We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general,...

Two simple methods for approximate determination of eigenvalues and eigenvectors of linear self-adjoint operators are considered in the following two cases: (i) lower-upper bound ${\lambda }_1$ of the spectrum $\sigma \left(A\right)$ of $A$ is an isolated point of $\sigma \left(A\right)$; (ii) ${\lambda }_1$ (not necessarily an isolated point of $\sigma \left(A\right)$ with finite multiplicity) is an eigenvalue of $A$.

2000 Mathematics Subject Classification: 58C06, 47H10, 34A60.The classical Filippov’s Theorem on existence of a local trajectory of the differential inclusion [x](t) О F(t,x(t)) requires the right-hand side F(·,·) to be Lipschitzian with respect to the Hausdorff distance and then to be bounded-valued. We give an extension of the quoted result under a weaker assumption, used by Ioffe in [J. Convex Anal. 13 (2006), 353-362], allowing unbounded right-hand side.

We consider the existence of extremal solutions to second order discontinuous implicit ordinary differential equations with discontinuous implicit boundary conditions in ordered Banach spaces. We also study the dependence of these solutions on the data, and cases when the extremal solutions are obtained as limits of successive approximations. Examples are given to demonstrate the applicability of the method developed in this paper.

In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T:X\to 2^{X*}$ and $S:C\to 2^D$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a kind of Karmardian’s...

We study conditions of discreteness of spectrum of the functional-differential operator $$ℒu=-u^{\text{'}\text{'}}+p\left(x\right)u\left(x\right)+{\int }_{-\infty }^{\infty }(u\left(x\right)-u\left(s\right)){\mathrm{d}}_{s}r(x,s)$$ on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum.

Let $K$ be a closed convex subset of a complete convex metric space $X$ and $T,I:K\to K$ two compatible mappings satisfying following contraction definition: $Tx,Ty)\le (Ix,Iy)+(1-a)max\{Ix.Tx),Iy,Ty)\}$ for all $x,y$ in $K$, where $0<a<1/2^{p-1}$ and $p\ge 1$. If $I$ is continuous and $I\left(K\right)$ contains $\left[T\right(K\left)\right]$ , then $T$ and $I$ have a unique common fixed point in $K$ and at this point $T$ is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of $I$ in their Theorem [3]...