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An operator theoretical approach to a class of fractional order differential equations. (English) Zbl 1226.47048

Summary: We propose a general method for obtaining the representation of solutions for linear fractional order differential equations based on the theory of \((a,k)\)-regularized families of operators. We illustrate the method for the case of the fractional order differential equation
\[ D^\alpha_tu'(t)+\mu D^\alpha_t u(t)=Au(t)+\frac{t^{-\alpha}}{\Gamma(1-\alpha)}\,(u'(0)+\mu u(0))+f(t),\quad t>0,\;0<\alpha\leq 1,\;\mu\geq 0, \]
where \(A\) is an unbounded closed operator defined on a Banach space \(X\) and \(f\) is an \(X\)-valued function.

MSC:

47D60 \(C\)-semigroups, regularized semigroups
34A08 Fractional ordinary differential equations
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References:

[1] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear Anal. TMA, 69, 3692-3705 (2008) · Zbl 1166.34033
[2] 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.
[3] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific Publ. Co.: World Scientific Publ. Co. Singapore · Zbl 0998.26002
[4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[5] 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)
[6] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[7] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach New York, Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987) · Zbl 0818.26003
[8] R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in: CIMS Lecture Notes, 1997, http://arxiv.org/0805.3823; R. Gorenflo, F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in: CIMS Lecture Notes, 1997, http://arxiv.org/0805.3823
[9] Baeumer, B.; Meerschaert, M. M.; Nane, E., Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc., 361, 7, 3915-3930 (2009) · Zbl 1186.60079
[10] Nane, E., Higher order PDE’s and iterated processes, Trans. Amer. Math. Soc., 360, 5, 2681-2692 (2008) · Zbl 1157.60071
[11] V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems (submitted for publication).; V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems (submitted for publication). · Zbl 1336.47046
[12] Prüss, J., (Evolutionary Integral Equations and Applications. Evolutionary Integral Equations and Applications, Monographs Math., vol. 87 (1993), Birkhäuser Verlag) · Zbl 0784.45006
[13] Gorenflo, R.; Mainardi, F., On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118, 283-299 (2000) · Zbl 0970.45005
[14] Gorenflo, R.; Mainardi, F., (Carpinteri, A.; Mainardi, F., Fractional Calculus: Integral and Differential Equations of Fractional Order. Fractional Calculus: Integral and Differential Equations of Fractional Order, Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag Vienna, New York), 223-276 · Zbl 1438.26010
[15] 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
[16] Lizama, C., Regularized solutions for abstract Volterra equations, J. Math. Anal. Appl., 243, 278-292 (2000) · Zbl 0952.45005
[17] Kostic, M., \((a, k)\)-regularized \(C\)-resolvent families: regularity and local properties, Abstr. Appl. Anal., 2009 (2009), Article ID 858242, 27 pages · Zbl 1200.47059
[18] Lizama, C., On approximation and representation of \(k\)-regularized resolvent families, Integral Equations Operator Theory, 41, 2, 223-229 (2001) · Zbl 1011.45006
[19] 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
[20] Lizama, C.; Sánchez, J., On perturbation of \(k\)-regularized resolvent families, Taiwanese J. Math., 7, 2, 217-227 (2003) · Zbl 1051.45009
[21] Lizama, C.; Prado, H., On duality and spectral properties of \((a, k)\)-regularized resolvents, Proc. Roy. Soc. Edinburgh Sect. A, 139, 3, 505-517 (2009) · Zbl 1206.47037
[22] 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
[23] Engel, K. J.; Nagel, R., (One-Parameter Semigroups for Linear Evolution Equations. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194 (2000), Springer: Springer New York, Berlin, Heidelberg) · Zbl 0952.47036
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