On the existence for diffeo-integral inclusion of Sobolev-type of fractional order with applications

Rabha W Ibrahim

Abstract


By using a suitable fixed point theorems, we study the existence of solutions for fractional diffeo-integral inclusion of Sobolev-type. The study arises in the case when the set-valued function has convex and non-convex values.

References
  • R. Hilfer, Fractional diffusion based on Riemann--Liouville fractional derivatives, J. Phys. Chem. Bio. 104(2000) 3914--3917.
  • R. Hilfer, The continuum limit for self-similar Laplacians and the Green function localization exponent, 1989, UCLA-Report 982051.
  • B. Ross, Fractional Calculus and its Applications , Vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Pub.Co.: Singapore, 2000.
  • K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, John-Wily and Sons, Inc., 1993.
  • I. Podlubny, Fractional Differential Equations, Acad.Press, London, 1999.
  • V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Res. Notes Math. Ser., Vol. 301, Longman/Wiley, New York, 1994.
  • S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives (Theory and Applications), Gorden and Breach, New York, 1993.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Math. in Science and Engineering, Acad. Press, New York/London, 1974.
  • A. M. A. El-Sayed, A. G. Ibrahim, Multi-valued fractional differential equations, Appl. Math. Comput. 68(1995) 15--25.
  • A. G. Ibrahim, A. M. A. El-Sayed, Define integral of fractional order for set valued function, J. Frac. Calculus 11 (May 1997).
  • A. M. A. El-Sayed, A. G. Ibrahim, Set valued integral equations of fractional-orders, Appl. Math. Comp. 118(2001) 113--121.
  • N. S. Papageeorgion, On integral inclusion of Volterra type in Banach spaces, Czechoslovak Math. J. 42(1992) 693--714.
  • N. S. Papageeorgion, On non convex valued Volterra integral inclusions in Banach spaces, Czechoslovak Math. J. 44(1994).
  • S. Aizicovici, V. Staicu, Continuous selections of solutions sets to Volterra integral inclusions in Banach spaces, Elec. J. Diffe. Equa. Vol. 2006(2006) 1--11.
  • M. Kanakaraj, K. Balachadran, Existence of solutions of Sobolev-type semilinear mixed integrodifferential inclusions in Banach spaces, J. of Applied and Stochastic Analysis 16:2(2003) 163--170.
  • K. Balachandar and J. P. Dauer, Elements of Control Theory, Narosa Publishing House, 1999.
  • L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Oxford,1982.
  • K. Deimling, Nonlinear Functional Analysis, Springer-Verlag,1985.
  • D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.
  • C. Avramescu, A fixed point theorem for multivalued mappings, Electronic. J. Qualitative Theory of Differential Equations. Vol. 17 (2004) 1--10.
  • K. Demling, Multivalued Differential Equations, Walter de Gruyter, New York, 1992.
  • J. P. Aubin, A. Cellina. Differential Inclusions. Springer, Berlin, 1984.
  • V. Barbu. Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff international Pupl. Leyden, 1976.
  • S. Hu, N. S. Papageeorgion. Handbook of Multivalued Analysis, Vol. I: Theory. Kluwer, Dordrecht, 1997.
  • S. Hu, N. S. Papageeorgion. Handbook of Multivalued Analysis, Vol. II: Applications. Kluwer, Dordrecht, 2000.
  • A. G. Kartsatos, K. Y. Shin. Solvability of functional evolutions via compactness methods in general Banach spaces. Nonlinear Anal., 21(1993) 517--535.
  • N. H. Pavel. Nonlinear Evolution Operators and Semigroups, Lecture Notes in Mathematics, Vol. 1260. Springer, Berlin, 1987.
  • I. I. Vrabie, Compactness Methods for Nonlinear Evolutions. Longman, Harlow, 1987.
  • M. Kisielewicz. Differential Inclusions and Optimal Control. Dordrecht, The Netherlands, 1991.
  • C. Avramescu, A fixed point theorem for multivalued mappings, Electronic. J. Qualitative Theory of Differential Equations. Vol. 17 (2004) 1--10.
  • A. M. A. El-Sayed, F. M. Gaafar, Fractional calculus and some intermediate physical processes, Appl. Math. and Comp. 144(2003) 117--126.
  • R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, Fractional telegraph equations, J. Math. Anal. Appl. 276 (2002) 145--159.
  • R. W. Ibrahim, Continuous solutions for fractional integral inclusion in locally convex topological space, Appl. Math. J. Chinese Univ. 24(2)(2009) 175--183.

Keywords


fractional calculus; set-valued function; diffeo-integral inclusion

Full Text:

PDF BibTeX


DOI: http://dx.doi.org/10.21914/anziamj.v52i0.1161



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.