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.

中文翻译：

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渐近接近高阶非线性微分方程

对于被认为是某些线性微分方程的扰动的高阶微分方程，研究了无界解的存在及其渐近行为。特别地，证明了具有多项式或非整数幂律渐近行为的解的存在性。这些结果给出了非线性和相应线性方程的解之间的关系，粗略地说，可以将其解释为线性情况和非线性情况之间的渐近接近。我们的方法基于归纳法、迭代过程和线性方程解的合适估计。