ON SOME GRONWALL TYPE INEQUALITIES

INVOLVING ITERATED INTEGRALS

Yeol Je Cho Department of Mathematics Education The Research Institute of Natural Sciences College of Education, Gyeongsang National University Chinju 660-701, Republic of Korea E-mail: yjcho@nongde.gsnu.ac.kr

Sever S. Dragomir School of Computer Science and Mathematics Victoria University of Technology PO Box 14428, MCMC, Melbourne Victoria 8001, Australia E-mail: sever.dragomir@vu.edu.au

Young-Ho Kim Department of Applied Mathematics Changwon National University Changwon 641-773, Republic of Korea E-mail: yhkim@sarim.changwon.ac.kr

Abstract: In this paper, some new Gronwall type inequalities involving iterated integrals are given.

1. Introduction

Let u : [α, α + h] → R be a continuous real-valued function satisfying the

Typeset by AMS-TEX

2000 AMS Subject Classification: 26D15, 35A05.

Key Words and Phrases: Gronwall-Bellman inequalities, integral inequality, iterated integrals, nondecreasing function.

inequality

0 ≤ u(t) ≤ ∫_{α}^{t}[a + bu(s)] ds, t ∈ [α, α + h],

where a, b are nonnegative constants. Then u(t) ≤ ahe^{bh} for t ∈ [α, α + h]. This result was proved by T. H. Gronwall [9] in the year 1919, and is the prototype for the study of several integral inequalities of Volterra type, and also for obtaining explicit bounds of the unknown function. Among the several results on this subject, the inequality of Bellman [3] is very well known:

Let x(t) and k(t) be real valued nonnegative continuous functions for t ≥ α. If a is a constant, a ≥ 0, and

x(t) ≤ a + ∫_{α}^{t} k(s)x(s) ds, t ≥ α,

then

x(t) ≤ a exp(∫_{α}^{t} k(s) ds), t ≥ α.

It is clear that Bellman’s result contains that of Gronwall. This is the reason why inequalities of this type were called “Gronwall-Bellman inequalities” or “Inequalities of Gronwall type”. The Gronwall type integral inequalities provide a necessary tool for the study of the theory of differential equations, integral equations and inequalities of various types (see Gronwall [9] and Guiliano [10]). Some applications of this result to the study of stability of the solution of linear and nonlinear differential equations may be found in Bellman [3]. Some applications to existence and uniqueness theory of differential equations may be found in Nemyckii-Stepanov [14], Bihari [4], and Langenhop [11]. During the past few years several authors (see references below and some of the references cited therein) have established several Gronwall type integral inequalities in two or more independent real variables. Of course, such results have application in the theory of partial differential equations and Volterra integral equations.

Bykov proved the following interesting integral inequality, which appear in [1, p. 98]:

Let u(t), b(t), k(t, s) and h(t, s, σ) be nonnegative continuous functions for α ≤ τ ≤ s ≤ t ≤ β and suppose that

(1.1)  \begin{aligned}u(t) & ≤ a + ∫_{α}^{t} b(s)u(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u(τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ)u(σ) dσ dτds \end{aligned}

for any t ∈ [α, β], where a ≥ 0 is a constant. Then

\begin{aligned}u(t) & ≤ a exp(∫_{α}^{t} b(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ) dσ dτ ds), t ∈ [α, β]. \end{aligned}

In this paper, we consider simple inequalities involving iterated integrals in the inequality (1.1) for the case when the function u in the right-hand side of the inequality (1.1) is replaced by the function u^{p} for some p, and the constant a is replaced by a nonnegative, nondecreasing function a(t). We also provide some related integral inequalities involving iterated integrals.

2. The case p > 1

In this section, we state and prove some new nonlinear integral inequalities involving iterated integrals. Throughout the paper, all the functions which appear in the inequalities are assumed to be real-valued. Before considering our first integral inequality involving iterated integrals, we need the following lemma, which appears in [1, p. 2].

