Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type f+ T, where f is a mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X, and T is a maximal monotone mapping from X into X. This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type f+ T. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings.

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We study the behavior of a quantum particle confined to a hard-wall strip of a constant width in which there is a finite number<i>N</i>of point perturbations. Constructing the resolvent of the corresponding Hamiltonian by means of Krein's formula, we analyze its spectral and scattering properties. The bound state problem is analogous to that of point interactions in the plane: since a two-dimensional point interaction is never repulsive, there are<i>m</i>discrete eigenvalues, 1<i>m</i><i>N</i>, the lowest of which is nondegenerate. On the other hand, due to the presence of the boundary the point interactions give rise to infinite series of resonances; if the coupling is weak they approach the thresholds of higher transverse modes. We derive also spectral and scattering properties for point perturbations in several related models: a cylindrical surface, both of a finite and of an infinite height, threaded by a magnetic flux, and a straight strip which

The subject of the paper is Schrdinger operators on tree graphs which are radial, having the branching number bn at all the vertices at the distance tn from the root. We consider a family of coupling conditions at the vertices characterized by (bn1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {tn+1tn} growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrdinger operator can be purely absolutely continuous.

Let <i>X</i> be a locally compact Hausdorff space and <i>C</i> ₀(<i>X</i>) the Banach space of continuous functions on <i>X</i> vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on <i>C</i> ₀(<i>X</i>). They arise from prime ideals of <i>C</i> ₀(<i>X</i>), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of <i>c</i> ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on <i>C</i> ₀(<i>X</i>) for arbitrary <i>X</i>. We also make some remarks where <i>C</i> ₀(<i>X</i>) is replaced by a non-commutative C*-algebra.