On the convergence rate of the method of successive approximations for the solution of nonlinear integral equations of Volterra type. (Russian) Zbl 0533.65092

Golovach, G. P.; Kalajda, A. F.

Vychisl. Prikl. Mat., Kiev 44, 21-31 (1981).

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

Keywords:

convergence rate; method of successive approximations; Volterra type