Note on Some Nonlinear Integral Inequalities and Applications to Differential Equations

Abstract

Using ideas from Boukerrioua and Guezane-Lakoud (2008), some nonlinear integral inequalities are established.

1. Introduction

Integral inequalities provide a very useful and handy device for the study of qualitative as well as quantitative properties of solutions of differential equations. The Gronwall-Bellman type (see, e.g., [1–4]) is particularly useful in that they provide explicit bounds for the unknown functions. One of the most useful inequalities in the development of the theory of diferential equations is given in the following theorem.

Theorem 1.1 (see [3]). If 𝑢(𝑡) and 𝑓(𝑡) are non-nonnegative continuous functions on [0,∞[ satisfying 𝑢²(𝑡)≤𝑐²+2^𝑡₀𝑓(𝑠)𝑢(𝑠)𝑑𝑠, (1.1) for some constant 𝑐≥0, then 𝑢(𝑡)≤𝑐+^𝑡₀𝑓(𝑠)𝑑𝑠,𝑡∈[0,∞[. (1.2)

The importance of this inequality lies in its successful utilization of the situation for which the other available inequalities do not apply directly. It has been frequently used to obtain global existence, uniqueness, stability, boundedness, and other properties of the solution for wide classes of nonlinear differential equations. The aim of this paper is to give other results on nonlinear integral inequalities and their applications.

2. Main Results

In this section, we begin by giving some material necessary for our study. We denote by ℝ, the set of real numbers and ℝ₊ the nonnegative real numbers.

Lemma 2.1. For 𝑥∈ℝ₊,𝑦∈ℝ₊,1/𝑝+1/𝑞=1, one has 𝑥^1/𝑝𝑦^1/𝑞≤𝑥𝑝+𝑦𝑞. (2.1)

Lemma 2.2 (see [1]). Let 𝑏(𝑡) and 𝑓(𝑡) be continuous functions for 𝑡≥𝛼, let 𝑣(𝑡) be a differentiable function for 𝑡≥𝛼 and suppose 𝑣^′(𝑡)(𝑡)𝑣(𝑡)(𝑡)𝑣(𝛼)≤𝑏+𝑓,𝑡≥𝛼,≤𝑣₀. (2.2) Then for 𝑡≥𝛼, 𝑣(𝑡)≤𝑣₀exp^𝑡_𝛼𝑏(𝑠)𝑑𝑠+^𝑡_𝛼𝑓(𝑠)exp^𝑡_𝑠𝑏(𝜎)𝑑𝜎𝑑𝑠. (2.3)

Now we state the main results of this work

Theorem 2.3. Let 𝑢,𝑎,𝑏,ℎ_𝑖(𝑖=1,…,𝑛) be real-valued nonnegative continuous functions and there exists a series of positive real numbers 𝑝₁,𝑝₂,…,𝑝_𝑛 and 𝑢(𝑡) satisfy the following integral inequality, 𝑢^𝑝(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.4) for 𝑡∈ℝ₊ then 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑝_𝑖𝑝𝑎(𝑠)+𝑝−𝑝_𝑖𝑝×exp^𝑡_𝑠𝑏(𝜎)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝜎)𝑑𝜎𝑑𝑠^1/𝑃, (2.5) for 𝑝≥𝑝^⋆=max{𝑝_𝑖,𝑖=1,…,𝑛}.

Proof. Define a function 𝑣(𝑡) by 𝑣(𝑡)=^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.6) then 𝑣(0)=0 and (2.4) can be written as 𝑢^𝑝(𝑡)≤𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡). (2.7) By (2.7) and Lemma 2.1, we get 𝑢^𝑝𝑖(𝑡)=𝑢^𝑝(𝑡)^{𝑝𝑖/𝑝}≤(𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡))^{𝑝𝑖/𝑝}≤𝑝_𝑖𝑝(𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡))+𝑝−𝑝_𝑖𝑝. (2.8) Differentiating (2.6), we get 𝑣^′(𝑡)=^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑢^𝑝𝑖(𝑡). (2.9) Using (2.8) and (2.9), it yields 𝑣^′(𝑡)≤^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑝)^𝑖𝑝(𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡))+𝑝−𝑝_𝑖𝑝=𝐴(𝑡)+𝐵(𝑡)𝑣(𝑡), (2.10) where 𝐵(𝑡)=𝑏(𝑡)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑡),𝐴(𝑡)=^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑝_𝑖𝑝𝑎(𝑡)+𝑝−𝑝_𝑖𝑝. (2.11) By Lemma 2.2, we have 𝑣(𝑡)≤^𝑡₀𝐴(𝑠)exp^𝑡_𝑠𝐵(𝜎)𝑑𝜎𝑑𝑠. (2.12) Using (2.7) and (2.12), we get 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑝_𝑖𝑝𝑎(𝑠)+𝑝−𝑝_𝑖𝑝×exp^𝑡_𝑠𝑏(𝜎)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝜎)𝑑𝜎𝑑𝑠^1/𝑃. (2.13)
This achieves the proof of the theorem.

