Periodic solutions of nonlinear differential equations involving the method of scalar nonlinearities

Summary The recently developed method of scalar nonlinearities is applied to establish a new type of existence proof for periodic solutions of nonlinear differential equations. It is proved that given a periodic solution of a certain linear differential equation whose coefficients are subject to some nonlinear constraint, a nonlinear differential equation, which is closely related to the linear one, has a periodic solution (of the same period) as well. While, in general, the nonlinear equation will not be explicitly resolvable, the linear equation (with constraint) will allow for explicitly given solutions.The proof is carried out by constructing a homotopy (between appropriately chosen integral operators) and is based on Leray-Schauder theory. Thus, an essential hypothesis is the a-priori boundedness of certain intermediate problems. The very definition of the homotopy, which seems to be unprecedented in the literature, bears resemblance with the introduction of Dirac's-function.The theory is applied to Duffing's equation, resulting in an abstract existence statement as well as the explicit construction of numerically tractable intermediate problems.     

Synthetic applications of 2-chlorooxiranes: preparation of thiazoles,dihydrothiazoles and selenazoles

The reaction of 2-chlorooxiranes 1 with thioamides and thioureas provides access to thiazoles, 4-hydroxy-4,5-dihydrothiazoles and 2-imino-2,3-dihydrothiazoles under mild conditions and with excellent yields. With 1 and selenourea, near quantitative yields of selenazoles are obtained.     

Intramolecular meerwein reactions of the anthraquinone system. Synthesis,X-ray structural analysis and spectroscopic properties of anthra[9,1-bc:10,5-b'c']tetrahydrodipyran derivatives

Meerwein reactions of 1,5-anthraquinone bis(diazonium hydrogen sulfate) with sufficiently activated olefins, such as acrylonitrile, acrylic esters, methacrylonitrile and its esters, styrene and α-methylstyrene yielded derivatives of the new heterocyclic system anthra[9,1-bc:10,5-b'c']-2,3,7,8-tetrahydrodipyran. Isolation of derivatives was realized and yields enhanced by using dimethyl methylphosphonate as reaction medium. Compounds 5–18 were isolated as mixtures of diastereomers, some of which were separated by crystallization. Proof of structure and stereochemistry was obtained for some of the compounds by X-ray crystallographic analysis. The ¹H- and ¹³C-nmr data provided further support. Surprisingly large chemical shift differences between some of the proton signals of isomers were observed although the two asymmetric centers are far apart through bonds and through space. The mechanism of formation is explained on the basis of redox modulation of a charge transfer complex in which the aliphatic radical attacks the oxygen atom of the anthraquinone. It is proposed to classify this new type of reaction as an intramolecular Meerwein reaction.