Abstract
We formulate a continuous function F∶R×H→H, where H is a separable Hilbert space such that the Cauchy problem. x′(t)=F(t, x(t)), x(t0)=x0 has no solution in any neighborhood of the point t0, no matter what t0∃ R and x0∃ H are considered.
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A. N. Godunov, “A counterexample to Peano's Theorem in an infinite-dimensional Hilbert space,” Vestnik Mosk. Gosud. Univ., Ser. Matern, i Mekhan., No. 5, 31–34 (1972).
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Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 467–477, March, 1974.
In conclusion, the author thanks O. G. Smolyanov and V. I. Averbukh for their constant interest and for a number of useful remarks.
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Godunov, A.N. Peano's theorem in an infinite-dimensional Hilbert space is false even in a weakened formulation. Mathematical Notes of the Academy of Sciences of the USSR 15, 273–279 (1974). https://doi.org/10.1007/BF01438383
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DOI: https://doi.org/10.1007/BF01438383