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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
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[1] C. J. Himmelberg and F. S. Van Vleck, ?Existence of solutions for generalized differential equations with unbounded right-hand side,? J. Diff. Eq.,61, No. 3, 295-320 (1986). · Zbl 0582.34002
[2] C. J. Himmelberg, ?Measurable relations,? Fundam. Math.,87, No. 1, 53-72 (1975). · Zbl 0296.28003
[3] A. Fryszkovski, ?Continuous selections for a class of nonconvex multivalued maps,? Stud. Math.,76, 163-174 (1983).
[4] F. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970). · Zbl 0214.09101
[5] G. Goodman, ?Subfunctions and the initial value problem for differential equations satisfying Carathéodory’s hypotheses,? J. Diff. Eq.,7, No. 2, 232-242 (1970). · Zbl 0276.34003
[6] J. Distel, ?Remarks on weak compactness in L1(?, x),? Glasgow Math. J.,18, No. 1, 87-91 (1977). · Zbl 0342.46020
[7] N. Bourbaki, Topological Vector Spaces [Russian translation], IL, Moscow (1971). · Zbl 0223.01001
[8] L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977). · Zbl 0127.06102
[9] A. F. Filippov, ?The existence of solutions of multivalued differential equations,? Mat. Zametki,10, No. 3, 307-313 (1971).
[10] C. Olech, ?Existence of solutions of nonconvex orientor fields,? Boll. Un. Math. It.,11, Suppl. 3, 189-197 (1975). · Zbl 0322.34002
[11] A. Bressan, ?On differential relations with lower continuous right-hand side. An existence theorem,? J. Diff. Eq.,37, No. 1, 89-97 (1980). · Zbl 0432.34007
[12] S. Lojasiewicz, ?Some theorems of Scorza-Dragoni type for multifunctions with applications to the problem of existence of solutions for differential multivalued equations,? Preprint No. 255, Institute of Mathematics, Polish Academy of Sciences (1982).
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