Mixed boundary-value problems for singular second-order ordinary differential equations
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 246-266
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It is proved that the boundary-value problem \begin{gather*} -u''+p_0(t)u(t)+\sum^m_{k=2}q_k(t) u^{2k+1}(t)+f_0(t)\varphi(u(t))=f(t), \quad 0<t<1, \\ u(a)=0, \quad u'(b)=0, \end{gather*} has a solution, provided that the following conditions are fulfilled: \begin{gather*} |p_0(t)|(t-a)\in L(a,b), \quad f(t)\sqrt{t-a}\in L(a,b), \\ 0\le f_0(t)\sqrt{t-a}\in L(a,b), \quad 0\le q_k(t)(t-a)^{k+1}\in L(a,b), \\ -c|u|\le\varphi(u)u, \quad c>0, \\ 1-\int^b_a p^-_0(t)(t-a)\,dt>0, \end{gather*} and, for $\varphi(u)\equiv 0$, the Galerkin method converges in the norm of the space $H^1(a,b;a)$. Several theorems of a similar kind are presented.
@article{ZNSL_2006_334_a17,
author = {M. N. Yakovlev},
title = {Mixed boundary-value problems for singular second-order ordinary differential equations},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {246--266},
year = {2006},
volume = {334},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a17/}
}
M. N. Yakovlev. Mixed boundary-value problems for singular second-order ordinary differential equations. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XIX, Tome 334 (2006), pp. 246-266. http://geodesic.mathdoc.fr/item/ZNSL_2006_334_a17/
[1] E. Lib, M. Loss, Analiz, Nauchnaya kniga, Novosibirsk, 1998
[2] M. N. Yakovlev, “Razreshimost nelineinykh uravnenii v konuse prostranstva Banakha”, Zap. nauchn. semin. POMI, 248, 1998, 225–230 | MR | Zbl
[3] M. N. Yakovlev, “Razreshimost singulyarnykh kraevykh zadach dlya obyknovennykh differentsialnykh uravnenii poryadka $2m$”, Zap. nauchn. semin. POMI, 309, 2004, 174–188 | MR | Zbl
[4] M. M. Vainberg, Variatsionnyi metod i metod monotonnykh operatorov, Nauka, M., 1972 | MR | Zbl