Abstract

The object of this paper is to establish some nonlinear integrodifferential integral inequalities in 𝑛 independent variables. These new inequalities represent a generalization of the results obtained by Pachpatte in the case of a function with one and two variables. Our results can be used as tools in the qualitative theory of a certain class of partial integrodifferential equation.

1. Introduction

It is well known that the integral inequalities involving functions of one and more than one independent variables, which provide explicit bounds on unknown functions, play a fundamental role in the development of the theory of differential equations. In the past few years, a number of integral inequalities had been established by many scholars, which are motivated by certain applications. For example, we refer the reader to (see [15]) and the references therein.

The study of integrodifferential inequalities for functions of one or 𝑛 independent variables is also a very important tool in the study of stability, existence, bounds, and other qualitative properties of differential equation solutions, integrodifferential equations, and in the theory of hyperbolic partial differential equations (see [69]).

One of the most useful inequalities is given in the following lemma (see [1, 10]).

Lemma 1.1 (see [1]). Let Φ(𝑥,𝑦) and 𝑐(𝑥,𝑦) be nonnegative continuous functions defined for 𝑥0, 𝑦0, for which the inequality Φ(𝑥,𝑦)𝑎(𝑥)+𝑏(𝑦)+𝑥0𝑦0𝑐(𝑠,𝑡)Φ(𝑠,𝑡)𝑑𝑠𝑑𝑡(1.1) holds for 𝑥0, 𝑦0, where 𝑎(𝑥),𝑏(𝑦)>0; 𝑎(𝑥),𝑏(𝑦)0 are continuous functions defined for 𝑥0, 𝑦0. Then []Φ(𝑥,𝑦)𝑎(0)+𝑏(𝑦)][𝑎(𝑥)+𝑏(0)[]𝑎(0)+𝑏(0)exp𝑥0𝑦0,𝑐(𝑠,𝑡)𝑑𝑠𝑑𝑡(1.2) for 𝑥0, 𝑦0.

Wendroff's inequality has recently evoked a lively interest, as may be seen from the papers of Pachpatte [10]. In [10], Pachpatte considered some new integrodifferential inequalities of the Wendroff type for functions of two independent variables. Our aim in this paper is to establish some integrodifferential inequalities in 𝑛 independent variables, an application of our results is also given.

2. Results

Throughout this paper, we will assume that 𝑆 in any bounded open set in the dimensional Euclidean space 𝑛 and that our integrals are on 𝑛 (𝑛1).

For 𝑥=(𝑥1,𝑥2,𝑥𝑛), 𝑡=(𝑡1,𝑡2,𝑡𝑛), 𝑥0=(𝑥01,𝑥02,,𝑥0𝑛)𝑆, we will denote 𝑥𝑥0𝑑𝑡=𝑥1𝑥01𝑥2𝑥02𝑥𝑛𝑥0𝑛𝑑𝑡𝑛𝑑𝑡1.(2.1)

Furthermore, for 𝑥,𝑡𝑛, we will write 𝑡𝑥 whenever 𝑡𝑖𝑥𝑖, 𝑖=1,2,,𝑛and𝑥𝑥00, for 𝑥,𝑥0𝑆.

We note 𝐷=𝐷1𝐷2𝐷𝑛, where𝐷𝑖=𝜕/𝜕𝑥𝑖,for𝑖=1,2,,𝑛.

We use the usual convention of writing 𝑠Ψ𝑢(𝑠)=0 if Ψ is the empty set.

Our main results are given in the following theorems.

