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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

34A60 Ordinary differential inclusions
Full Text: DOI
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