*(English)*Zbl 1007.39018

The authors consider a couple of functional equations related to the realm of second-order linear difference equations with continuous arguments, for which previous work in the last ten years has concentrated mainly on necessary and sufficient conditions entailing the oscillatory nature of all solutions, and little attention has been given to the nonoscillatory ones.

The paper is devoted to consider sharp conditions guaranteeing the existence of nonoscillatory solutions for these equations. In one of the two cases dealt with, the authors improve and generalize the incipient known results using arguments relying on a couple of ground results of functional analysis. For the other case the study is reduced to the detailed consideration of several first-order nonlinear functional equations.

##### MSC:

39B22 | Functional equations for real functions |

39A11 | Stability of difference equations (MSC2000) |