Remark 2.4. if we take 𝑛=2,𝑝≥1,𝑝₁=𝑝,and𝑝₂=1, then the inequality established in Theorem 2.3 become the inequality given in [4,Theorem 1(𝑎₁)].

Theorem 2.5. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function (𝑎(𝑡)+𝑝^⋆/𝑝)/𝑏(𝑡) is nondecreasing and 𝑢^𝑝(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.14) for 𝑡∈ℝ₊ then 𝑢(𝑡)≤𝑎(𝑡)+𝑝^⋆𝑝^1/𝑝×𝑝1−^⋆𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/𝑝−𝑝⋆}, (2.15) for 𝑝_⋆=min{𝑝_𝑖,𝑖=1,…,𝑛}≤𝑝<𝑝^⋆ and 𝑡<𝛽_𝑝,𝑝⋆,where 𝛽_𝑝,𝑝⋆=sup𝑡∈ℝ₊∶𝑝^⋆/𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1. (2.16)

Proof. For 𝑝_⋆≤𝑝<𝑝^⋆
By (2.7) and the fact that (𝑝^⋆/𝑝)>1, one gets: 𝑢^𝑝𝑖(𝑡)=𝑢^𝑝(𝑡)^{𝑝𝑖/𝑝}≤(𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡))^{(𝑝𝑖/𝑝)}≤𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡)+𝑝^⋆/𝑝^𝑝⋆/𝑝. (2.17) Differentiating (2.6) and using (2.17), we obtain 𝑣^′(𝑡)≤^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)(𝑡)𝑣(𝑡)+𝑝+𝑏^⋆𝑝^𝑝⋆/𝑝, (2.18) then 𝑣^′(𝑡)≤𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆/𝑝𝑏(𝑡)(𝑡)𝑎(𝑡)+𝑝+𝑣^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^{(𝑝⋆/𝑝)−1}. (2.19) Since the function (𝑎(𝑡)+𝑝^⋆/𝑝)/𝑏(𝑡) is nondecreasing, for 0≤𝑡≤𝜏 then, 𝑣^′(𝑡)(𝑡)a(𝜏)+𝑝≤𝑀^⋆/𝑝𝑏(𝜏)+𝑣(𝑡), (2.20) where 𝑀(𝑡)=𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^{(𝑝⋆/𝑝)−1}. (2.21) Consequently 𝑣(𝑡)+𝑎(𝜏)+𝑝^⋆/𝑝𝑏(𝜏)≤𝑎(𝜏)+𝑝^⋆/𝑝𝑏(𝜏)exp^𝑡₀𝑀(𝑠)𝑑𝑠. (2.22) For 𝜏=𝑡, we can see that 𝑎(𝑡)+𝑝^⋆𝑝(𝑡)𝑣(𝑡)≤𝑎(𝑡)+𝑝+𝑏^⋆𝑝exp^𝑡₀𝑀(𝑠)𝑑𝑠, (2.23) then the function 𝑀(𝑡) can be estimated as 𝑀(𝑡)≤𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}⋅exp^𝑡₀𝑝^⋆𝑝−1𝑀(𝑠)𝑑𝑠. (2.24) Let 𝐿(𝑡)=𝑝^⋆𝑝−1𝑀(𝑡). (2.25) Now we estimate the expression 𝐿(𝑡)exp(−∫^𝑡₀𝐿(𝑠)𝑑𝑠) by using (2.24) to get 𝐿(𝑡)exp^𝑡₀(𝑠)≤𝑝−𝐿𝑑𝑠^⋆𝑝−1𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}. (2.26) Remarking that 𝐿(𝑡)exp^𝑡₀(𝑠)=𝑑−𝐿𝑑𝑠𝑑𝑡−exp^𝑡₀(𝑠)≤𝑝−𝐿𝑑𝑠_⋆𝑝−1𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}, (2.27) we integrate (2.27) from 0 to 𝑡 to get 1−exp^𝑡₀−𝐿(𝑠)𝑑𝑠≤^𝑡₀𝑝^⋆𝑝−1𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠, (2.28) replacing 𝐿(𝑡) by its value in (2.28), we obtain 1−exp^𝑡₀𝑝1−^⋆𝑝𝑀(𝑠)𝑑𝑠≤^𝑡₀𝑝^⋆𝑝−1𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠, (2.29) then exp^𝑡₀𝑀(𝑠)𝑑𝑠≤1−^𝑡₀𝑝^⋆𝑝−1𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(1−(𝑝⋆/𝑝))}. (2.30) Using (2.7), (2.23), and (2.30) we have, 𝑢(𝑡)≤𝑎(𝑡)+𝑝^⋆𝑝^1/𝑝⋅𝑝1−^⋆𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}. (2.31) This achieves the proof of the theorem.

