Synthesis of BEDT-TTF derivatives with carboxylic ester and amide functionalities

Synthetic routes to BEDT-TTF derivatives bearing side chain carboxylic ester and amide groups are reported. Methyl ET-ethanoate was prepared in five steps from vinylacetic acid; amide groups were installed early in the synthesis by mixed anhydride methods before the final coupling reaction.     

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.     

3,4‐[2,2‐Bis(methoxyethoxymethoxymethyl)propylenedithio]‐3′,4′‐(ethylenedithio)tetrathiafulvalene: a spiro‐substituted BEDT–TTF analogue

The crystal structure of the title compound [systematic name: 2‐(1,3‐dithiolo[4,5‐b][1,4]dithiin‐2‐ylidene)‐6,6‐bis(methoxyethoxymethoxymethyl)‐1,3‐dithiolo[4,5‐b][1,4]dithiepine], C₂₁H₃₀O₆S₈, a spiro‐substituted BEDT–TTF analogue [BEDT–TTF is bis(ethylenedithio)tetrathiafulvalene], has a strongly bent heterocyclic framework. The seven‐membered ring adopts a pseudo‐chair conformation with notably widened ring bond angles, especially at the methylene C atoms [119.49 (11) and 117.60 (11)°]. The axial side chain adopts an extended conformation, but the equatorial side chain curls back on itself and the O atom nearest the ring system is involved in three short contacts to H atoms (2.45–2.53 Å). The molecules pack in centrosymmetrically related pairs, which are isolated from each other by columns of the polyether side chains. This study emphasizes the ease of distortion of the neutral bis(propylenedithio)tetrathiafulvalene ring structure, and how the need to accommodate side chains can easily override the tendency of these donor systems to form stacks in the crystalline state.     

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