Convergence results for solutions of a first-order differential equation

Volume 6, Issue 1, pp 18--28 http://dx.doi.org/10.22436/jnsa.006.01.04

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Authors

Liviu C. Florescu - Faculty of Mathematics, ''Al. I. Cuza'' University, Carol I, 11, 700506, Iaşi, Romania.

Abstract

We consider the first order differential problem: \[ (P_n) \begin{cases} u'(t) = f_n(t, u(t)),\,\,\,\,\, \texttt{for almost every} \quad t \in [0, 1],\\ u(0) = 0. \end{cases} \] Under certain conditions on the functions \(f_n\), the problem \((P_n)\) admits a unique solution \(u_n \in W^{1;1}([0; 1];E)\). In this paper, we propose to study the limit behavior of sequences \((u_n)_{n\in \mathbb{N}}\) and \((u'_n)_{n\in \mathbb{N}}\), when the sequence \((f_n)_{n\in \mathbb{N}}\) is subject to different growing conditions.

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Keywords

• Tight sets
• Jordan finite-tight sets
• Young measure
• fiber product
• Prohorov's theorem.

MSC

•  28A20
•  28A33
•  46E30
•  46N10

References

• [1] D. L. Azzam, Ch. Castaing, L. Thibault, Three boundary value problems for second order differential inclusions in Banach spaces, Control Cybernet. , 31 (2002), 659-693.


• [2] Ch. Castaing, P. Raynaud de Fitte, On the fiber product of Young measures with application to a control problem with measures, Adv. Math. Econ., 6 (2004), 1-38.


• [3] Ch. Castaing, P. Raynaud de Fitte, M. Valadier, Young measures on topological spaces , With applications in control theory and probability theory, Kluwer Academic Publ., Dordrecht, Boston, London (2004)


• [4] Ch. Castaing, P. Raynaud de Fitte, A. Salvadori, Some variational convergence results for a class of evolution inclusions of second order using Young measures, Adv. Math. Econ. , 7 (2005), 1-32.


• [5] J. Hoffmann-Jørgensen, Convergence in law of random elements and random sets, High dimensional probability (Oberwolfach, 1996), 151-189, Progress in Probability no. 43, Birkhäuser, Basel (1998)


• [6] L. C. Florescu, C. Godet-Thobie, A version of biting lemma for unbounded sequences in \(L^1\) with applications, Mathematical Analysis and Applications, 58-73, AIP Conference Proceedings, New York (2006)


• [7] L. C. Florescu, Finite-tight sets , Central European Journal of Mathematics, 5 (2007), 619-638.


• [8] L. C. Florescu, Existence results in relaxed variational calculus, Analele Universităţii de Vest, Timişoara, Seria Matematică-Informatică XLV, 1 (2007), 231-244.


• [9] L. C. Florescu, C. Godet-Thobie, Young Measures and Compactness in Measure Spaces, Walter de Gruyter, Berlin, Boston (2012)


• [10] M. Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste XXVI, Supplemento, (1994), 349-394.