Asymptotic behavior of solutions of a class of delay differential systems

The asymptotic behavior of the bounded solutions of the delayed system { x ′ ( t ) = − F ( x ( t ) ) + G ( y ( t − r ) ) , y ′ ( t ) = − F ( y ( t ) ) + G ( x ( t − r ) ) , is investigated, where r > 0 is a given constant, F , G ∈ C ( R 1 ) , F is strictly increasing on R 1 , and either G ( x ) ≥ F ( x ) for all x ∈ R 1 or G ( x ) ≤ F ( x ) for all x ∈ R 1 . It is shown that every bounded solution of the system tends to a constant vector as t → ∞ . The result obtained extends the existing ones in the literature.