Differential-difference equations reducible to difference and q-difference equations

✍ Scribed by E.Yu. Romanenko

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 620 KB
• Volume: 42
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(01)00181-x

✦ Synopsis

The paper considers quasi-nonlinear differential-difference equations (DDE) of the form

which is a representative example of so-cMled completely integrable DDEs (i.e., DDEs that are reducible to functional equations). This equation is shown to exhibit the "nonstandard" (from the viewpoint of differential equations theory) behavior of solutions. Namely, for its smooth bounded solution x(t), either x'(t) tends to zero as t ---* oc or, on the contrary, the maximum of Ixl(t)[ on [0, T] increases ad infinitum as T -* oz. Solutions with the latter property are not uniformly continuous on R + and are infeasible for differential equations. Such solutions are referred to as asymptotically discontinuous. Investigation of asymptotically discontinuous solutions shows that for every such solution x(t), there exists a sequence ti -~ oc such that the graph of x(t) in the vicinity of t~ approaches as i --~ oo to a certain vertical segment.