“…In fact, condition (10) in our main result (see Theorem 4) refers an, eventually, opposite monotony to (A 2 ) and improves the existent results in the literature for periodic higher order boundary value problems. In short, our technique is based on lower and upper solutions not necessarily ordered, in the topological degree theory, like it was suggested, for example, in [23,24], and has the following key points:…”