Nonlocal Cauchy problem for fractional evolution equations

Nonlinear Analysis: Real World Applications

Volumn 11, Issue 5, 2010, Pages 4465-4475

  Zhou, Yong ^a   Jiao, Feng ^a  

^a XIANGTAN UNIVERSITY   (China)

Author keywords

C0 semigroup; Fractional evolution equations; Laplace transform; Mild solution; Nonlocal Cauchy problem; Probability density

Indexed keywords

EVOLUTION EQUATIONS; FRACTIONAL EVOLUTION EQUATIONS; MILD SOLUTION; NONLOCAL CAUCHY PROBLEM; PROBABILITY DENSITIES; SEMI-GROUP;

BANACH SPACES; BIOLOGY; DIFFERENTIAL EQUATIONS; LAPLACE TRANSFORMS; PROBABILITY DENSITY FUNCTION; TOPOLOGY; WAVELET TRANSFORMS;

MATHEMATICAL TRANSFORMATIONS;

EID: 77955515765     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2010.05.029     Document Type: Article

References (43)

1. K. Diethelm, and A.D. Freed On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity F. Keil, W. Mackens, H. Voss, J. Werther, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties 1999 Springer-Verlag Heidelberg 217 224
2. L. Gaul, P. Klein, and S. Kempfle Damping description involving fractional operators Mech. Syst. Signal Process. 5 1991 81 88
3. W.G. Glockle, and T.F. Nonnenmacher A fractional calculus approach of self-similar protein dynamics Biophys. J. 68 1995 46 53
4. R. Hilfer Applications of Fractional Calculus in Physics 2000 World Scientific Singapore
5. F. Mainardi Fractional calculus: some basic problems in continuum and statistical mechanics A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics 1997 Springer-Verlag Wien 291 348
6. F. Metzler, W. Schick, H.G. Kilian, and T.F. Nonnenmacher Relaxation in filled polymers: a fractional calculus approach J. Chem. Phys. 103 1995 7180 7186
7. A.A. Kilbas, Hari M. Srivastava, and J. Juan Trujillo Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies vol. 204 2006 Elsevier Science B.V. Amsterdam
8. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Differential Equations 1993 John Wiley New York
9. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego
10. V. Lakshmikantham, S. Leela, and J. Vasundhara Devi Theory of Fractional Dynamic Systems 2009 Cambridge Academic Publishers Cambridge
11. B. Ahmad, and J.J. Nieto Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions Bound. Value Probl. 2009 11 pages. Art. ID 708576
12. B. Baeumer, S. Kurita, and M.M. Meerschaert Inhomogeneous fractional diffusion equations J. Frac. Appl. Anal. 8 2005 375 397
13. Bashir Ahmad, and Juan J. Nieto Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations Abstr. Appl. Anal. 2009 Art. no. 494720
14. Bashir Ahmad, and Juan J. Nieto Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions Comput. Math. Appl. 58 2009 1838 1843
15. Moustafa El-Shahed, Juan J. Nieto, Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order. Comput. Math. Appl., in press (doi:10.1016/j.camwa.2010.03.031 ).
16. M. Belmekki, J.J. Nieto, and R. Rodriguez-Lopez Existence of periodic solution for a nonlinear fractional differential equation Bound. Value Probl. 2009 18 pages. Art. ID 324561
17. Y.K. Chang, and J.J. Nieto Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators Numerical Funct. Anal. Optim. 30 2009 227 244
18. Y.K. Chang, and J.J. Nieto Some new existence results for fractional differential inclusions with boundary conditions Math. Comput. Model. 49 2009 605 609
19. S.D. Eidelman, and A.N. Kochubei Cauchy problem for fractional diffusion equations J. Differential Equations 199 2004 211 255
20. M.M. El-Borai Semigroups and some nonlinear fractional differential equations Appl. Math. Comput. 149 2004 823 831
21. M.M. El-Borai Some probability densities and fundamental solutions of fractional evolution equations Chaos Solutions Fractals 149 2004 823 831
22. A.M.A. El-Sayed Fractional order diffusion-wave equations Internat. J. Theoret. Phys. 35 1996 311 322
23. A.M.A. El-Sayed, and A.G. Ibrahim Multivalued fractional differential equations Appl. Math. Comput. 68 1995 15 25
24. A.M.A. El-Sayed Nonlinear functional differential equations of arbitrary orders Nonlinear Anal. 33 1998 181 186
25. O.K. Jardat, A. Al-Omari, and S. Momani Existence of the mild solution for fractional semilinear initial value problems Nonlinear Anal. 69 9 2008 3153 3159
26. J. Henderson, and A. Ouahab Fractional functional differential inclusions with finite delay Nonlinear Anal. 70 2009 2091 2105
27. V. Lakshmikantham, and A.S. Vatsala Basic theory of fractional differential equations Nonlinear Anal. 69 2008 2677 2682
28. F. Mainardi, P. Paradisi, and R. Gorenflo Probability distributions generated by fractional diffusion equations J. Kertesz, I. Kondor, Econophysics: An Emerging Science 2000 Kluwer Dordrecht
29. M.M. Meerschaert, D.A. Benson, H. Scheffler, and B. Baeumer Stochastic solution of spacetime fractional diffusion equations Phys. Rev. E 65 2002 1103 1106
30. M. Muslim Existence and approximation of solutions to fractional differential equations Math. Comput. Model. 49 2009 1164 1172
31. Nickolai Kosmatov Integral equations and initial value problems for nonlinear differential equations of fractional order Nonlinear Anal. 70 2009 2521 2529
32. W.R. Schneider, and W. Wayes Fractional diffusion and wave equation J. Math. Phys. 30 1989 134 144
33. G. Zaslavsky Fractional kinetic equation for hamiltonian chaos, chaotic advection, tracer dynamics and turbulent dispersion Physica D 76 1994 110 122
34. Yong Zhou, Feng Jiao, and Jing Li Existence and uniqueness for p -type fractional neutral differential equations Nonlinear Anal. 71 2009 2724 2733
35. Yong Zhou, Feng Jiao, and Jing Li Existence and uniqueness for fractional neutral differential equations with infinite delay Nonlinear Anal. 71 2009 3249 3256
36. Shuqin Zhang Monotone iterative method for initial value problem involving RiemannLiouville fractional derivatives Nonlinear Anal. 71 2009 2087 2093
37. B. Bonilla, M. Rivero, L. Rodriguez-Germa, and J.J. Trujillo Fractional differential equations as alternative models to nonlinear differential equations Appl. Math. Comput. 187 2007 79 88 (Pubitemid 46635706)
38. R.P. Agarwal, V. Lakshmikantham, and J.J. Nieto On the concept of solution for fractional differential equations with uncertainty Nonlinear Anal. 72 2010 2859 2862
39. Y.F. Luchko, M. Rivero, J.J. Trujillo, and M.P. Velasco Fractional models, non-locality, and complex systems Comput. Math. Appl. 59 2010 1048 1056
40. L. Byszewski Theorems about existence and uniqueness of solutions of a semi-linear evolution nonlocal Cauchy problem J. Math. Anal. Appl. 162 1991 494 505
41. L. Byszewski, and V. Lakshmikantham Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space Appl. Anal. 40 1991 11 19
42. A. Pazy Semigroups of Linear Operators and Applications to Partial Differential Equations 1983 Springer New York
43. X. Fu, and K. Ezzinbi Existence of solutions for neutral differential evolution equations with nonlocal conditions Nonlinear Anal. 54 2003 215 227