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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∈&#x1D49E;0:=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.