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Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. (English) Zbl 1248.34004

Summary: We study the existence and uniqueness of a weighted pseudo-almost periodic (mild) solution to the semilinear fractional equation \[ \partial ^{\alpha}_t u = Au + \partial ^{\alpha - 1}_t f(\cdot, u), 1< \alpha <2, \] where \(A\) is a linear operator of sectorial negative type. This article also deals with the existence of these types of solutions to abstract partial evolution equations.

MSC:

34A08 Fractional ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
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[1] C. Zhang, Pseudo almost periodic functions and their applications, Thesis, The University of Western Ontario, 1992.
[2] Zhang, C., Pseudo almost periodic solutions of some differential equations, J. math. anal. appl., 151, 62-76, (1994) · Zbl 0796.34029
[3] Zhang, C., Integration of vector-valued pseudo almost periodic functions, Proc. amer. math. soc., 121, 167-174, (1994) · Zbl 0818.42003
[4] Zhang, C., Pseudo almost periodic solutions of some differential equations II, J. math. anal. appl., 192, 543-561, (1995) · Zbl 0826.34040
[5] Zhang, C., Almost periodic type and ergodicity, (2003), Kluwer Academic Publishers and Science Press · Zbl 1068.34001
[6] Ait Dads, E.; Arino, O., Exponential dichotomy and existence of pseudo almost periodic solutions of some differential equations, Nonlinear anal., 27, 4, 361-386, (1996) · Zbl 0855.34055
[7] Ait Dads, E.; Ezzinbi, K.; Arino, O., Pseudo-almost periodic solutions for some differential equations in a Banach space, Nonlinear anal., 28, 7, 1141-1155, (1997) · Zbl 0874.34041
[8] Amir, B.; Maniar, L., Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain, Ann. math. blaise Pascal, 6, 1, 1-11, (1999) · Zbl 0941.34059
[9] Cuevas, C.; Pinto, M., Existence and uniqueness of pseudo almost periodic solutions of semilinar Cauchy problems with non dense domain, Nonlinear anal., 45, 73-83, (2001) · Zbl 0985.34052
[10] Cuevas, C.; Hernández, H., Pseudo almost periodic solutions for abstract partial functional differential equations, Appl. math. lett., 22, 534-538, (2009) · Zbl 1170.35551
[11] Diagana, T., Pseudo almost periodic solutions to some differential equations, Nonlinear anal., 60, 7, 1277-1286, (2005) · Zbl 1061.34040
[12] Diagana, T.; Mahop, C.M.; N’Guérékata, G.M., Pseudo almost periodic solutions to some semilinear differential equations, Math. comput. modelling, 43, 1-2, 89-96, (2006) · Zbl 1096.34038
[13] Diagana, T.; Mahop, C.M.; N’Guérékata, G.M.; Toni, B., Existence and uniqueness of pseudo almost periodic solutions to some classes of semilinear differential equations and applications, Nonlinear anal., 64, 11, 2442-2453, (2006) · Zbl 1102.34043
[14] Diagana, T.; Mahop, C.M., Pseudo almost periodic solutions to a neutral delay integral equation, Cubo, 9, 1, 47-55, (2007) · Zbl 1122.45002
[15] Diagana, T., Existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equation, Ejqtde, 3, 1-12, (2007) · Zbl 1108.35122
[16] Diagana, T.; Hernández, E., Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional-differential equations and applications, J. math. anal. appl., 327, 2, 776-791, (2007) · Zbl 1123.34060
[17] Diagana, T., Pseudo almost periodic functions in Banach spaces, (2007), Nova Science Publishers, Inc. New-York · Zbl 1234.43002
[18] Zhao, Z.-H.; Chang, Y.-K.; Nieto, J.J., Almost automorphic and pseudo-almost automorphic mild solutions to an abstract differential equation in Banach spaces, Nonlinear anal., 72, 1886-1894, (2010) · Zbl 1189.34116
[19] Agarwal, R.P.; Diagana, T.; Hernández, E., Weighted pseudo almost periodic solutions to some partial neutral functional differential equations, J. nonlinear convex anal., 8, 3, 397-415, (2007) · Zbl 1155.35104
[20] Diagana, T., Weighted pseudo almost periodic solutions to some differential equations, Nonlinear anal., 68, 8, 2250-2260, (2008) · Zbl 1131.42006
[21] Diagana, T., Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations, J. math. anal. appl., 350, 18-28, (2009) · Zbl 1167.34023
[22] Diagana, T., Weighted pseudo almost periodic functions and applications, C. R. acad. sci. Paris ser. I, 343, 10, 643-646, (2006) · Zbl 1112.43005
[23] D.A. Benson, The fractional advection – dispersion equation, Ph.D. Thesis, University of Nevada, Reno, NV, 1998.
