It is well known that, if in addition to usual smoothness conditions, the functions f and the prescribed data u0(x) are periodic in x with the same period T, then there exists a solution u(x, y) of (1.1) satisfying u(x, 0)=u0(x), u(0, y)=vo(y)+uo(0), where vo(0)=0 for xC-(-oo,oo) and yCE[-a, a]. However, this solution is periodic only on the x-axis, but not necessarily in the strip xC(-oo,oo), yC [-a, a]. Cesari in [ 1] has raised the question whether it is possible to choose v0(y) in such a way that u(x, y) is periodic in x for all yC [-a, a]. Thus, we observe that the periodicity requirement u(x+T, y) =u(x, y) leads to the integral equation: