A Theorem Concerning Uniform Simplification at a Transition Point and the Problem of Resonance

Given sectors S = \ ε ;aⱼ < ε < bⱼ ,0 < | ε | < ρ \(1 j ν ) and functions δ ⱼ (1 j ν ) such that (i) ∪ ⱼ Sⱼ = \ ε ;0 < | ε | < ρ \, (ii) δ ⱼ is holomorphic in Sⱼ, (iii) δ ⱼ is asymptotically zero as ε → 0 in Sⱼ (iv) | δ ⱼ (ε ) - δ ₖ (ε ) | c₀ exp \ - c₁ / | ε |^λ \ in Sⱼ ∩ Sₖ for some positive numbers c₀, c₁ and λ whenever Sⱼ ∩ Sₖ ≠ ∅, we prove that | δ ⱼ (ε ) | c₂ exp \ - c₁ / | ε |^λ \ in Sⱼ for some positive number c₂. Then, utilizing this result, we prove that the Matkowsky condition implies resonance in the sense of N. Kopell under a reasonable assumption. The sufficiency of the Matkowsky condition with regard to Ackerberg–O’Malley resonance has been an open question. This work gives an affirmative answer to this question in a reasonably general case.