Poisson Stable Solutions of Semi-Linear Differential Equations

David Cheban

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2024), pp. 17-43.

Résumé

We study the problem of existence of Poisson stable (in particular, almost periodic, almost automorphic, recurrent) solutions to the semi-linear differential equation $$ x'=(A_0+A(t))x+F(t,x) $$ with unbounded closed linear operator $A_0$, bounded operators $A(t)$ and Poisson stable functions $A(t)$ and $F(t,x)$. Under some conditions we prove that there exists a unique (at least one) solution which possesses the same recurrence property as the coefficients.

Bibliographie

[1] Group, Sijthoff Noordhoff, Alphen aan den Rijn, 1979, viii+319 pp. | MR

[2] Cheban D. N., “On the Comparability of Points of Dynamical Systems with regard to the Character of Recurrence Property in the Limit”, Mathematical Sciences, 1, "Shtiintsa", Kishinev, 1977, 66-71

[3] Cheban D. N., “Levitan Almost Periodic and Almost Automorphic Solutions of $V$-monotone Differential Equations”, J.Dynamics and Differential Equations, 20:3 (2008), 669–697 | DOI | MR | Zbl

[4] David N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, 2009, ix+186 pp. | MR | Zbl

[5] David N. Cheban, Global Attractors of Nonautonomous Dynamical and Control Systems, Interdisciplinary Mathematical Sciences, 18, 2nd Edition, World Scientific, River Edge, NJ, 2015, xxv+589 pp. | MR

[6] David N. Cheban, Nonautonomous Dynamics: Nonlinear Oscillations and Global Attractors, Springer Nature Switzerland AG, 2020, xxii+434 pp. | MR | Zbl

[7] Chicone C. and Latushkin Yu., Evolution Semigroups in Dynamicals Systems and Differential Equations, Amer. Math. Soc., Providence, RI, 1999, 361 pp. | MR

[8] Shui-Nee Chow and Hugo Leiva, “Dynamical Spectrum for Time Dependent Linear Systems in Banach Spaces”, Japan J. Indust. Appl. Math., 11 (1994), 379-415 | DOI | MR | Zbl

[9] Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, New York, 1981 | DOI | MR | Zbl

[10] Levitan B. M., Almost Periodic Functions, Gos.Izdat.Tekhn-Teor. Lit., M., 1953, 396 pp. (in Russian) | MR | Zbl

[11] Levitan B. M. and Zhikov V. V., Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982 | MR | Zbl

[12] Lusternik L. A. and Sobolev V. L., Elements of Functional Analysis, Hindustan Publishing Corporation, India, 1975, xvi+360 pp. | MR

[13] Pazy A., Semigroup of Linear Operators and applications to Partial Differential Equations, Springer-Verlag New York Inc., 1983, x+279 pp. | MR

[14] Sell G. R., Topological Dynamics and Ordinary Differential Equations, Van Nostrand-Reinhold, 1971 | MR | Zbl

[15] Sov. Math., Dokl., 1962, no. 3, 1320-1322 | MR | Zbl

[16] Shcherbakov B. A., “On Classes of Poisson Stability of Motion. Pseudorecurrent motions”, Bul. Akad. Stiince RSS Moldoven, 1963, no. 1, 58–72 (in Russian) | MR | Zbl

[17] Shcherbakov B. A., Topologic Dynamics and Poisson Stability of Solutions of Differential Equations, Ştiinţa, Chişinău, 1972, 231 pp. (in Russian) | MR

[18] Shcherbakov B. A., “The Compatible Recurrence of Bounded Solutions of First Order Differential Equations”, Differencial'nye Uravnenija, 10 (1974), 270–275 (in Russian) | MR | Zbl

[19] Shcherbakov B. A., “The Nature of Recurrence of Solutions of Linear Differential Systems”, An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.), 21 (1975), 57–59 (in Russian) | MR | Zbl

[20] Differential Equations, 11:7 (1975), 937–943 | MR | Zbl

[21] Shcherbakov B. A., Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations, Shtiintsa, Kishinev, 1985, 147 pp. (in Russian) | MR | Zbl

[22] Shcherbakov B. A. and Koreneva L.V., “The Preservation of the Nature of the Recurrence of the Solutions of Differential Equations under Perturbations”, Differencial'nye Uravnenija, 10:3 (1974), 470–475 (in Russian) | MR | Zbl

[23] Introduction to Topological Dynamics, Noordhoff, Leyden, 1975 | MR | MR | Zbl