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We consider the sufficiency of the Matkowsky condition concerning the differential equation      ε  y + f ( x , ε  ) y ′ + g ( x , ε  ) y = 0 ( −  a ⩽  x ⩽  b ) \varepsilon y + f(x,\varepsilon )y’ + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)       under the assumption that      f ( 0 , ε  ) = 0 f(0,\varepsilon ) = 0       identically in      ε  , f x ( 0 , ε  ) ≠  0 \varepsilon ,{f_x}(0,\varepsilon ) \ne 0       with      f &gt; 0 f &gt; 0       for      x &gt; 0 x &gt; 0       and      f &gt; 0 f &gt; 0       for      x &gt; 0 x &gt; 0      . Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if      f f       and      g g       are convergent power series for      | ε  | &gt; ρ  ( ρ  &gt; 0 ) , f ( x , 0 ) = −  2 x |\varepsilon | &gt; \rho \;(\rho &gt; 0),f(x,0)=-2x       and the interval      [ −  a , b ] [ - a,b]       is contained in a disc      D D       with center at      0 0      . The main problem in this work is to remove from Sibuya’s result the assumption that      D D       is a disc.

The sufficiency of the Matkowsky condition in the problem of resonance