# On Some Nonlinear Integrodifferential Inequalities for Functions ofn Independent Variables

Authors: Hassane Khellaf; Mohamed El-Hadi Smakdji
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

The object of this paper is to establish some nonlinear integrodifferential integral inequalities inn 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.


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## Body

## 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 [1–5]) and the references therein.The study of integrodifferential inequalities for functions of one orn 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 [6–9]).One of the most useful inequalities is given in the following lemma (see [1, 10]).Lemma 1.1 (see [1]).
LetΦ(x,y) and c(x,y) be nonnegative continuous functions defined for x≥0, y≥0, for which the inequality
(1.1)Φ(x,y)≤a(x)+b(y)+∫0x∫0yc(s,t)Φ(s,t)dsdt
holds for x≥0, y≥0, where a(x),b(y)>0; a′(x),b′(y)≥0 are continuous functions defined for x≥0, y≥0. Then
(1.2)Φ(x,y)≤[a(0)+b(y)][a(x)+b(0)][a(0)+b(0)]exp⁡(∫0x∫0yc(s,t)dsdt),
for x≥0, y≥0.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 thatS in any bounded open set in the dimensional Euclidean space ℝn and that our integrals are on ℝn (n≥1).Forx=(x1,x2,…xn), t=(t1,t2,…tn), x0=(x10,x20,…,xn0)∈S, we will denote (2.1)∫x0xdt=∫x10x1∫x20x2⋯∫xn0xn⋯dtn⋯dt1.Furthermore, forx,t∈ℝn, we will write t≤x whenever ti≤xi, i=1,2,…,nandx≥x0≥0, for x,x0∈S.We noteD=D1D2⋯Dn, whereDi=∂/∂xi,fori=1,2,…,n.We use the usual convention of writing∑s∈Ψu(s)=0 if Ψ is the empty set.Our main results are given in the following theorems.Theorem 2.1.
LetΦ(x) and c(x) be nonnegative continuous functions defined on S, for which the inequality
(2.2)Φ(x)≤∑i=1nai(xi)+∫x0xc(t)Φ(t)dt
holds for all x∈S with x≥x0≥0, where ai(xi)>0,ai′(xi) are continuous functions defined for xi≥0 for all i=1,2,…,n. Then
(2.3)Φ(x)≤A(x)exp⁡(∫x0xc(t)dt),
for x∈S with x≥x0≥0, where
(2.4)A(x)=[a1(x1)+a2(x20)+∑s=3nas(xs)][a1(x10)+a2(x2)+∑s=3nas(xs)][a1(x10)+a2(x20)+∑s=3nas(xs)].
Proof.
We define the functionu(x) by the right member of (2.2), Then
(2.5)Du(x)=c(x)Φ(x),(2.6)u(x10,x2,…,xn)=a1(x10)+a2(x2)+∑s=3nas(xs),(2.7)u(x1,x20,x3,…,xn)=a1(x1)+a2(x20)+∑s=3nas(xs).
Using Φ(x)≤u(x)  in (2.5), we have
(2.8)Du(x)≤c(x)u(x).
From (2.8), we observe that
(2.9)u(x)Du(x)u2(x)≤c(x),
that is
(2.10)u(x)Du(x)u2(x)≤c(x)+(Dnu(x))(D1⋯Dn-1u(x))u2(x),
hence
(2.11)Dn(D1⋯Dn-1u(x)u(x))≤c(x).
Integrating (2.11) with respect to xn from xn0 to xn, we have
(2.12)(D1⋯Dn-1u(x))u(x)≤∫xn0xnc(x1,…,xn-1,tn)dtn,
thus
(2.13)u(x)D1⋯Dn-1u(x)u2(x)≤∫xn0xnc(x1,…,xn-1,tn)dtn+(Dn-1u(x))(D1⋯Dn-2u(x))u2(x),
that is
(2.14)Dn-1(D1⋯Dn-2u(x)u(x))≤∫xn0xnc(x1,…,xn-1,tn)dtn.
Integrating (2.14) with respect to xn-1 from xn-10 to xn-1, we have
(2.15)D1⋯Dn-2u(x)u(x)≤∫xn-10xn-1∫xn0xnc(x1,…xn-2,tn-1,tn)dtndtn-1.
