Global attractivity for half-linear differential systems with periodic coefficients

The system to be considered in this paper is ( x ′ y ′ ) = A ( t ) ( x y ) + B ( t ) ( ϕ p ( x ) ϕ p ∗ ( y ) ) . Here, A ( t ) is a 2 × 2 diagonal matrix and B ( t ) is a 2 × 2 anti-diagonal matrix, and ϕ q ( z ) = | z | q − 2 z with q > 1 . The coefficients of B ( t ) are assumed to be periodic, but the coefficients of A ( t ) are not necessarily periodic. The system is of half-linear type. Sufficient conditions are given for the zero solution of the half-linear system to be globally asymptotically stable. The zero solution of the system ( x ′ y ′ ) = B ( t ) ( ϕ p ( x ) ϕ p ∗ ( y ) ) is stable, but is not attractive. Our concern is to clarify a positive effect of the diagonal part A ( t ) ( x y ) on the global asymptotic stability for the half-linear system. Some simple examples are included to illustrate the main result.