Copyright © 2009 Wei Li and Ping Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Consider the half-eigenvalue problem (ϕp(x′))′+λa(t)ϕp(x+)−λb(t)ϕp(x−)=0 a.e. t∈[0,1], where 1<p<∞, ϕp(x)=|x|p−2x, x±(⋅)=max⁡{±x(⋅), 0} for x∈𝒞0:=C([0,1],ℝ), and a(t) and b(t) are indefinite integrable weights in the Lebesgue space ℒγ:=Lγ([0,1],ℝ),1≤γ≤∞. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in (a,b)∈(ℒγ,wγ)2, where wγ denotes the weak topology in ℒγ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in (a,b)∈(ℒγ,‖⋅‖γ)2, where ‖⋅‖γ is the Lγ norm of ℒγ.