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The relativistic $S$-matrix formalism of Feynman is applied to the bound-state problem for two interacting Fermi-Dirac particles. The bound state is described by a wave function depending on separate times for each of the two particles. Two alternative integral equations for this wave function are derived with kernels in the form of an expansion in powers of ${g}^{2}$, the dimensionless coupling constant for the interaction. Each term in these expansions gives Lorentz-invariant equations. The validity and physical significance of these equations is discussed. In extreme nonrelativistic approximation and to lowest order in ${g}^{2}$ they reduce to the appropriate Schr\"odinger equation.One of these integral equations is applied to the deuteron ground state using scalar mesons of mass $\ensuremath{\mu}$ with scalar coupling. For neutral mesons the Lorentz-invariant interaction is transformed into the sum of the instantaneous Yukawa interaction and a retarded correction term. The value obtained for ${g}^{2}$ differs only by a fraction proportional to ${(\frac{\ensuremath{\mu}}{M})}^{2}$ from that obtained by using a phenomenological Yukawa potential. For a purely charged meson theory a correction term is obtained by a direct solution of the relativistic integral equation using only the first term in the expansion of the kernel. This correction is due to the fact that a nucleon can emit, or absorb, positive and negative mesons only alternately. The constant ${g}^{2}$ is increased by a fraction of $1.1(\frac{\ensuremath{\mu}}{M})$ or 15 percent.