The existence of mild and classical solutions for a second-order abstract fractional problem

Nonlinear Analysis, Theory, Methods and Applications

Volumn 73, Issue 9, 2010, Pages 3130-3139

  Tatar, Nasser Eddine ^a  

^a KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS   (Saudi Arabia)

Author keywords

Cauchy problem; Cosine family; Fractional derivative; Mild solutions; Second order abstract problem

Indexed keywords

CAUCHY PROBLEMS; COSINE FAMILIES; FRACTIONAL DERIVATIVES; MILD SOLUTION; SECOND ORDERS;

ABSTRACTING;

EID: 77955657759     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2010.06.085     Document Type: Article

References (27)

1. C.C. Travis, and G.F. Webb Cosine families and abstract nonlinear second order differential equations Acta Math. Acad. Sci. Hungaricae 32 1978 76 96
2. M.E. Hernandez Existence of solutions to a second order partial differential equation with nonlocal conditions Electron J. Differential Equations 2003 51 2003 1 10
3. C.C. Travis, and G.F. Webb Compactness, regularity and uniform continuity properties of strongly continuous cosine families Houston J. Math. 3 4 1977 555 567
4. M. Kirane, M. Medved', N.-e. Tatar, Semilinear Volterra integrodifferential problem with less regularity (submitted for publication).
5. C.C. Travis, and G.F. Webb An abstract second order semilinear Volterra integrodifferential equation SIAM J. Math. Anal. 10 2 1979 412 424
6. M. Bahaj Remarks on the existence results for second-order differential inclusions with nonlocal conditions J. Dyn. Control Syst. 15 1 2009 27 43
7. M. Benchohra, and S.K. Ntouyas Existence of mild solutions of second order initial value problems for delay integrodifferential inclusions with nonlocal conditions Math. Bohem. 4 127 2002 613 622
8. M. Benchohra, and S.K. Ntouyas Existence results for the semi-infinite interval for first and second order integrodifferential equations in Banach spaces with nonlocal conditions Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 41 2002 13 19
9. M. Benchohra, and S.K. Ntouyas Existence results for multivalued semilinear functional differential equations Extracta Math. 18 1 2003 1 12
10. L. Byszewski, and V. Lakshmikantham Theorems about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space Appl. Anal. 40 1 1991 11 19
11. E.M. Hernandez, H.R. Henrquez, and M.A. McKibben Existence of solutions for second order partial neutral functional differential equations Integral Equations Operator Theory 62 2008 191 217
12. R.L. Bagley, and P.J. Torvik A different approach to the analysis of viscoelastically damped structures AIAA J. 21 1983 741 748
13. W. Chen, and S. Holm Modified Szabo's wave equation models for lossy media obeying frequency power law J. Acoust. Soc. Am. 114 5 2003 2570 2575
14. M. Fukunaga, and N. Shimizu Role of prehistories in the initial value problem of fractional viscoelastic equations Nonlinear Dyn. 38 2004 207 220
15. K.S. Hedrich, and A. Filipovski Longitudinal creep vibrations of a fractional derivative order rheological rod with variable cross section Facta Univ. Ser. Mech. Automat. Control Robot. 3 12 2002 327 349
16. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Jan van Mill, Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies vol. 204 2006 Elsevier Amsterdam, The Netherlands
17. A.A. Lokshin Wave equation with singular delayed time Dokl. Akad. Nauk. SSSR 240 1978 43 46 (in Russian)
18. A.A. Lokshin, and V.E. Rok Fundamental solutions of the wave equation with delayed time Dokl. Akad. Nauk. SSSR 239 1978 1305 1308 (in Russian)
19. D. Matignon, Asymptotic stability of the WebsterLokshin model, Available online.
20. D. Matignon, J. Audounet, and G. Montseny Energy decay for wave equations with damping of fractional order Proc. Fourth International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena 1998 INRIA-SIAM Golden, Colorado 638 640
21. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 John Wiley and Sons New York
22. K.B. Oldham, and J. Spanier The Fractional Calculus 1974 Academic Press New York, London
23. E. Orsingher, and L. Beghin Time-fractional telegraph equations and telegraph processes with Brownian time Probab. Theory Related Fields 128 2004 141 160
24. I. Podlubny Fractional Differential Equations Mathematics in Sciences and Engineering 198 1999 Academic Press San Diego, CA
25. S.G. Samko, A.A. Kilbas, and O.I. Marichev Fractional Integrals and Derivatives. Theory and Applications 1993 Gordon and Breach Yverdon
26. R. Gorenflo, and S. Vessella A. Dold, B. Eckmann, F. Takens, Abel Integral Equations: Analysis and Applications Lecture Notes in Mathematics vol. 1461 1991 Springer-Verlag Berlin, Heidelberg, New York
27. H. Haddar, J.-R. Li, and D. Matignon Efficient solution of a wave equation with fractional-order dissipative terms J. Comput. Appl. Math. 234 6 2010 2003 2010