Our aim is to study fractional order differential inclusions with infinite delays in Banach spaces. We impose the regularity condition on multivalued nonlinearity in terms of measures of noncompactness to get the existence result. Some properties of the solution map are proved.

We study the following generalized quasivariational inequality problem: given a closed convex set <i>X</i> in a normed space <i>E</i> with the dual <i>E</i> ^*, a multifunction \Phi :Xightarrow 2^{E^{*}} and a multifunction :<i>X</i>2^<i>X</i>, find a point \Phi :Xightarrow 2^{E^{*}} such that \Phi :Xightarrow 2^{E^{*}} , \Phi :Xightarrow 2^{E^{*}} . We prove some existence theorems in which may be discontinuous, <i>X</i> may be unbounded, and is not assumed to be Hausdorff lower semicontinuous.

Let and be a real uniformly smooth Banach space. Let be a nonempty closed convex subset of and be a strictly pseudocontractive mapping in the sense of FE Browder and WV Petryshyn [1]. Let be a bounded sequence in and be real sequences in satisfying some restrictions. Let be the bounded sequence in generated from any given by the Ishikawa iteration method with errors: , , . It is shown in this paper that if is compact or demicompact, then converges strongly to a fixed point of .

A not necessarily continuous, linear or multiplicative function from an algebra A into itself is called a 2-local automorphism if agrees with an automorphism of A at each pair of points in A. In this paper, we study when a 2-local automorphism of a C-algebra, or a standard operator algebra on a locally convex space, is an automorphism.

Recently, the invertibility of linear combinations of two idempotents has been studied by several authors. Let P and Q be idempotents in a Banach algebra. It was shown that the invertibility of P+ Q is equivalent to that of aP+ bQ for nonzero a, b with a+ b= 0. In this note, we obtain a similar result for square zero operators and those operators satisfying x2= dx for some scalar d. More generally, we show that if P, Q satisfy a quadratic polynomial (x c)(x d) then the linear combination aP+ bQ c (a+ b) being invertible or Fredholm (and the index) is independent of the choice of the nonzero scalars a, b. However, this is not the case when P and Q are involutions, unitaries, partial isometries, k-potents (k 3) and other nilpotents, as counterexamples are provided.

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

We analyze a family of singular Schrdinger operators describing a Neumann waveguide with a periodic array of singular traps of a delta\prime type. We show that in the limit when perpendicular size of the guide tends to zero and the delta\prime interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {H} has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer \mathsf {H} at infinity.

We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.

We analyze the Hamiltonian proposed by Smilansky to describe irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics of the ground state and add new insights by finding the discrete spectrum numerically in the subcritical case. Furthermore, we show that the model then has a rich resonance structure.