We study bounded solutions of a nonlinear Lyapunov-type problem in Banach and Hilbert spaces. Necessary and sufficient conditions for the existence of bounded solutions on the entire axis are obtained under the assumption that the homogeneous equation admits exponential dichotomy on the semiaxes. Conditions for the existence of homoclinic chaos in nonlinear evolution equations are presented.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 6, pp. 761–773, June, 2019
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Boichuk, O.A., Pokutnyi, O.O. Bounded Solutions of the Nonlinear Lyapunov Equation and Homoclinic Chaos. Ukr Math J 71, 869–882 (2019). https://doi.org/10.1007/s11253-019-01685-w
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DOI: https://doi.org/10.1007/s11253-019-01685-w