Theorem 2.6. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the function (𝑎(𝑡)+𝑝_⋆/𝑝)/𝑏(𝑡), is nondecreasing and 𝑢^𝑝(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.32) for 𝑡∈ℝ₊ then 𝑢(𝑡)≤𝑎(𝑡)+𝑝_⋆𝑝^1/𝑝×𝑝1−^⋆𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝_⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}, (2.33) for 𝑝<𝑝_⋆ and 𝑡<𝛽_𝑝,𝑝⋆, 𝛽_𝑝,𝑝⋆=sup𝑡∈ℝ₊∶𝑝^⋆/𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑎(𝑡)+𝑝)^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1. (2.34)

Proof. for 𝑝<𝑝_⋆=min{𝑝_𝑖,𝑖=1,…,𝑛}.
Using (2.7), the fact that the function (𝑎(𝑡)+𝑝_⋆/𝑝)/𝑏(𝑡) is nondecreasing, and 𝑝_⋆/𝑝≻1, we have 𝑢^𝑝𝑖(𝑡)=𝑢^𝑝(𝑡)^{𝑝𝑖/𝑝}≤(𝑎(𝑡)+𝑏(𝑡)𝑣(𝑡))^{𝑝𝑖/𝑝}≤𝑎(𝑡)(𝑡)𝑣(𝑡)+𝑝+𝑏^⋆𝑝^𝑝⋆/𝑝. (2.35) Differentiating (2.6) and using (2.35), we obtain 𝑣^′(𝑡)≤^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑎(𝑡)(𝑡)𝑣(𝑡)+𝑝)+𝑏^⋆𝑝^𝑝⋆/𝑝, (2.36) then 𝑣^′(𝑡)≤𝑏(𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑎(𝑡)+𝑝)^⋆/𝑝𝑏(𝑡)(𝑡𝑎(𝑡)+𝑝+𝑣)^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^{(𝑝⋆/𝑝)−1}, (2.37) from the proof of Theorem 2.5, we get the required inequality in (2.33).

Remark 2.7. if we take 𝑛=2, then the inequalities established in Theorems 2.5 and 2.6 become the inequalities given in [5, Theorem 1.2].

