Szufla, Stanisław On the equation \(x'=f(t,x)\) in locally convex spaces. (English) Zbl 0569.34052 Math. Nachr. 118, 179-185 (1984). 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 Cited in 1 ReviewCited in 5 Documents MSC: 34G10 Linear differential equations in abstract spaces Keywords:quasicomplete locally convex topological vector space; Kamke function PDFBibTeX XMLCite \textit{S. Szufla}, Math. Nachr. 118, 179--185 (1984; Zbl 0569.34052) Full Text: DOI References: [1] Ambrosetti, Rend. Sem. Mat. Univ. Padova 39 pp 349– (1967) [2] Cellina, Funkcial. Ekvac. 14 pp 129– (1971) [3] Ordinary differential equations in Banach spaces, Lecture Notes 596, Springer-Verlag 1977 · doi:10.1007/BFb0091636 [4] Goebel, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 18 pp 367– (1970) [5] Hukuhara, J. Fac. Sci. Univ. Tokyo, Sec. I 8 pp 111– (1959) [6] General topology, Toronto–New York–London 1957 [7] Topology, vol. II, New York–London–Warszawa 1968 [8] [Russian Text Ignored.], [Russian Text Ignored.] AH CCCP 131 pp 510– (1960) [9] Mönch, J. Nonlin. Anal. 4 pp 985– (1980) [10] Olech, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys. 8 pp 667– (1960) [11] [Russian Text Ignored.], [Russian Text Ignored.] 27 pp 81– (1972) [12] Topological vector spaces, New York–London 1966 [13] Szufla, Boll. Un. Mat. Ital. 15-A pp 535– (1978) [14] Vidossich, J. Math. Anal. Appl. 36 pp 581– (1971) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.