Using a generalization of the semigroup theory of linear operators, we prove existence and uniqueness of mild solutions for the semilinear fractional order differential equation $${D}^{\alpha+1}_t u(t) + \mu {D}_t^{\beta} u(t) - Au(t) = f(t,u(t)), t\in (0,\infty), \alpha \in (0,\infty), \alpha \leq \beta \leq 1, \, \mu \geq 0, $$ with the property that the solution can be written as $u=f+h$ where $f$ belongs to the space of periodic (resp. almost periodic, compact almost automorphic, almost automorphic) functions and $h$ belongs to the space $ P_0(\mathbb{R}_{+},X):= \{ \phi\in BC(\mathbb{R}_{+},X) \, :\,\, \lim_{T \to \infty}\frac{1}{T} \int_{0}^{T}||\phi(s)||ds=0 \}$. Moreover, this decomposition is unique.
References
W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser, Basel, 2001. MR1886588 W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96. Birkhäuser, Basel, 2001. MR1886588
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. MR1868564 0989.34002 E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. MR1868564 0989.34002
S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. USA 52 (1964), 907–910. MR168997 0134.30102 10.1073/pnas.52.4.907 S. Bochner, Continuous mappings of almost automorphic and almost periodic functions, Proc. Nat. Acad. Sci. USA 52 (1964), 907–910. MR168997 0134.30102 10.1073/pnas.52.4.907
S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. USA 47 (1961), 582–585. MR126094 10.1073/pnas.47.4.582 S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. USA 47 (1961), 582–585. MR126094 10.1073/pnas.47.4.582
S. Bochner, A new approach in almost-periodicity, Proc. Nat. Acad. Sci. USA 48 (1962), 2039–2043. MR145283 0112.31401 10.1073/pnas.48.12.2039 S. Bochner, A new approach in almost-periodicity, Proc. Nat. Acad. Sci. USA 48 (1962), 2039–2043. MR145283 0112.31401 10.1073/pnas.48.12.2039
S. Bochner and J. von Neumann, On compact solutions of operational-differential equations I, Ann. Math. 36 (1935), 255–290. MR1503222 10.2307/1968678 S. Bochner and J. von Neumann, On compact solutions of operational-differential equations I, Ann. Math. 36 (1935), 255–290. MR1503222 10.2307/1968678
B. De Andrade, C. Cuevas and E. Henriquez, Almost automorphic solutions of hyperbolic evolution equations, Banach J. Math. Anal. 6 (2012), no. 1, 90–100. MR2862545 1253.34054 B. De Andrade, C. Cuevas and E. Henriquez, Almost automorphic solutions of hyperbolic evolution equations, Banach J. Math. Anal. 6 (2012), no. 1, 90–100. MR2862545 1253.34054
R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, CIMS Lecture Notes, http://arxiv.org/0805.3823. R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, CIMS Lecture Notes, http://arxiv.org/0805.3823.
N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta 45 (2006), no. 5, 765–771. N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta 45 (2006), no. 5, 765–771.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. MR2218073 1092.45003 A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. MR2218073 1092.45003
J. Liang, J. Zhang and T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 1493–1499. MR2390946 1134.43001 10.1016/j.jmaa.2007.09.065 J. Liang, J. Zhang and T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2008), no. 2, 1493–1499. MR2390946 1134.43001 10.1016/j.jmaa.2007.09.065
J.H. Liu and X.Q. Song, Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. Funct. Anal. 258 (2010), 196–207. MR2557959 1194.47047 10.1016/j.jfa.2009.06.007 J.H. Liu and X.Q. Song, Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations, J. Funct. Anal. 258 (2010), 196–207. MR2557959 1194.47047 10.1016/j.jfa.2009.06.007
C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl. 243 (2000), 278–292. MR1741524 0952.45005 10.1006/jmaa.1999.6668C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl. 243 (2000), 278–292. MR1741524 0952.45005 10.1006/jmaa.1999.6668
C. Lizama, An operator theoretical approach to a class of fractional order differential equations, Appl. Math. Lett. 24 (2011), no. 2, 184–190. MR2735138 10.1016/j.aml.2010.08.042 1226.47048 C. Lizama, An operator theoretical approach to a class of fractional order differential equations, Appl. Math. Lett. 24 (2011), no. 2, 184–190. MR2735138 10.1016/j.aml.2010.08.042 1226.47048
C. Lizama and G.M. N'Guérékata, Bounded mild solutions for semilinear integro-differential equations in Banach spaces, Integral Equations and Operator Theory 68 (2010), 207–227. MR2721083 1209.45007 10.1007/s00020-010-1799-2 C. Lizama and G.M. N'Guérékata, Bounded mild solutions for semilinear integro-differential equations in Banach spaces, Integral Equations and Operator Theory 68 (2010), 207–227. MR2721083 1209.45007 10.1007/s00020-010-1799-2
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993) [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)]. MR1347689 S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York (1993) [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)]. MR1347689
M. Stojanović and R. Gorenflo, Nonlinear two-term time fractional diffusion-wave problem, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3512–3523. MR2683809 10.1016/j.nonrwa.2009.12.012 1203.35289 M. Stojanović and R. Gorenflo, Nonlinear two-term time fractional diffusion-wave problem, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 3512–3523. MR2683809 10.1016/j.nonrwa.2009.12.012 1203.35289
T.J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), no. 3, 518–524. MR2395199 1154.46023 10.1007/s00233-007-9011-y T.J. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum 76 (2008), no. 3, 518–524. MR2395199 1154.46023 10.1007/s00233-007-9011-y
C.Y. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Kluwer Academic Publishers, New York, 2003. MR2000981 1068.34001 C.Y. Zhang, Almost Periodic Type Functions and Ergodicity, Science Press, Kluwer Academic Publishers, New York, 2003. MR2000981 1068.34001
C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 151 (1994), 62–76. MR1257954 10.1006/jmaa.1994.1005 C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. Appl. 151 (1994), 62–76. MR1257954 10.1006/jmaa.1994.1005
C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192 (1995), 543–561. MR1332227 0826.34040 10.1006/jmaa.1995.1189 C.Y. Zhang, Pseudo almost periodic solutions of some differential equations II, J. Math. Anal. Appl. 192 (1995), 543–561. MR1332227 0826.34040 10.1006/jmaa.1995.1189