Continuing this process, we obtain
(2.16)D1D2u(x)u(x)≤∫x30x3⋯∫xn0xnc(x1,x2,t3,…,tn)dtndtn-1⋯dt3,
from this we obtain
(2.17)D2(D1u(x)u(x))≤∫x30x3⋯∫xn0xnc(x1,x2,t3,…,tn)dtndtn-1⋯dt3.
Integrating (2.17) with respect to x2 from x20 to x2 and by (2.7), we have
(2.18)D1u(x)u(x)≤a1′(x1)a2(x00)+a1(x1)+∑s=3nas(xs)+∫x20x2⋯∫xn0xnc(x1,t2,t3,…,tn)dtndtn-1⋯dt2.
Integrating (2.18) with respect to x1 from x10 to x1 and by (2.6), we have
(2.19)log⁡u(x)u(x10,x2,…,xn)≤∫x10x1a1′(t1)a2(x20)+a1(t1)+∑s=3nas(xs)dt1+∫x0xc(t)dt,
that is
(2.20)u(x)≤A(x)exp⁡(∫x0xc(t)dt).
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 casen=2, x∈ℝ+2 and x0=(x10,x20)=(0,0) in Theorem 2.1. our estimate reduces to Lemma 1.1 (see [10]).Theorem 2.3.
LetΦ(x), c(x), DiΦ(x), and DΦ(x) be nonnegative continuous functions for all i=1,2,…,n defined for x∈S, Φ(x10,x2,x3,…,xn)=0 and Φ(x1,…,xi-1,xi0,xi+1,…xn)=0 for any i=2,3,…,n. If
(2.21)DΦ(x)≤∑i=1nai(xi)+∫x0xc(t)[Φ(t)+DΦ(t)]dt
holds for x∈S, where ai(xi)>0; ai′(xi)≥0 are continuous functions defined for xi≥0 for all i=1,2,…,n. Then
(2.22)DΦ(x)≤∑i=1nai(xi)+∫x0xc(t)[A(t)exp⁡(∫x0t[1+c(τ)]dτ)]dt.
For x∈S with x≥t≥τ≥x0≥0, where A(x) is defined in (2.4).
Proof.
We define the function(2.23)u(x)=∑i=1nai(xi)+∫x0xc(t)[Φ(t)+DΦ(t)]dt,(2.24)u(x10,x2,x3,…,xn)=a1(x10)+∑i=2nai(xi).
Then, (2.21) can be restated as
(2.25)DΦ(x)≤u(x).
Differentiating (2.23),
(2.26)Du(x)=c(x)[Φ(x)+DΦ(x)].
Integrating both sides of (2.26) to x from x0 to x, we have
(2.27)Φ(x)≤∫x0xu(t)dt.
Now, using (2.27) and (2.25) in (2.26) we obtain
(2.28)Du(x)≤c(x)(u(x)+∫x0xu(t)dt).
If we put
(2.29)v(x)=u(x)+∫x0xu(t)dt,(2.30)v(x1,…,xi-1,xi0,xi+1,…,xn)=u(x1,…,xi-1,xi0,xi+1,…,xn),
then by (2.29), we have
(2.31)Dv(x)=Du(x)+u(x).
Using the facts that Du(x)≤c(x)v(x) and u(x)≤v(x), we have
(2.32)Dv(x)≤[1+c(x)]v(x).
Which, by following an argument similar to that in the proof of Theorem 2.1, yields the estimate for v(x) such that
(2.33)v(x)≤A(x)exp⁡(∫x0x[1+c(t)]dt).
By (2.33) and (2.28), we have
(2.34)Du(x)≤c(x)A(x)exp⁡(∫x0x[1+c(t)]dt),(2.35)D1D2⋯Dn-1u(x1,…,xn-1,xn0)=0.
Integrating both sides of (2.34) to xn from xn0 to xn and by (2.35), we have
(2.36)D1D2⋯Dn-1u(x)≤∫xn0xnc(x1,…xn-1,tn)A(x1,…xn-1,tn)exp⁡(∫x0t[1+c(τ)]dτ)dtn.
By (2.23), we have
(2.37)D1D2u(x1,x2,x30,x4,…,xn)=0.
Continuing this process, and by (2.37), we obtain
(2.38)D1D2u(x)≤∫x30x3⋯∫xn0xnc(x1,x2,t3,…,tn)A(x1,x2,t3,…,tn)exp⁡(∫x0t[1+c(τ)]dτ)dtn⋯dt3.
By (2.23), we have
(2.39)D1u(x1,x20,x3,x4,…,xn)=a1′(x1).
