Shooting method in the application of boundary value problems for differential equations with sign-changing weight function

Abstract

Abstract In this paper, we use the shooting method to study the solvability of the boundary value problem of differential equations with sign-changing weight function: u ″ ( t ) + ( λ a + ( t ) − μ a − ( t ) ) g ( u ) = 0 , 0 < t < T , u ′ ( 0 ) = 0 , u ′ ( T ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}^{^{\prime\prime} }\left(t)+\left(\lambda {a}^{+}\left(t)-\mu {a}^{-}\left(t))g\left(u)=0,\hspace{1.0em}0\lt t\lt T,\\ u^{\prime} \left(0)=0,\hspace{1.0em}u^{\prime} \left(T)=0,\end{array}\right. where a ∈ L [ 0 , T ] a\in L\left[0,T] is sign-changing and the nonlinearity g : [ 0 , ∞ ) → R g:{[}0,\infty )\to {\mathbb{R}} is continuous such that g ( 0 ) = g ( 1 ) = g ( 2 ) = 0 g\left(0)=g\left(1)=g\left(2)=0 , g ( s ) > 0 g\left(s)\gt 0 for s ∈ ( 0 , 1 ) s\in \left(0,1) , g ( s ) < 0 g\left(s)\lt 0 for s ∈ ( 1 , 2 ) s\in \left(1,2) .