A remark on absolutely continuous functions in ℝ n

We introduce the notion ofα, λ-absolute continuity for functions of several variables and we compare it with the Hencl’s definition. We obtain that eachα, λ-absolutely continuous function isn, λ-absolutely continuous in the sense of Hencl and hence is continuous, differentiable almost everywhere and satisfies change of variables results based on a coarea formula and an area formula.     

A linear second-order ODE with only polynomial solutions

A linear second-order ODE featuring an arbitrary number of free parameters is identified, all solutions of which are polynomials.     

Fixed Point Theorems in Partially Ordered Metric Spaces and Existence Results for Integral Equations

We derive some new coincidence and common fixed point theorems for self-mappings satisfying a generalized contractive condition in partially ordered metric spaces. As applications of the presented theorems, we obtain fixed point results for generalized contraction of integral type and we prove an existence theorem for solutions of a system of integral equations.     

Synthesis of chromium(0) and molybdenum(0) bis (eta(6)-arene) derivatives and their monoelectronic oxidation to [M(eta(6)-arene)(2)](+) cations

M(eta(6)-arene)(2) species (M = Cr, arene = 1,3,5-Me(3)C(6)H(3); M = Mo, arene = 1,3,5-Me(3)C(6)H(3), 1,3,5-(i)Pr(3)C(6)H(3)), have been prepared by a modified Fischer-Hafner synthesis or by metal vapour techniques. The reaction of Cr(eta(6)-1,3,5-Me(3)C(6)H(3))(2) with the fulvene derivatives pentacarbomethoxycyclopentadiene (pcmcpH), 1-benzoyl-6-hydroxy-6-phenylfulvene (dbcpH), or 1-benzoyl-3-nitro-6-hydroxy-6-phenylfulvene (dbncpH) proceeds with evolution of dihydrogen and formation of the ionic derivatives [Cr(eta(6)-1,3,5-Me(3)C(6)H(3))(2)][E], where E = pcmcp, dbcp, or dbncp. Mo(eta(6)-arene)(2) derivatives (arene = toluene, 1,3,5-Me(3)C(6)H(3), 1,3,5-(i)Pr(3)C(6)H(3)) are oxidized to [Mo(eta(6)-arene)(2)](+) by pcmcpH. The crystal and molecular structures of [M(eta(6)-1,3,5-R(3)C(6)H(3))(2)][pcmcp] (M = Cr, R = Me; M = Mo, R = Me, (i)Pr) have been solved by X-ray single crystal diffraction.     

On Branciari’s theorem for weakly compatible mappings

In a recent paper B. Samet and H. Yazidi [B. Samet, H. Yazidi, An extension of Banach fixed point theorem for mappings satisfying a contractive condition of integral type, Ital. J. Pure Appl. Math. (in press)] have obtained an interesting theorem for mappings satisfying a contractive condition of integral type. The aim of this note is to present a generalization of their main result.     

Three Solvable Many-Body Problems in the Plane

Three solvable many-body problems in the plane are introduced. They are characterized by equations of motion of Newtonian type ('mass times acceleration equals force'), and are solvable for an arbitrary number n of particles. The equations of motion are rotation-invariant. The forces are velocity-dependent. These models give rise to a rich phenomenology of motions, whose detailed exploration is postponed to future papers.