奇异非线性动力方程可数个正解的存在性

Existence of Countably Many Positive Solutions to Nonlinear Dynamic Equations with Singularity

考虑时标上二阶奇异动力方程:{[φ(u△(t))]▽+a(t)f(u(t))=0,t∈(0,T),u(0)-βu△(0)=γu△(η),u△(T)=0,其中:φ:→是增同胚和正同态,且φ(0)=0;β,γ≥0;0<η<ρ(T);a:[0,T]→[0,+∞)在[0,T]上有可数多个奇点.利用可积性处理奇性,并使用不动点指数定理证明了上述方程存在可数多个正解。

The present paper focuses on the following singular second-order dynamic equation on time scales：{[φ（u△（t））]▽＋a（t）f（u（t））=0,t∈（0,T）,u（0）-βu△（0）=γu△（η）,u△（T）=0,where φ：→ is the increasing homeomorphism and positive homomorphism and φ（0） = 0,β,γ ≥0,0 η ρ（T）,a：[0,T]→[0,＋ ∞） has countably many singularities in [0,T].We dealt with the singularity utilizing the integration,and investigated the existence of countably many positive solutions to the equation with the fixed point index theory.