Studies in Applied Mathematics

Lebovitz, N. R.; Schaar, R. J.

doi: 10.1002/sapm19775611pmid: N/A

In the theory of singularly perturbed initial‐value problems, the principal assumption concerns a certain Jacobian matrix: all its eigenvalues should have negative real parts at each point of the reduced (or degenerate) path. If the reduced path contains a point of bifurcation, this assumption is violated. The simplest kind of bifurcation with exchange of stabilities involves just two smooth curves intersecting at a single point. The analysis of the singular perturbation theory in the case when bifurcation is present depends on whether or not both curves have finite slopes at the point of bifurcation. The case when both slopes are finite was treated in (1); the case when the bifurcating curve has a vertical tangent is treated in the present paper.

Abrahamson, Leif R.

doi: 10.1002/sapm197756151pmid: N/A

A priori estimates are derived for the solutions of the boundary value problem εy″ + a(x)y′ + b(x)y = f(x), c ⩽ x ⩽ d, y(c) = α, y(d) = β. Here 0 < ε ≪ 1 is a small parameter and a(x) has a single simple zero in (c,d) (the turning point). It is shown that the solutions of this problem are uniformly bounded for ε→0 by the norms of f, α and β if and only if b(x)<0 at the turning point. However, in certain cases there are weak a priori estimates for the solutions even if this condition is not fulfilled.

Cain, B. E.; Saunders, B. D.; Schneider, Hans

doi: 10.1002/sapm197756171pmid: N/A

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non‐zero vector has a positive multiple which is the sum of two v‐dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc‐connectedness of the numerical range determined by v.

Benney, D. J.

doi: 10.1002/sapm197756181pmid: N/A

Nonlinear partial differential equations which permit both short and long wave solutions are studied. The theory reveals a variety of phenomena relevant to a broad class of physical problems. Among the properties investigated are resonances, instabilities and steady state solutions.