On the distance between adjacent zeros of solutions of first order differential equations with distributed delays

Upper Bounds for the Distance between Adjacent Zeros of First-Order Linear Differential Equations with Several Delays

“…This equation has the form of Equation (1) with q 1 (t) = 0.001, q 2 (t) = 0.667, υ 1 = 1, υ 2 = 1.5. q 2 (ω)dω = 1.005, for all t. Theorem 2 with j = 2 and n = 1 implies that D(x) ≤ 3υ 2 = 4.5. However, all the results of [14] cannot give this estimation, as we will show. Let P = 0.001 + 0.667, h(t) = t − υ 2 and g r (t) = t − δ r υ 1 , r = 1, 2, 3.…”

“…However, only a few works have considered the distance between zeros of Equation ( 1) and its general forms. For more details about this topic, we refer to the works of El-Morshedy and Attia [14] and McCalla [20]. This encourages us to study this property for Equation (1) and clarify the influence of the several delays in the distribution of zeros of Equation (1).…”

“…where υ > 0, q ∈ C([t 0 , ∞), [0, ∞)), has attracted the interest of many mathematicians; for example, [4][5][6][7][11][12][13][14][20][21][22][24][25][26][27][28][29][30]. The purpose of most of these works was to obtain new estimations for the upper bound (UB) between successive zeros of all solutions.…”

“…x(t) ; for example, see [13,14,22,25,27,28]. Furthermore, the authors [13,14,26] obtained new criteria of iterative types for UB of all solutions of Equation ( 2).…”

“…Assume that α 1 = β 2 = 3 and α 2 = β 1 = 0.001, using Maple software, we obtain Motivated by the recent contributions of [3,13,14,28], in this work, we obtain new estimations for the UB of all solutions of Equation (1). Furthermore, our results improve upon many previous results for both Equations ( 1) and ( 2).…”