Mild solutions for abstract fractional differential equations

References

• Baeumer , B , Meerschaert , MM and Nane , E . 2009 . Brownian subordinators and fractional Cauchy problems . Trans. Am. Math. Soc. , 361 : 3915 – 3930 .
   Web of Science ®Google Scholar
• Clment , Ph , Gripenberg , G and Londen , S-O . 2000 . Schauder estimates for equations with fractional derivatives . Trans. Am. Math. Soc. , 352 : 2239 – 2260 .
   Web of Science ®Google Scholar
• Eidelman , SD and Kochubei , AN . 2004 . Cauchy problem for fractional diffusion equations . J. Differ. Eqns , 199 ( 2 ) : 211 – 255 .
   Web of Science ®Google Scholar
• Giannantoni , C . 2003 . The problem of the initial conditions and their physical meaning in linear differential equations of fractional order . Appl. Math Comput. , 141 ( 10 ) : 87 – 102 .
   Web of Science ®Google Scholar
• Hilfer , R . 2000 . Applications of Factional Calculus in Physics , River Edge , , NJ : World Scientific .
   Google Scholar
• Heymans , N and Podlubny , I . 2006 . Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives . Rheol. Acta , 45 : 765 – 771 .
   Web of Science ®Google Scholar
• Kilbas , AA , Srivastsava , HM and Trujillo , JJ . Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Vol. 204, Elsevier Science, Amsterdam, 2006
   Google Scholar
• Li , M , Chen , C and Li , FB . 2010 . On fractional powers of generators of fractional resolvent families . J. Funct. Anal. , 259 ( 10 ) : 2702 – 2726 .
   Web of Science ®Google Scholar
• Mainardi , F . 1997 . “ Fractional calculus: some basic problems in continuum mechanics ” . In Fractals and Fractional Calculus in Mechanics , Edited by: Carpinteri , A and Mainardi , F . 291 – 348 . New York : Springer .
   Google Scholar
• Podlubny , I . 1999 . Fractional Differential Equations , Vol. 198 , San Diego, CA , , USA : Academic Press .
   Google Scholar
• Samko , S , Kilbas , AA and Marichev , OI . 1993 . Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach Science Publishers, Yverdo, 1993
   Google Scholar
• Hernández , E , O' Regan , D and Balachandran , K . 2010 . On recent developments in the theory of abstract differential equations with fractional derivatives . Nonlinear Anal. , 73 ( 10 ) : 3462 – 3471 .
   Web of Science ®Google Scholar
• Lizama , C . 2000 . Regularized solutions for abstract Volterra equations . J. Math. Anal. Appl. , 243 : 278 – 292 .
   Web of Science ®Google Scholar
• Prüss , J . 1993 . Evolutionary Integral Equations and Applications, Monographs Mathematics , Vol. 87 , Basel : Birkhäuser Verlag .
   Google Scholar
• Lizama , C and Sánchez , J . 2003 . On perturbation of k-regularized resolvent families . Taiwanese J. Math. , 7 : 217 – 227 .
   Web of Science ®Google Scholar
• Lizama , C and Prado , H . 2009 . On duality and spectral properties of (a, k)-regularized resolvents . Proc. R. Soc. Edinb.: Sect. A , 139 ( 3 ) : 505 – 517 .
   Google Scholar
• Kellermann , H and Hieber , M . 1989 . Integrated semigroups . J. Funct. Anal. , 84 : 160 – 180 .
   Web of Science ®Google Scholar
• Arendt , W and Kellermann , H . 1987 . Integrated Solutions of Volterra Integrodifferential Equations and Applications, Pitman Research Notes in Mathematics Series , Vol. 190 , 21 – 51 . New York : Longman, Harlowr .
   Google Scholar
• Lizama , C . 2001 . On approximation and representation of k-regularized resolvent families . Integr. Eqns Oper. Theory , 41 : 223 – 229 .
   Web of Science ®Google Scholar
• Lizama , C and Prado , H . 2003 . Rates of approximations and ergodic limits of regularized operator families . J. Approx. Theory , 122 : 42 – 61 .
   Web of Science ®Google Scholar
• Li , YC and Shaw , SY . 2004 . Mean ergodicity and mean stability of regularized solution families . Mediterr. J. Math. , 1 : 175 – 193 .
