On oscillation and nonoscillation properties of Emden-Fowler difference equations

A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation

Comparison results for oscillation of nonlinear differential equations

((without abstract.))     

An equivalence theorem on properties A, B for third order differential equations

Differential equations are often classified according to oscillatory/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.The second author wishes to thank C.N.R. of Italy and Grant Agency of Czech Republic (grant 201/96/0410) which made this research possible.     

On intermediate solutions and the Wronskian for half-linear differential equations

The asymptotic behavior of nonoscillatory solutions of the half-linear differential equation is studied. In particular, two Wronskian-type functions, which have some interesting properties, similar to the one of the Wronskian in the linear case, are given. Using these properties and suitable integral inequalities, the existence of the so-called intermediate solutions is examined and an open problem is solved.     

Nucleophilic aromatic substitutions on 4,5-dicyanopyridazine. Pyrrole and indole systems as carbon nucleophiles

4,5-Dicyanopyridazine was found to react with pyrrole and indole counterparts not only as heterocyclic azadiene in inverse electron-demand Hetero Diels–Alder reactions, as previously evidenced, but even as a very reactive heterocyclic electrophile at C-4 carbon, in formal S_NAr2 processes where a CN group acts as leaving group. In particular, operating in acetic acid as solvent, nucleophilic addition–elimination sequences allowed a facile access to pyrrolyl- and indolylpyridazines, through the corresponding 1,4-dihydropyridazine adducts.     

On oscillatory solutions of quasilinear differential equations

Necessary and sufficient conditions for the existence of at least one oscillatory solution of a second-order quasilinear differential equation are presented. These results yield also new conditions guaranteeing the coexistence of oscillatory and nonoscillatory solutions. Our approach is based on the asymptotic representation of solutions by means of a periodic function and of a suitable zero-counting function.