Global attractivity in monotone and subhomogeneous almost periodic systems
The paper is devoted to the study of global dynamics in monotone and subhomogeneous almost periodic systems. By using the theory of skew-product semiflows and the concept of part metric, the global attractivity theorem is proved for a class of skew-product semiflows \(\Pi _t(x,y)=(u(x,y,t),\sigma (t)y)\) on \(P\times Y\), where \(P\) is a the positive cone of a Banach space \(X\) with nonempty interior \(\text{int}(P)\), and \(\sigma (t):Y\to Y\) is a compact minimal and distal flow. It is shown that the skew-product semiflow is contracting with respect to the part metric \(\rho\) on \(\text{int}(P)\). It should be noted that the monotonicity is not assumed for the nonautonomous system. The global attractivity theorem proved in the paper is applied to certain types of monotone and subhomogeneous almost periodic systems. The following results are obtained: 1. A threshold dynamics is established for Kolmogorov reaction-diffusion equations under general boundary conditions. 2. The global attractivity of a unique positive almost periodic solution for either irreducible cooperative and strictly subhomogeneous, or cooperative and strongly subhomogeneous ordinary differential equations is shown. 3. The global attractivity of a unique positive almost periodic solution for cooperative and strictly subhomogeneous delay differential equations is shown.
scientific article; zbMATH DE number 1884233
scientific article