Theorem 2.1. Let Φ(𝑥) and 𝑐(𝑥) be nonnegative continuous functions defined on 𝑆, for which the inequality Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0𝑐(𝑡)Φ(𝑡)𝑑𝑡(2.2) holds for all 𝑥𝑆 with 𝑥𝑥00, where 𝑎𝑖(𝑥𝑖)>0,𝑎𝑖(𝑥𝑖) are continuous functions defined for 𝑥𝑖0 for all 𝑖=1,2,,𝑛. Then Φ(𝑥)𝐴(𝑥)exp𝑥𝑥0,𝑐(𝑡)𝑑𝑡(2.3) for 𝑥𝑆 with 𝑥𝑥00, where 𝐴𝑎(𝑥)=1𝑥1+𝑎2𝑥02+𝑛𝑠=3𝑎𝑠𝑥𝑠𝑎1𝑥01+𝑎2𝑥2+𝑛𝑠=3𝑎𝑠𝑥𝑠𝑎1𝑥01+𝑎2𝑥02+𝑛𝑠=3𝑎𝑠𝑥𝑠.(2.4)
Proof. We define the function 𝑢(𝑥) by the right member of (2.2), Then 𝐷𝑢(𝑥)=𝑐(𝑥)Φ(𝑥),(2.5)𝑢𝑥01,𝑥2,,𝑥𝑛=𝑎1𝑥01+𝑎2𝑥2+𝑛𝑠=3𝑎𝑠𝑥𝑠,(2.6)𝑢𝑥1,𝑥02,𝑥3,,𝑥𝑛=𝑎1𝑥1+𝑎2𝑥02+𝑛𝑠=3𝑎𝑠𝑥𝑠.(2.7) Using Φ(𝑥)𝑢(𝑥) in (2.5), we have 𝐷𝑢(𝑥)𝑐(𝑥)𝑢(𝑥).(2.8) From (2.8), we observe that 𝑢(𝑥)𝐷𝑢(𝑥)𝑢2(𝑥)𝑐(𝑥),(2.9) that is 𝑢(𝑥)𝐷𝑢(𝑥)𝑢2𝐷(𝑥)𝑐(𝑥)+𝑛𝑢𝐷(𝑥)1𝐷𝑛1𝑢(𝑥)𝑢2,(𝑥)(2.10) hence 𝐷𝑛𝐷1𝐷𝑛1𝑢(𝑥)𝑢(𝑥)𝑐(𝑥).(2.11) Integrating (2.11) with respect to 𝑥𝑛 from 𝑥0𝑛 to 𝑥𝑛, we have 𝐷1𝐷𝑛1𝑢(𝑥)𝑢(𝑥)𝑥𝑛𝑥0𝑛𝑐𝑥1,,𝑥𝑛1,𝑡𝑛𝑑𝑡𝑛,(2.12) thus 𝑢(𝑥)𝐷1𝐷𝑛1𝑢(𝑥)𝑢2(𝑥)𝑥𝑛𝑥0𝑛𝑐𝑥1,,𝑥𝑛1,𝑡𝑛𝑑𝑡𝑛+𝐷𝑛1𝑢𝐷(𝑥)1𝐷𝑛2𝑢(𝑥)𝑢2,(𝑥)(2.13) that is 𝐷𝑛1𝐷1𝐷𝑛2𝑢(𝑥)𝑢(𝑥)𝑥𝑛𝑥0𝑛𝑐𝑥1,,𝑥𝑛1,𝑡𝑛𝑑𝑡𝑛.(2.14) Integrating (2.14) with respect to 𝑥𝑛1 from 𝑥0𝑛1 to 𝑥𝑛1, we have 𝐷1𝐷𝑛2𝑢(𝑥)𝑢(𝑥)𝑥𝑛1𝑥0𝑛1𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑥𝑛2,𝑡𝑛1,𝑡𝑛𝑑𝑡𝑛𝑑𝑡𝑛1.(2.15) Continuing this process, we obtain 𝐷1𝐷2𝑢(𝑥)𝑢(𝑥)𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑥2,𝑡3,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡𝑛1𝑑𝑡3,(2.16) from this we obtain 𝐷2𝐷1𝑢(𝑥)𝑢(𝑥)𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑥2,𝑡3,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡𝑛1𝑑𝑡3.(2.17) Integrating (2.17) with respect to 𝑥2 from 𝑥02 to 𝑥2 and by (2.7), we have 𝐷1𝑢(𝑥)𝑎𝑢(𝑥)1𝑥1𝑎2𝑥00+𝑎1𝑥1+𝑛𝑠=3𝑎𝑠𝑥𝑠+𝑥2𝑥02𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,𝑡3,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡𝑛1𝑑𝑡2.(2.18) Integrating (2.18) with respect to 𝑥1 from 𝑥01 to 𝑥1 and by (2.6), we have log𝑢(𝑥)𝑢𝑥01,𝑥2,,𝑥𝑛𝑥1𝑥01𝑎1𝑡1𝑎2𝑥02+𝑎1𝑡1+𝑛𝑠=3𝑎𝑠𝑥𝑠𝑑𝑡1+𝑥𝑥0𝑐(𝑡)𝑑𝑡,(2.19) that is 𝑢(𝑥)𝐴(𝑥)exp𝑥𝑥0.𝑐(𝑡)𝑑𝑡(2.20) By (2.20) and Φ(𝑥)𝑢(𝑥), we obtain the desired bound in (2.3).

