Communications in Mathematical Analysis

On the existence of solutions for nonconvex hyperbolic differential inclusions

Aurelian Cernea

Full-text: Open access

Abstract

We establish some Filippov type existence theorems for solutions of certain nonconvex fractional hyperbolic differential inclusions involving Caputo's fractional derivative.

Article information

Source
Commun. Math. Anal., Volume 9, Number 1 (2010), 109-120.

Dates
First available in Project Euclid: 21 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1271890722

Mathematical Reviews number (MathSciNet)
MR2640309

Zentralblatt MATH identifier
1193.35003

Subjects
Primary: 34A60: Differential inclusions [See also 49J21, 49K21] 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx]
Secondary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]

Keywords
fractional derivative hyperbolic differential inclusion decomposable set

Citation

Cernea , Aurelian. On the existence of solutions for nonconvex hyperbolic differential inclusions. Commun. Math. Anal. 9 (2010), no. 1, 109--120. https://projecteuclid.org/euclid.cma/1271890722


Export citation

References

  • S. Abbas and M. Benchohra, Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7 (2009), pp 62-72.
  • S. Abbas and M. Benchohra, Darboux problem for perturbed partial hyperbolic differential equations of fractional order with finite. Nonlinear Anal., Hybrid Syst. 3 (2009), pp 597-604.
  • E. Ait Dads, M. Benchohra and S. Hamani, Impulsive fractional differential inclusions involving Caputo fractional derivative. Fract. Calc. Appl. Anal. 12 (2009), pp 15-38.
  • M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338 (2008), pp 1340-1350.
  • M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential inclusions with infinite delay and applications to control theory. Fract. Calc. Appl. Anal. 11 (2008), pp 35-56.
  • A. Bressan, A. Cellina and A. Fryskowski, A class of absolute retracts in spaces of integrable functions, Proc. Amer. Math. Soc. 111 (1991), pp 413-418.
  • A. Bressan and G. Colombo, Extensions and selections of maps with decomposable values Studia Math., 90 (1988), pp 69-86.
  • M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969.
  • A. Cernea, Continuous version of Filippov's theorem for fractional differential inclusions. Nonlinear Anal. 72 (2010), pp 204-208.
  • C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer, Berlin, 1977.
  • R. M. Colombo, A. Fryszkowski, T. Rzezuchowski and V. Staicu, Continuous selections of solution sets of Lipschitzean differential inclusions. Funkcial. Ekvac. 34 (1991), pp 321-330.
  • A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations of arbitrary orders. Appl. Math. Comput. 68 (1995), pp 15-25.
  • A. F. Filippov, Classical solutions of differential equations with multivalued right hand side. SIAM J. Control 5 (1967), pp 609-621.
  • J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal. 70 (2009), pp 2091-2105.
  • J. Henderson and A. Ouahab, Impulsive differential inclusions with fractional order, Comput. Math. Appl. doi:10.1016/j.camwa.2009.05.011.
  • A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  • K. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  • A. Ouahab, Some results for fractional boundary value problem of differential inclusions. Nonlinear Anal. 69 (2009), pp 3871-3896.
  • N. Papageorgiu and N. Shahzad, On functional-differential inclusions of Volterra-type. New Zealand J. Math. 28 (1999), pp 77-87.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • I. Podlubny, I. Petrǎs, B. M. Vinagre, P. O'Leary and L. Dorčak, Analoque realizations of fractional order controllers, fractional order calculus and its applications. Nonlinear Dynam. 29 (2002), pp 281-296.
  • V. Staicu, On a nonconvex hyperbolic differential inclusion. Proc. Edinburgh Math. Soc. 35 (1992), pp 375-382.
  • H. D. Tuan, On local controllability of hyperbolic inclusions. J. Math. Systems Estim. Control 4 (1994), pp 319-339.
  • H. D. Tuan, On solution sets of nonconvex Darboux problems and applications to optimal control with end point constraints J. Austral. Math. Soc. B - 37 (1996), pp 354-391.