We consider a nonlinear system on the direct product \( {{\mathbb{R}}^m}\times {{\mathbb{R}}^n} \). For this system, under the conditions of indefinite coercivity and indefinite monotonicity, we establish the existence of a bounded Lipschitz invariant section over \( {{\mathbb{R}}^m} \).
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References
A. M. Samoilenko, “On preservation of the invariant torus under perturbations,” Izv. Akad. Nauk SSSR, Ser. Mat., 34, No. 6, 1219–1240 (1970).
A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations, Kluwer, Dordrecht (1991).
A. M. Samoilenko, “Perturbation theory of smooth invariant tori of dynamical systems,” Nonlin. Anal., 30, No. 5, 3121–3133 (1997).
I. M. Grod, “On the smoothness of bounded invariant manifolds of linear inhomogeneous extensions of dynamical systems,” Ukr. Math. J., 48, No. 1, 154–157 (1996).
S. B. Bodnaruk and V. L. Kulik, “On the parameter dependence of bounded invariant manifolds of autonomous systems of differential equations,” Ukr. Math. J., 48, No. 6, 838–845 (1996).
A. M. Samoilenko, Yu. V. Teplins’kyi, and I. V. Semenyshyna, “On the existence of a smooth bounded semiinvariant manifold for a degenerate nonlinear system of difference equations in the space m,” Nonlin. Oscillat., 6, No. 3, 371–392 (2003).
M. O. Perestyuk and V. Yu. Slyusarchuk, “Green-Samoilenko operator in the theory of invariant sets of nonlinear differential equations,” Ukr. Mat. Zh., 60, No. 7, 948–957 (2008); English translation: Ukr. Math. J., 60, No. 7, 1123–1136 (2008).
É. Mukhamadiev, Kh. Nazhmiddinov, and B. N. Sadovskii, “Application of the Schauder–Tikhonov principle to the problem of bounded solutions of differential equations,” Funkts. Anal. Prilozh., 6, No. 6, 83–84 (1972).
V. E. Slyusarchuk, “Invertibility of almost periodic c-continuous functional operators,” Mat. Sb., 116, No. 4, 483–501 (1981).
T. Ważewski, “Sur un principe topologique de l’examen de l’allure asymptotique des integrales des equations differentielles ordinaires,” Ann. Pol. Math., 20, 279–313 (1947).
R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
A. M. Samoilenko, I. O. Parasyuk, and V. A. Lahoda, “Lipschitz invariant tori of indefinite-monotone systems,” Ukr. Mat. Zh., 64, No. 3, 363–383 (2012); English translation: Ukr. Math. J., 64, No. 3, 408–432 (2012).
A. M. Samoilenko, M. O. Perestyuk, and I. O. Parasyuk, Differential Equations. A Textbook [in Ukrainian], Kyivs’kyi Universytet, Kyiv (2010).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 1, pp. 103–118, January, 2013.
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Lahoda, V.A., Parasyuk, I.O. Theorem on the existence of an invariant section over \( {{\mathbb{R}}^m} \) for the indefinite monotone system in \( {{\mathbb{R}}^m}\times {{\mathbb{R}}^n} \) . Ukr Math J 65, 114–131 (2013). https://doi.org/10.1007/s11253-013-0768-8
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DOI: https://doi.org/10.1007/s11253-013-0768-8