Remark 2.2. We note that in the special case 𝑛=2, 𝑥2+ and 𝑥0=(𝑥01,𝑥02)=(0,0) in Theorem 2.1. our estimate reduces to Lemma 1.1 (see [10]).

Theorem 2.3. Let Φ(𝑥), 𝑐(𝑥), 𝐷𝑖Φ(𝑥), and 𝐷Φ(𝑥) be nonnegative continuous functions for all 𝑖=1,2,,𝑛 defined for 𝑥𝑆, Φ(𝑥01,𝑥2,𝑥3,,𝑥𝑛)=0 and Φ(𝑥1,,𝑥𝑖1,𝑥0𝑖,𝑥𝑖+1,𝑥𝑛)=0 for any 𝑖=2,3,,𝑛. If 𝐷Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0[]𝑐(𝑡)Φ(𝑡)+𝐷Φ(𝑡)𝑑𝑡(2.21) holds for 𝑥𝑆, where 𝑎𝑖(𝑥𝑖)>0; 𝑎𝑖(𝑥𝑖)0 are continuous functions defined for 𝑥𝑖0 for all 𝑖=1,2,,𝑛. Then 𝐷Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0𝑐(𝑡)𝐴(𝑡)exp𝑡𝑥0[]1+𝑐(𝜏)𝑑𝜏𝑑𝑡.(2.22) For 𝑥𝑆 with 𝑥𝑡𝜏𝑥00, where 𝐴(𝑥) is defined in (2.4).
Proof. We define the function 𝑢(𝑥)=𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0[]𝑐(𝑡)Φ(𝑡)+𝐷Φ(𝑡)𝑑𝑡,(2.23)𝑢𝑥01,𝑥2,𝑥3,,𝑥𝑛=𝑎1𝑥01+𝑛𝑖=2𝑎𝑖𝑥𝑖.(2.24) Then, (2.21) can be restated as 𝐷Φ(𝑥)𝑢(𝑥).(2.25) Differentiating (2.23), [].𝐷𝑢(𝑥)=𝑐(𝑥)Φ(𝑥)+𝐷Φ(𝑥)(2.26) Integrating both sides of (2.26) to 𝑥 from 𝑥0 to 𝑥, we have Φ(𝑥)𝑥𝑥0𝑢(𝑡)𝑑𝑡.(2.27) Now, using (2.27) and (2.25) in (2.26) we obtain 𝐷𝑢(𝑥)𝑐(𝑥)𝑢(𝑥)+𝑥𝑥0.𝑢(𝑡)𝑑𝑡(2.28) If we put 𝑣(𝑥)=𝑢(𝑥)+𝑥𝑥0𝑢(𝑡)𝑑𝑡,(2.29)𝑣𝑥1,,𝑥𝑖1,𝑥0𝑖,𝑥𝑖+1,,𝑥𝑛𝑥=𝑢1,,𝑥𝑖1,𝑥0𝑖,𝑥𝑖+1,,𝑥𝑛,(2.30) then by (2.29), we have 𝐷𝑣(𝑥)=𝐷𝑢(𝑥)+𝑢(𝑥).(2.31) Using the facts that 𝐷𝑢(𝑥)𝑐(𝑥)𝑣(𝑥) and 𝑢(𝑥)𝑣(𝑥), we have []𝐷𝑣(𝑥)1+𝑐(𝑥)𝑣(𝑥).(2.32) Which, by following an argument similar to that in the proof of Theorem 2.1, yields the estimate for 𝑣(𝑥) such that 𝑣(𝑥)𝐴(𝑥)exp𝑥𝑥0[].1+𝑐(𝑡)𝑑𝑡(2.33) By (2.33) and (2.28), we have 𝐷𝑢(𝑥)𝑐(𝑥)𝐴(𝑥)exp𝑥𝑥0[],1+𝑐(𝑡)𝑑𝑡(2.34)𝐷1𝐷2𝐷𝑛1𝑢𝑥1,,𝑥𝑛1,𝑥0𝑛=0.(2.35) Integrating both sides of (2.34) to 𝑥𝑛 from 𝑥0𝑛 to 𝑥𝑛 and by (2.35), we have 𝐷1𝐷2𝐷𝑛1𝑢(𝑥)𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑥𝑛1,𝑡𝑛𝐴𝑥1,𝑥𝑛1,𝑡𝑛exp𝑡𝑥0[]1+𝑐(𝜏)𝑑𝜏𝑑𝑡𝑛.(2.36) By (2.23), we have 𝐷1𝐷2𝑢𝑥1,𝑥2,𝑥03,𝑥4,,𝑥𝑛=0.(2.37) Continuing this process, and by (2.37), we obtain 𝐷1𝐷2𝑢(𝑥)𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑥2,𝑡3,,𝑡𝑛𝐴𝑥1,𝑥2,𝑡3,,𝑡𝑛exp𝑡𝑥0[]1+𝑐(𝜏)𝑑𝜏𝑑𝑡𝑛𝑑𝑡3.(2.38) By (2.23), we have 𝐷1𝑢𝑥1,𝑥02,𝑥3,𝑥4,,𝑥𝑛=𝑎1𝑥1.(2.39) Integrating both sides of (2.38) to 𝑥2 from 𝑥02 to 𝑥2 and by (2.39), we have 𝐷1𝑢(𝑥)𝑎1𝑥1+𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,,𝑡𝑛𝐴𝑥1,𝑡2,,𝑡𝑛×exp𝑡𝑥0[]1+𝑐(𝜏)𝑑𝜏𝑑𝑡𝑛𝑑𝑡2.(2.40) Integrating (2.40) with respect to 𝑥1 from 𝑥01 to 𝑥1, and by (2.24), we have 𝑢(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0𝑐(𝑡)𝐴(𝑡)exp𝑡𝑥0[]1+𝑐(𝜏)𝑑𝜏𝑑𝑡.(2.41) By (2.41) and (2.25), we obtain the desired bound in (2.22).

