Abstract
Let X be a real Banach space and I a nonempty interval. Let \(K:I\rightsquigarrow X\) be a multi-function with the graph \(\mathcal {K} \). We give here a characterization for \(\mathcal {K} \) to be approximate/near weakly invariant with respect to the differential inclusion \(x^{\prime }(t)\in F(t, x(t))\) by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârjă et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârjă et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion \(x^{\prime }(t)\in F(t,x(t))\).
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Acknowledgments
The first author was supported by a grant of the Ministry of Higher Education and Scientific Research Algerian, project number 265/PNE/Roumanie/2014-2015. The second author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0154.
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Benniche, O., Cârjă, O. Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions. J Dyn Control Syst 23, 249–268 (2017). https://doi.org/10.1007/s10883-016-9312-0
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DOI: https://doi.org/10.1007/s10883-016-9312-0