Geodesic
On the absence of exact means for some bounded functions
Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 431-447.

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It is shown that there exist almost periodic functions $f(t)$ such that the function $$ \varphi(t)=\frac1{c+\int^t_0f(t)\,dt} $$ is bounded but does not have an exact mean value. This fact implies that there are first-order Bernoulli differential equations with almost periodic coefficients such that some bounded solutions do not have exact mean values. Bibliography: 2 titles. 
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I. N. Blinov. On the absence of exact means for some bounded functions. Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 431-447. http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a1/

[1] Blinov I. N., “Effekt ischeznoveniya pochti periodicheskikh reshenii u nelineinogo differentsialnogo uravneniya s kvaziperiodicheskimi koeffitsientami”, Izv. AN SSSR. Ser. matem., 46:6, 1333–1341 | MR | Zbl

[2] Zigmund A., Trigonometricheskie ryady, Mir, M., 1965 | MR