Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations

Summary Conditions are given for the nonlinear differential equation (1)L _n y+f(t, y, ..., ...,y ^(n–1)=0to have solutions which exist on a given interval t₀, )and behave in some sense like specified solutions of the linear equation (2)L _n z=0as t.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on t ₀, ),and a corollary of the latter applies to the case where Lz=z ⁿ.