Nonlinear integral inequality with power and its application in delay integro-differential equations

*Correspondence: tianyazhou369@163.com 1School of Mathematical Sciences, University of Jinan, Jinan, P.R. China Full list of author information is available at the end of the article Abstract New nonlinear integral inequalities (NII) are presented in this paper. Based on mathematical analysis technique, several estimation results are obtained, which not only complement the aforementioned results, but also generalize the inequalities to the more general nonlinearities. As an application, they can be employed to estimate the bound on the solutions of power integro-differential equations (IDE).


Introduction
As everyone knows, there exists a class of mathematical models described by differential equations, such as Malthus population model. However, a lot of differential equations do not possess the exact solution. Under this case, integral inequalities are significant for investigating the boundedness, stability, asymptotic behavior of solutions to differential equations. Gronwall [1] put forward the well-known Gronwall inequality to estimate the solution of linear differential equation. Bihari inequality [2] extended [1] to nonlinear one, and many authors have been devoted to studying NII in recent years . For example, based on the generalized Gronwall inequality, Tian et al. [3] investigated the asymptotic behavior of switched delay systems that represent a class of systems in practical engineering and have wide application in automated highways, power systems, and so on. Pachpatte [4] considered a linear integral inequality (1.1). Theorem 1.1 ([4]) Let c 0 ≥ 0 and u, b, c, d ∈ C(R + , R + ), R + = [0, +∞). If After that, Abdeldaim and El-Deeb [12] generalized (1.1) and investigated the delay integral inequality (1.2).
Note that inequalities (1.2) and (1.3) have been proved in the cases p = 1 and p ∈ (0, 1], respectively, how about p > 1? The aforementioned results are not covered, and it would also be interesting to generalize the inequalities considered in [4,12,21] to the more general nonlinearities, which is the motivation why we further study the above inequalities and their general cases. We study some new power NII and establish several estimation results under the condition of p > 1, which not only complement the ones established in [4,12,21] but also generalize inequalities (1.1)-(1.3) to the more general nonlinearities. The obtained results can be employed to study the boundedness of the delay IDE. As an application, two illustrative examples are also presented.

Main results
Throughout the paper, R = (-∞, +∞), R + = [0, +∞), C(D, E) and C 1 (D, E) defined on D with range in E are continuous functions and continuously differentiable function sets, respectively. The three lemmas are essential to proving the main results.

Lemma 2.3 Suppose that p > 0 is a constant and α(t) is a nondecreasing function with
Then v(t) is a nondecreasing function, and Next we will prove the following two cases 0 < p ≤ 1 and p > 1, respectively.
where h(t) and g(t) are defined by i.e., (2.6) Multiplying (2.6) by exp(- Integrating the above inequality from 0 to t, we have This together with (2.4) produces Define Then w(0) = k(T), w is a nondecreasing function, and (2.7) Differentiating w and using (2.7), we get and The above inequality multiplied by 1p gives By simple calculation of (2.8), Letting t = T in the above inequality, we have Because T is arbitrary, This together with (2.4), (2.7) implies Based on Cases 1 and 2, we can draw a conclusion that u(t) satisfies (2.2).

Examples
Now, we study the boundedness of the integral equation and IDE with delay.