## On the existence of $$\psi$$-minimal viable solutions for a class of differential inclusions.(English)Zbl 0759.34014

Let $$X$$ be a Banach space, $$\psi: X\to R$$ a convex, continuous function, $$K\subset X$$, $$\text{int}(K)\neq\emptyset$$. $$F$$ is a continuous multifunction on $$K$$ with convex values in $$X$$ and satisfies standard conditions required for the existence of solutions of (1) $$\dot x\in F(x)$$, including $$\alpha(F(B))\leq k'\alpha(B)$$, where $$\alpha$$ denotes the measure of noncompactness. In his main result the author proves that if $$F(x)\cap\text{int}(T_ K(x))\neq\emptyset$$ ($$T_ K$$ — Bouligand’s tangent cone) then there is a solution $$x(\cdot)$$ of (1) for which $$\psi(\dot x(t))=\inf\{\psi(z)$$: $$z\in F(x(t))\cap T_ K(x(t))\}$$. If $$X$$ is finite dimensional then the tangency condition is weakened to $$F(x)\cap T_ K(x)\neq\emptyset$$.

### MSC:

 34A60 Ordinary differential inclusions
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