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Existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. (English) Zbl 1260.34016
Summary: This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of the abstract results.

MSC:
34A08Fractional differential equations
34G20Nonlinear ODE in abstract spaces
WorldCat.org
Full Text: DOI
References:
[1] L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81-88, 1991.
[2] W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46-53, 1995.
[3] R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” Journal of Chemical Physics, vol. 103, no. 16, pp. 7180-7186, 1995.
[4] F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, pp. 291-348, Springer, Vienna, Austria, 1997. · Zbl 0917.73004
[5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0998.26002 · doi:10.1142/9789812817747
[6] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[7] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433-440, 2002. · Zbl 1005.34051 · doi:10.1016/S0960-0779(01)00208-9
[8] M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 823-831, 2004. · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[9] V. Lakshmikantham, S. Leela J, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, 2009. · Zbl 1188.37002
[10] R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 6, pp. 2859-2862, 2010. · Zbl 1188.34005 · doi:10.1016/j.na.2009.11.029
[11] Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1063-1077, 2010. · Zbl 1189.34154 · doi:10.1016/j.camwa.2009.06.026
[12] J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 262-272, 2011. · Zbl 1214.34010 · doi:10.1016/j.nonrwa.2010.06.013
[13] R.-N. Wang, T.-J. Xiao, and J. Liang, “A note on the fractional Cauchy problems with nonlocal initial conditions,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1435-1442, 2011. · Zbl 1251.45008 · doi:10.1016/j.aml.2011.03.026
[14] H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075-1081, 2007. · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
[15] M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 823-831, 2004. · Zbl 1046.34079 · doi:10.1016/S0096-3003(03)00188-7
[16] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[17] A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1442-1450, 2011. · Zbl 1228.45013 · doi:10.1016/j.camwa.2011.03.075
[18] A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 232-236, 2009. · Zbl 1175.26004 · doi:10.1016/j.jmaa.2009.04.012
[19] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[20] A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 66-79, 2002. · Zbl 1005.26002 · doi:10.1016/S0022-247X(02)00048-3
[21] M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,” Fixed Point Theory and Applications, vol. 2011, Article ID 867932, 2011. · Zbl 1219.34102 · doi:10.1155/2011/867932 · eudml:223289
[22] M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,” Abstract and Applied Analysis, vol. 2011, Article ID 161246, 25 pages, 2011. · Zbl 1217.34124 · doi:10.1155/2011/161246 · eudml:226612
[23] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012. · Zbl 1248.26011 · doi:10.1142/9789814355216
[24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[25] H. Amann, “Periodic solutions of semilinear parabolic equations,” in Nonlinear Analysis, pp. 1-29, Academic Press, New York, NY, USA, 1978. · Zbl 0464.35050
[26] P. E. Sobolevskii, “Equations of parabolic type in a Banach space,” American Mathematical Society Translations Series 2, vol. 49, pp. 1-62, 1966.
[27] H. Liu and J.-C. Chang, “Existence for a class of partial differential equations with nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3076-3083, 2009. · Zbl 1185.34112 · doi:10.1016/j.na.2009.02.035
[28] C. C. Travis and G. F. Webb, “Existence, stability, and compactness in the \alpha -norm for partial functional differential equations,” Transactions of the American Mathematical Society, vol. 240, pp. 129-143, 1978. · Zbl 0414.34080 · doi:10.2307/1998809