Remark on a paper by Barbuti
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Abstract
Barbuti derived some condition for the boundedness of solutions of equation (1) (see the article) and showed the relation of this condition to the results of Caccioppoli and Gusarov. Barbuti's results are interesting, but the formulation of the main theorem and the proof are rather cumbersome. Meanwhile, the main proofs can be carried out in a very elementary way, using the usual deductions that are always used in the study of equation (1); they are available, for example, in B. M. Levitan's book “Decomposition by Eigenfunctions". In the present note we give simplified proofs of some results of Barbuti and Gusarov.
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References
- U. Barbuti. On some theorems of stability. Coll. of Scuola Normale Superiore. Pisa, 1954, 8, No. 1‑2, 81‑91.
- Caccioppoli. A question of stability. Rend. R. Acc. Naz. 1930, of the Lincei, 6, II, 251‑254.
- Gusarov L.A. On the boundedness of solutions of a linear equation of the second order, Doklady AN SSSR, 1949, XVIII, No. 2. (in Russian)
- Ascoli. “On asymptotic computing...”. Rend. Acc. Naz. 1935, of the Lincei (6), 22, 234‑43.
- Ascoli. “On a case of stability...” Ann. of Pure and Applied Matem. Ser. 1947, IV., m. XVI, 199‑201.
- Nemytsky V.V. and Stepanov V.I. Qualitative theory of differential equations Gostekhizdat, 1949. (in Russian)
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