×

Almost automorphic mild solutions to fractional differential equations. (English) Zbl 1166.34033

Authors’ abstract: We introduce the concept of \(\alpha\)-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation
\[ D^\alpha_t u(t) = Au(t) + t^n f(t), 1 \leq \alpha \leq 2, n\in \mathbb Z \]
considered in a Banach space \(X\), where \(f: R \rightarrow X\) is almost automorphic. We also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation
\[ D^{\alpha}_t u(t) = Au(t) + f(t, u(t)), \quad 1 \leq \alpha \leq 2 \]
assuming \(f(t, x)\) is almost automorphic in \(t\) for each \(x \in X\), satisfies a global Lipschitz condition and takes values on \(X\). Finally, we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation \[ D^{\alpha}_t u(t) = Au(t) + f(t, u(t), u'(t)),\quad 1 \leq \alpha \leq 2 \]
under analogous conditions as in the previous case.

MSC:

34G20 Nonlinear differential equations in abstract spaces
26A33 Fractional derivatives and integrals
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arendt, W.; Batty, C.; Hieber, M.; Neubrander, F., (Vector-valued Laplace Transforms and Cauchy Problems. Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, vol. 96 (2001), Birkhäuser: Birkhäuser Basel) · Zbl 0978.34001
[2] Basit, B.; Pryde, A. J., Asymptotic behavior of orbits of \(C_0\)-semigroups and solutions of linear and semilinear abstract differential equations, Russ. J. Math. Phys., 13, 1, 13-30 (2006) · Zbl 1123.34043
[3] E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001; E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001 · Zbl 0989.34002
[4] Bugajewski, D.; Diagana, T., Almost automorphy of the convolution operator and applications to differential and functional differential equations, Nonlinear Stud., 13, 2, 129-140 (2006) · Zbl 1102.44007
[5] Cuesta, E., Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Sys. (Suppl.), 277-285 (2007) · Zbl 1163.45306
[6] Diagana, T., Some remarks on some second-order hyperbolic differential equations, Semigroup Forum, 68, 357-364 (2004) · Zbl 1083.34042
[7] Diagana, T.; N’Guérékata, G. M., Almost automorphic solutions to semilinear evolution equations, Funct. Differ. Equ., 13, 2, 195-206 (2006) · Zbl 1102.34044
[8] Fattorini, H. O., (Second Order Linear Differential Equations in Banach Spaces. Second Order Linear Differential Equations in Banach Spaces, North-Holland Math. Studies, vol. 108 (1985), North-Holland: North-Holland Amsterdam, New York, Oxford) · Zbl 0564.34063
[9] Goldstein, J. A.; N’Guérékata, G. M., Almost automorphic solutions of semilinear evolution equations, Proc. Amer. Math. Soc., 133, 8, 2401-2408 (2005) · Zbl 1073.34073
[10] Gorenflo, R.; Mainardi, F., Fractional calculus: Integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer Verlag: Springer Verlag Wien, New York), 223-276 · Zbl 1438.26010
[11] Gorenflo, R.; Mainardi, F., On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118, 283-299 (2000) · Zbl 0970.45005
[12] N’Guérékata, G. M., Existence and uniqueness of almost automorphic mild solutions of some semilinear abstract differential equations, Semigroup Forum, 69, 80-86 (2004) · Zbl 1077.47058
[13] N’Guérékata, G. M., Topics in Almost Automorphy (2005), Springer Verlag: Springer Verlag New York · Zbl 1073.43004
[14] N’Guerekata, G. M., Almost Automorphic and Almost Periodic Functions in Abstract Spaces (2001), Kluwer Acad., Plenum: Kluwer Acad., Plenum New York, Boston, Moscow, London · Zbl 1001.43001
[15] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 45, 5, 765-771 (2006)
[16] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific Publ. Co.: World Scientific Publ. Co. Singapore · Zbl 0998.26002
[17] Liu, J.; N’Guérékata, G. M.; van Minh, N., Almost automorphic solutions of second order evolution equations, Appl. Anal., 84, 11, 1173-1184 (2005) · Zbl 1085.34045
[18] Lizama, C., Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl., 243, 278-292 (2000) · Zbl 0952.45005
[19] Lizama, C., On approximation and representation of \(k\)-regularized resolvent families, Integral Equ. Oper. Theory, 41, 2, 223-229 (2001) · Zbl 1011.45006
[20] Lizama, C.; Prado, H., Rates of approximation and ergodic limits of regularized operator families, J. Approx. Theory, 122, 1, 42-61 (2003) · Zbl 1032.47024
[21] Lizama, C.; Sánchez, J., On perturbation of \(k\)-regularized resolvent families, Taiwanese J. Math., 7, 2, 217-227 (2003) · Zbl 1051.45009
[22] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[23] Prüss, J., (Evolutionary Integral Equations and Applications. Evolutionary Integral Equations and Applications, Monographs Math., vol. 87 (1993), Birkhäuser Verlag) · Zbl 0784.45006
[24] Sato, R.; Shaw, S. Y., Strong and uniform mean stability of cosine and sine operator functions, J. Math. Anal. Appl., 330, 1293-1306 (2007) · Zbl 1123.47035
[25] Shaw, S. Y.; Chen, J. C., Asymptotic behavior of \((a, k)\)-regularized families at zero, Taiwanese J. Math., 10, 2, 531-542 (2006) · Zbl 1106.45004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.