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

2010; Volume: 52; Linguagem: Inglês

10.21914/anziamj.v52i0.1161

1445-8810

Rabha W. Ibrahim,

Numerical methods in engineering

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.