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On the convergence rate of the method of successive approximations for the solution of nonlinear integral equations of Volterra type. (Russian) Zbl 0533.65092
Let f(x), K(x,s,u) be continuous in $$D=\{(x,s)\in [a,b]:\quad | u- f(a)| \leq l,\quad l>0\}; | K(x,s,u)-K(x,s,v)| \leq N(x,s)| u-v|$$ in D, then for $$a\leq x\leq \min(b,a+(1-\max_{a\leq x\leq b}| f(x)-f(a)|)\cdot(\max_{D}| K(x,s,u)|)^{-1}$$ there exists a unique and continuous solution of $$\phi(x)=f(x)+\int^{x}_{a}K(x,s,\phi(s))ds,$$ and the sequence $$\phi_{n+1}(x)=f(x)+\int^{x}_{a}K(x,s,\phi_ n(s))ds (n=0,1,...)$$ converges to it.
Moreover, $\max_{a\leq x\leq h}| \phi_{n+1}(x)-\phi_ n(x)| \leq \frac{1}{n!}\| \int^{x}_{a}K(x,s,f(s))ds\|_ C(\int^{h}_{a}\max_{x\in [s,h]}N(x,s)ds)^ n$ $$(n=1,2,...)$$. An example is given.
Reviewer: J.Albrycht
##### MSC:
 65R20 Numerical methods for integral equations 45G10 Other nonlinear integral equations