Advertisement

Journal of Evolution Equations

, Volume 12, Issue 2, pp 367–392 | Cite as

Generalized Cauchy problems involving nonlocal and impulsive conditions

  • Nguyen Minh Chuong
  • Tran Dinh KeEmail author
Article

Abstract

We study the abstract Cauchy problem involving a class of nonlinear differential inclusions, with impulsive and nonlocal conditions. By using MNC estimates, the existence result and continuous dependence on initial data of the solution set are proved.

Mathematics Subject Classification (2000)

35K90 47G20 47H04 47H08 47H10 

Keywords

Generalized Cauchy problem Nonlocal condition Impulsive condition Nondensely defined operator Condensing map Fixed point theorem Multimap Measure of noncompactness MNC estimate 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abada N., Benchohra M., Hammouche H.: Existence and Controllability Results for Nondensely Defined Impulsive Semilinear Functional Differential Inclusions. J. Differ. Equ. 246, 3834–3863 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Akhmerov R. R., Kamenskii M. I., Potapov A. S., Rodkina A. E., Sadovskii B. N., Measures of Noncompactness and Condensing Operators. Birkhäuser, Boston, MA, Basel, Berlin, 1992.Google Scholar
  3. 3.
    Anguraj A., Karthikeyan K.: Existence of Solutions for Impulsive Neutral Functional Differential Equations with Nonlocal Conditions. Nonlinear Anal. 70, 2717–2721 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arendt W.: Vector-valued Laplace Transforms and Cauchy Problem. Israel J. Math. 59, 327–352 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Arendt W., Batty C. J. K., Hieber M., Neubrander F., Vector-valued Laplace Transforms and Cauchy Problems, In: Monographs in Mathematics, Vol. 96, Birkhauser Verlag, Basel, 2001.Google Scholar
  6. 6.
    Aubin J.-P, Frankowska H., Set-valued Analysis. Reprint Of The 1990 Edition. Modern Birkhäuser Classics. Birkhäuser, Boston, MA, 2009.Google Scholar
  7. 7.
    Benchohra M., Henderson J., Ntouyas S., Impulsive Differential Equations and Inclusions, In: Contemporary Mathematics and its Applications, Vol. 2. Hindawi, New York, 2006.Google Scholar
  8. 8.
    Benchohra M., Gatsori E.P., Górniewicz L., Ntouyas S.K.: Nondensely Defined Evolution Impulsive Differential Equations with Nonlocal Conditions. Fixed Point Theory 4(2), 185–204 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Benedetti I., Obukhovskii V., Zecca P.: Controllability For Impulsive Semilinear Functional Differential Inclusions with a Non-compact Evolution Operator. Discuss. Math. Differ. Inclusions Control Optim. 31(1), 39–69 (2011)CrossRefGoogle Scholar
  10. 10.
    Borisovich Y.G., Gelman B.D., Myshkis A.D., Obukhovskii V.V.: Introduction to Theory of Multivalued Maps and Differential Inclusions, 2nd edn. Librokom, Moscow (2011) (in Russian)Google Scholar
  11. 11.
    Byszewski L.: Theorems About the Existence and Uniqueness of Solutions of a Semilinear Evolution Nonlocal Cauchy Problem. J. Math. Anal. Appl. 162, 494–505 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Byszewski L., Akca H.: On a Mild Solution of a Semilinear Functional-differential Evolution Nonlocal Problem. J. Appl. Math. Stoch. Anal. 10(3), 265–271 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Byszewski L., Lakshmikantham V.: Theorem About the Existence and Uniqueness of a Solution of a Nonlocal Abstract Cauchy Problem in a Banach Space. Appl. Anal. 40, 11–19 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cardinali T., Rubbioni P.: Mild Solutions for Impulsive Semilinear Evolution Differential Inclusions. J. Appl. Funct. Anal. 1(3), 303–325 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Da Prato G., Sinestrari E.: Differential Operators with Nondense Domain. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(4), 285–344 (1987) (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Deimling K., Multivalued Differential Equations. De Gruyter Series In Nonlinear Analysis And Applications, 1. Walter De Gruyter & Co., Berlin, 1992.Google Scholar
  17. 17.
    Fan Z.: Impulsive Problems for Semilinear Differential Equations with Nonlocal Conditions. Nonlinear Anal. 72, 1104–1109 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Fan Z., Li G.: Existence Results for Semilinear Differential Equations with Nonlocal and Impulsive Conditions. J. Func. Anal. 258, 170–1727 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fu X.: On Solutions of Neutral Nonlocal Evolution Equations with Nondense Domain. J. Math. Anal. Appl. 299, 392–410 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fu X., Cao Y.: Existence for Neutral Impulsive Differential Inclusions with Nonlocal Conditions. Nonlinear Anal. 68, 3707–3718 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gatsori E.P., Górniewicz L., Ntouyas S.K., Sficas G.Y.: Existence Results for Semilinear Functional Differential Inclusions with Infinite Delay. Fixed Point Theory 6(1), 47–58 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Górniewicz L.: Topological Fixed Point Theory of Multivalued Mappings, 2nd edn. Springer, Dordrecht (2006)zbMATHGoogle Scholar
  23. 23.
    Hernandez E., Pierri M., Goncalves G.: Existence Results for an Impulsive Abstract Partial Differential Equation with State-dependent. Delay. Comp. Math. Appl. 52, 411–420 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hu S., Papageorgiou N. S., Handbook of Multivalued Analysis, Vol. I: Theory. Kluwer, Dordrecht, Boston, London, 1997.Google Scholar
  25. 25.
    Jackson D.: Existence and Uniqueness of Solutions to Semilinear Nonlocal Parabolic Equations. J. Math. Anal. Appl. 172, 256–265 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ji S., Wen S.: Nonlocal Cauchy Problem for Impulsive Differential Equations in Banach Spaces. Int. J. Nonlinear Sci. 10(1), 88–95 (2010)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kamenskii M., Obukhovskii V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. In: De Gruyter Series in Nonlinear Analysis and Applications, Vol. 7. Walter De Gruyter, Berlin, New York, NY, 2001.Google Scholar
  28. 28.
    Kellerman H., Hieber M.: Integrated Semigroups. J. Funct. Anal. 84, 160–180 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Liu H., Chang J.-C.: Existence for a Class of partial Differential Equations with Nonlocal Conditions. Nonlinear Anal. 70, 3076–3083 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lin Y., Liu J.H.: Semilinear Integrodifferential Equations with Nonlocal Cauchy Problem. Nonlinear Anal. TMA, 26(5), 1023–1033 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Liu J. H.: A Remark on the Mild Solutions of Non-local evolution Equations. Semigroup Forum 66, 63–67 (2003)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Ntouyas S.K., Tsamatos P.C.: Global Existence for Semilinear Evolution Equations with Nonlocal Conditions. J. Math. Anal. Appl. 210, 679–687 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Obukhovskii V., Yao J.-C.: On Impulsive Functional Differential Inclusions with Hille-yosida Operators in Banach Spaces. Nonlinear Anal. 73, 1715–1728 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Obukhovskii V., Zecca P.: On Semilinear Differential Inclusions in Banach Spaces with Nondensly Defined Operators. J. Fixed Point Theory Appl. 9(1), 85–100 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Thieme H.: Integrated Semigroups and Integrated Solutions to Abstract Cauchy Problems. J. Math. Anal. Appl. 152, 416–447 (1990)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.Department of MathematicsHanoi National University of EducationHanoiVietnam

Personalised recommendations