In this paper we show that for some specific initial data, the Riemann problem for a simple model of $2\times 2$ hyperbolic conservation laws has solutions containing delta-shock waves.

A linear map $T:\mathcal{A}\to \mathcal{B}$ between algebras (or spaces of functions) $\mathcal{A}$ and $\mathcal{B}$ is called separating if $x \cdot y=0$ implies $Tx\cdot Ty=0$ for all $x,y\in \mathcal{A}$. It is well known that a separating map between certain commutative semisimple Banach algebras is very close to being a weighted composition operator on the maximal ideal spaces. In this paper, after introducing the notion of the cozero set for the elements of a Banach module, we first extend the notion of the separating maps to Banach module case. Our approach depends on the notion of point multipliers on a Banach module $\mathcal{X}$ and the relation between hyper maximal submodules of $\mathcal{X}$ and point multipliers on it. Then we generalize some well known results about separating maps between certain subspaces of continuous functions to Banach module case. In particular, we show that, imposing some additional assumptions on Banach modules, such map can be represented as a variation of a weighted composition operator. We also obtain a result concerning the automatic continuity of a bijective separating map whose inverse is also separating.

Recently, we constructed a new calculation method for the Dunkl-Williams constant $DW(X)$ of a normed linear space $X$. In this paper, we determine the Dunkl-Williams constant of symmetric octagonal norms on $\mathbb{R}^2$ by using our method.

Let $H$ be a real Hilbert space and let $C$ be a nonempty closed convex subset of $H$. Let $\alpha >0$ and let $A$ be an $\alpha$-inverse-strongly monotone mapping of $C$ into $H$. Let $B$ be a maximal monotone operator on $H$ and let $F$ be a maximal monotone operator on $H$ such that the domain of $F$ is included in $C$. Let $(A+B)^{-1}0$ and $F^{-1}0$ be the sets of zero points of $A+B$ and $F$, respectively. Let $0< k<1$ and let $g$ be a $k$-contraction of $H$ into itself. Let $G$ be a strongly positive bounded linear self-adjoint operator on $H$ with coefficient $\overline{\gamma}>0$ and let $0< \gamma <\frac{\overline{\gamma}}{k}$. In this paper, under the assumption $(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset$, In this paper, we prove a strong convergence theorem for finding a point $z_0\in (A+B)^{-1}0\cap F^{-1}0$ which is a unique fixed point of a nonlinear operator and also a unique solution of a variational inequality. $z_0\in (A+B)^{-1}0\cap F^{-1}0$ is a unique fixed point of $P_{(A+B)^{-1}0\cap F^{-1}0}(I-G+\gamma g)$. This point $z_0\in (A+B)^{-1}0\cap F^{-1}0$ is also a unique solution of a variational inequality. Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in Nonlinear Analysis and Optimization.