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.

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 demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.