Broucke, Mireille; Arapostathis, Ari Continuous interpolation of solutions of Lipschitz inclusions. (English) Zbl 0984.34009 J. Math. Anal. Appl. 258, No. 2, 565-572 (2001). Let \(F\) be a set-valued map defined on \([0,T]\times \mathbb{R}^{n}\) and taking values on closed and nonempty subsets of \(\mathbb{R}^{n}\). The authors show that if \(F\) is measurable in \(t\) and Lipschitzian continuous in \(x\), then for a given finite set of trajectories to the problem \[ \dot x(t)\in F(t,x), \quad x(0)=\xi, \tag{1} \] starting from distinct initial points, there exists a continuous selection from the set of solutions to problem (1) that interpolates these trajectories. An argument from the paper of A. Cellina and A. Ornelas [Rocky Mt. J. Math. 22, 117-124 (1992; Zbl 0752.34012)] is used by the authors. A second result presented in this paper concerns the existence of Lipschitzian selections when \(F\) has compact and convex values. Reviewer: Mouffak Benchohra (Sidi Bel Abbes) Cited in 3 Documents MSC: 34A60 Ordinary differential inclusions Keywords:differential inclusions; continuous and Lipschitzian selection; trajectories; interpolation Citations:Zbl 0752.34012 PDFBibTeX XMLCite \textit{M. Broucke} and \textit{A. Arapostathis}, J. Math. Anal. Appl. 258, No. 2, 565--572 (2001; Zbl 0984.34009) Full Text: DOI Link References: [1] Aubin, J.; Cellina, A., Differential Inclusions: Set-Valued Maps and Viability Theory (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0538.34007 [2] Bressan, A., Selections of Lipschitz multifunctions generating a continuous flow, Differential Integral Equations, 4, 483-490 (1991) · Zbl 0722.34009 [3] Broucke, M., Regularity of solutions and homotopic equivalence for hybrid systems, Proc. 37th IEEE CDC (1988), p. 4283-4288 [4] Cellina, A., On the set of solutions to lipschitzian differential inclusions, Differential Integral Equations, 1, 495-500 (1988) · Zbl 0723.34009 [5] Deimling, K., Multivalued Differential Equations (1992), de Gruyter: de Gruyter Berlin · Zbl 0760.34002 [6] Filippov, A. F., Differential Equations with Discontinuous Righthand Sides (1988), Kluwer Academic: Kluwer Academic Boston · Zbl 0664.34001 [7] Cellina, A.; Ornelas, A., Representation of the attainable set for lipschitzian differential inclusions, Rocky Mountain J. Math., 22, 117-124 (1992) · Zbl 0752.34012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.