On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations

The n-dimensional cyclic systems of first order nonlinear differential equations(A) x i ′ + p i ( t ) x i + 1 α i = 0 , i = 1 , … , n ( x n + 1 = x 1 ) , (B) x i ′ = p i ( t ) x i + 1 α i , i = 1 , … , n ( x n + 1 = x 1 ) , are analyzed in the framework of regular variation. Under the assumption that α 1 ⋯ α n < 1 and p i ( t ) , i = 1 , … , n , are regularly varying functions, it is shown that the situation in which system (A) (resp. (B)) possesses decreasing (resp. increasing) regularly varying solutions of negative (resp. positive) indices can be completely characterized, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of decay (resp. growth) precisely. Examples are presented to demonstrate that the main results for (A) and (B) can be applied effectively to some higher order scalar nonlinear differential equations to provide new accurate information about the existence and the asymptotic behavior of their positive strongly monotone solutions.