Theorem 2.8. Suppose that the hypothesis of Theorem 2.3 holds and moreover the function 𝑏(𝑡) is decreasing. Let 𝑐 be a real valued nonnegative continuous and nondecreasing function for 𝑡∈ℝ₊. If 𝑢^𝑝(𝑡)≤𝑐^𝑝(𝑡)+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.38) then(1) 𝑢(𝑡)≤𝑐(𝑡)1+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠)exp^𝑡_𝑠𝑏(𝜎)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝𝐻_𝑖(𝜎)𝑑𝜎𝑑𝑠^1/𝑝, (2.39) for 𝑝≥𝑝^⋆, where 𝐻_𝑖(𝑠)=ℎ_𝑖(𝑠)𝑐(𝑠)^{𝑝𝑖−𝑝}. (2.40)(2) 𝑢(𝑡)(𝑡)𝑝≤𝑐1+^⋆𝑝^1/𝑝×𝑝1−^⋆𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠)𝑝1+^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}, (2.41) for 𝑝_⋆≤𝑝<𝑝^⋆ and𝑡<𝛽_𝑝,𝑝⋆where 𝛽_𝑝,𝑝⋆=sup𝑡∈ℝ₊∶𝑝^⋆/𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠)𝑝1+^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1. (2.42)(3) 𝑢(𝑡)(𝑡)𝑝≤𝑐1+^⋆𝑝^1/𝑝×𝑝1−^⋆𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠)𝑝1+^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}, (2.43) for and 𝑡<𝛽_𝑝,𝑝⋆where 𝛽_𝑝,𝑝⋆=sup𝑡∈ℝ₊∶𝑝^⋆/𝑝−1^𝑡₀𝑏(𝑠)^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠𝑝)1+^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1. (2.44)

Proof. Since 𝑐(𝑡) is a nonnegative, continuous, and nondecreasing function, for 𝑡∈ℝ₊, from (2.38) we observe that 𝑢(𝑡)𝑐(𝑡)^𝑝≤1+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑐(𝑠)^{𝑝𝑖−𝑝}𝑢(𝑠)𝑐(𝑠)^𝑝𝑖𝑑𝑠, (2.45) we put 𝑤(𝑡)=𝑢(𝑡)𝑐(𝑡),𝐻_𝑖(𝑡)=ℎ_𝑖(𝑡)𝑐(𝑡)^{𝑝𝑖−𝑝}, (2.46) then, we have 𝑤^𝑝(𝑡)≤1+𝑏(𝑡)^𝑡₀^𝑖=𝑛_𝑖=1𝐻_𝑖(𝑠)𝑤^𝑝𝑖(𝑠)𝑑𝑠. (2.47)
Then a direct application of the inequalities established in Theorems 2.3, 2.5, and 2.6 gives the required results.

Theorem 2.9. Suppose that the hypothesis of Theorem 2.3 holds. Assume that the functions (𝑎(𝑡)+𝑝^⋆/𝑝)/𝑏(𝑡),(𝑎(𝑡)+𝑝_⋆/𝑝)/𝑏(𝑡) are nondecreasing and let 𝑘(𝑡,𝑠) and its derivative partial 𝜕/𝜕𝑠𝑘(𝑡,𝑠) be real-valued nonnegative continuous functions, for 0≤𝑠≤𝑡≤∞. If 𝑢^𝑝(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝐾(𝑡,𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.48) then(1) 𝑢(𝑡)≤𝑎(𝑡)+𝑏(𝑡)^𝑡₀𝐴(𝜎)exp^𝑡_𝜎𝐵(𝜏)𝑑𝜏)𝑑𝜎^1/𝑝, (2.49) where 𝐴(𝑡)=𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑝)^𝑖𝑝𝑎(𝑡)+𝑝−𝑝_𝑖𝑝+^𝑡₀𝜕𝑘(𝜕𝑠𝑡,𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠𝑝)^𝑖𝑝𝑎(𝑠)+𝑝−𝑝_𝑖𝑝𝑑𝑠, (2.50)𝐵(𝑡)=𝐾(𝑡,𝑡)𝑏(𝑡)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑡)+^𝑡₀𝜕𝑘()𝑏(𝑠)𝜕𝑠𝑡,𝑠^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑠)𝑑𝑠, (2.51) for 𝑝≥𝑝^⋆.
(2) 𝑢(𝑡)≤𝑎(𝑡)+𝑝^⋆𝑝^1/𝑝𝑝1−^⋆𝑝−1^𝑡₀𝑙(𝑠)𝑎(𝑠)+𝑝^⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}. (2.52) For 𝑝_⋆≤𝑝≤𝑝^⋆and𝑡<𝛽_𝑝,𝑝⋆,where𝛽_𝑝,𝑝⋆=sup𝑡∈𝑅₊∶𝑝^⋆/𝑝−1^𝑡₀𝑙(𝑠)𝑎(𝑠)+𝑝_⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1. (2.53)(3) 𝑢(𝑡)≤𝑎(𝑡)+𝑝_⋆𝑝^1/𝑝𝑝1−^⋆𝑝−1^𝑡₀𝑙(𝑠)𝑎(𝑠)+𝑝_⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠^{1/(𝑝−𝑝⋆)}. (2.54) For 𝑝<𝑝_⋆ and 𝑡<𝛽_𝑝,𝑝⋆, where 𝛽_𝑝,𝑝⋆=sup𝑡∈𝑅₊∶𝑝^⋆/𝑝−1^𝑡₀𝑙(𝑠)𝑎(𝑠)+𝑝_⋆𝑝^{(𝑝⋆/𝑝)−1}𝑑𝑠≤1, (2.55)𝑙(𝑡)=𝑏(𝑡)𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)+𝑏(𝑡)^{1−(𝑝⋆/𝑝)𝑡}₀𝜕𝑘()𝑏(𝑠)𝜕𝑠𝑡,𝑠^{𝑝⋆/𝑝𝑖=𝑛}_𝑖=1ℎ_𝑖(𝑠)𝑑𝑠. (2.56)

