Relativistic and non-relativistic theories of bound states mediated by a finite mass meson are discussed. The Fredholm formula is applied to the Bethe-Salpeter equations. Difficulties concerned with divergent Fredholm traces are avoided by factorizing divergent terms which are independent of the eigenvalue problem. An eigenvalue of the coupling constant is found for a given binding energy of two scalar nucleons as well as spinor nucleons. In the nonrelativistic limit the same method is applied to the Schrödinger equation with the static Yukawa potential. The eigenvalue of the coupling constant is found for the real deuteron, and the eigenfunction turns out to be very similar to the Hulthén wave function.