Remark 2.4. We note that in the special case 𝑛=2, 𝑥2+ and 𝑥0=(𝑥01,𝑥02)=(0,0) in Theorem 2.3, then our result reduces to Theorem 1 obtained in [10].

Theorem 2.5. Let Φ(𝑥), 𝑐(𝑥), 𝐷𝑖Φ(𝑥), and 𝐷Φ(𝑥) be nonnegative continuous functions for all 𝑖=1,2,,𝑛 defined for 𝑥𝑆, Φ(𝑥01,𝑥2,𝑥3,,𝑥𝑛)=0 andΦ(𝑥1,,𝑥𝑖1,𝑥0𝑖,𝑥𝑖+1,𝑥𝑛)=0 for any 𝑖=2,3,,𝑛. If 𝐷Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑀Φ(𝑥)+𝑥𝑥0[]𝑐(𝑡)Φ(𝑡)+𝐷Φ(𝑡)𝑑𝑡(2.42) holds for 𝑥𝑆, where 𝑎𝑖(𝑥𝑖)>0; 𝑎𝑖(𝑥𝑖)0 are continuous functions defined for 𝑥𝑖0 for all 𝑖=1,2,,𝑛. and 𝑀0 is constant. Then 𝐷Φ(𝑥)𝐴(𝑥)exp𝑥𝑥0[],𝑀+𝑐(𝑡)+𝑀𝑐(𝑡)𝑑𝑡(2.43) for 𝑥𝑆, with 𝑥𝑡𝑥00, where 𝐴(𝑥) is defined in (2.4).
Proof. We define the function 𝑢(𝑥)=𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑀Φ(𝑥)+𝑥𝑥0[]𝑐(𝑡)Φ(𝑡)+𝐷Φ(𝑡)𝑑𝑡(2.44) with 𝑢𝑥01,𝑥2,𝑥3,,𝑥𝑛=𝑎1𝑥01+𝑛𝑖=2𝑎𝑖𝑥𝑖.(2.45) Differentiating (2.44), we have [[.𝐷𝑢(𝑥)=𝑀𝐷Φ(𝑥)+𝑐(𝑥)Φ(𝑥)+𝐷Φ(𝑥)]](2.46) Using the fact that 𝐷Φ(𝑥)𝑢(𝑥) and 𝑀Φ(𝑥)𝑢(𝑥), we have []𝐷𝑢(𝑥)𝑀+𝑐(𝑥)+𝑀𝑐(𝑥)𝑢(𝑥),(2.47) by (2.47), we have 𝑢(𝑥)𝐴(𝑥)exp𝑥𝑥0[],𝑀+𝑐(𝑡)+𝑀𝑐(𝑡)𝑑𝑡(2.48) where 𝐴(𝑥) is defined in (2.4).
By (2.48) and using the fact that 𝐷Φ(𝑥)𝑢(𝑥) from (2.42), we obtain the desired bound in (2.43).

