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

The paper considers quasi-nonliner differential-difference equations (DDE) of the form which is a representative example of so-called 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 solutions x(t), and x′(t) tends to zero as t → ∞ or, on the contrary, the maximum of x′(t) on [0,T] increases ad infinitum as T → ∞. 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 → ∞ such that the graph of x(t) in the vicinity of ti approaches as i → ∞ to a certain vertical segment.