Lemma 2.1. Let b(t) and f(t) be continuous function for t ≥ α, let v(t) be a differentiable function for t ≥ α and suppose that

v′(t) ≤ b(t)v(t) + f(t), t ≥ α,

and v(α) ≤ v_{0}. Then we have

v(t) ≤ v_{0} exp(∫_{α}^{t} b(s) ds) + ∫_{α}^{t} f(s) exp(∫_{s}^{t} b(τ) dτ) ds, t ≥ α.

Theorem 2.2. Let u(t), b(t), k(t, s) and h(t, s, τ) be nonnegative continuous functions for α ≤ τ ≤ s ≤ t ≤ β and let p > 1 be a constant. Suppose a(t) ≥ 0 is nondecreasing in J = [α, β] and

(2.1)  \begin{aligned}u(t) & ≤ a(t) + ∫_{α}^{t} b(s)u^{p}(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u^{p}(τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ)u^{p}(σ) dσ dτds, t ∈ [α, β]. \end{aligned}

Then we have

(2.2)  u(t) ≤ a(t)[1 − (p − 1) ∫_{α}^{t} B(s)a^{p−1}(s) ds]^{\frac{1}{1−p}} , t ∈ [α, β_{p}),

where

β_{p} = sup {t ∈ J : (p − 1) ∫_{α}^{t} B(s)a^{p−1}(s) ds < 1}

and

B(t) = b(t) + ∫_{α}^{t} k(t, s) ds + ∫_{α}^{t} ∫_{α}^{s} h(t, s, τ) dτ ds.

Proof. We denote the right-hand side of (2.1) by a(t) + v(t). Then, for α ≤ t ≤ T < β_{p}, (2.1) implies v(α) = 0, the function v(t) is nondecreasing in t ∈ [α, β],

(2.3)  u(t) ≤ a(t) + v(t)

and

v′(t) = b(t)u^{p}(t) + ∫_{α}^{t} k(t, τ)u^{p}(τ) dτ + ∫_{α}^{t} ∫_{α}^{τ} h(t, τ, σ)u^{p}(σ) dσ dτ
≤ B(t)[a(t) + v(t)]^{p}
≤ B(t)[a(t) + v(t)]^{p−1}[a(T) + v(t)],

that is,

(2.4)  v′(t) ≤ R(t)[a(T) + v(t)],

ON SOME GRONWALL TYPE INEQUALITIES 5

where R(t) = B(t)[a(t) + v(t)]^{p−1}. Lemma 2.1 and (2.4) imply

v(t) ≤ a(T) ∫_{α}^{t} R(s) exp(∫_{s}^{t} R(τ) dτ) ds

and so

v(t) + a(T) ≤ a(T) exp(∫_{α}^{t} R(s) ds), α ≤ t ≤ T.

Hence, for t = T,

(2.5)  v(t) + a(t) ≤ a(t) exp(∫_{α}^{t} R(s) ds).

From (2.5), we successively obtain [v(t) + a(t)]^{p−1} ≤ a^{p−1}(t) exp(∫_{α}^{t}(p − 1)R(s) ds),

R(t) ≤ B(t)a^{p−1}(t) exp(∫_{α}^{t}(p − 1)R(s) ds),
Z(t) ≤ (p − 1)B(t)a^{p−1}(t) exp(∫_{α}^{t} Z(s) ds),

where Z(t) = (p − 1)R(t). Consequently, we have

Z(t) exp(− ∫_{α}^{t} Z(s) ds) ≤ (p − 1)B(t)a^{p−1}(t)

or

\frac{d}{dt}[− exp(− ∫_{α}^{t} Z(s) ds)] ≤ (p − 1)B(t)a^{p−1}(t).

Integrating this from α to t yields

1 − exp(− ∫_{α}^{t} Z(s) ds) ≤ ∫_{α}^{t}(p − 1)B(s)a^{p−1}(s) ds,

from which we conclude that

exp(∫_{α}^{t} R(s) ds) ≤ [1 − (p − 1) ∫_{α}^{t} B(s)a^{p−1}(s) ds]^{\frac{1}{1−p}} .

