Spectral analysis of sturm-liouville boundary value problem with retarted argument

In this study , we examined the spectral analysis of Sturm-Liouville problem with retarded argument formed by y^″(t)+p(t)λy(t)+μ(t)y(t-∆(t)) differential expression in [0,π] which has a discontinuous solution at t=π/2 , with boundary conditions y(0) cos⁡〖α+(y^' (0))/p sin⁡〖α=0〗 〗 y(π) cos⁡〖β+y^' (π) sin⁡〖β=0〗 〗 and y(π/2-0)-δ_1 y(π/2+0)=0 y^' (π/2-0)-δ_2 y^' (π/2+0)=0 transmission conditions. Here p(t)={█(a^2 0≤t≤π/2@1 π/2≤t≤π)┤ is a continuous real valued function, μ(t) is continuous in [0,├ π/2)∪(π/2,π]┤ and has a finite limit μ(π/2±0)=lim┬(t→π/2±0)⁡〖μ(t)〗 , the real-valued function ∆(t)≥0 is continuous in [0,├ π/2)∪(π/2,π]┤ and has a finite limit ∆(π/2±0)=lim┬(t→π/2±0)⁡〖∆(t)〗 t-∆(t)={█(0 t∈ [0,π/2) @ π/2 t∈ (π/2,π] )┤ In addition, λ is a real eigenparameter and α,β,δ_1,δ_2≠0 are arbitrary real numbers. Keywords: Eigenvalue, eigenfuction, delayed differantial equation.