Remark 2.6. We note that in the special case 𝑛=2, 𝑥2+ and 𝑥0=(𝑥01,𝑥02)=(0,0) in Theorem 2.5, then our result reduces to Theorem 2 obtained in [10].

Theorem 2.7. Let Φ(𝑥), 𝑝(𝑥), and 𝑞(𝑥) be nonnegative continuous functions defined for 𝑥𝑆. If Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0𝑝(𝑡)Φ(𝑡)𝑑𝑡+𝑥𝑥0𝑝(𝑡)𝑡𝑥0𝑞(𝑠)Φ(𝑠)𝑑𝑠𝑑𝑡(2.49) holds for 𝑥𝑥00, where 𝑎𝑖(𝑥𝑖)>0; 𝑎𝑖(𝑥𝑖)0 are continuous functions defined for 𝑥𝑖0 for all 𝑖=1,2,,𝑛. Then Φ(𝑥)𝑛𝑖=1𝑎𝑖𝑥𝑖+𝑥𝑥0𝑝(𝑡)𝑄(𝑡)𝑑𝑡,(2.50) for all 𝑥𝑥00, where 𝑄(𝑥)=𝐴(𝑥)exp𝑥𝑥0,(𝑝(𝑡)+𝑞(𝑡))𝑑𝑡(2.51) with 𝐴(𝑥) defined in (2.4).
Proof. The proof of this Theorem follows by an argument similar to that in Theorem 2.1, We omit the details.

Remark 2.8. We note that in the special case 𝑛=2, 𝑥2+ and 𝑥0=(𝑥01,𝑥02)=(0,0) in Theorem 2.7, our result reduces to Theorem 2 obtained in [10].

3. Nonlinear Integrodifferential in 𝑛 Independents Variables

In this section, we will give some new nonlinear integrodifferential inequalities for the functions of n-independent variables.

We can also give the following lemma.

Lemma 3.1 (see [2, 11]). Let 𝑢(𝑥), 𝑎(𝑥), and 𝑏(𝑥) be nonnegative continuous functions, defined for 𝑥𝑆.
Assume that 𝑎(𝑥) is a positive, continuous function and nondecreasing in each of the variables 𝑥𝑆. If 𝑢(𝑥)𝑎(𝑥)+𝑥𝑥0𝑏(𝑡)𝑢(𝑡)𝑑𝑡(3.1) holds for all 𝑥𝑆, with 𝑥𝑥00. Then 𝑢(𝑥)𝑎(𝑥)exp𝑥𝑥0.𝑏(𝑡)𝑑𝑡(3.2)

