Upper and lower solutions method for regular singular differential equations with quasi-derivative boundary conditions

作者：Verma Amit K; Agarwal Ravi P^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2012, 17(12): 4551-4558.

DOI：10.1016/j.cnsns.2012.03.027

• Monotone iterative technique
• Lower and upper solutions
• Quasi derivative
• Pseudo derivative
• Regular singular point

In this paper we consider the class of nonlinear singular differential equations of the type
-(p(x)y'(x))' + q(x)f(x,y(x),p(x)y'(x)) = 0, 0 < x < 1,
subject to the boundary conditions
limp(x)y'(x) 0(x -> 0), limp(x)y'(x) = 0(x -> 1).
Conditions on pd(x) and qd(x) are imposed so that x = 0 is a regular singular point. An approximation scheme which is iterative in nature is proposed to generate two monotonic sequences. To start the iteration we use upper and lower solutions which can be ordered in one way (v(0) <= u(0)) or the other (u(0) <= v(0)). We prove some new existence results and determine the region of multiple solutions.

• 出版日期2012-12