Abstract :
Let U(t,λ) be a solution of the Dirichlet problem y"+(λt-q(t))y=0 -1⩽t⩽1 y(-1)=0=y(x), with variable t on (-1,x), for fixed x, which satisfies the initial condition U(-1,λ)=0, ∂U/∂t(-1,λ)=1. In this paper, the asymptotic representation of the corresponding eigenfunctions of the eigenvalues has been investigated. Furthermore, the leading term of the asymptotic formula for ∂U/∂λ(x,λn(x)),λ\´n (x) and ∫-1x (υ,λn)dυ is obtained where λn (x) is a negative eigenvalue of the Dirichlet problem on [-1,x] with fixed x<0
Keywords :
Sturm-Liouville equation; differential equations; eigenvalues and eigenfunctions; Dirichlet problem; asymptotic formula; asymptotic representation; eigenfunctions; eigenvalues; initial condition; negative eigenvalue; second order differential equation; Boundary conditions; Differential equations; Eigenvalues and eigenfunctions;