   Google Scholar
• Shaw , SY and Chen , JC . 2006 . Asymptotic behavior of (a,k)-regularized resolvent families at zero . Taiwanese J. Math. , 10 ( 2 ) : 531 – 542 .
   Web of Science ®Google Scholar
• Engel , KJ and Nagel , R . 2000 . One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics , Vol. 194 , New York : Springer-Verlag .
   Google Scholar
• Nagy , B . 1974 . On cosine functions on Banach spaces . Acta Sci. Math. Szeged , 36 : 281 – 290 .
   Google Scholar
• Li , M and Zheng , Q . 2004 . On spectral inclusions and approximations of α-times resolvent families . Semigroup Forum , 69 : 356 – 368 .
   Web of Science ®Google Scholar
• Kostic , M . 2009 . (a, k)-regularized C-resolvent families: Regularity and local properties . Abstract Appl. Anal. , 2009 Article ID 858242, 27pp., doi:10.1155/2009/858242
   Web of Science ®Google Scholar
• Bazhlekova , E . Fractional Evolution Equations in Banach Spaces, Thesis, Technische Universiteit Eindhoven, 2001
   Google Scholar
• Araya , D and Lizama , C . 2008 . Almost automorphic mild solutions to fractional differential equations . Nonlinear Anal. , 69 ( 11 ) : 3692 – 3705 .
   Web of Science ®Google Scholar
• Arendt , W , Batty , CJK , Hieber , M and Neubrander , F . 2001 . Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics , Vol. 96 , Basel : Birkhäuser Verlag .
   Google Scholar
• Diagana , T , Mophou , GM and N'Guérékata , GM . 2010 . On the existence of mild solutions to some semilinear fractional integro-differential equations . Electron. J. Qual. Theory Differ. Eqns. , 58 : 1 – 17 .
   Google Scholar
• Zhou , Y and Jiao , F . 2010 . Existence of mild solutions for fractional neutral evolution equations . Comput. Math. Appl. , 59 ( 3 ) : 1063 – 1077 .
   Web of Science ®Google Scholar
• Karczewska , A and Lizama , C . 2009 . Solutions to stochastic fractional relaxation equations . Phys. Scr. , T136 014030, 7pp
   Web of Science ®Google Scholar
• Bagley , RL and Torvik , PJ . 1984 . On the appearance of the fractional derivative in the behavior of real materials . J. Appl. Mech. , 51 : 294 – 298 .
   Web of Science ®Google Scholar
• Saha Ray , S and Bera , RK . 2005 . Analytical solution of the Bagley Torvik equation by Adomian decomposition method . Appl. Math. Comput. , 168 ( 1 ) : 398 – 410 .
   Web of Science ®Google Scholar
• Karczewska , A and Lizama , C . 2010 . Solutions to stochastic fractional oscillation equations . Appl. Math. Lett. , 23 ( 11 ) : 1361 – 1366 .
   Web of Science ®Google Scholar
• Cuesta , E . 2007 . Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations . Discrete Contin. Dyn. Syst. (Suppl.) , : 277 – 285 .
   Google Scholar
• Cuevas , C and Lizama , C . 2008 . Almost automorphic solutions to a class of semilinear fractional differential equations . Appl. Math. Lett. , 21 ( 12 ) : 1315 – 1319 .
   Web of Science ®Google Scholar
• Lizama , C and N'Guérékata , GM . 2010 . Bounded mild solutions for semilinear integro differential equations in Banach spaces . Integr. Eqns Oper. Theory , 68 : 207 – 227 .
   Web of Science ®Google Scholar
• Henríquez , HR , Pierri , M and Táboas , P . 2008 . On S-asymptotically ω-periodic functions on Banach spaces and applications . J. Math. Anal. Appl. , 343 : 1119 – 1130 .
   Web of Science ®Google Scholar
• Henríquez , HR , Pierri , M and Táboas , P . 2008 . Existence of S-asymptotically ω-periodic solutions for abstract neutral equations . Bull. Aust. Math. Soc. , 78 : 365 – 382 .
   Web of Science ®Google Scholar
• Naber , M . 2010 . Linear fractionally damped oscillator . Int. J. Differ. Eqns , 2010 Article ID 197020, 12pp., doi:10.1155/2010/197020
   Google Scholar