A new class of nonlinear Gronwall–Bellman delay integral inequalities with power and its applications

In this paper, we establish some new delay Gronwall–Bellman integral inequalities with power, which can be used as a convenient tool to study the qualitative properties of solutions to differential and integral equations. We also give some examples to illustrate the application of our results to obtain the estimation for the solution of the integral and differential equations.


Introduction
The field of differential equations has developed a perfect structure. Since the 20th century, inequality theory has been an active research field, a series of basic theories of inequalities have been also established [1][2][3][4]. Since for most differential equations it is difficult to find the exact form of expression, people turn to studying the qualitative nature of the solutions of differential equations, for example, the existence, uniqueness, asymptotic property, boundedness and vibration of solutions of differential equations and difference equations; inequalities have become important tools to study the qualitative properties of differential equations. In recent decades, related studies on integral inequalities have produced many results (see  and the references therein). In 1919, Gronwall [1] established the following important integral inequality for a continuous function u: In 1943, Bellman [2] obtained the estimation of the unknown function u for some constant c ≥ 0, In 1975, Pachpatte [3] studied the following integral inequality: where u, f , and g are real-valued nonnegative continuous functions defined on I = [0, ∞), and a(t) is a positive, monotonic, nondecreasing continuous function defined on I.
In 1999, Owaidy et al. [5] discussed the following inequality: where u, f and g are real-valued nonnegative continuous functions defined on I = [0, ∞).
In 2011, Abdeldaim et al. [8] studied the following Gronwall-Bellman type inequality with power: where u, f and h are nonnegative real-valued continuous functions defined on [0, ∞), and u 0 and p are positive constants.
In this paper, inspired by the above work, we mainly establish the following nonlinear Gronwall-Bellman inequalities: The structure of this paper is as follows: In Sect. 2, we illustrates some basic lemmas, which will be used in later sections. In Sect. 3, we give some new nonlinear Gronwall-Bellman inequalities. In Sect. 4, we give two examples to illustrate the application of the obtained results in the qualitative research of differential equation solutions. In Sect. 5, we conclude our results.

Preliminaries
First, we explain some symbols will to be used: R denotes the set of real numbers and R + = [0, ∞), and C(M, S) denotes the class of all continuous functions on the set M with range in the set S.
Here are some very useful lemmas.
We can get the following exceptional cases.
Let K = 1, p = 1, we have a q ≤ qa + (1q), a ≥ 0, 0 < q ≤ 1. then Proof We assume that and Differentiating with respect to t of (9) and using (6), we get Multiplying by exp(-r Next, integrating t from t 0 to t for the above inequality, we get Since v(σ (t 0 )) = u 0 , we can get the estimation This completes the proof.

Main result and proof
In this section, we establish and prove a new class of nonlinear Gronwall-Bellman type delay integral inequalities with power. where Proof Using (6), we have Plugging (11) into (1), we can get then v(t) is a nondecreasing function, using (12) and (13), we obtain Using (5), from the above inequality we get and Plugging (14) and (15) into (13), we can obtain where t ∈ [t 0 , T], T ∈ R + , and Let Then we can get y(t) is a nondecreasing and positive function, and v(t) ≤ y(t), y(t 0 ) = B(T).
Differentiating y(t) with respect to t and using σ (t) ≤ t, we have From the above inequality we get Integrating both side of the above inequality from t 0 to t, then we can obtain the estimation for y(t): 1 q , we can obtain Because of the arbitrariness of T, we can obtain The proof is complete.
Remark 1 If q = 1, Theorem 3.1 reduces to Theorem 2.1 in [9]. If q = p, a(t) = x 0 , σ (t) = t, p = β = 1, α = q, Theorem 3.1 reduces to Theorem 2.3 in [10]. If q = 1, a(t) = x 0 , σ (t) = t, p = β = 1, α = 2p, β = q, Theorem 3.1 reduces to Theorem 2.5 in [10]. If q = 1, a(t) = x 0 , σ (t) = t, α = p, β = 2p -1, Theorem 3.1 reduces to Theorem 2.8 in [10]. where Proof First, we denote and J(t) is a nondecreasing and nonnegative continuous function, then and u(t) ≤ J(σ (t)) ≤ J(t), J(σ (t 0 )) = J(t 0 ) = u 0 . Differentiating with respect to t of the above equation, we get where Differentiating with respect to t of Y (σ (t)), we get from the above inequality, we can get Multiplying by exp( r+mp-1 m σ (t) t 0 h(ξ ) dξ ) on both sides of the above inequality, we get Integrating both sides of the above inequality from t 0 to t, we can get By the definition of β(t), plugging the above inequality into (18), we can get Integrating both sides of the above inequality from t to t 0 , we get Therefore, from Lemma 2.2 we can get The proof is completed.
In the following, we discuss the inequality (3). First we assume that the following conditions are satisfied; ( thus y j (t) are nondecreasing functions, y 1 (t) = 1 and then y j+1 (t) y j (t) are nondecreasing, positive and continuous functions. (C 3 ) We define the following functions: , j = 1, 2, 3, 4.
Then H j are positive continuous and strictly increasing functions on [0, ∞). We assume that H -1 j define the inverse function of H j , which are also continuous nondecreasing functions.
= ln (a which means that u(t) is bounded, for t ∈ [0, ∞). The proof is completed.

Conclusion
In this paper, we first give a new lemma about the nonlinear Gronwall-Bellman delay integral inequality, then we establish some new delay Gronwall-Bellman integral inequalities with power. And the inequalities obtained in this paper are further generalizations of some results obtained by Li et al. [9]. The results of this paper contribute to the study of the qualitative properties of solutions of differential and integral equations. By the method of Theorem 3.3 in this paper, we can further generalize Eq. (3) to ϕ u(t) ≤ a(t) + n j=1 t t 0 g j (t, ξ )h j u(ξ ) dξ then we can get similar results for the estimations on u(t).