[24] Schumer, R.; Benson, D.A., Eulerian derivative of the fractional advection – dispersion equation, J. contaminant, 48, 69-88, (2001)
[25] Henry, B.I.; Wearne, S.L., Existence of Turing instabilities in a two-species fractional reaction – diffusion system, SIAM J. appl. math., 62, 870-887, (2002) · Zbl 1103.35047
[26] Ahn, V.V.; McVinisch, R., Fractional differential equations driven by levy noise, J. appl. math. stoch. anal., 16, 2, 97-119, (2003) · Zbl 1042.60034
[27] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276
[28] Hilfer, H., Applications of fractional calculus in physics, (2000), World Scientific Publ. Co. Singapore · Zbl 0998.26002
[29] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., ()
[30] Kiryakova, V., (), (John Wiley, New York, NY, USA)
[31] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002
[32] ()
[33] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[34] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[35] Agarwal, R.P.; Belmekki, M.; Benchohra, M., A survey on semilinear differential equations and inclusions involving riemann – liouville fractional derivative, Adv. difference equ., (2009), Article ID 981728, 47 pages · Zbl 1182.34103
[36] R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. doi:10.1007/sl10440-008-9356-6. · Zbl 1198.26004
[37] R.P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations, Georgian Math. J. (in press). · Zbl 1179.26011
[38] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal., 72, 6, 2859-2862, (2010) · Zbl 1188.34005
[39] Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J. math. anal. appl., 338, 1340-1350, (2008) · Zbl 1209.34096
[40] Diethelm, K.; Ford, N.J., Analysis of fractional equations, J. math. anal. appl., 265, 2, 229-248, (2002) · Zbl 1014.34003
[41] Diethelm, K.; Freed, A.D., On the solution of nonlinear fractional order equations used in the modeling of viscoplasticity, (), 217-224
[42] El-Borai, M.M., Some probability densities and fundamental solutions of fractional evolutions equations, Chaos solitons fractals, 14, 433-440, (2002) · Zbl 1005.34051
[43] El-Borai, M.M., Semigroup and some nonlinear fractional differential equations, Appl. math. comput., 149, 823-831, (2004) · Zbl 1046.34079
[44] El-Borai, M.M., The fundamental solutions for fractional evolution equations of parabolic type, J. appl. math. stoch. anal., 3, 197-211, (2004) · Zbl 1081.34053
[45] El-Sayed, A.M.A., Fractional order evolution equations, J. fract. calc., 7, 89-100, (1995) · Zbl 0839.34069
[46] El-Sayed, A.M.A., Fractional-order diffusion-wave equation, Internat. J. theoret. phys., 35, 2, 311-322, (1996) · Zbl 0846.35001
[47] El-Sayed, A.M.A., Nonlinear functional-differential equations of arbitrary orders, Nonlinear anal., 33, 2, 181-186, (1998) · Zbl 0934.34055
[48] Chen, J.; Liu, F.; Turner, I.; Anh, V., The fundamental and numerical solutions of the riez space-fractional reaction-dispersion equation, Anziam, 50, 45-57, (2008) · Zbl 1179.35029
[49] Gaul, L.; Klein, P.; Kempfle, S., Damping discription involving fractional operators, Mech. syst. signal process., 5, 2, 81-88, (1991)
[50] Hu, T.; Wang, Y., Numerical detection of the lowest “efficient dimensions” for chaotic fractional differential system, Open math. J., 1, 11-18, (2008) · Zbl 1185.34006
[51] Mophou, G.M.; N’Guérékata, G.M., Mild solutions for semilinear fractional differential equations, Electron. J. differential equations, 2009, 21, 1-9, (2009) · Zbl 1179.34002
[52] Mophou, G.M.; N’Guérékata, G.M., Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup forum, 79, 315-322, (2009) · Zbl 1180.34006
[53] G.M. Mophou, G.M. N’Guérékata, On integral solutions of some nonlocal fractional differential equations with nondense domain, Nonlinear Anal. doi:101016/j.na.2009.03.029. · Zbl 1178.34005
[54] Ahmad, B.; Nieto, J.J., Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. value probl. 2009, (2009), Article ID 708576, 11 pages · Zbl 1167.45003
[55] Belmekki, M.; Nieto, J.J.; Rodriguez-Lopez, R., Existence of periodic solution for a nonlinear fractional differential equation, Bound. value probl., 2009, (2009), Article ID 324561, 18 pages · Zbl 1181.34006
[56] N’Guérékata, G.M., Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear anal., 70, 5, 1873-1876, (2009) · Zbl 1166.34320
[57] Lakshmikantham, V., Theory of fractional differential equations, Nonlinear anal., 60, 10, (2008), 3337-334 · Zbl 1162.34344
[58] Lakshmikantham, V.; Vatsala, A., Basic theory of fractional differential equations, Nonlinear anal., 69, 8, 2677-2682, (2008) · Zbl 1161.34001
[59] V. Lahshmikantham, A. Vatsala, Theory of fractional differential inequalities and applications, Commun. Appl. Anal..
