We study boundary value problems of the type Ax = r, φ(x) = φ(b) (φ ∈ M ⊆ E*) in ordered Banach spaces.

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 $p\ge 1$.

Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that $E={\mathbf{R}}^{m+1}$ is given the presheaf of solutions of$$\sum _{i=1}^m\frac{\partial u}{\partial x_i}=\frac{\partial u}{\partial x_{m+1}}$$and $B$ is a subset of ${\mathbf{R}}^m\times [-\infty ,0]$ satisfying$$\left\{\right(\alpha x,{\alpha }^{2}t):(x,t)\in B,x\in {\mathbf{R}}^m,t\in \mathbf{R}\}\subset B$$for $\alpha \>0$ arbitrarily small. Then $B$ is thin at 0 if and only if $B$ is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...