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Existence of mild solutions for fractional neutral evolution equations. (English) Zbl 1189.34154

Summary: By using the fractional power of operators and some fixed point theorems, we discuss a class of fractional neutral evolution equations with nonlocal conditions and obtain various criteria on the existence and uniqueness of mild solutions. In the end, we give an example to illustrate the applications of the abstract results.

MSC:

34K37 Functional-differential equations with fractional derivatives
26A33 Fractional derivatives and integrals
34K40 Neutral functional-differential equations
45J05 Integro-ordinary differential equations
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[1] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity, (Keil, F.; Mackens, W.; Voss, H.; Werther, J., Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg), 217-224
[2] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991)
[3] Glockle, W. G.; Nonnenmacher, T. F., A fractional calculus approach of self-similar protein dynamics, Biophys. J., 68, 46-53 (1995)
[4] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002
[5] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag Wien), 291-348 · Zbl 0917.73004
[6] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995)
[7] Kilbas, A. A.; Srivastava, Hari M.; Juan Trujillo, J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam) · Zbl 1092.45003
[8] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley: John Wiley New York · Zbl 0789.26002
[9] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[10] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of Fractional Dynamic Systems (2009), Cambridge Scientific Publishers · Zbl 1188.37002
[11] Baeumer, B.; Kurita, S.; Meerschaert, M. M., Inhomogeneous fractional diffusion equations, J. Frac. Appl. Anal., 8, 375-397 (2005) · Zbl 1202.86005
[12] Eidelman, S. D.; Kochubei, A. N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199, 211-255 (2004) · Zbl 1129.35427
[13] El-Borai, M. M., Semigroups and some nonlinear fractional differential equations, Applied Math. Comput., 149, 823-831 (2004) · Zbl 1046.34079
[14] El-Borai, M. M., Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals, 149, 823-831 (2004) · Zbl 1046.34079
[15] El-Sayed, A. M.A., Fractional order diffusion-wave equations, Internat. J. Theoret. Phys., 35, 311-322 (1996) · Zbl 0846.35001
[16] El-Sayed, A. M.A.; Ibrahim, A. G., Multivalued fractional differential equations, Appl. Math. Comput., 68, 15-25 (1995) · Zbl 0830.34012
[17] El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33, 181-186 (1998) · Zbl 0934.34055
[18] Jardat, O. K.; Al-Omari, A.; Momani, S., Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal., 69, 9, 3153-3159 (2008) · Zbl 1160.34300
[19] Kochubei, A. N., A Cauchy problem for evolution equations of fractional order, Differ. Equ., 25, 967-974 (1989) · Zbl 0696.34047
[20] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal., 69, 2677-2682 (2008) · Zbl 1161.34001
[21] Mainardi, F.; Paradisi, P.; Gorenflo, R., Probability distributions generated by fractional diffusion equations, (Kertesz, J.; Kondor, I., Econophysics: An Emerging Science (2000), Kluwer: Kluwer Dordrecht)
[22] Meerschaert, M. M.; Benson, D. A.; Scheffler, H.; Baeumer, B., Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E, 65, 1103-1106 (2002) · Zbl 1244.60080
[23] Muslim, M., Existence and approximation of solutions to fractional differential equations, Math. Comput. Modelling, 49, 1164-1172 (2009) · Zbl 1165.34304
[24] Schneider, W. R.; Wayes, W., Fractional diffusion and wave equation, J. Math. phys., 30, 134-144 (1989) · Zbl 0692.45004
[25] Zaslavsky, G., Fractional kinetic equation for hamiltonian chaos, chaotic advection, tracer dynamics and turbulent dispersion, Physica D, 76, 110-122 (1994) · Zbl 1194.37163
[26] Zhou, Yong; Jiao, Feng; Li, Jing, Existence and uniqueness for \(p\)-type fractional neutral differential equations, Nonlinear Anal., 71, 2724-2733 (2009) · Zbl 1175.34082
[27] Zhou, Yong; Jiao, Feng; Li, Jing, Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear Anal., 71, 3249-3256 (2009) · Zbl 1177.34084
[28] Byszewski, L., Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162, 494-505 (1991) · Zbl 0748.34040
[29] Byszewski, L.; Lakshmikantham, V., Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40, 11-19 (1991) · Zbl 0694.34001
[30] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer New York · Zbl 0516.47023
[31] Fu, X.; Ezzinbi, K., Existence of solutions for neutral differential evolution equations with nonlocal conditions, Nonlinear Anal., 54, 215-227 (2003) · Zbl 1034.34096
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