Integrating both sides of (2.38) to x2 from x20 to x2 and by (2.39), we have
(2.40)D1u(x)≤a1′(x1)+∫x20x2∫x30x3⋯∫xn0xnc(x1,t2,…,tn)A(x1,t2,…,tn)×exp⁡(∫x0t[1+c(τ)]dτ)dtn⋯dt2.
Integrating (2.40) with respect to x1 from x10 to x1, and by (2.24), we have
(2.41)u(x)≤∑i=1nai(xi)+∫x0xc(t)A(t)exp⁡(∫x0t[1+c(τ)]dτ)dt.
By (2.41) and (2.25), we obtain the desired bound in (2.22).Remark 2.4.
We note that in the special casen=2, x∈ℝ+2 and x0=(x10,x20)=(0,0) in Theorem 2.3, then our result reduces to Theorem 1 obtained in [10].Theorem 2.5.
LetΦ(x), c(x), DiΦ(x), and DΦ(x) be nonnegative continuous functions for all i=1,2,…,n defined for x∈S, Φ(x10,x2,x3,…,xn)=0 andΦ(x1,…,xi-1,xi0,xi+1,…xn)=0 for any i=2,3,…,n. If
(2.42)DΦ(x)≤∑i=1nai(xi)+M[Φ(x)+∫x0xc(t)[Φ(t)+DΦ(t)]dt]
holds for x∈S, where ai(xi)>0; ai′(xi)≥0 are continuous functions defined for xi≥0 for all i=1,2,…,n. and M≥0 is constant. Then
(2.43)DΦ(x)≤A(x)exp⁡(∫x0x[M+c(t)+Mc(t)]dt),
for x∈S, with x≥t≥x0≥0, where A(x) is defined in (2.4).
Proof.
We define the function(2.44)u(x)=∑i=1nai(xi)+M[Φ(x)+∫x0xc(t)[Φ(t)+DΦ(t)]dt]
with
(2.45)u(x10,x2,x3,…,xn)=a1(x10)+∑i=2nai(xi).
Differentiating (2.44), we have
(2.46)Du(x)=M[DΦ(x)+c(x)[Φ(x)+DΦ(x)]].
Using the fact that DΦ(x)≤u(x) and MΦ(x)≤u(x), we have
(2.47)Du(x)≤[M+c(x)+Mc(x)]u(x),
by (2.47), we have
(2.48)u(x)≤A(x)exp⁡(∫x0x[M+c(t)+Mc(t)]dt),
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 casen=2, x∈ℝ+2 and x0=(x10,x20)=(0,0) in Theorem 2.5, then our result reduces to Theorem 2 obtained in [10].Theorem 2.7.
LetΦ(x), p(x), and q(x) be nonnegative continuous functions defined for x∈S. If
(2.49)Φ(x)≤∑i=1nai(xi)+∫x0xp(t)Φ(t)dt+∫x0xp(t)(∫x0tq(s)Φ(s)ds)dt
holds for x≥x0≥0, where ai(xi)>0; ai′(xi)≥0 are continuous functions defined for xi≥0 for all i=1,2,…,n. Then
(2.50)Φ(x)≤∑i=1nai(xi)+∫x0xp(t)Q(t)dt,
for all x≥x0≥0, where
(2.51)Q(x)=A(x)exp⁡(∫x0x(p(t)+q(t))dt),
with A(x) defined in (2.4).
Proof.
The proof of this Theorem follows by an argument similar to that in Theorem2.1, We omit the details.Remark 2.8.
We note that in the special casen=2, x∈ℝ+2 and x0=(x10,x20)=(0,0) in Theorem 2.7, our result reduces to Theorem 2 obtained in [10].


## 3. Nonlinear Integrodifferential inn Independents Variables

In this section, we will give some new nonlinear integrodifferential inequalities for the functions ofn-independent variables.We can also give the following lemma.Lemma 3.1 (see [2, 11]).
Letu(x), a(x), and b(x) be nonnegative continuous functions, defined for x∈S.
Assume thata(x) is a positive, continuous function and nondecreasing in each of the variables x∈S. If
(3.1)u(x)≤a(x)+∫x0xb(t)u(t)dt
holds for all x∈S, with x≥x0≥0. Then
(3.2)u(x)≤a(x)exp⁡(∫x0xb(t)dt).Theorem 3.2.
