Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities

Abstract

In this paper, by arithmetic-geometric mean inequality and Riccati transformation, interval oscillation criteria are established for second-order forced impulsive differential equation with mixed nonlinearities of the formwhere t t 0 , k ∈ N; Φ * (u) = |u| * −1 u; {τ k } is the impulse moments sequence with 0 t 0 = τ 0 < τ 1 < τ 2 < · · · < τ k < · · · and lim k→∞ τ k = ∞; α = p/q, p, q are odds, and the exponents satisfy β 1 > · · · > β m > α > β m+1 > · · · > β n > 0.Some known results are generalized and improved. Examples are also given to illustrate the effectiveness and non-emptiness of our results.

Interval oscillation criteria for nonlinear impulsive delay differential equations with damping term

“…When the case kðc 1 Þ < kðd 1 Þ holds, from Lemmas 2.1, 2.3 and 2.5 we easily obtain [15,20] we know that when the delay term, damping term or impulses disappears, our Theorem 2.1 reduces to the main results of [12][13][14][15][16][17][18][19]21] respectively. In the following we will establish a Kemenev type interval oscillation criteria for (1.1) by the ideas of Philos [22] and Kong [24].…”

“…In this paper, motivated mainly by [15,20], we study the interval oscillation of second order nonlinear impulsive delay differential equations with damping term (1.1). By using Riccati transformation and H functions (introduced first by Philos [22]), we establish some interval oscillation criteria which generalize or improve some known results of [12][13][14][15][16][17][18][19][20][21]23]. Moreover, we also give two examples to illustrate the effectiveness and non-emptiness of our results.…”

“…In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers, see, for example, [10][11][12][13][14]. In [15], Özbekler and Zafer considered the following equations ðrðtÞu a ðx 0 ÞÞ 0 þ pðtÞu a ðx 0 Þ þ qðtÞu b ðxÞ ¼ eðtÞ; t -h i ; DðrðtÞu a ðx 0 ÞÞ þ q i u b ðxÞ ¼ e i ; t ¼ h i ; i 2 N:…”

The Kamenev type interval oscillation criteria of mixed nonlinear impulsive differential equations under variable delay effects

“…, n and obtained several interval oscillation results which recovered the early ones in [8] and [14]. When σ (t) is a nonnegative constant, i.e., σ (t) = σ 0 (σ 0 ≥ 0), by idea of [23], Guo et al [2] studied (1) and developed the results of [1,22,24].…”

“…Combining (23) with (24), we obtain estimation of x α (tσ (t))/x α (t) on (τ k , τ k+1 ] for k = k(c 1 ) + 1, . .…”