We present an approximation method for Picard second order boundary value problems with Carathéodory righthand side. The method is based on the idea of replacing a measurable function in the right-hand side of the problem with its Kantorovich polynomial. We will show that this approximation scheme recovers essential solutions to the original BVP. We also consider the corresponding finite dimensional problem. We suggest a suitable mapping of solutions to finite dimensional problems to piecewise constant functions so that the later approximate a solution to the original BVP. That is why the presented idea may be used in numerical computations.
@article{bwmeta1.element.doi-10_2478_s11533-013-0323-8,
author = {Jacek Gulgowski},
title = {Approximation of solutions to second order nonlinear Picard problems with Carath\'eodory right-hand side},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {155-166},
zbl = {1297.34017},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0323-8}
}
Jacek Gulgowski. Approximation of solutions to second order nonlinear Picard problems with Carathéodory right-hand side. Open Mathematics, Tome 12 (2014) pp. 155-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0323-8/
[1] Besov K.O., On the continuity of the generalized Nemytskii operator on spaces of differentiable functions, Math. Notes, 2002, 71(1–2), 154–165 http://dx.doi.org/10.1023/A:1013998928829 | Zbl 1026.47050
[2] Capietto A., Mawhin J., Zanolin F., On the existence of two solutions with a prescribed number of zeros for a superlinear two-point boundary value problem, Topol. Methods Nonlinear Anal., 1995, 6(1), 175–188 | Zbl 0849.34018
[3] Cui Y., Sun J., Zou Y., Global bifurcation and multiple results for Sturm-Liouville problems, J. Comput. Appl. Math., 2011, 235(8), 2185–2192 http://dx.doi.org/10.1016/j.cam.2010.10.014 | Zbl 1223.34030
[4] DeVore R.A., Lorentz G.G., Constructive Approximation, Grundlehren Math. Wiss., 303, Springer, Berlin, 1993 http://dx.doi.org/10.1007/978-3-662-02888-9
[5] Erbe L., Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems, Math. Comput. Modelling, 2000, 32(5–6), 529–539 http://dx.doi.org/10.1016/S0895-7177(00)00150-3 | Zbl 0970.34019
[6] Granas A., Guenther R.B., Lee J.W., On a theorem of S. Bernstein, Pacific J. Math., 1978, 74(1), 67–82 http://dx.doi.org/10.2140/pjm.1978.74.67 | Zbl 0377.34003
[7] Granas A., Guenther R.B., Lee J.W., Nonlinear boundary value problems for some classes of ordinary differential equations, Rocky Mountain J. Math., 1980, 10(1), 35–58 http://dx.doi.org/10.1216/RMJ-1980-10-1-35 | Zbl 0476.34017
[8] Granas A., Guenther R.B., Lee J., Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissertationes Math. (Rozprawy Mat.), 244, Polish Acad. Sci. Inst. Math., Warsaw, 1985 | Zbl 0615.34010
[9] Jentzen A., Neuenkirch A., A random Euler scheme for Carathéodory differential equations, J. Comput. Appl. Math., 2009, 224(1), 346–359 http://dx.doi.org/10.1016/j.cam.2008.05.060 | Zbl 1160.65003
[10] Krasnoselskii M.A., The continuity of the operator fu(x) = f[x; u(x)], Doklady Akad. Nauk SSSR, 1951, 77(2), 185–188 (in Russian)
[11] Krasnoselskii M.A., Topological Methods in the Theory of Nonlinear Integral Equations, International Series of Monographs on Pure and Applied Mathematics, 45, Pergamon Press, New York, 1964
[12] Lorentz G.G., Bernstein Polynomials, Chelsea Publishing, New York, 1986
[13] Rynne B.P., Second-order Sturm-Liouville problems with asymmetric, superlinear nonlinearities, Nonlinear Anal., 2003, 54(5), 939–947 http://dx.doi.org/10.1016/S0362-546X(03)00119-6 | Zbl 1037.34019
[14] Rynne B.P., Second order, Sturm-Liouville problems with asymmetric, superlinear nonlinearities. II, Nonlinear Anal., 2004, 57(7–8), 905–916 http://dx.doi.org/10.1016/j.na.2004.03.021 | Zbl 1065.34021
[15] Stengle G., Numerical methods for systems with measurable coefficients, Appl. Math. Lett., 1990, 3(4), 25–29 http://dx.doi.org/10.1016/0893-9659(90)90040-I | Zbl 0725.65071
[16] Stengle G., Error analysis of a randomized numerical method, Numer. Math., 1995, 70(1), 119–128 http://dx.doi.org/10.1007/s002110050113 | Zbl 0817.65058
[17] Vainberg M.M., Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco, 1964