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On the difference property of families of measurable functions

Rafał Filipów (2003)

Colloquium Mathematicae

We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.

On the existence of solutions of some second order nonlinear difference equations

Małgorzata Migda, Ewa Schmeidel, Małgorzata Zbąszyniak (2005)

Archivum Mathematicum

We consider a second order nonlinear difference equation Δ 2 y n = a n y n + 1 + f ( n , y n , y n + 1 ) , n N . ( E ) The necessary conditions under which there exists a solution of equation (E) which can be written in the form y n + 1 = α n u n + β n v n , are given. Here u and v are two linearly independent solutions of equation Δ 2 y n = a n + 1 y n + 1 , ( lim n α n = α < and lim n β n = β < ) . A special case of equation (E) is also considered.

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