This, together with (2.3) and (2.5), implies (2.2). This completes the proof. In the same manner, we can prove the following theorem:

Theorem 2.3. Let u(t), b(t), k(t, s) and σ(t) be nonnegative continuous functions for α ≤ s ≤ t ≤ β and let p > 1 be a constant. Suppose that σ(t) is nondecreasing in J = [α, β] and

u(t) ≤ σ(t){a_{1} + ∫_{α}^{t} b(s)u^{p}(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u^{p}(τ) dτds}

for any t ∈ [α, β], where a_{1} ≥ 0 is a constant. Then we have

u(t) ≤ a_{1}σ(t) exp(σ(t))[1 − (p − 1)a_{1}^{p−1} ∫_{α}^{t} B_{1}(s)σ^{p−1}(s) exp(σ(s)) ds]^{\frac{1}{1−p}}

for any t ∈ [α, β_{p}), where B_{1}(t) = b(t) + ∫_{α}^{t} k(t, τ) dτ and

β_{p} = sup{t ∈ J : (p − 1)a_{1}^{p−1} ∫_{α}^{t} B_{1}(s)σ^{p−1}(s) exp(σ(s)) ds < 1}.

Let α < β, and set J_{i} = {(t_{1}, t_{2}, …, t_{i}) ∈ R^{i} : α ≤ t_{i} ≤ ⋯ ≤ t_{1} ≤ β}, i = 1, ⋯, n.

Theorem 2.4. Let u(t), a(t) and b(t) be nonnegative continuous functions in J = [α, β] and let p > 1 be a constant. Suppose that \frac{a(t)}{b(t)} is nondecreasing in J and

(2.6) \begin{aligned}u(t) & ≤ a(t) + b(t)[∫_{α}^{t} k_{1}(t, t_{1})u^{p}(t_{1}) dt_{1} + ⋯ \\ & + ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{n−1}} k_{n}(t, t_{1}, ⋯, t_{n})u^{p}(t_{n}) dt_{n}) ⋯)dt_{1}] \end{aligned}

for any t ∈ J, where k_{i}(t, t_{1}, …, t_{i}) are nonnegative continuous functions in J_{i+1} for i = 1, 2, ⋯, n. Suppose thta the partial derivatives \frac{∂k_{i}}{∂t} (t, t_{1}, ⋯, t_{i}) exist and are nonnegative and continuous in J_{i+1} for i = 1, 2, ⋯, n. Then, for any t ∈ J,

(2.7) u(t) ≤ a(t)[1 − (p − 1) ∫_{α}^{t}(\frac{a(s)}{b(s)})^{p−1}(R[b^{p}](s) + Q[b^{p}](s)) ds]^{\frac{1}{1−p}}

ON SOME GRONWALL TYPE INEQUALITIES 7

for any t ∈ [α, β̃_{p}), where β̃_{p} = sup{t ∈ J : (p − 1)a_{1}^{p−1} ∫_{α}^{t}(a(s)/b(s))^{p−1}(R[b^{p}](s) + Q[b^{p}](s) ds < 1},

R[w](t) = k_{1}(t, t)w(t) + ∫_{α}^{t} k_{2}(t, t, t_{2})w(t_{2})dt_{2}

+ ∑_{i=3}^{n} ∫_{α}^{t}(∫_{α}^{t_{2}} ⋯ (∫_{α}^{t_{i−1}} k_{i}(t, t, t_{2}, ⋯, t_{i})w(t_{i}) dt_{i}) ⋯) dt_{2},

Q[w](t) = ∫_{α}^{t} \frac{∂k_{1}}{∂t} (t, t_{1})w(t_{1}) dt_{1}

+ ∑_{i=2}^{n} ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{i−1}} \frac{∂k_{i}}{∂t} (t, t_{1}, ⋯, t_{i})w(t_{i}) dt_{i}) ⋯) dt_{1}

for each continuous function w(t) in J. Proof. First, we note that R[w] and Q[w] are linear functionals and

