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

• We study cyclic systems of differential equations in the framework of regular variation.
• Existence and asymptotic behavior of strongly monotone solutions is established.
• Applications to scalar differential equations of the higher order are presented.

The n-dimensional cyclic systems of first order nonlinear differential equationsequation(A)xi′+pi(t)xi+1αi=0,i=1,…,n(xn+1=x1),equation(B)xi′=pi(t)xi+1αi,i=1,…,n(xn+1=x1), are analyzed in the framework of regular variation. Under the assumption that α1⋯αn<1α1⋯αn<1 and pi(t)pi(t), i=1,…,ni=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.