The author considers the Newton method for the Volterra integral equation [F(x)](t):=x(t)−φ(t)+∫taf(t,s,x(s))ds=0 (t∈[a,a+δ]). He shows that Čaplygin's method for Cauchy problems of ordinary differential equations or parabolic equations like ut=uxx+f(t,x,u), u(t,0)=u(t,2π), u(0,x)=v0(x) defines exactly the same sequences as Newton's method applied to the equivalent integral equations. Therefore the convergence of Čaplygin's method can be proved in a simpler way and under weaker assumptions than was done until now. Further applications of Čaplygin's method refer to functional equations as well as nonlinear parabolic equations uxy=f(x,y,u), u(x,α)=σ(x), u(a,y)=τ(y). The author remarks that Čaplygin's method seems to be of little practical value, since the resulting linear problems are, in general, not easier to solve than the given nonlinear problem. For the Nicoletti boundary value problem in ordinary differential equations x′(t)=f(t,x(t)), (t∈[a,b]), x(t)=(xi(t))m1, xi(t)=yi, where a≤t1≤⋯≤tm≤b, the author proposes an iteration method of the form x′n+1=αxn+1−αxn+f(t,xn(t)), xn+1,i(ti)=yi with α>0 sufficiently small, which is, however, only linearly convergent. Since numerical examples are missing in this paper, there remains some doubt whether a shooting method for solving the Nicoletti boundary value problem would not be more effective.
Chaplygin's method is Newton's method / Vidossich, Giovanni. - In: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS. - ISSN 0022-247X. - 66:1(1978), pp. 188-206. [10.1016/0022-247X(78)90277-9]