Asymptotic proximity to higher order nonlinear differential equations

The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.