Asymptotic assessment of distribution voltage profile using a nonlinear ODE model

The paper formulates the TPBV system (θ', v', s', w') with boundary data (θ(0)=0, v(0)=1, s(L)=0, w(L)=0) and small forcing p=ε p̃, q=ε q̃ under constant line parameters G,B, and proves existence of a C1 solution for sufficiently small ε by a one-parameter shooting argument and the implicit function theorem: reduce to an IVP for (v,w) with parameter η=w(0), define φ(η,λ)=w(L;η,λ), note φ(η,0)=η so ∂φ/∂η|_(0,0)=1, and invoke IFT to solve φ(η*(λ),λ)=0 for λ≈0 (i.e., ε small) . The candidate solution instead constructs a fixed-point map T on the order interval K⊂C[0,L] for v by eliminating s (via its linear equation) and integrating w backward from x=L, then shows T is a contraction for small ε, yielding a unique fixed point and hence a C1 solution; this is a standard Banach contraction proof. Both approaches correctly obtain existence for small ε; the paper’s IFT proof gives local existence (and implicit local uniqueness of the shooting parameter) while the model’s proof yields a quantitative contraction and uniqueness within K. Minor gaps in the paper (e.g., explicit control ensuring v(x)>0 on [0,L] and extension to x=L) could be clarified, but do not overturn the main existence claim .