Theorem 3.2. Let Φ(𝑥), 𝑎(𝑥), 𝑏(𝑥), 𝑐(𝑥), 𝑓(𝑥), 𝐷𝑖Φ(𝑥), and 𝐷Φ(𝑥) be nonnegative continuous functions for all 𝑖=1,2,,𝑛 defined for 𝑥𝑆, Φ(𝑥01,𝑥2,𝑥3,,𝑥𝑛)=0andΦ(𝑥1,,𝑥𝑖1,𝑥0𝑖,𝑥𝑖+1,𝑥𝑛)=0 for any 𝑖=2,3,,𝑛. Let 𝐾(Φ(𝑥)) be a real-valued, positive, continuous, strictly nondecreasing, subadditive, and submultiplicative function for Φ(𝑥)0, and let 𝐻(Φ(𝑥)) be a real-valued, continuous positive, and nondecreasing function defined for 𝑥𝑆. Assume that 𝑎(𝑥) and 𝑓(𝑥) are positive and nondecreasing in each of the variables 𝑥𝑆. If 𝐷Φ(𝑥)𝑎(𝑥)+𝑓(𝑥)𝐻𝑥𝑥0+𝑐(𝑡)𝐾(Φ(𝑡))𝑑𝑡𝑥𝑥0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡(3.3) holds, for 𝑥𝑆 with 𝑥𝑥00. Then 𝐺𝐷Φ(𝑥)𝑎(𝑥)+𝑓(𝑥)𝐻1𝐺(𝜉)+𝑥𝑥0𝑐(𝑡)𝐾(𝑝(𝑡)𝑓(𝑡))𝑑𝑡exp𝑥𝑥0,𝑏(𝑡)𝑑𝑡(3.4) for 𝑥𝑆, where 𝑝(𝑥)=𝑥𝑥0exp𝑡𝑥0𝑏(𝑠)𝑑𝑠𝑑𝑡,𝜉=𝑥0𝑐(𝑡)𝐾(𝑎(𝑡)𝑝(𝑡))𝑑𝑡,𝐺(𝑧)=𝑧𝑧0𝑑𝑠𝐾(𝐻(𝑠)),𝑧𝑧0>0,(3.5) where 𝐺1 is the inverse function of 𝐺, and 𝐺(𝜉)+𝑥𝑥0𝑐(𝑡)𝐾(𝑝(𝑡)𝑓(𝑡))𝑑𝑡(3.6) is in the domain of 𝐺1 for 𝑥𝑆.
Proof. We define the function 𝑧(𝑥)=𝑎(𝑥)+𝑓(𝑥)𝐻𝑥𝑥0,𝑐(𝑡)𝐾(Φ(𝑡))𝑑𝑡(3.7) then (2.4) can be restated as 𝐷Φ(𝑥)𝑧(𝑥)+𝑥𝑥0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡.(3.8) Clearly, 𝑧(𝑥) is a positive, continuous function and nondecreasing in each of the variables 𝑥𝑆, using (3.1) of Lemma 3.1 to (3.8), we have 𝐷Φ(𝑥)𝑧(𝑥)exp𝑥𝑥0.𝑏(𝑡)𝑑𝑡(3.9) Integrating to 𝑥 from 𝑥0 to 𝑥, we have Φ(𝑥)𝑧(𝑥)𝑝(𝑥),(3.10) where 𝑝(𝑥)=𝑥𝑥0exp𝑡𝑥0𝑏(𝑠)𝑑𝑠𝑑𝑡.(3.11) By (3.7), we have 𝑧(𝑥)=𝑎(𝑥)+𝑓(𝑥)𝐻(𝑣(𝑥)),(3.12) where 𝑣(𝑥)=𝑥𝑥0𝑐(𝑡)𝐾(Φ(𝑡))𝑑𝑡.(3.13) By (3.10) and (3.13), we have Φ(𝑥){𝑎(𝑥)+𝑓(𝑥)𝐻(𝑣(𝑥))}𝑝(𝑥).(3.14) From (3.14) and (3.13) and since 𝐾 is a subadditive and submultiplicative function, we notice that 𝑣(𝑥)𝑥𝑥0[]𝑐(𝑡)𝐾{𝑎(𝑡)+𝑓(𝑡)𝐻(𝑣(𝑡))}𝑝(𝑡)𝑑𝑡,𝑥𝑥0𝑐(𝑡)𝐾(𝑎(𝑡)𝑝(𝑡))𝑑𝑡+𝑥𝑥0𝑐(𝑡)𝐾(𝑓(𝑡)𝑝(𝑡))𝐾(𝐻(𝑣(𝑡)))𝑑𝑡,𝑥0𝑐(𝑡)𝐾(𝑎(𝑡)𝑝(𝑡))𝑑𝑡+𝑥𝑥0𝑐(𝑡)𝐾(𝑓(𝑡)𝑝(𝑡))𝐾(𝐻(𝑣(𝑡)))𝑑𝑡.(3.15) We define Ψ(𝑥) as the right side of (3.14), then Ψ𝑥01,𝑥2,𝑥3,,𝑥𝑛=𝑥0𝑐(𝑡)𝐾(𝑎(𝑡)𝑝(𝑡))𝑑𝑡,(3.16)𝑣(𝑥)Ψ(𝑥).(3.17)Ψ(𝑥) is positive and nondecreasing in each of the variables 𝑥2,,𝑥𝑛𝑅+𝑛1,then 𝐷1Ψ(𝑥)=𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,,𝑡𝑛𝐾𝑝𝑥1,𝑡2,,𝑡𝑛𝑓𝑥1,𝑡2,,𝑡𝑛𝐻𝑣𝑥×𝐾1,𝑡2,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡2,𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑑𝑥1,𝑡2,,𝑡𝑛𝐾𝑝𝑥1,𝑡2,,𝑡𝑛𝑓𝑥1,𝑡2,,𝑡𝑛𝐻Ψ𝑥×𝐾1,𝑡2,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡2,𝐾(𝐻(Ψ(𝑥)))𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,,𝑡𝑛𝑝𝑥×𝐾1,𝑡2,,𝑡𝑛𝑓𝑥1,𝑡2,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡2.(3.18) Dividing both sides of (3.18) by 𝐾(𝐻(Ψ(𝑥))), we get 𝐷1Ψ(𝑥)𝐾(𝐻(Ψ(𝑥)))𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,,𝑡𝑛𝐾𝑝𝑥1,𝑡2,,𝑡𝑛𝑓𝑥1,𝑡2,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡2.(3.19) We note that 𝐺(𝑧)=𝑧𝑧0𝑑𝑠𝐾(𝐻(𝑠)),𝑧𝑧0>0.(3.20) Thus, it follows that 𝐷1𝐷𝐺(Ψ(𝑥))=1Ψ(𝑥).𝐾(𝐻(Ψ(𝑥)))(3.21) From (3.19), (3.20), and (3.21), we have 𝐷1𝐺(Ψ(𝑥))𝑥2𝑥02𝑥3𝑥03𝑥𝑛𝑥0𝑛𝑐𝑥1,𝑡2,,𝑡𝑛𝐾𝑝𝑥1,𝑡2,,𝑡𝑛𝑓𝑥1,𝑡2,,𝑡𝑛𝑑𝑡𝑛𝑑𝑡2.(3.22) Now, setting 𝑥1=𝑠 in (3.22) and then integrating with respect from 𝑥01 to 𝑥1, we obtain Ψ𝑥𝐺(Ψ(𝑥))𝐺01,𝑥2,,𝑥𝑛+𝑥𝑥0𝑐(𝑡)𝐾(𝑝(𝑡)𝑓(𝑡))𝑑𝑡,(3.23) by (3.23), we have Ψ(𝑥)𝐺1𝐺Ψ𝑥01,𝑥2,,𝑥𝑛+𝑥𝑥0.𝑐(𝑡)𝐾(𝑝(𝑡)𝑓(𝑡))𝑑𝑡(3.24) The required inequality in (3.4) follows from the fact (3.9), (3.12), (3.17), and (3.24).

