Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions

References

[1] S. Abbas, M. Benchohra, A. Petrusel, Ulam stability for Hilfer 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/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml.Search in Google Scholar

[2] B. Ahmad, S. Ntouyas, Nonlocal fractional boundary value problems with slit-strips boundary conditions, Fract. Calc. Appl. Anal. 18, No 1 (2015), 261–280; 10.1515/fca-2015-0017; https://www.degruyter.com/view/j/fca.2015.18.issue-1/issue-files/fca.2015.18.issue-1.xml.Search in Google Scholar

[3] J. Banaś, K. Goebel, Measure of Noncompactness in Banach Space. Marcel Dekker Inc., New York, 1980.Search in Google Scholar

[4] M. Belmekki, M. Benchohra, Existence results for fractional order semilinear functional differential equations with nondense domain. Nonlinear Anal. 72, No 2 (2010), 925–932.10.1016/j.na.2009.07.034Search in Google Scholar

[5] D. Bothe, Multivalued perturbation of m-accretive differential inclusions. Israel J. Math. 108 (1998), 109–138.10.1007/BF02783044Search in Google Scholar

[6] L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40, No 1 (1991), 11–19.10.1080/00036819008839989Search in Google Scholar

[7] K. Diethelm, A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity. In: Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (Eds. F. Keil, W. Mackens, H. Voss, J.Werther), Springer-Verlag, Heidelberg (1999), 217–224.10.1007/978-3-642-60185-9_24Search in Google Scholar

[8] K.M. Furati, M.D. Kassim, N.e.-Tatar, Existence and uniqueness for a problem involving Hilfer factional derivative. Comput. Math. Appl. 64, No 6 (2012), 1612–1626.Search in Google Scholar

[9] H.B. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257 (2015), 344–354.10.1016/j.amc.2014.10.083Search in Google Scholar

[10] D.J. Guo, V. Lakshmikantham, X.Z. Liu, Nonlinear Integral Equations in Abstract Spaces. Kluwer Academic Publishers Group, Dordrecht, 1996.10.1007/978-1-4613-1281-9Search in Google Scholar

[11] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.10.1142/3779Search in Google Scholar

[12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science B.V., Amsterdam, 2006.Search in Google Scholar

[13] V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. 69, No 8 (2008), 2677–2682.10.1016/j.na.2007.08.042Search in Google Scholar

[14] K.X. Li, J.G. Peng, J.X. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives. J. Funct. Anal. 263, No 2 (2012), 476–510.10.1016/j.jfa.2012.04.011Search in Google Scholar

[15] Y.L. Liu, J.Y. Lv, Existence results for Riemann–Liouville fractional neutral evolution equations. Adv. Differ. Equ. 83 (2014), 16 pp.10.1186/1687-1847-2014-83Search in Google Scholar

[16] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Fractals and Fractional Calculus in Continuum Mechanics (Eds. A. Carpinteri and F. Mainardi), Springer-Verlag, Wien, 1997, 291–348.10.1007/978-3-7091-2664-6_7Search in Google Scholar

[17] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York, 1993.Search in Google Scholar

[18] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 4, No 5 (1980), 985–999.10.1016/0362-546X(80)90010-3Search in Google Scholar

[19] A. Ouahab, Fractional semilinear differential inclusions. Comput. Math. Appl. 64, No 10 (2012), 3235–3252.10.1016/j.camwa.2012.03.039Search in Google Scholar

[20] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.Search in Google Scholar

[21] M.H.M. Rashid, Y. El-Qaderi, Semilinear fractional integro-differential equations with compact semigroup. Nonlinear Anal. 71, No 12 (2009), 6276–6282.10.1016/j.na.2009.06.035Search in Google Scholar

[22] K. Rykaczewski, Approximate controllability of differential inclusions in Hilbert spaces. Nonlinear Anal. 75, No 5 (2012), 2701–2712.10.1016/j.na.2011.10.049Search in Google Scholar

[23] R. Sakthivel, P. Revathi, Y. Ren, Existence of solutions for nonlinear fractional stochastic differential equations. Nonlinear Anal. 81 (2013), 70–86.10.1016/j.na.2012.10.009Search in Google Scholar

[24] T. Sandev, R. Metzler, Z. Tomovski, Fractional diffusion equation with a generalized Riemann-Liouville time fractional derivative. J. Phys. A: Math. Theor. 44, No 25 (2011), ID 255203.10.1088/1751-8113/44/25/255203Search in Google Scholar

[25] W.R. Schneider, W. Wyss, Fractional diffusion and wave equation. J. Math. Phys. 30, No 1 (1989), 134–144.10.1063/1.528578Search in Google Scholar

[26] X.B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations. Nonlinear Anal. 74, No 5 (2011), 2003–2011.10.1016/j.na.2010.11.007Search in Google Scholar

[27] R.N. Wang, D.H. Chen, T.J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators. J. Differential Equations252, No 1 (2012), 202–235.10.1016/j.jde.2011.08.048Search in Google Scholar

[28] J.R. Wang, Y.R. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative. Appl. Math. Comput. 266 (2015), 850–859.10.1016/j.amc.2015.05.144Search in Google Scholar

[29] J.R. Wang, Y. Zhou, Complete controllability of fractional evolution systems. Commun. Nonlinear Sci. Numer. Simul. 17, No 11 (2012), 4346–4355.10.1016/j.cnsns.2012.02.029Search in Google Scholar

[30] J.R. Wang, Y. Zhou, W. Wei, A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, No 10 (2011), 4049–4059.10.1016/j.cnsns.2011.02.003Search in Google Scholar

[31] M. Yang, Q.R. Wang, Approximate controllability of Riemann-Liouville fractional differential inclusions. Appl. Math. Comput. 274 (2016), 267–281.10.1016/j.amc.2015.11.017Search in Google Scholar

[32] M. Yang, Q.R. Wang, Approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. Math. Methods Appl. Sci. 40, No 4 (2017), 1126–1138.10.1002/mma.4040Search in Google Scholar

[33] Y. Zhou, Basic Theory of Fractional Differential Equations. World Scientific, Singapore, 2014.10.1142/9069Search in Google Scholar

[34] Y. Zhou, F. Jiao, Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. Real World Appl. 11, No 5 (2010), 4465–4475.10.1016/j.nonrwa.2010.05.029Search in Google Scholar

[35] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, No 3 (2010), 1063–1077.10.1016/j.camwa.2009.06.026Search in Google Scholar

[36] Y. Zhou, L. Peng, On the time-fractional Navier-Stokes equations, Comput. Math. Appl. 73, No 6 (2017), 874–891.10.1016/j.camwa.2016.03.026Search in Google Scholar

[37] Y. Zhou, L. Zhang, X.H. Shen, Existence of mild solutions for fractional evolution equations. J. Integral Equations Appl. 25, No 3 (2013), 557–585.10.1216/JIE-2013-25-4-557Search in Google Scholar