×

A constructive proof of the Cauchy-Kovalevskaya theorem for ordinary differential equations. (English) Zbl 1464.34028

Summary: We give a constructive proof of the classical Cauchy-Kovalevskaya theorem for ordinary differential equations which provides a sufficient condition for an initial value problem to have a unique, analytic solution. Our proof is inspired by a modern numerical technique for rigorously solving nonlinear problems known as the radii polynomial approach. The main idea is to recast the existence and uniqueness of analytic solutions as a fixed point problem on an appropriately chosen Banach space, and then prove a fixed point exists via a constructive version of the Banach fixed point theorem. A key aspect of this method is the use of an approximate solution which plays a crucial role in the theoretical proof. Our proof is constructive in the sense that we provide an explicit recipe for constructing the fixed point problem, an approximate solution, and the bounds necessary to prove the existence of the fixed point.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34A45 Theoretical approximation of solutions to ordinary differential equations

Software:

navierstokes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ebert, MR; Reissig, M., Basics for Partial Differential Equations (2018), New York: Springer, New York · doi:10.1007/978-3-319-66456-9_3
[2] Day, S., Lessard, J.-P., Mischaikow, K.: Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal. 45(4), 1398-1424 (2007) (electronic) · Zbl 1151.65074
[3] Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed-point theorem. SIAM J. Numer. Anal. 35(5), 2004-2013 (1998) (electronic) · Zbl 0972.65084
[4] van den Berg, J.B., Breden, M., Lessard, J.-P., van Veen, L.: Spontaneous periodic orbits in the Navier-Stokes flow (2019). https://arxiv.org/abs/1902.00384
[5] Kepley, S.; Mireles James, JD, Chaotic motions in the restricted four body problem via Devaney’s saddle-focus homoclinic tangle theorem, J. Differ. Equ., 266, 4, 1709-1755 (2015) · Zbl 1422.37062 · doi:10.1016/j.jde.2018.08.007
[6] van den Berg, JB; Deschênes, A.; Lessard, J-P; Mireles James, JD, Stationary coexistence of hexagons and rolls via rigorous computations, SIAM J. Appl. Dyn. Syst., 14, 2, 942-979 (2015) · Zbl 1371.37036 · doi:10.1137/140984506
[7] van den Berg, JB; Jaquette, J., A proof of Wright’s conjecture, J. Differ. Equ., 264, 12, 7412-7462 (2018) · Zbl 1388.34068 · doi:10.1016/j.jde.2018.02.018
[8] Gameiro, M.; Lessard, J-P, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differ. Equ., 249, 9, 2237-2268 (2010) · Zbl 1256.35196 · doi:10.1016/j.jde.2010.07.002
[9] van den Berg, J.B., Queirolo, E.: A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. J. Comput. Dynamics. 8(1), 59-97 (2021) doi:10.3934/jcd.2021004 · Zbl 1469.37015
[10] Murray, M., Mireles James, J.D.: Chebyshev-Taylor parameterization of stable/unstable manifolds for periodic orbits: implementation and applications. Int. J. Bifurc. Chaos 27(14), 1-32 (2017) (submitted) · Zbl 1382.34053
[11] Lessard, J-P; Reinhardt, C., Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal., 52, 1, 1-22 (2014) · Zbl 1290.65060 · doi:10.1137/13090883X
[12] Mireles James, JD; Mischaikow, K., Rigorous a-posteriori computation of (un)stable manifolds and connecting orbits for analytic maps, SIAM J. Appl. Dyn. Syst., 12, 2, 957-1006 (2013) · Zbl 1330.37029 · doi:10.1137/12088224X
[13] Kalies, WD; Kepley, S.; Mireles James, JD, Analytic continuation of local (un)stable manifolds with rigorous computer assisted error bounds, SIAM J. Appl. Dyn. Syst., 17, 1, 157-202 (2018) · Zbl 1409.65110 · doi:10.1137/17M1135888
[14] van den Berg, JB; Mireles James, JD; Reinhard, C., Computing (un)stable manifolds with validated error bounds: non-resonant and resonant spectra, J. Nonlinear Sci., 26, 1055-1095 (2016) · Zbl 1360.37176 · doi:10.1007/s00332-016-9298-5
[15] van den Berg, JB; Mireles James, JD, Parameterization of slow-stable manifolds and their invariant vector bundles: theory and numerical implementation, Discrete Contin. Dyn. Syst. Ser. A, 36, 9, 4637-4664 (2016) · Zbl 1366.37070 · doi:10.3934/dcds.2016002
[16] Gonzalez, JL; MirelesJames, JD, High-order parameterization of stable/unstable manifolds for long periodic orbits of maps, SIAM J. Appl. Dyn. Syst., 16, 3, 1748-1795 (2017) · Zbl 1378.37057 · doi:10.1137/16M1090041
[17] van den Berg, JB; Lessard, J-P, Rigorous numerics in dynamics, Not. Am. Math. Soc., 62, 9, 1057-1061 (2015) · Zbl 1338.68301 · doi:10.1090/noti1276
[18] van den Berg, JB; Mireles James, JD; Lessard, J-P; Wanner, T.; Day, S.; Mischaikow, K., Rigorous Numerics in Dynamics (2018), Providence: American Mathematical Society, Providence · Zbl 1396.37003 · doi:10.1090/psapm/074
[19] Mireles James, J.D.: Validated numerics for equilibria of analytic vector fields: invariant manifolds and connecting orbits. Rigorous Numerics in Dynamics. Proceedings of Symposia in Applied Mathematics, vol 74. pp. 80 (2018). doi:10.1090/psapm/074/02 · Zbl 1409.65109
[20] Arioli, G.; Koch, H., Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113, 51-70 (2015) · Zbl 1304.35181 · doi:10.1016/j.na.2014.09.023
[21] Scheidemann, V.: Introduction to complex analysis in several variables (2005). https://www.springer.com/us/book/9783764374907 · Zbl 1085.32001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.