Transf. Spec. Funct. 24, No 10 (2013), 773–782.10.1080/10652469.2012.756875Suche in Google Scholar

[13] W.G. Glöckle, T.F. Nonnenmacher, A fractional calculus approach to selfsimilar protein dynamics. Biophys. J. 68, No 1 (1995), 46–53.10.1016/S0006-3495(95)80157-8Suche in Google Scholar

[14] A. Iomin, V. Mendez, W. Horsthemke, Fractional Dynamics in Comb-Like Structures. World Scientific Publishing Co. Pte. Ltd, Singapore (2018).10.1142/11076Suche in Google Scholar

[15] M.D. Kassim, K.M. Furati, N.-E. Tatar, Asymptotic behavior of solutions to nonlinear fractional differential equations. Math. Model Anal. 21, No 5 (2016), 610–629.10.3846/13926292.2016.1198279Suche in Google Scholar

[16] M.D. Kassim, K.M. Furati, N.-E. Tatar, Asymptotic behavior of solutions to nonlinear initial-value fractional differential problems. Electron. J. Differ. Equ. 2016, No 291 (2016), 1–14.Suche in Google Scholar

[17] M.D. Kassim, N.-E. Tatar, Stability of logarithmic type for a Hadamard fractional differential problem. J. Pseudodiffer. Oper. Appl. 11, No 1 (2020), 447–466.10.1007/s11868-019-00285-3Suche in Google Scholar

[18] M.D. Kassim, N.-E. Tatar, Convergence of solutions of fractional differential equations to power-type functions. Electron. J. Differ. Equ. 2020, No 111 (2020), 1–14.Suche in Google Scholar

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

[20] A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: methods results and problem–I. Appl. Anal. 78, No 1-2 (2001), 153–192.10.1080/0003681021000022032Suche in Google Scholar

[21] T. Kusano, W.F. Trench, Global existence of second order differential equations with integrable coefficients. J. London Math. Soc. 31, No 1 (1985), 478–486.10.1112/jlms/s2-31.3.478Suche in Google Scholar

[22] T. Kusano, W.F. Trench, Existence of global solutions with prescribed asymptotic behavior for nonlnear ordinary differential equations. Ann. di Mat. Pura Appl. 142, No 1 (1985), 381–392.10.1007/BF01766602Suche in Google Scholar

[23] Y. Liang, S. Wang, W. Chen, Z. Zhou, R.L. Magin, A survey of models of ultraslow diffusion in heterogeneous materials. Appl. Mech. Rev. 71, No 4 (2019), 1–16.10.1115/1.4044055Suche in Google Scholar

[24] O. Lipovan, On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations. Glasgow Math. J. 45, No 1 (2003), 179–187.10.1017/S0017089502001143Suche in Google Scholar

[25] R.L. Magin, O. Abdullah, D. Băleanu, J.Z. Xiaohong, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. Magn. Reson. 190, No 2 (2008), 255–270.10.1016/j.jmr.2007.11.007Suche in Google Scholar PubMed

[26] M. Medved, On the asymptotic behavior of solutions of nonlinear differential equations of integer and also of non-integer order. Electron. J. Qual. Theory Differ. Equ. 2012, No 10 (2012), 1–9.10.14232/ejqtde.2012.3.10Suche in Google Scholar

[27] M. Medved, Asymptotic integration of some classes of fractional differential equations. Tatra Mt. Math. Publ. 54, No 1 (2013), 119–132.10.2478/tmmp-2013-00010Suche in Google Scholar

[28] M. Medved, M. Pospišsil, Asymptotic integration of fractional differential equations with integrodifferential right-hand side. Math. Model. Anal. 20, No 4 (2015), 471–489.10.3846/13926292.2015.1068233Suche in Google Scholar

[29] R. Metzler, W. Schick, H.G. Kilian, T.F. Nonennmacher, Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, No 16 (1995), 7180–7186.10.1063/1.470346Suche in Google Scholar

[30] O.G. Mustafa, Y.V. Rogovchenko, Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. Nonlinear Anal. Theory Methods Appl. 51, No 2 (2002), 339–368.10.1016/S0362-546X(01)00834-3Suche in Google Scholar

[31] B.G. Pachpatte, Inequalities for Differential and Integral Equations. Academic Press, San-Diego (1998).Suche in Google Scholar

[32] I. Podlubny, Fractional Differential Equations. Academic Press, San-Diego (1999).Suche in Google Scholar

[33] Y.V. Rogovchenko, On asymptotics behavior of solutions for a class of second order nonlinear differential equations. Collect. Math. 49, No 1 (1998), 113–120.Suche in Google Scholar

[34] S.P. Rogovchenko, Y.V. Rogovchenko, Asymptotics of solutions for a class of second order nonlinear differential equations. Univ. Iagellonicae Acta Math. 38, No 1 (1998), 157–164.Suche in Google Scholar

[35] J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007).10.1007/978-1-4020-6042-7Suche in Google Scholar

[36] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993).Suche in Google Scholar

[37] J. Tong, The asymptotic behavior of a class of nonlinear differential equations of second order. Proc. Amer. Math. Soc. 84, No 2 (1982), 235–236.10.1090/S0002-9939-1982-0637175-4Suche in Google Scholar

[38] W.F. Trench, On the asymptotic behavior of solutions of second order linear differential equations. Proc. Amer. Math. Soc. 14, No 1 (1963), 12–14.10.1090/S0002-9939-1963-0142844-7Suche in Google Scholar

[39] P. Waltman, On the asymptotic behavior of solutions of a nonlinear equation. Proc. Amer. Math. Soc. 15, No 6 (1964), 918–923.10.1090/S0002-9939-1964-0176170-8Suche in Google Scholar