Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space Lp[0, 2π]2 was studied by M.I.Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in
Pringsheim's sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space LP[0, 1]2, p=(p1, p2), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect
to each of the indices.
We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.