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Abstract. We prove the existence of solutions to first-order, local, time-dependent mean-field games with periodic boundary conditions on the d-dimensional torus. Our approach extends the Banach space monotone operator framework, previously developed for the stationary case, to the substantially harder time-dependent setting. We reformulate the MFG system as a variational inequality for a monotone operator and introduce a low-order p-Laplacian regularization that restores coercivity without high-order smoothing. For the regularized problems, existence follows from an abstract monotone operator theorem. We then derive uniform a priori estimates, including energy bounds, higher integrability via the nonlinear adjoint method, and, by passing to the limit using Minty's method, obtain variational inequality solutions. We further show that these solutions are strong solutions to the original MFG system, adapted to the BV framework. Compared to earlier Hilbert space approaches that rely on high-order elliptic regularization, our method operates in natural Banach spaces, yields stronger solutions, and provides a framework better suited to numerical algorithms.