The author considers the existence and approximation of a solution for the following Chandrasekhar H-equation with perturbation: \[ (*)\quad H(t)=1+H(t)\int^{1}_{0}K(t,s)\Psi (s)H(s)ds+\int^{1}_{0}P(t,s,H(t),H(s))ds. \] Under certain conditions, theorem 1 gives the existence of a positive solution in a subset \(D^ r_{\delta}\) of \(C^ 0([0,1])\) using a fixed point theorem for strict set contractions. Adding some further assumptions the author proves in theorem 2 the monotone convergence of an iteration method to a solution of (*).
[Remark of the referee: (a) In the proof of theorem 2, the author states that \(H_ 0\in D^ r_{\delta}\) with \(H_ 0(t)=1\), but this is incorrect, as \(1=\| H_ 0\|_ C\leq r\leq \beta \delta_ 2\leq 1/4(1+\epsilon)<1,\) where the definition of \(D^ r_{\delta}\) and \(\beta\) and the conditions (II) and (V) are used; therefore theorem 2 is false. Starting with \(H_ 0(t)=0\) and replacing condition (4.1) by a suitable other one, it is perhaps possible to prove the convergence of the iteration method. (b) The assumptions of theorem 1 are incomplete. Setting \(P(t,s,u,v)=-2\), then assumption I is satisfied with \(\gamma_ 1=\gamma_ 2=0\) and \(\epsilon =2\), but for \(H(t)=0\) we get \((TH)(t)=-2\), contracting \(TD^ r_{\delta}\subset D^ r_{\delta}\). Adding the assumption \(\epsilon\leq 1\) the result of theorem 1 seems to be correct.]