Proof. Let 𝑣(𝑡)=^𝑡₀𝐾(𝑡,𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠, (2.57)  (1) for 𝑝≥𝑝^⋆.
Differentiating(2.57) we get 𝑣^′(𝑡)=𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑢^𝑝𝑖(𝑡)+^𝑡₀𝜕𝑘()𝜕𝑠𝑡,𝑠^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)𝑑𝑠. (2.58) Using (2.8) and (2.58) and the fact that 𝑣(𝑡) is nondecreasing, we obtain, for 0≤𝑠≤𝑡, 𝑣^′(𝑡)≤𝐾(𝑡,𝑡)𝑏(𝑡)^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑡)+^𝑡₀𝜕𝑘()𝑏(𝑠)𝜕𝑠𝑡,𝑠^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑠)𝑣(𝑡)(𝑑𝑠+𝐾𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑝_𝑖𝑝𝑎(𝑡)+𝑝−𝑝_𝑖𝑝+^𝑡₀𝜕𝑘(𝜕𝑠𝑡,𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑝_𝑖𝑝𝑎(𝑠)+𝑝−𝑝_𝑖𝑝𝑑𝑠, (2.59) then 𝑣^′(𝑡)≤𝐵(𝑡)𝑣(𝑡)+𝐴(𝑡). (2.60)
By Lemma 2.2, we have 𝑣(𝑡)≤^𝑡₀𝐴(𝑠)exp^𝑡_𝑠𝐵(𝜎)𝑑𝜎𝑑𝑠, (2.61) where 𝐴(𝑡)=𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑝)^𝑖𝑝𝑎(𝑡)+𝑝−𝑝_𝑖𝑝+^𝑡₀𝜕𝑘(𝜕𝑠𝑡,𝑠)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠𝑝)^𝑖𝑝𝑎(𝑠)+𝑝−𝑝_𝑖𝑝𝐵(𝑡)()𝑏(𝑡)𝑑𝑠,=𝐾𝑡,𝑡^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑡)+^𝑡₀𝜕𝑘()𝑏(𝑠)𝜕𝑠𝑡,𝑠^𝑖=𝑛_𝑖=1𝑝_𝑖𝑝ℎ_𝑖(𝑠)𝑑𝑠. (2.62) Finally using (2.61) in (2.7), we get the required inequality.(2)For 𝑝_⋆≤𝑝<𝑝^⋆. Using (2.17) and (2.58), we get 𝑣^′(𝑡)≤𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^𝑝⋆/𝑝+^𝑡₀𝜕𝑘()𝑎(𝑠)+𝑝𝜕𝑠𝑡,𝑠^⋆𝑝+𝑏(𝑠)𝑣(𝑠)^𝑝⋆/𝑝𝑑𝑠, (2.63)𝑣^′(𝑡)≤𝑏(𝑡)𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑎(𝑡)+𝑝^⋆/𝑝𝑏(𝑡)(𝑡)𝑎(𝑡)+𝑝+𝑣^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^{(𝑝⋆/𝑝)−1}+^𝑡₀𝜕𝑘()×𝑏(𝑠)𝜕𝑠𝑡,𝑠_{𝑝⋆/𝑝𝑖=𝑛𝑖=1}ℎ_𝑖(𝑠)𝑎(𝑠)+𝑝^⋆/𝑝𝑏(𝑠)(𝑠)𝑎(𝑠)+𝑝+𝑣^⋆/𝑝𝑏(𝑠)+𝑣(𝑠)^{(𝑝⋆/𝑝)−1}𝑑𝑠. (2.64) For 0≤𝑠≤t≤𝜏, and the fact that (𝑎(𝑡)+𝑝^⋆/𝑝)/𝑏(𝑡)is nondecreasing, we have 𝑎(𝑠)+𝑝^⋆/𝑝𝑏(𝑠)(𝑠)≤𝑎(𝑡)+𝑝+𝑣^⋆/𝑝𝑏(𝑡)(𝑡)≤𝑎(𝜏)+𝑝+𝑣^⋆/𝑝𝑏(𝜏)(𝑡),𝑎(𝑠)+𝑝+𝑣_⋆/𝑝𝑏(𝑠)+𝑣(𝑠)^{(𝑝⋆/𝑝)−1}≤𝑎(𝑡)+𝑝^⋆/𝑝𝑏(𝑡)+𝑣(𝑡)^{(𝑝⋆/𝑝)−1}=𝑏(𝑡)^{1−(𝑝⋆/𝑝)}𝑎(𝑡)+𝑝^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^{(𝑝⋆/𝑝)−1}, (2.65) then, 𝑣^′(𝑡)(𝑡𝑎(𝜏)+𝑝≤𝑀)^⋆/𝑝𝑏(𝜏)+𝑣(𝑡), (2.66) where 𝑀(𝑡)=𝑏(𝑡)𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)+𝑏(𝑡)^{1−(𝑝⋆/𝑝)𝑡}₀𝜕𝑘()𝑏(𝑠)𝜕𝑠𝑡,𝑠^{𝑝⋆/𝑝𝑖=𝑛}_𝑖=1ℎ_𝑖(𝑠)×𝑎(𝑡)(𝑡)𝑣(𝑡)𝑑𝑠+𝑝+𝑏^{(𝑝⋆/𝑝)−1}. (2.67) From the proof of Theorem 2.5, we get the required inequality.(3)Using (2.35) and (2.58), we get𝑣^′(𝑡)≤𝐾(𝑡,𝑡)^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡𝑎(𝑡)+𝑝)^⋆𝑝+𝑏(𝑡)𝑣(𝑡)^𝑝⋆/𝑝+^𝑡₀𝜕𝑘(𝑎(𝑠)+𝑝𝜕𝑠𝑡,𝑠)^⋆𝑝+𝑏(𝑠)𝑣(𝑠)^𝑝⋆/𝑝𝑑𝑠. (2.68) taking account the fact that (𝑎(𝑡)+(𝑝_⋆/𝑝))/𝑏(𝑡) is nondecreasing and from the proof of Theorem 2.6, we get the required inequality.

