Oscillation of first-order differential equations with several non-monotone retarded arguments

Abstract

Consider the first-order linear differential equation with several non-monotone retarded arguments {x^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0} , {t\geq t_{0}} , where the functions {p_{i},\tau_{i}\in C([t_{0},\infty),\mathbb{R}^{+})} , for every {i=1,2,\ldots,m} , {\tau_{i}(t)\leq t} for {t\geq t_{0}} and {\lim_{t\to\infty}\tau_{i}(t)=\infty} . New oscillation criteria which essentially improve the known results in the literature are established. An example illustrating the results is given.

Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

“…In addition, new practical lower limit-upper limit type criteria similar to those in [8,9,15,16] are obtained. These new conditions improve some results in [2,5,8,9,11,13,[16][17][18][19]. An illustrative example is given to show the strength and applicability of our results.…”

“…where λ(k) is the smaller real root of the equation λ = e λk . The same problem has been considered for Equation (1) with non-monotone delays, see [2,4,11,[17][18][19]. The latter case is much more complicated than the monotone delays case.…”

“…Very recently, Bereketoglu et al [18] proved that Equation (1) oscillates if for some ∈ N the following criterion holds lim sup t→∞ t g(t) p(s)e g(t)…”

Oscillation tests for first-order linear differential equations with non-monotone delays

“…Bereketoglu et al [3] improved (11), and proved that Eq. (1) oscillates if there exists n ∈ N such that…”

A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

“…In 2019 Bereketoglu et al [25] derived the following oscillation conditions: Assume that there exist non-decreasing functions σ i ∈ C ([t 0 , ∞) , R + ) such that ( 18) is satisfied and for some k ∈ N…”