Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

Quotient probabilistic normed spaces and completeness results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Efficient synthesis of aryl vinyl ethers exploiting 2,4,6-trivinylcyclotriboroxane as a vinylboronic acid equivalent

The synthesis of functionalized aryl vinyl ether derivatives can be readily achieved utilizing a room-temperature copper(II) acetate mediated coupling of substituted phenols with 2,4,6-trivinylcyclotriboroxane-pyridine complex in the presence of a suitable base. The scope of the procedure was demonstrated by the generation of an array of substituted aryl vinyl ethers. The reaction was seen to be tolerant of a diverse range of functional groups yielding products in high isolated yields. We have shown that one role of an amine base in the reaction sequence is the in situ generation of an amine coordinated boroxine ring. An X-ray crystal structure and low temperature (11)B NMR study of 2,4,6-trivinylcyclotriboroxane-pyridine complex demonstrated the nature of the tetracoordinate boron species, which may have a key role to play within the reaction sequence.