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A Priori Estimates for Solutions of Singular Perturbations with a Turning Point
Author(s) -
Abrahamson Leif R.
Publication year - 1977
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm197756151
Subject(s) - a priori and a posteriori , turning point , bounded function , mathematics , a priori estimate , zero (linguistics) , simple (philosophy) , mathematical analysis , point (geometry) , boundary value problem , boundary (topology) , singular point of a curve , physics , geometry , philosophy , linguistics , epistemology , acoustics , period (music)
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.
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