[60] Lahshmikantham, V.; Devi, J.V., Theory of fractional differential equations in Banach spaces, Eur. J. pure appl. math., 1, 38-45, (2008) · Zbl 1146.34042
[61] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanic, (), 291-348
[62] Cuesta, E.; Lubich, Ch.; Palencia, C., Convolution quadrature time discretization of fractional diffusion-wave equations, Math. comput., 75, 673-696, (2006) · Zbl 1090.65147
[63] Cuesta, E.; Palencia, C., A numerical method for an integro-differential equation with memory in Banach spaces: qualitative properties, SIAM J. numer. anal., 41, 1232-1241, (2003) · Zbl 1054.65131
[64] Eidelman, S.D.; Kochubei, A.N., Cauchy problem for fractional diffusion equations, J. differential equations, 199, 211-255, (2004) · Zbl 1068.35037
[65] R.P. Agarwal, B. de Andrade, C. Cuevas, On type of periodicity and ergodicity to a class of fractional order differential equations, Adv. Difference Equ. (in press). · Zbl 1194.34007
[66] Cuevas, C.; de Souza, J.C., S-asymptotically \(\omega\)-periodic solutions of semilinear fractional integro-differential equations, Appl. math. lett., 22, 865-870, (2009) · Zbl 1176.47035
[67] Cuevas, C.; de Souza, J.C., Existence of S-asymptotically \(\omega\)-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear anal., (2009)
[68] Banás, J.; O’Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. math. anal. appl., 345, 573-582, (2008) · Zbl 1147.45003
[69] Banás, J.; Zajac, T., Solvability of a functional integral equation of fractional order in the class of functions having limits at infinite, Nonlinear anal., (2009)
[70] Rzepka, B., On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order, Topol. methods nonlinear anal., 32, (2008), 573-102 · Zbl 1173.45003
[71] Haase, M., ()
[72] Cuesta, E., Asymptotically behavior of the solutions of fractional integro-differential equations and some discretizations, Discrete contin. dyn. syst. (suppl.), 277-285, (2007) · Zbl 1163.45306
[73] Prüss, J., ()
[74] Gripenberg, G.; Londen, S.O.; Staffans, O., ()
[75] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., ()
[76] Fattorini, O., ()
[77] Henríquez, H.; Lizama, C., Compact almost automorphic solutions to integral equations with infinite delay, Nonlinear anal., 71, 12, 6029-6037, (2009) · Zbl 1179.43004
[78] Lunardi, A., ()
[79] Lions, J.L.; Peetre, J., Sur une classe d’espaces d’interpolation, Publ. math. inst. hautes études sci., 19, 5-68, (1964) · Zbl 0148.11403
[80] Da Prato, G.; Grisvard, P., Equations d’évolution abstraites nonlinéaires de type parabolique, Ann. mat. pura appl., 4, 120, 329-396, (1979) · Zbl 0471.35036
[81] Liang, J.; Zhang, J.; Xiao, T.J., Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. math. anal. appl., 340, 1493-1499, (2008) · Zbl 1134.43001
[82] Li, H.; Huang, F.; Li, J., Composition of pseudo almost periodic functions and semilinear differential equations, J. math. anal. appl., 255, 436-446, (2001) · Zbl 1047.47030
[83] Granas, A.; Dugundji, J., Fixed point theory, (2003), Springer-Verlag New York · Zbl 1025.47002
[84] Fink, A.M., ()
[85] Martin, R.H., Nonlinear operators and differential equations in Banach spaces, (1987), Robert E. Krieger Publ. Co. Florida
[86] Engel, K.J.; Nagel, R., ()
[87] Simon, J., Compact sets in the space \(L^p(0, T; \mathcal{B})\), Ann. mat. pure appl., CXLVI, 65-96, (1987) · Zbl 0629.46031
[88] Diagana, T., Existence of solutions to some classes of partial fractional differential equations, Nonlinear anal., (2009) · Zbl 1196.34008
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