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Approximate Weak Invariance and Relaxation for Fully Nonlinear Differential Inclusions

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Abstract

In this paper we prove that a given set K is approximately weakly invariant with respect to the fully nonlinear differential inclusion

$${x^\prime (t) \in Ax (t) + F (x (t))}$$

, where A is an m-dissipative operator, and F is a given multi-function in a Banach space, if and only if the set \({F(\xi)}\) is A-quasi-tangent to the set K, for every \({{\xi \in K}}\) . As an application, we establish that the approximate solutions of the given differential inclusion approximate the solutions of the relaxed (convexified) nonlinear differential inclusion

$${x^\prime (t) \in Ax (t) + \overline{co}F (x (t))}$$

, with no hypotheses of Lipschitz type for multi-function F.

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References

  1. P. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Thése, Orsay, 1972.

  2. Cârjă O., Necula M., Vrabie I. I.: Viability, Invariance and Applications. Elsevier Science B. V., Amsterdam (2007)

    MATH  Google Scholar 

  3. Cârjă O., Necula M., Vrabie I. I.: Necessary and sufficient conditions for viability for nonlinear evolution inclusions. Set-Valued Anal. 16, 700–731 (2008)

    Google Scholar 

  4. Clarke F. H., Ledyaev Yu. S., Radulescu M. L.: Approximate invariance and differential inclusions in Hilbert spaces. J. Dynam. Control Systems 3((4), 493–518 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Frankowska H.: A priori estimates for operational differential inclusions. J. Differential Equations 84((1), 100–128 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Lakshmikantham and S. Leela, Nonlinear differential equations in abstract spaces, International Series in Nonlinear Mathematics: Theory, Methods and Applications 2, Pergamon Press, 1981.

  7. Lazu A. I., Postolache V.: Approximate weak invariance for semilinear differential inclusions in Banach spaces. Cent. Eur. J. Math. 9((5), 1143–1155 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. A.A. Tolstonogov Properties of integral solutions of differential inclusions with m-accretive operators, Mat. Zametki 49 (6) (1991), 119–131, 159 (in Russian); translation in Math. Notes 49 (5–6) (1991), 636-644.

  9. I. I. Vrabie, Compactness methods for nonlinear evolutions, Second Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, 75, Longman, 1995.

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Authors and Affiliations

  1. Faculty of Mathematics, “Al. I. Cuza” University, Bd. Carol I, no. 11, 700506, Iasi, Romania

    Irina Căpraru

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  1. Irina Căpraru
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Correspondence to Irina Căpraru.

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Căpraru, I. Approximate Weak Invariance and Relaxation for Fully Nonlinear Differential Inclusions. Mediterr. J. Math. 10, 201–212 (2013). https://doi.org/10.1007/s00009-012-0194-4

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  • DOI: https://doi.org/10.1007/s00009-012-0194-4

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