Remark 2.10. if we take 𝑛=2,𝑝≥1,𝑝₁=𝑝,𝑝₂=1, then the inequality established in Theorem 2.9 (part 1) becomes the inequality given in [4, Theorem 1(𝑎₃)].

3. Further Results

In this section, we investigate some Gronwall-type inequalities.

Theorem 3.1. Assume that 𝑢(𝑡) and 𝑓(𝑡) are non-nonnegative continuous functions on [0,∞[ and 𝑐≥0 is a constant. If 𝑘(𝑡,𝑠) is defined as in Theorem 2.9, then 𝑢^𝑝(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝜏)𝑢^𝑞(𝜏)+^𝜏_𝑡0𝑘(𝜏,𝑠)𝑢^𝑟(𝑠)𝑑𝑠𝑑𝜏, (3.1) implies 𝑢(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝜏𝑞)𝑝𝑐+𝑝−𝑞𝑝exp^𝑡_𝑡0𝐴^∗(𝑠)𝑑𝑠+^𝑡_𝑡0𝐵^∗(𝑠)exp^𝑡_𝑠𝐴^∗(𝜎)𝑑𝜎𝑑𝑠𝑑𝜏^1/𝑝 (3.2) where 𝐴^∗(𝑡)=𝑞𝑝𝑓(𝑡)+𝑟𝑞𝐾(𝑡,𝑠)+^𝑡_𝑡0𝜕𝐾(),𝐵𝜕𝑠𝑡,𝑠𝑑𝑠_∗(𝑡)=𝑝−𝑟𝑝𝐾(𝑡,𝑠)+^𝑡_𝑡0𝜕𝐾(),𝜕𝑠𝑡,𝑠𝑑𝑠 (3.3) where 𝑝≠0, 0≤𝑞≤𝑝 and 0≤𝑟≤𝑝.

