Let $x_{k+1}=T{x}_k+b$ be an iterative process for solving the operator equation $x=Tx+b$ in Hilbert space $X$. Let the sequence ${\left\{x_k\right\}}_{k=o}^{\infty }$ formed by the above described iterative process be convergent for some initial approximation $x_o$ with a limit $x^{*}=T{x}^{*}+b$. For given $l>1,m_0,m_1,\cdots ,m_l$ let us define a new sequence ${\left\{y_k\right\}}_{k=m_1}^{\infty }$ by the formula $y_k={\alpha }_0^{\left(k\right)}{x}_k+{\alpha }_1^{\left(k\right)}{x}_{k-m_1}+...+{\alpha }_l^{\left(k\right)}{x}_{k-m_l}$, where ${\alpha }_i^{\left(k\right)}$ are obtained by solving a minimization problem for a given functional. In this paper convergence properties of ${\alpha }_i^{\left(k\right)}$ are investigated and on the basis of the results thus obtainded it is proved that ${lim}_{k\to \infty }∥x^{*}-y_k∥/{∥x^{*}-x_k∥}^p=0$ for some $p\ge 1$.