LetΦ(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 x∈S, Φ(x10,x2,x3,…,xn)=0andΦ(x1,…,xi-1,xi0,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 x∈S. Assume that a(x) and f(x) are positive and nondecreasing in each of the variables x∈S. If
(3.3)DΦ(x)≤a(x)+f(x)H(∫x0xc(t)K(Φ(t))dt)+∫x0xb(t)DΦ(t)dt
holds, for x∈S with x≥x0≥0. Then
(3.4)DΦ(x)≤{a(x)+f(x)H(G-1[G(ξ)+∫x0xc(t)K(p(t)f(t))dt])}exp⁡(∫x0xb(t)dt),
for x∈S, where
(3.5)p(x)=∫x0xexp⁡(∫x0tb(s)ds)dt,ξ=∫x0∞c(t)K(a(t)p(t))dt,G(z)=∫z0zdsK(H(s)),z≥z0>0,
where G-1 is the inverse function of G, and
(3.6)G(ξ)+∫x0xc(t)K(p(t)f(t))dt
is in the domain of G-1 for x∈S.
Proof.
We define the function(3.7)z(x)=a(x)+f(x)H(∫x0xc(t)K(Φ(t))dt),
then (2.4) can be restated as
(3.8)DΦ(x)≤z(x)+∫x0xb(t)DΦ(t)dt.
Clearly, z(x) is a positive, continuous function and nondecreasing in each of the variables x∈S, using (3.1) of Lemma 3.1 to (3.8), we have
(3.9)DΦ(x)≤z(x)exp⁡(∫x0xb(t)dt).
Integrating to x from x0 to x, we have
(3.10)Φ(x)≤z(x)p(x),
where
(3.11)p(x)=∫x0xexp⁡(∫x0tb(s)ds)dt.
By (3.7), we have
(3.12)z(x)=a(x)+f(x)H(v(x)),
where
(3.13)v(x)=∫x0xc(t)K(Φ(t))dt.
By (3.10) and (3.13), we have
(3.14)Φ(x)≤{a(x)+f(x)H(v(x))}p(x).
From (3.14) and (3.13) and since K is a subadditive and submultiplicative function, we notice that
(3.15)v(x)≤∫x0xc(t)K[{a(t)+f(t)H(v(t))}p(t)]dt,≤∫x0xc(t)K(a(t)p(t))dt+∫x0xc(t)K(f(t)p(t))K(H(v(t)))dt,≤∫x0∞c(t)K(a(t)p(t))dt+∫x0xc(t)K(f(t)p(t))K(H(v(t)))dt.
We define Ψ(x) as the right side of (3.14), then
(3.16)Ψ(x10,x2,x3,…,xn)=∫x0∞c(t)K(a(t)p(t))dt,(3.17)v(x)≤Ψ(x).Ψ(x) is positive and nondecreasing in each of the variables x2,…,xn∈R+n-1,then
(3.18)D1Ψ(x)=∫x20x2∫x30x3⋯∫xn0xnc(x1,t2,…,tn)K(p(x1,t2,…,tn)f(x1,t2,…,tn))×K(H(v(x1,t2,…,tn)))dtn⋯dt2,≤∫x20x2∫x30x3⋯∫xn0xnd(x1,t2,…,tn)K(p(x1,t2,…,tn)f(x1,t2,…,tn))×K(H(Ψ(x1,t2,…,tn)))dtn⋯dt2,≤K(H(Ψ(x)))∫x20x2∫x30x3⋯∫xn0xnc(x1,t2,…,tn)×K(p(x1,t2,…,tn)f(x1,t2,…,tn))dtn⋯dt2.
Dividing both sides of (3.18) by K(H(Ψ(x))), we get
(3.19)D1Ψ(x)K(H(Ψ(x)))≤∫x20x2∫x30x3⋯∫xn0xnc(x1,t2,…,tn)K(p(x1,t2,…,tn)f(x1,t2,…,tn))dtn⋯dt2.
We note that
(3.20)G(z)=∫z0zdsK(H(s)),z≥z0>0.
Thus, it follows that
(3.21)D1G(Ψ(x))=D1Ψ(x)K(H(Ψ(x))).
From (3.19), (3.20), and (3.21), we have
(3.22)D1G(Ψ(x))≤∫x20x2∫x30x3⋯∫xn0xnc(x1,t2,…,tn)K(p(x1,t2,…,tn)f(x1,t2,…,tn))dtn⋯dt2.
