DOI QR코드

DOI QR Code

NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE AND CONDENSING TYPE IN HILBERT SPACES

  • Abedi, Hossein ;
  • Jahanipur, Ruhollah
  • Received : 2013.09.28
  • Published : 2015.03.31

Abstract

In this paper, we study the existence of classical and generalized solutions for nonlinear differential inclusions $x^{\prime}(t){\in}F(t,x(t))$ in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condensing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper.

Keywords

differential inclusions;set-valued integral;semimonotone and hemicontinuous multifunctions;condensing multifunctions

References

  1. R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambrige, 2001.
  2. J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, 1984.
  3. J. P. Aubin and H. Frankowska, Set Valued Analysis, Birkhauser, 1990.
  4. R. Aumann, Integral of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. https://doi.org/10.1016/0022-247X(65)90049-1
  5. D. Barcenas and W. Urbina, Measurable multifunctions in nonseparable Banach spaces, SIAM J. Math. Anal. 28 (1997), no. 5, 1212-1226. https://doi.org/10.1137/S0036141095296005
  6. A. Bressan, On the qualitative theory of lower semicontinuous differential inclusions, J. Differential Equations 77 (1989), no. 2, 379-391. https://doi.org/10.1016/0022-0396(89)90150-2
  7. A. Bressan, Selections of Lipschitz multifunctions generating a continuous flow, Differential Integral Equations 4 (1991), no. 3, 483-490.
  8. A. Bressan and Z. Wang, Classical solutions to differential inclusions with totaly disconnected right-hand side, J. Differential Equations 246 (2009), no. 2, 629-640. https://doi.org/10.1016/j.jde.2008.07.001
  9. F. E. Browder, Non-linear equations of evolution, Ann. of Math. 80 (1964), 485-523. https://doi.org/10.2307/1970660
  10. F. E. Browder, Nonlinear initial value problems, Ann. of Math. 81 (1965), 51-87.
  11. A. Cellina, Multivalued differential equations and ordinary differential equations, SIAM J. Appl. Math. 18 (1970), 533-538. https://doi.org/10.1137/0118046
  12. A. Cellina, A view on differential inclusions, Rend. Semin. Mat. Univ. Politec. Torino. 63 (2005), no. 3, 197-209.
  13. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985.
  14. N. Dunford and J. T. Schwartz, Linear Operators. I. General Theorey, Interscience Publishers, Inc., New York, 1958.
  15. A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control 5 (1967), 609-621. https://doi.org/10.1137/0305040
  16. H. Frankowska, Set-valued analysis and control theory (monograph), Birkhauser, (in prepatation).
  17. A. Granas and J. Dugundji, Fixed point Theory, Springer-Verlag, 2003.
  18. S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I, Kluwer Aca-demic, 1997.
  19. R. Jahanipur, Stability of stochastic delay evolution equations with monotone nonlin-earity, Stoch. Anal. Appl. 21 (2003), no. 1, 161-181. https://doi.org/10.1081/SAP-120017537
  20. R. Jahanipur, Nonlinear functional differential equations of monotone-type in Hilbert spaces, Nonlinear Anal. 72 (2010), no. 3-4, 1393-1408. https://doi.org/10.1016/j.na.2009.08.023
  21. M. Kamenskii, V. Obukhovski, and P. Zecca, Condensing multivalued maps and semi-linear differential inclusion in Banach spaces, De Gruyter Ser. Nonlinear Anal. Appl. 7, Walter de Gruyter, Berlin-New York, 2001.
  22. A. Lasota and Z. Opial, An application for the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781-786.
  23. A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
  24. X. Xue and J. Yu, Periodic solutions for semi-linear evolution inclusions, J. Math. Anal. Appl. 331 (2007), no. 2, 1246-1262. https://doi.org/10.1016/j.jmaa.2006.09.056
  25. B. Z. Zangeneh, An energy-type inequality, Math. Inequal. Appl. 1 (1998), no. 3, 453-461.