Proof. Define a function 𝑧(𝑡) by the right side of (3.1) then 𝑧^′(𝑡)=𝑓(𝑡)𝑢^𝑞(𝑡)+^𝑡_𝑡0𝐾(𝑡,𝑠)𝑢^𝑟(𝑠)(𝑡𝑞𝑑𝑠≤𝑓)𝑝𝑍(𝑡)+𝑝−𝑞𝑝+^𝑡_𝑡0𝑘()𝑟𝑡,𝑠𝑝𝑍(𝑡)+𝑝−𝑟𝑝𝑑𝑠. (3.4) Define a function 𝑣(𝑡) by 𝑣(𝑡)=𝑞𝑝𝑍(𝑡)+𝑝−𝑞𝑝+^𝑡_𝑡0𝐾()𝑟𝑡,𝑠𝑝𝑍(𝑠)+𝑝−𝑟𝑝𝑑𝑠. (3.5) Then 𝑣𝑡₀=𝑞𝑝𝑐+𝑝−𝑞𝑝,𝑞𝑝𝑍(𝑡)(𝑡),𝑧≤𝑣_′(𝑡)≤𝑓(𝑡)𝑣(𝑡), (3.6) and 𝑣(𝑡) is nondecreasing for 𝑡∈ℝ₊.
Then, 𝑣^′(𝑡)=𝑞𝑝𝑧^′(𝑡)()𝑟+𝐾𝑡,𝑠𝑝𝑍(𝑡)+𝑝−𝑟𝑝+^𝑡_𝑡0𝜕𝑘()𝑟𝜕𝑠𝑡,𝑠𝑝𝑍(𝑠)+𝑝−𝑟𝑝𝑑𝑠, (3.7) its follow from (3.4), (3.6), and (3.7) that 𝑣^′(𝑡)≤𝑞𝑝𝑓(𝑡)+𝑟𝑞𝐾(𝑡,𝑠)+^𝑡_𝑡0𝜕𝐾()𝑣(𝑡)+𝜕𝑠𝑡,𝑠𝑑𝑠𝑝−𝑟𝑝𝐾(𝑡,𝑠)+^𝑡_𝑡0𝜕𝐾()𝜕𝑠𝑡,𝑠𝑑𝑠, (3.8) then (3.8) can be written as 𝑣^′(𝑡)≤𝐴^∗(𝑡)𝑣(𝑡)+𝐵^∗(𝑡), (3.9) where 𝐴^∗(𝑡)=𝑞𝑝𝑓(𝑡)+𝑟𝑞𝐾(𝑡,𝑠)+^𝑡_𝑡0𝜕𝐾(),𝐵𝜕𝑠𝑡,𝑠𝑑𝑠_∗(𝑡)=𝑝−𝑟𝑝𝐾(𝑡,𝑠)+^𝑡₀𝜕𝐾(),𝜕𝑠𝑡,𝑠𝑑𝑠 (3.10) by Lemma 2.2 we obtain 𝑣(𝑡)≤𝑞𝑝𝑐+𝑝−𝑞𝑝exp^𝑡_𝑡0𝐴^∗(𝑠)𝑑𝑠+^𝑡_𝑡0𝐵^∗(𝑠)exp^𝑡_𝑠𝐴^∗(𝜎)𝑑𝜎𝑑𝑠, (3.11) from (3.6) and (3.11), it follows that 𝑧^′(𝑡)(𝑡𝑞≤𝑓)𝑝𝑐+𝑝−𝑞𝑝exp^𝑡_𝑡0𝐴^∗(𝑠)𝑑𝑠+^𝑡_𝑡0𝐵^∗(𝑠)exp^𝑡_𝑠𝐴^∗(𝜎)𝑑𝜎𝑑𝑠, (3.12) integrating (3.12), we obtain 𝑧(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝜏𝑞)𝑝𝑐+𝑝−𝑞𝑝exp^𝑡_𝑡0𝐴^∗(𝑠)𝑑𝑠+^𝑡_𝑡0𝐵^∗(𝑠)exp^𝑡_𝑠𝐴^∗(𝜎)𝑑𝜎𝑑𝑠𝑑𝜏, (3.13) but 𝑢(𝑡)≤(𝑧(𝑡))^1/𝑝, (3.14) then the result required is found.

