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On the equation \(x'=f(t,x)\) in locally convex spaces. (English) Zbl 0569.34052

Let E be a quasicomplete locally convex topological vector space and let P be a family of continuous seminorms generating the topology of E. Assume that \(x_ 0\in E\) and \(B=\{x\in E: p_ i(x-x_ 0)\leq b\) for \(i=1,...,r\}\) where \(p_ 1,...,p_ r\in P\). Theorem. The problem \(x'(t)=f(t,x(t))\), \(x(0)=x_ 0\), where f: [0,a]\(\times B\to E\) is a bounded continuous function, has at least one solution on \(J=[0,d]\), where \(d=\min (a,b/M)\) and \(M=\sup \{p_ i(f(t,x)): x\in B,t\in [0,a]\), \(i=1,...,r\}\), if one of the following conditions holds. a) \(\beta_ p(f(J\times X))\leq h_ p(\beta_ p(X))\) for any \(p\in P\) and any bounded subset X of B; b) for every \(p\in P\) the space E is p-separable and \(\beta_ p(f(t,X))\leq h_ p(t,\beta_ p(X))\) for any \(p\in P\), \(t\in J\) and any bounded subset X of B; c) the function f is uniformly continuous and \(\beta_ p(f(t,X))\leq h_ p(t,\beta_ p(X))\) for any \(p\in P\), \(t\in J\) and any bounded subset X of B, where \(\beta_ p\) is a ”ball” measure of noncompactness and \(h_ p\) is a Kamke function.
Reviewer: G.Bottaro

MSC:

34G10 Linear differential equations in abstract spaces
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