M. Ait Mansour, J. Lahrache, A. El Ayoubi, The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited

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DOI: 10.23952/jano.5.2023.1.07
Volume 5, Issue 1, 1 April 2023, Pages 111-124

Abstract. In this paper, given a function $f: I\times V \rightarrow R^m,$ where $V$ is an open subset of $R^m,$ $x_0\in V$ , and $I=[0,T]$ is the interval of interest, we consider the Cauchy ordinary differential equation initial-value problem $\dot{x}(f,x_0)=f(t,x(t)),$ $x(0)=x_0.$ We first present a new quantitative stability result under a partial and/or global variation of the data of the problem by involving exact and/or approximate fixed points for which we apply Lim’s Lemma either in its exact format or in its very recent approximate version. Our main result is then applied to parametric linear control systems. Finally, we demonstrate that our treatment is coherent with the management of perturbations generated in the classic one-step numerical method. A numerical example written in Scilab 6.1 illustrates the obtained stability.crete examples of quasi-point-separable topological vector spaces, which are not locally convex.

How to Cite this Article:
M. Ait Mansour, J. Lahrache, A. El Ayoubi, The stability of the parametric Cauchy problem of initial-value ordinary differential equations revisited, J. Appl. Numer. Optim. 5 (2023), 111-124.