A viable result for nonconvex differential inclusions with memory

Vasile Lupulescu and Mihai Necula

Abstract: Let $X$ be a separable Banach space, $\sigma>0$ and $\mathcal{C}_{\sigma}:=\mathcal{C}([-\sigma,0],X)$} the Banach space of the continuous functions from $[-\sigma,0]$ into $X$, $K$ a locally closed set in $X$ and $F\dpt[a,b)\times\mathcal{C}_{\sigma}\to 2^{X}$} a closed valued and locally integrable bounded multifunction, with $F(.,\varphi)$ measurable and $F(t,.)$ Lipschitz continuous in the Hausdorff--Pompeiu metric. In this paper we establish some sufficient conditions in order that, for each $\tau\in\lbrack a,b)$ and for each $\varphi\in\mathcal{C}_{\sigma}$ with $\varphi(0)\in K$, there exist at least one solution $u:[\tau-\sigma,T]\to X$ of the differential inclusion $u^{\prime}(t)\in F(t,u_{t})$, such that $u_{\tau}=\varphi$ on $[-\sigma,0]$ and $u(t)\in K$ for every $t\in\lbrack\tau,T]$.

Keywords: differential inclusions with memory; viability result.

Classification (MSC2000): 34A60, 49K25.

Full text of the article:

• Compressed PostScript file (184 kilobytes)
• PDF file (200 kilobytes)