The existence of a competitive equilibrium is shown, without using the well-known fixed point technique, under weak gross substitutability. The price domain is limited to the strictly positive orthant in order that the excess demand functions may not reduce to the trivally vanishing one under Walras's law and the gross substitutability conditions. We use instead a boundary condition such that a zero price will result in infinite demand. The case of an open subcone price domain for which the ordinary fixed point theorem is not applicable is also treated.