A principle for study of quasi-gradient methods of approximate solving operator equations in Hilbert spaces

The article deals with nonlinear operator equations $f(x)=0$ with operators $f$ defined on a ball $B(x_0,R)$ in a Hilbert space $X$ and taking values from $X$. It is considered iterative methods of type $x_{n+1}=x_n-\Lambda(x_n)T(x_n)$, $n=0,1,2,ldots$, where $T(\xi)$ is an operator from $B(x_0,R)$ into $X$ and $\Lambda(\xi)$ a real functional on on $B(x_0,R)$. It is described conditions under that there is a phenomenon of relaxation of residuals: $\|f(x_{n+1}\|<\|f(x_n)\|$. The study of the convergence of iterations and its rate us reduce to the analysis of a scalar function; the graph of this function determines as the conditions of the convergence of iterations well as the rate of this convergence; moreover, it allows to write simple a priori and a posteriori estimates of errors. The general scheme covers classical methods of minimal residuals, of steepest descent, of minimal errors and some others.