4. Application

In this section we present some applications of Theorems 2.3, 2.5, 2.6 and 3.1 to investigate certain properties of solutions of differential equation.

Example 4.1. We consider a nonlinear differential equation 𝑢^𝑝′(𝑡)((𝑡,𝑢(0)=𝐻𝑡,𝑢))=𝑢₀. (4.1) Assume that 𝑝,𝑝_𝑖(𝑖=1,…,𝑛)≥0, are fixed real numbers,𝑢₀ is a real constant, and 𝐻∶ℝ₊×ℝ₊→ℝ₊, |𝐻(𝑡,𝑢(𝑡))|≤𝑙(𝑡)+^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑡)𝑢^𝑝𝑖(𝑡), (4.2)ℎ_𝑖,𝑙∶ℝ₊→ℝ₊ are continuous functions.

Integrating (4.1), from 0 to 𝑡 and using (4.2) we obtain |𝑢(𝑡)|^𝑝≤𝑎(𝑡)+^𝑡₀^𝑖=𝑛_𝑖=1ℎ_𝑖(𝑠)𝑢^𝑝𝑖(𝑠)(𝑠)𝑑𝑠, (4.3) where 𝑎(𝑡)=|𝑢₀|^𝑝+∫^𝑡₀𝑙(𝑠)𝑑𝑠. By applying Theorems 2.3, 2.5, and 2.6, we estimate the solution 𝑢(𝑡) of the equation, that gives us a bound of the solution.

Example 4.2. Consider the following initial value problem: 𝑢^𝑝(𝑡)^′=𝑓(𝑡)𝑢^𝑞(𝑡)+^𝑡_𝑡0𝐾(𝑡,𝑠)𝑢^𝑟(𝑠)𝑑𝑠,𝑢𝑡₀=𝑐, (4.4) where 𝑓(𝑡) and 𝑘(𝑡,𝑠) are as defined in Theorem 3.1, and 𝑝≠0,0≤𝑞≤𝑝 and 0≤𝑟≤𝑝,and𝑐 is a constant.

Theorem 4.3. Assume 𝑢(𝑡) is a solution of (4.4), then |𝑢(𝑡)|≤|𝑐|^𝑝+^𝑡_𝑡0𝑓(𝜏𝑞)𝑝𝑐+𝑝−𝑞𝑝exp^𝑡_𝑡0𝐴^∗(𝑠)𝑑𝑠+^𝑡_𝑡0𝐵^∗(𝑠)exp^𝑡_𝑠𝐴^∗(𝜎)𝑑𝜎𝑑𝑠𝑑𝜏^1/𝑝 (4.5) where 𝐴^∗(𝑡) and 𝐵^∗(𝑡) are defined in (3.10).

Proof. The solution 𝑢(𝑡) of (4.4) satisfies the following equivalent equation: 𝑢^𝑝(𝑡)=𝑐^𝑝+^𝑡_𝑡0𝑓(𝜏)𝑢^𝑞(𝜏)+^𝑡_𝑡0𝑘(𝜏,𝑠)𝑢^𝑟(𝑠)𝑑𝑠𝑑𝜏. (4.6) It follows from (4.6) that |𝑢(𝑡)|^𝑝≤|𝑐|^𝑝+^𝑡_𝑡0𝑓(𝜏)|𝑢(𝜏)|^𝑞+^𝑡_𝑡0𝑘(𝜏,𝑠)|𝑢(𝜏)|^𝑟𝑑𝑠𝑑𝜏. (4.7)
Using Theorem 3.1, we obtain (4.5).