Now, setting x1=s in (3.22) and then integrating with respect from x10 to x1, we obtain
(3.23)G(Ψ(x))≤G(Ψ(x10,x2,…,xn))+∫x0xc(t)K(p(t)f(t))dt,
by (3.23), we have
(3.24)Ψ(x)≤G-1[G(Ψ(x10,x2,…,xn))+∫x0xc(t)K(p(t)f(t))dt].
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 Theorem3.2.Corollary 3.3.
LetΦ(x), a(x), b(x), c(x), DiΦ(x), DΦ(x), and K(Φ(x)) be as defined in Theorem 3.2. If
(3.25)DΦ(x)≤a(x)+∫x0xc(t)g(Φ(t))dt+∫x0xb(t)DΦ(t)dt
holds, for x∈ℝ+n with x≥x0≥0. Then
(3.26)DΦ(x)≤{a(x)+T-1[T(ξ)+∫x0xc(t)K(p(t))dt]}exp⁡(∫x0xb(t)dt),
for x∈ℝ+n with x≥x0≥0, where
(3.27)p(x)=∫x0xexp⁡(∫x0tb(s)ds)dt,ξ=∫x0∞c(t)K(a(t)p(t))dt,T(z)=∫z0zdsK(s),z≥z0>0,
where T-1 is the inverse function of T and
(3.28)T(ξ)+∫x0xc(t)K(p(t))dt
is in the domain of T-1 for x∈ℝ+n.
Proof.
The proof of this Corollary follows by an argument similar to that in Theorem3.2. We omit the details.Corollary 3.4.
LetΦ(x), b(x), c(x), DiΦ(x), and DΦ(x) be as defined in Theorem 3.2. If
(3.29)DΦ(x)≤M+∫x0xc(t)Φ(t)dt+∫x0xb(t)DΦ(t)dt
holds, for x∈ℝ+n with x≥x0≥0, where M>0 is a constant, then
(3.30)Φ(x)≤M{1+exp⁡[log⁡(∫x0∞c(t)p(t)dt)+∫x0xc(t)p(t)dt]}p(x),
for x∈ℝ+n with x≥x0≥0, where
(3.31)p(x)=∫x0xexp⁡(∫x0tb(s)ds)dt.
Proof.
Settingg(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 Theorem3.2 to the study of boundedness of the solution of a partial integrodifferential equation.Consider the nonlinear partial integrodifferential equation(4.1)Du(x)=f(x)+∫0xh(x,t,u(t),Du(t))dt,u(…,xi,0,xi+2,…)=0,∀i=1,2,…,n,
for x∈ℝ+n, where h:ℝ+n×ℝ×ℝ→ℝ, f(x):ℝ+n→ℝ are continuous functions.Assume that functions are defined and continuous on their respective domains of definition, such that(4.2)|f(x)|≤M,|h(x,t,u(t),Du(t))|≤c(t)|u(t)|+b(t)|Du(t)|
for x∈ℝ+n, where M>0 is a constant and c(x) and b(x) are nonnegative, continuous functions defined for x∈ℝ+n. If Φ(x) is any solution of boundary value problem (4.1), then(4.3)DΦ(x)=f(x)+∫0xh(x,t,Φ(t),DΦ(t))dt
for x∈ℝ+n, by (4.2), we have(4.4)|DΦ(x)|=M+∫0xc(t)|Φ(x)|+b(t)|DΦ(x)|dt.
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). (4.5)|Φ(x)|≤Mp(x){1+exp⁡[log⁡(∫0∞c(t)p(t)dt)+∫0xc(t)p(t)dt]}
or x∈ℝ+n, where (4.6)p(x)=∫0xexp⁡(∫0tb(s)ds)dt.Remark 4.1.
Using a similar method of those in the proof of the theorems above, we can also obtainnew 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:(4.7)DΦ(x)≤a(x)+∫α(x0)α(x)c(t)K(Φ(t))dt+∫β(x0)β(x)b(t)DΦ(t)dt.DΦ(x)≤a(x)+f(x)∫α(x0)α(x)c(t)Φ(t)K(Φ(t))dt+∫β(x0)β(x)b(t)DΦ(t)dt.DΦ(x)≤a(x)+f(x)H(∫α(x0)α(x)c(t)n(Φ(t))K(Φ(t))dt)+∫β(x0)β(x)b(t)g(DΦ(t))dt.



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