# The boundary layer problem for certain non-linear ordinary differential equations

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## Abstract

The boundary layer problem for certain non-linear ordinary differential equations COMPOSITIO MATHEMATICA HOWARDG. BERGMANN The boundary layer problem for certain non- linear ordinary differential equations Compositio Mathematica, tome 11 (1953), p. 119-169. <http://www.numdam.org/item?id=CM_1953__11__119_0> © Foundation Compositio Mathematica, 1953, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ The boundary layer problem for certain non-linear ordinary differential equations by Howard G. Bergmann (New York) Introduction. The theory of buckling of circular plates leads to a boundary value problem for a pair of non-linear ordinary differential equa- tions of second order depending upon a parameter. When this parameter approaches zero as a limit, the solution will approach a limit function which is no longer satisfied by all boundary con- ditions. The non-uniform convergence in the "boundary layer" can be studied by an appropriate stretching process. These phenomena have so far been treated only for the special equations resulting from the theory of plates, in particular in [1]. It is the intention of this paper to establish similar phenomena for differential equations of a simpler type, which lend themselves more easily to generalization. The differential equations considered are where p and q are functions of x defined, without loss of generality, in the interval 20131 ~ x ~ 1, with the associated boundary con- ditions pl, P2 being given constants. We shall distinguish three cases: We a

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