Skip to main content
SpringerLink
Log in

Multivalued perturbations ofm-accretive differential inclusions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2X, we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x 0. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L 1([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ4 in which no integral solution exists.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Attouch and A. Damlamian,On multivalued evolution equations in Hilbert spaces, Israel Journal of Mathematics12 (1972), 373–390.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Baras,Compacité de l’opérateur f→u solution d’une équation non linéaire (du/dt)+Au∋f, Comptes Rendus de l’Académie des Sciences, Paris286 (1978), 1113–1116.

    MATH  MathSciNet  Google Scholar 

  3. V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.

    MATH  Google Scholar 

  4. Ph. Benilan, M. G. Crandall and A. Pazy,Nonlinear Evolution Equations in Banach Spaces, monograph in preparation.

  5. D. Bothe,Upper semicontinuous perturbations of m-accretive operators and differential inclusions with dissipative right-hand side, inTopology in Nonlinear Analysis (K. Geba and L. Górniewicz, eds.), Banach Center Publications35, 1996, pp. 139–148.

  6. D. Bothe,Flow invariance for perturbed nonlinear evolution equations, Abstract and Applied Analysis1 (1996), 417–433.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Bothe,Reaction-diffusion systems with discontinuities. A viability approach, Proceedings of the Second World Congress of Nonlinear Analysts, Nonlinear Analysis30 (1997), 677–686.

    MATH  MathSciNet  Google Scholar 

  8. A. Bressan and V. Staicu,On nonconvex perturbations of maximal monotone differential inclusions, Setvalued Analysis2 (1994), 415–437.

    MATH  MathSciNet  Google Scholar 

  9. H. Brezis,Operateurs Maximaux Monotones, North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  10. H. Brezis,New results concerning monotone operators and nonlinear semigroups, inAnalysis of Nonlinear Problems, RIMS, 1974, pp. 2–27.

  11. A. Cellina and M. V. Marchi,Non-convex perturbations of maximal monotone differential inclusions, Israel Journal of Mathematics46 (1983), 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Colombo, A. Fonda and A. Ornelas,Lower semicontinuous perturbations of maximal monotone differential inclusions, Israel Journal of Mathematics61 (1988), 211–218.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. G. Crandall and T. M. Liggett,Generation of semi-groups of nonlinear transformations on general Banach spaces, American Journal of Mathematics93 (1971), 265–298.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Deimling,Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

    MATH  Google Scholar 

  15. K. Deimling,Multivalued Differential Equations, de Gruyter, Berlin, 1992.

    MATH  Google Scholar 

  16. J. Diestel,Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics485, Springer-Verlag, Berlin, 1975.

    MATH  Google Scholar 

  17. J. Diestel, W. M. Ruess and W. Schachermayer,Weak compactness in L 1 μ,X), Proceedings of the American Mathematical Society118 (1993), 447–453.

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Górniewicz, A. Granas and W. Kryszewski,Sur la méthode de l’homotopie dans la théorie des point fixes pour les applications multivoques. Partie 2: L’indice dans les ANRs compactes, Comptes Rendus de l’Académie des Sciences, Paris308 (1989), 449–452.

    MATH  Google Scholar 

  19. S. Gutman,Evolutions governed by m-accretive plus compact operators, Nonlinear Analysis7 (1983), 707–715.

    Article  MATH  MathSciNet  Google Scholar 

  20. D. M. Hyman,On decreasing sequences of compact absolute retracts, Fundamenta Mathematicae64 (1969), 91–97.

    MATH  MathSciNet  Google Scholar 

  21. E. Mitidieri and I. I. Vrabie,Differential inclusions governed by non convex perturbations of m-accretive operators, Differential and Integral Equations2 (1989), 525–531.

    MATH  MathSciNet  Google Scholar 

  22. V. V. Obukhovskij,Semilinear functional-differential inclusions in a Banach space and controlled parabolic systems, Soviet Journal of Automation and Information Sciences24 (1991), 71–79.

    MathSciNet  Google Scholar 

  23. A. Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.

    MATH  Google Scholar 

  24. S. Schmidt,Existenzsätze für gewöhnliche Differentialgleichungen in Banachräumen, Funkcialaj Ekvacioj35 (1992), 199–222.

    MATH  MathSciNet  Google Scholar 

  25. A. A. Tolstonogov,Properties of integral solutions of differential inclusions with m-accretive operators, Mathematical Notes49 (1991), 636–644.

    MATH  MathSciNet  Google Scholar 

  26. A. A. Tolstonogov and Y. I. Umanskii,Solutions of evolution inclusions II, Siberian Mathematical Journal33 (1992), 693–702.

    Article  MathSciNet  Google Scholar 

  27. P. Volkmann,Ein Existenzsatz für gewöhnliche Differentialgleichungen in Banachräumen, Proceedings of the American Mathematical Society80 (1980), 297–300.

    Article  MATH  MathSciNet  Google Scholar 

  28. I. I. Vrabie,Compactness Methods for Nonlinear Evolutions, Pitman, London, 1987.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Informatik, Universität-GH Paderborn, Warburger Str. 100, D-33098, Paderborn, Germany

    Dieter Bothe

Authors
  1. Dieter Bothe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dieter Bothe.

Additional information

The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bothe, D. Multivalued perturbations ofm-accretive differential inclusions. Israel J. Math. 108, 109–138 (1998). https://doi.org/10.1007/BF02783044

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02783044

Keywords

Search

Navigation