We introduce a new class of almost periodic operators and establish conditions for the existence of almost periodic solutions of nonlinear equations that are not necessarily almost periodic in Bochner’s sense.
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S. Bochner, “Beitrage zur Theorie der fastperiodischen. I Teil. Funktionen einer Variablen,” Math. Ann., 96, 119–147 (1927); “Beitrage zur Theorie der fastperiodischen. II Teil. Funktionen mehrerer Variablen,” Math. Ann., 96, 383–409 (1927).
B. M. Levitan, Almost Periodic Functions [in Russian], Gostekhizdat, Moscow (1953).
L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).
L. A. Lyusternik and V. I. Sobolev, Elements of Functional Analysis [in Russian], Nauka, Moscow (1965).
V. Yu. Slyusarchuk, “Conditions of almost periodicity of bounded solutions for nonlinear difference equations with continuous argument,” Nelin. Kolyv., 16, No. 1, 118–124 (2013).
V. Yu. Slyusarchuk, “Criterion for the existence of almost periodic solutions of nonlinear difference equations without using the H-classes of these equations,” Bukovyn. Mat. Zh., 1, No. 1-2, 136–138 (2013).
V. Yu. Slyusarchuk, “Conditions for the existence of almost periodic solutions of nonlinear difference equations with discrete argument,” Nelin. Kolyv., 16, No. 3, 416–425 (2013).
V. Yu. Slyusarchuk, “Conditions for the existence of almost periodic solutions of nonlinear difference equations in Banach spaces,” Ukr. Mat., Zh., 65, No. 2, 307–312 (2013); English translation: Ukr. Math. J., 65, No. 2, 341–347 (2013).
L. Amerio, “Soluzioni quasiperiodiche, o limital, di sistemi differenziali non lineari quasi-periodici, o limitati,” Ann. Mat. Pura Appl., 39, 97–119 (1955).
B. P. Demidovich, Lectures on the Mathematical Theory of Stability [in Russian], Nauka, Moscow (1967).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
J. Favard, “Sur les équations différentielles à coefficients presquepériodiques,” Acta Math., 51, 31–81 (1927).
É. Mukhamadiev, “On the invertibility of functional operators in the space of functions bounded on the axis,” Mat. Zametki, 11, No. 3, 269–274 (1972).
É. Mukhamadiev, “Investigations into the theory of periodic and bounded solutions of differential equations,” Mat. Zametki, 30, No. 3, 443–460 (1981).
V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116(158), No. 4(12), 483–501 (1981).
V. E. Slyusarchuk, “Invertibility of nonautonomous functional-differential operators,” Mat. Sb., 130(172), No. 1(5), 86–104 (1986).
V. E. Slyusarchuk, “Necessary and sufficient conditions for the invertibility of nonautonomous functional-differential operators,” Mat. Zametki, 42, No. 2, 262–267 (1987).
L. Amerio, “Sull equazioni differenziali quasi-periodiche astratte,” Ric. Mat., 30, 288–301 (1960).
V. V. Zhikov, “Proof of the Favard theorem on the existence of almost periodic solutions for an arbitrary Banach space,” Mat. Zametki, 23, No. 1, 121–126 (1978).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 2, pp. 230–244, February, 2015.
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Slyusarchuk, V.Y. Almost Periodic Solutions of Nonlinear Equations that are not Necessarily Almost Periodic in Bochner’s Sense. Ukr Math J 67, 267–282 (2015). https://doi.org/10.1007/s11253-015-1078-0
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DOI: https://doi.org/10.1007/s11253-015-1078-0