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Solutions of a differential inclusion with unbounded right-hand side. (English. Russian original) Zbl 0702.34019
Sib. Math. J. 29, No. 5, 857-868 (1988); translation from Sib. Mat. Zh. 29, No. 5(171), 212-225 (1988).
Let the multivalued mapping F: $$T\times X\to X$$ $$(T=(0,1)$$, X an Euclidean space) have the following properties: 1) F is $${\mathcal L}\times {\mathcal B}_ x$$ measurable; 2) for almost all $$t\in T$$ and each x either F(t,.) has a closed graph at x and F(t,x) is convex or the restriction of F(t,.) to some neighbourhood of x is lower semicontinuous; 3) there exists a function f: $$T\times X\to {\mathbb{R}}^+$$ of Carathéodory type which is integrally bounded on bounded subsets of X and $$F(t,x)\cap \bar B(0,f(t,x))\neq \emptyset$$ almost everywhere on T for any x. Let g: $$T\times {\mathbb{R}}^+\to {\mathbb{R}}^+$$ be nondecreasing in the second argument and assume that $$\dot r=g(t,r)$$ has a maximal solution $$r(t),r(0)=r_ 0$$ on T for some $$r_ 0\geq 0$$. Consider the differential inclusion (1) $$\dot x\in F(t,x)$$. Theorem. let the mapping F: $$T\times X\to X$$ satisfy 1)-3) with $$f(t,x)=g(t,\| x\|)$$. Then for any $$x_ 0$$, $$\| x_ 0\| \leq r_ 0$$, (1) has a solution x(t), $$x(0)=x_ 0$$ defined on T.
Reviewer: I.Vrkoč

##### MSC:
 34A60 Ordinary differential inclusions
Full Text:
##### References:
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