R[w_{1}] ≤ R[w_{2}], Q[w_{1}] ≤ Q[w_{2}]

if w_{1}(t) ≤ w_{2}(t) for any t ∈ J and

R[w_{1}w_{2}] ≤ R[w_{1}]w_{2}, Q[w_{1}w_{2}] ≤ Q[w_{1}]w_{2}

if w_{1}(t) is nonnegative in J and w_{2}(t) is nondecreasing in J. We set

\begin{aligned}v(t) & = ∫_{α}^{t} k_{1}(t, t_{1})u^{p}(t_{1}) dt_{1} + ⋯ \\ & + ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{n−1}} k_{n}(t, t_{1}, ⋯, t_{n})u^{p}(t_{n}) dt_{n}) ⋯)dt_{1}. \end{aligned}

Then, for α ≤ t ≤ T < β_{p}, (2.6) implies v(α) = 0, the function v(t) is nondecreasing,

(2.8)  u(t) ≤ a(t) + b(t)v(t)

and we have

v′(t) = R[u^{p}](t) + Q[u^{p}](t) ≤ (R[b^{p}](t) + Q[b^{p}](t))(\frac{a(t)}{b(t)} + v(t))^{p},

8 Y. J. CHO, S. S. DRAGOMIR AND Y. H. KIM

that is,

(2.9)  v′(t) ≤ R(t)[a(T)/b(T) + v(t)],

where R(t) = (R[b^{p}](t) + Q[b^{p}](t))[a(t)/b(t) + v(t)]^{p−1}. Lemma 2.1 and (2.9) imply

v(t) + \frac{a(T)}{b(T)} ≤ \frac{a(T)}{b(T)} exp(∫_{α}^{t} R(s) ds), α ≤ t ≤ T.

Hence, for t = T,

(2.10)  v(t) + \frac{a(t)}{b(t)} ≤ \frac{a(t)}{b(t)} exp(∫_{α}^{t} R(s) ds).

From (2.10), we successively obtain

[v(t) + \frac{a(t)}{b(t)}]^{p−1} ≤ [\frac{a(t)}{b(t)}]^{p−1} exp(∫_{α}^{t}(p − 1)R(s) ds),

R(t) ≤ (R[b^{p}](t) + Q[b^{p}](t))[\frac{a(t)}{b(t)}]^{p−1} exp(∫_{α}^{t}(p − 1)R(s) ds),

Z(t) ≤ (p − 1)(R[b^{p}](t) + Q[b^{p}](t))[\frac{a(t)}{b(t)}]^{p−1} exp(∫_{α}^{t}(p − 1)R(s) ds),

where Z(t) = (p − 1)R(t). Consequently, we have

\frac{d}{dt}[− exp(− ∫_{α}^{t} Z(s) ds)] ≤ (p − 1)(R[b^{p}](t) + Q[b^{p}](t))[\frac{a(t)}{b(t)}]^{p−1}.

Integrating this from α to t yields

\begin{matrix}1 − exp(− ∫_{α}^{t} Z(s) ds) \\ ≤ (p − 1) ∫_{α}^{t}(\frac{a(s)}{b(s)})^{p−1}(R[b^{p}](s) + Q[b^{p}](s)) ds, \end{matrix}

from which we conclude that

\begin{matrix}exp(∫_{α}^{t} R(s) ds) \\ ≤ [1 − (p − 1) ∫_{α}^{t}(\frac{a(s)}{b(s)})^{p−1}(R[b^{p}](s) + Q[b^{p}](s)) ds]^{\frac{1}{1−p}} . \end{matrix}

This, together with (2.8) and (2.10), implies (2.7). This completes the proof.

3. The case p > 0 (p ≠ 1)

In this section, we use another method for studying nonlinear integral inequalities. Before considering the first result of the integral inequality, we need the following lemma, which appears in [1, p. 38].

Lemma 3.1. Let v(t) be a positive differential function satisfying the inequality

v′(t) ≤ b(t)v(t) + k(t)v^{p}(t), t ∈ J = [α, β],

where the functions b and k are continuous in J and p ≥ 0 (p ≠ 1) is a constant. Then we have

v(t) ≤ exp(∫_{α}^{t} b(s) ds)[v^{q}(α) + q ∫_{α}^{t} k(s) exp(−q ∫_{α}^{s} b(τ) dτ) ds]^{1/q}

for any t ∈ [α, β_{1}), where β_{1} is chosen so that the expression between [⋯] is positive in the subinterval [α, β_{1}).