Many interesting corollaries can be obtained from Theorem 3.2.

Corollary 3.3. Let Φ(𝑥), 𝑎(𝑥), 𝑏(𝑥), 𝑐(𝑥), 𝐷𝑖Φ(𝑥), 𝐷Φ(𝑥), and 𝐾(Φ(𝑥)) be as defined in Theorem 3.2. If 𝐷Φ(𝑥)𝑎(𝑥)+𝑥𝑥0𝑐(𝑡)𝑔(Φ(𝑡))𝑑𝑡+𝑥𝑥0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡(3.25) holds, for 𝑥𝑛+ with 𝑥𝑥00. Then 𝐷Φ(𝑥)𝑎(𝑥)+𝑇1𝑇(𝜉)+𝑥𝑥0𝑐(𝑡)𝐾(𝑝(𝑡))𝑑𝑡exp𝑥𝑥0,𝑏(𝑡)𝑑𝑡(3.26) for 𝑥𝑛+ with 𝑥𝑥00, where 𝑝(𝑥)=𝑥𝑥0exp𝑡𝑥0𝑏(𝑠)𝑑𝑠𝑑𝑡,𝜉=𝑥0𝑐(𝑡)𝐾(𝑎(𝑡)𝑝(𝑡))𝑑𝑡,𝑇(𝑧)=𝑧𝑧0𝑑𝑠𝐾(𝑠),𝑧𝑧0>0,(3.27) where 𝑇1 is the inverse function of 𝑇 and 𝑇(𝜉)+𝑥𝑥0𝑐(𝑡)𝐾(𝑝(𝑡))𝑑𝑡(3.28) is in the domain of 𝑇1 for 𝑥𝑛+.
Proof. The proof of this Corollary follows by an argument similar to that in Theorem 3.2. We omit the details.

