两类一阶奇异非线性微分方程初值问题解的精确渐近行为

Exact Asymptotic Behaviour of Solutions Near 0 to Two Classes of Initial Value Problems for First Order Singular Differential Equations

应用分离变量法和Karamata正规变化理论,在f和g满足适当的结构条件下,得到了两类一阶奇异非线性微分方程初值问题-u'(t)=b(t)f(u(t)),t>0,u(0):=limt→0+u(t)=+∞和v'(t)=b(t)g(v(t)),v(t)>0,t>0,v(0)=0解在0附近的精确渐近行为.其中,所给的结构条件隐含了f在无穷远处以指数p(p>1)正规变化或快速变化(快速趋于+∞);g在0处以指数-γ(γ>0)正规变化(隐含着lims→0+g(s)=+∞)或快速变化(快速趋于+∞);b在(0,∞)内非负非平凡,并且a>0,b∈L1(0,a).

Under the new structure conditions on nonlinear term f and singular term g,using Karamata regular variation theory and the separated variable method,we derive the exact asymptotic behaviour of solutions near 0 to a singular initial value problem for order differential equation-u (t)=b(t)f(u(t)),t>0,u(0) :=limt→0+u(t) =+∞and an initial value problem for a singular first order differential equation v (t) =b(t)g(v(t)),v(t)>0,t>0,v(0) = 0,where the new structure conditions imply that f is regularly var...