On Some New Nonlinear Integral Inequalities of Gronwall-Bellman Type

Abstract

Abstract. In this paper, we establish some new nonlinear integral inequalities of Gronwall-Bellman type. These inequalities generalize some famous inequalities which can be used in applications as handy tools to study the qualitative as well as quantitative properties of solutions of some nonlinear ordinary differential and integral equations. More accurately we extend certain results which have been proved in A. Abdeldaim and M. Yakout [1] and H. El-Owaidy, A. A. Ragab, A. Abdeldaim [7] too.

Further nonlinear version of inequalities and their applications

“…where B(l, u) be defined as in (5). By keeping l fixed, u = ζ, integrate first from 0 to u and then again keeping u fixed, l = ϑ and integrate the resulting inequality from 0 to l for l, u ∈ R + and using (16), we have…”

On some generalizations of nonlinear dynamic inequalities on time scales and their applications

“…for all t ∈ T 0 , where ζ and ξ are defined as in (22) and (23) respectively. Now a suitable application of Lemma 3.4 to (27) with v(t 0 ) = 0, yields (28) v(t) ≤ t t0 e ζ (t, σ(s))ξ(s)∆s.…”

“…We know that Gronwall-Bellman inequality [10,28], play a considerable role in the study of qualitative properties of the solutions of some certain differential equations (see Theorem 1.1), (e.g., [7,12,21,51]). Many other results on its generalizations may be seen in [1,3,27]. The discrete version of (2) studied by Baburao G. Pachpatte [46].…”

Analysis on some powered integral inequalities with retarded argument and application

“…In the ongoing years, Gronwall-Bellman type inequalities have been enormously improved by the affirmation of their importance and inherent value in numerous branches of the applied sciences. These inequalities with their linear and nonlinear generalizations have earned much consideration in the works of Abdeldaim et al [3], Bainov et al [4], Bihari [5], El-Owaidy et al [6], Jiang et al [7], Kim [8], Liu et al [9], Ngoc et al [10], Oguntuase [11], Pachpatte [12], Pachpatte [13], Wang et al [14], Zhou et al [15].…”