Affine-periodic solutions for generalized ODEs and other equations
DOI:
https://doi.org/10.12775/TMNA.2022.027Keywords
Affine-periodic solutions, Henstock-Kurzweil-Stieltjes integral, Krasnosel'skiĭ fixed point theorem, Banach fixed point theoremAbstract
It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible $n\times n$ matrix $Q$, with entries in $\mathbb C$, we establish conditions for the existence of a $(Q,T)$-affine-periodic solution within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ. We apply our main results to measure differential equations with Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.References
E. Alvarez, A. Gómez and M. Pinto, (ω, c)-periodic functions and mild solutions to abstract fractional integro-differential equations, Electronic J. Qualitative Theory of Diff. Eq. 16 (2018), 1–8.
H. Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, Gruyter Studies in Mathematics, vol. 13, Walter de Gruyter Co., Berlin, 1990.
sc T.M. Apostol, Mathematical Analysis, Addison Wesley Publishing Company, second edition, 1974.
T.S. Blyth and E.F. Robertson, Basic Linear Algebra, Springer, second edition, 2002.
A. Bounemoura, B. Fayad and L. Niederman, Superexponential stability of quasiperiodic motion in Hamiltonian systems, Commun. Math. Phys. 350 (2017), 361–386.
K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, Berlin, 1985.
M. Federson, J.G. Mesquita and A. Slavı́k, Measure functional differential equations and functional dynamic equations on time scales, J. Differential Equations 252 (2012), no. 6, 3816–3847.
M. Federson, J.G. Mesquita and A. Slavı́k, Basic results for functional differential and dynamic equations involving impulses, Math. Nachr. 286 (2013), no. 2–3, 181–204.
M. Federson, R. Grau and J.G. Mesquita, Prolongation of solutions of measure differential equations and dynamical equations on time scales, Math. Nachr. 292 (2019), no. 1, 22–55.
D. Franková, Regulated functions, Math. Bohem. 116 (1991), 20–59.
I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Oxford Science Publications, Oxford, New York, 1995.
R.E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lectures Notes in Math, vol. 568, Springer–Verlag, 1977.
B.C. Hall, Lie Groups, Lie Algebras, and Representations: an Elementary Introduction, Springer–Verlag, New York, 2003.
R. Henstock, The equivalence of generalized forms of the Ward, variational, Denjoy–Stieltjes, and Perron–Stieltjes integrals, Proc. London Math. Soc. 10 (1960), no. 3, 281–303.
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Springer–Verlag, Berlin, 1991.
C.S. Hönig, Volterra–Stieltjes Integral Equations. Functional Analytic Methods; Linear Constraints , Mathematics Studies, No. 16, Notas de Matemática, No. 56, [Notes on Mathematics, No. 56], North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., New York, 1975.
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd edition, World Scientific Publishing, New Jersey, London, Singapore, 2004.
A.N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk 98 (1954), 527–530. (Russian)
M.A. Krasnosel’skiı̆, Some problems of Nonlinear Analysis, Amer. Math. Soc. Transl. Ser. 2 10 (1958), no. 2, 345–409.
J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), no. 82, 418–449. (Russian)
J. Kurzweil, Generalized ordinary differential equations, Czechoslovak Math. J. 8 (1958), no. 83, 360–388.
E.L. Lima, Álgebra Linear, IMPA, Rio de Janeiro, 2008. (Portuguese)
J.G. Mesquita, Measure functional differential equations and impulse functional dynamic equations on time scales, Ph.D. thesis, University of São Paulo, São Carlos, 2012.
J.G. Mesquita and A. Slavı́k, Periodic averaging theorems for various types of equations, J. Math. Anal. Appl. 387 (2012), no. 2, 862–877.
G. Monteiro, A. Slavı́k and M. Tvrdý, Kurzweil–Stieltjes Integral Theory and Applications, Series in Real Analysis, vol. 14, World Scientific Publishing Co., River Edge, NJ, 2018.
P. Muldowney, A Modern Theory of Random Variation. With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration, John Wiley & Sons, Inc., Hoboken, NJ, 2012.
H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Tome I, Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotiques, Dover Publications, Inc., New York, N.Y., 1957. (French)
E.H. Rothe, Introduction to Various Aspects of Degree Theory in Banach Spaces, Mathematical Surveys and Monographs, vol. 23, American Mathematical Society, Providence, RI, 1986.
Š. Schwabik, Generalized Ordinary Differential Equations, World Scientific, Series in Real Anal., vol. 5, 1992.
C. Wang and Y. Li, Affine-periodic solutions for nonlinear dynamic equations on time scales, Affine-periodic solutions for nonlinear dynamic equations on time scales, Adv. Difference Equ. (2015), 2015:286, 16 pp. (English summary)
C. Wang, X. Yang and Y. Li, The existence of affine-periodic solutions for nonlinear differential equations, Rocky Mountain J. Math. 46 (2016), no. 5, 1717–1737.
H. Wang, X. Yang and Y. Li, Rotating-symmetric solutions for nonlinear systems with symmetry, Acta Math. Appl. Sin. Engl. Ser. 31 (2015), no. 2, 307–312. (English summary)
S. Wang, The existence of affine-periodic solutions for nonlinear impulsive differential equations, Bound. Value Probl. (2018), paper no. 113, 13 pp.
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Copyright (c) 2022 Márcia Federson, Rogelio Grau, Carolina Mesquita
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