Let $K$ be a closed convex subset of a Hilbert space $H$ and $T:K⊸K$ a nonexpansive multivalued map with a unique fixed point $z$ such that $\left\{z\right\}=T\left(z\right)$. It is shown that we can construct a sequence of approximating fixed points sets converging in the sense of Mosco to $z$.

The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space $X$. We investigate some approximation methods generated by sequences of forms $a_n$ and $b_n$ defined on a dense subspace of $X$. The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.

Weak solutions of given problems are sometimes not necessarily unique. Relevant solutions are then picked out of the set of weak solutions by so-called entropy conditions. Connections between the original and the numerical entropy condition were often discussed in the particular case of scalar conservation laws, and also a general theory was presented in the literature for general scalar problems. The entropy conditions were realized by certain inequalities not generalizable to systems of equations...

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 $p\ge 1$.

Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map $T$ with a fixed point $p$ and prove that these iterates converge to a fixed point $q$ of $T$ under certain conditions. This fixed point $q$ may be different from...