Global attractivity for half-linear differential systems with periodic coefficients

The system to be considered in this paper is(x′y′)=A(t)(xy)+B(t)(ϕp(x)ϕp∗(y)). Here, A(t)A(t) is a 2×22×2 diagonal matrix and B(t)B(t) is a 2×22×2 anti-diagonal matrix, and ϕq(z)=|z|q−2zϕq(z)=|z|q−2z with q>1q>1. The coefficients of B(t)B(t) are assumed to be periodic, but the coefficients of A(t)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)(ϕp(x)ϕp∗(y)) is stable, but is not attractive. Our concern is to clarify a positive effect of the diagonal partA(t)(xy) on the global asymptotic stability for the half-linear system. Some simple examples are included to illustrate the main result.