On the best constants in the solvability conditions for the periodic boundary value problem for higher-order functional differential equations

Minimal periods of solutions to higher-order functional differential equations

“…From Lemma 3 it follows that a non-constant component x j (from the proof of Lemma 3) of the solution x to (5) is a solution to (16) with p = p j , some constant C, some measurable function τ : [0, T ] → [0, T ]. If (17), it follows from Lemma 4 that the solution x j is unique: x j (t) ≡ −C. From ( 14) it follows that each component x i of the non-constant solution x is constant.…”

“…Let a positive number P be given. Problem For n = 1, n = 2, n = 3, n = 4 this Lemma is proved in [19,20,21,22], for arbitrary n in [23,24,17].…”

“…All these assertions are well known. Proofs of 1), 2), 6) one can see in [13,14,15,16,17], 3), 4), 5) in, for example, [17].…”

Lasota–Opial type conditions for periodic problem for systems of higher-order functional differential equations

“…Other interesting results about problems (6), (4) can be found also in the papers [10][11][12][13][14][15].…”

“…For the case i ≡ 0 (i = 1, n -1), these results are generalized in [9]. It is interesting that the numbers N n are in some connection with Favard's, Bernoulli's, and Euler's numbers (see [10] and [11]) and if the operator 0 is monotone, then condition (8) transforms to the condition 0 < b…”