Oscillatory and monotone solutions of first-order nonlinear delay differential equations

Abstract

For first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.MSC 2010: 34C15, 34K11, 34K12In the interval ℝ + = [ , +∞[, we consider the differential equationwhere f : ℝ + × ℝ → ℝ and τ : ℝ + → ℝ are continuous functions. Moreover, the function τ satisfies the following conditions:We use the following definitions: • Let a ≥ . A continuous function u : [a, +∞[ → ℝ is said to be a solution of the differential equation (1) if it is continuously differentiable in some interval ]a , +∞[ ⊂ ]a, +∞[, and in this interval it satisfies equation (1). • A solution u of the differential equation (1), defined on the interval [a, +∞[, is said to be proper if it is not identically zero in any neighborhood of +∞, i.e., there exists a sequence t k ∈ [a , +∞[ (k = , , . . .) such that lim k→+∞ t k = +∞ and u(t k ) ̸ = (k = , , . . . ).• A proper solution of equation (1) is said to be oscillatory if it has a sequence of zeros tending to +∞; otherwise it is said to be nonoscillatory. • A proper solution u of equation (1) is said to be a Kneser solution if there exists t > such that• A Kneser solution u of equation (1) is said to be vanishing at infinity if lim t→+∞ u(t) = , and it is said to be remote from zero if lim t→+∞ u(t) ̸ = .

On Kneser solutions of the $n$-th order nonlinear differential inclusions

“…The Kneser-type problems have been studied during last 120 years in detail, from the recent papers dealing with this topic, let us mention e.g. [10], [26], [27]. In the mentioned publications various generalizations of classical results have been obtained, including delay differential equations or differential equations with regularly varying coefficients.…”