A Priori Estimates for Solutions of Singular Perturbations with a Turning Point

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.