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In this paper we study the existence and stability of two-dimensional discrete gap breathers in a two-dimensionaldiatomic face-centered square lattice consisting of altating light and heavy atoms,with on-site potential and couplingpotential.This study is focused on two-dimensional breathers with their frequency in the gap that separates the acousticand optical bands of the phonon spectrum.We demonstrate the possibility of the existence of two-dimensional gapbreathers by using a numerical method.Six types of two-dimeusional gap breathers are obtained,i.e.,symmetric,mirror-symmetric and asymmetric,whether the center of the breather is on a light or a heavy atom.The differencebetween one-dimensional discrete gap breathers and two-dimensional discrete gap breathers is also discussed.We ttseAubry’s theory to analyze the stability of discrete gap breathers in the two-dimensional diatomic face-centered squarelattice.