S. A. Abramov, M. A. Barkatou, and M. Van-hoeij, Apparent singularities of linear difference equations with polynomial coefficients, Applicable Algebra in Engineering, Communication and Computing, vol.17, issue.2, pp.117-133, 2006.
DOI : 10.1007/s00200-005-0193-9
URL : https://hal.archives-ouvertes.fr/hal-00116031

R. P. Agarwal, Contraction and approximate contraction with an application to multi-point boundary value problems, Journal of Computational and Applied Mathematics, vol.9, issue.4, pp.315-325, 1983.
DOI : 10.1016/0377-0427(83)90003-1
URL : https://doi.org/10.1016/0377-0427(83)90003-1

P. R. Arantes-gilz, F. Bréhard, and C. Gazzino, Validated Semi-Analytical Transition Matrix for Linearized Relative Spacecraft Dynamics via Chebyshev Polynomials, 2018 Space Flight Mechanics Meeting, p.24, 2018.
DOI : 10.2514/2.4231

X. Bai, Modified Chebyshev-Picard iteration methods for solution of initial value and boundary value problems, 2010.

A. Benoit, M. Jolde¸sjolde¸s, and M. Mezzarobba, Rigorous uniform approximation of D-finite functions using Chebyshev expansions, Mathematics of Computation, vol.86, issue.305, pp.1303-1341, 2017.
DOI : 10.1090/mcom/3135
URL : https://hal.archives-ouvertes.fr/hal-01022420

J. P. Boyd, Chebyshev and Fourier spectral methods, 2001.
DOI : 10.1007/978-3-642-83876-7

F. Bréhard, N. Brisebarre, and M. Joldes, Validated and numerically efficient Chebyshev spectral methods for linear ordinary differential equations. Preprint (https://hal.archives-ouvertes, 2017.

C. Clenshaw and H. Norton, The solution of nonlinear ordinary differential equations in Chebyshev series, The Computer Journal, vol.6, issue.1, pp.88-92, 1963.
DOI : 10.1093/comjnl/6.1.88

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, Siam, vol.26, 1977.
DOI : 10.1137/1.9781611970425

A. Hungria, J. Lessard, J. D. Mireles, and . James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Mathematics of Computation, vol.85, issue.299, pp.1427-1459, 2016.
DOI : 10.1090/mcom/3046

L. Kantorovich, B. Vulikh, and A. Pinsker, Functional analysis in partially ordered spaces (in Russian), 1950.

Y. Katznelson, An introduction to harmonic analysis, 2004.

E. W. Kaucher and W. L. Miranker, Self-validating numerics for function space problems: Computation with guarantees for differential and integral equations, 1984.

J. Lessard and C. Reinhardt, Rigorous Numerics for Nonlinear Differential Equations Using Chebyshev Series, SIAM Journal on Numerical Analysis, vol.52, issue.1, pp.1-22, 2014.
DOI : 10.1137/13090883X
URL : http://www-m3.mathematik.tu-muenchen.de/foswiki/pub/M3/Allgemeines/Publications/chebsiam.pdf

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, 2002.
DOI : 10.1201/9781420036114

O. M. Nica-bolojan, Fixed point methods for nonlinear differential systems with nonlocal conditions, 2013.

S. Olver and A. Townsend, A Fast and Well-Conditioned Spectral Method, SIAM Review, vol.55, issue.3, pp.462-489, 2013.
DOI : 10.1137/120865458
URL : http://arxiv.org/pdf/1202.1347

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, 1970.
DOI : 10.1137/1.9780898719468

A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pribli?. Metod. Re?en. Differencial'. Uravnen. Vyp, vol.2, pp.115-134, 1964.

R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Mathematical and Computer Modelling, vol.49, issue.3-4, pp.703-708, 2009.
DOI : 10.1016/j.mcm.2008.04.006

]. F. Robert, ´ Etude et utilisation de normes vectorielles en analyse numérique linéaire (in French), 1968.

J. B. Van-den, J. Berg, and . Lessard, Rigorous Numerics in Dynamics, Notices of the American Mathematical Society, vol.62, issue.09, p.2015
DOI : 10.1090/noti1276

N. Yamamoto, A Numerical Verification Method for Solutions of Boundary Value Problems with Local Uniqueness by Banach's Fixed-Point Theorem, SIAM Journal on Numerical Analysis, vol.35, issue.5, pp.2004-2013, 1998.
DOI : 10.1137/S0036142996304498

T. Yamamoto, A unified derivation of several error bounds for Newton's process, Journal of Computational and Applied Mathematics, vol.12, issue.13, pp.179-191, 1985.
DOI : 10.1016/0377-0427(85)90015-9