Corollary 3.4. Let Φ(𝑥), 𝑏(𝑥), 𝑐(𝑥), 𝐷𝑖Φ(𝑥), and 𝐷Φ(𝑥) be as defined in Theorem 3.2. If 𝐷Φ(𝑥)𝑀+𝑥𝑥0𝑐(𝑡)Φ(𝑡)𝑑𝑡+𝑥𝑥0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡(3.29) holds, for 𝑥𝑛+ with 𝑥𝑥00, where 𝑀>0 is a constant, then Φ(𝑥)𝑀1+explog𝑥0+𝑐(𝑡)𝑝(𝑡)𝑑𝑡𝑥𝑥0𝑐(𝑡)𝑝(𝑡)𝑑𝑡𝑝(𝑥),(3.30) for 𝑥𝑛+ with 𝑥𝑥00, where 𝑝(𝑥)=𝑥𝑥0exp𝑡𝑥0𝑏(𝑠)𝑑𝑠𝑑𝑡.(3.31)
Proof. Setting 𝑔(𝑥)=𝑥 and 𝑎(𝑥)=𝑀 in Corollary 3.3, we obtain our result in this Corollary. We omit the details.

Similarly, we can obtain many other kinds of estimates.

4. An Application

In this section, we present an immediate simple example of application of Theorem 3.2 to the study of boundedness of the solution of a partial integrodifferential equation.

Consider the nonlinear partial integrodifferential equation𝐷𝑢(𝑥)=𝑓(𝑥)+𝑥0𝑢(𝑥,𝑡,𝑢(𝑡),𝐷𝑢(𝑡))𝑑𝑡,,𝑥𝑖,0,𝑥𝑖+2,=0,𝑖=1,2,,𝑛,(4.1) for 𝑥𝑛+, where 𝑛+××, 𝑓(𝑥)𝑛+ are continuous functions.

Assume that functions are defined and continuous on their respective domains of definition, such that||||||||||||||||𝑓(𝑥)𝑀,(𝑥,𝑡,𝑢(𝑡),𝐷𝑢(𝑡))𝑐(𝑡)𝑢(𝑡)+𝑏(𝑡)𝐷𝑢(𝑡)(4.2) for 𝑥𝑛+, where 𝑀>0 is a constant and 𝑐(𝑥) and 𝑏(𝑥) are nonnegative, continuous functions defined for 𝑥𝑛+. If Φ(𝑥) is any solution of boundary value problem (4.1), then𝐷Φ(𝑥)=𝑓(𝑥)+𝑥0(𝑥,𝑡,Φ(𝑡),𝐷Φ(𝑡))𝑑𝑡(4.3) for 𝑥𝑛+, by (4.2), we have||||𝐷Φ(𝑥)=𝑀+𝑥0||||||||𝑐(𝑡)Φ(𝑥)+𝑏(𝑡)𝐷Φ(𝑥)𝑑𝑡.(4.4) Now, by a suitable application of Corollary 3.4 of Theorem 3.2, we obtain the bound on the solution Φ(𝑥) of (4.1). ||||Φ(𝑥)𝑀𝑝(𝑥)1+explog0+𝑐(𝑡)𝑝(𝑡)𝑑𝑡𝑥0𝑐(𝑡)𝑝(𝑡)𝑑𝑡(4.5) or 𝑥𝑛+, where 𝑝(𝑥)=𝑥0exp𝑡0.𝑏(𝑠)𝑑𝑠𝑑𝑡(4.6)

Remark 4.1. Using a similar method of those in the proof of the theorems above, we can also obtain new reversed inequalities of our results. Our results also can be generalized to integrodifferential inequalities with a time delay for functions of one or 𝑛 independent variables, this is under study and will be addressed in a forthcoming work. Among these integrodifferential inequalities with a delay, we can quote:

𝐷Φ(𝑥)𝑎(𝑥)+𝛼𝑥𝛼(𝑥)0𝑐(𝑡)𝐾(Φ(𝑡))𝑑𝑡+𝛽𝑥𝛽(𝑥)0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡.𝐷Φ(𝑥)𝑎(𝑥)+𝑓(𝑥)𝛼𝑥𝛼(𝑥)0𝑐(𝑡)Φ(𝑡)𝐾(Φ(𝑡))𝑑𝑡+𝛽𝑥𝛽(𝑥)0𝑏(𝑡)𝐷Φ(𝑡)𝑑𝑡.𝐷Φ(𝑥)𝑎(𝑥)+𝑓(𝑥)𝐻𝛼𝑥𝛼(𝑥)0+𝑐(𝑡)𝑛(Φ(𝑡))𝐾(Φ(𝑡))𝑑𝑡𝛽𝑥𝛽(𝑥)0𝑏(𝑡)𝑔(𝐷Φ(𝑡))𝑑𝑡.(4.7)