An essential element in the investigation of the integral inequalities in the following theorems is the application of the result of Lemma 3.1.

Theorem 3.2. Let u(t), b(t), k(t, s), h(t, s, σ) be nonnegative continuous functions for α ≤ σ ≤ s ≤ t ≤ β and suppose that

(3.1)  \begin{aligned}u(t) & ≤ a + ∫_{α}^{t} b(s)u^{p}(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u^{p}(τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ)u^{p}(σ) dσ dτds \end{aligned}

for any t ∈ [α, β], where a > 0 and p ≥ 0 (p ≠ 1) are a constants. Then we have

(3.2)  \begin{aligned}u(t) & ≤ [a^{q} + q ∫_{α}^{t}(b(s) + ∫_{α}^{s} k(s, τ) dτ \\ & + ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ) dσ dτ) ds]^{1/q} \end{aligned}

for any t ∈ [α, β_{1}), where q = 1 − p and β_{1} is chosen so that the expression between [⋯] is positive in the subinterval [α, β_{1}).

Proof. We denote the right-hand side of (3.1) by the function v(t). Then the function v(t) is nondecreasing in t ∈ [α, β], u(t) ≤ v(t), v(α) = a and

v′(t) = b(t)u^{p}(t) + ∫_{α}^{t} k(t, τ)u^{p}(τ) dτ + ∫_{α}^{t} ∫_{α}^{τ} h(t, τ, σ)u^{p}(σ) dσ dτ
≤ b(t)v^{p}(t) + ∫_{α}^{t} k(t, τ)v^{p}(τ) dτ + ∫_{α}^{t} ∫_{α}^{τ} h(t, τ, σ)v^{p}(σ) dσ dτ
≤ (b(t) + ∫_{α}^{t} k(t, τ) dτ + ∫_{α}^{t} ∫_{α}^{τ} h(t, τ, σ) dσ dτ)v^{p}(t).

Therefore, applying Lemma 3.1, we arrive at (3.2). This completes the proof.

Theorem 3.3. Let u(t), b(t), k(t, s), h(t, s, σ) be nonnegative continuous functions for α ≤ σ ≤ s ≤ t ≤ β and suppose that

(3.3)  \begin{aligned}u(t) & ≤ a(t) + ∫_{α}^{t} b(s)u^{p}(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u^{p}(τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ)u^{p}(σ) dσ dτds \end{aligned}

for any t ∈ [α, β], where a(t) is a positive nondecreasing function and p ≥ 0 (p ≠ 1) is a constant. Then we have

(3.4)  \begin{aligned}u(t) & ≤ [A^{q}(t) + q ∫_{α}^{t}(b(s) + ∫_{α}^{s} k(s, τ) dτ \\ & + ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ) dσ dτ) ds]^{\frac{1}{q}} \end{aligned}

for any t ∈ [α, β_{1}), where q = 1 − p, A(t) = sup_{s∈[α,t]} a(s) and β_{1} is chosen so that the expression between [⋯] is positive in the subinterval [α, β_{1}).

Proof. The function A(t) is nondecreasing in t ∈ [α, β]. Thus (3.3) implies that, for all α ≤ t ≤ T ≤ β,

(3.5)  \begin{aligned}u(t) & ≤ A(T) + ∫_{α}^{t} b(s)u^{p}(s) ds + ∫_{α}^{t} ∫_{α}^{s} k(s, τ)u^{p}(τ) dτds \\ & + ∫_{α}^{t} ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ)u^{p}(σ) dσ dτds. \end{aligned}

We denote the right-hand side of (3.5) by the function v(t). Then the function v(t) is nondecreasing in t ∈ [α, β], u(t) ≤ v(t), v(α) = A(T) and

v′(t) ≤ (b(t) + ∫_{α}^{t} k(t, τ) dτ + ∫_{α}^{t} ∫_{α}^{τ} h(t, τ, σ) dσ dτ)v^{p}(t).

Consequently, Lemma 3.1 implies

u(t) ≤ [A^{q}(T) + q ∫_{α}^{t}(b(s) + ∫_{α}^{s} k(s, τ) dτ + ∫_{α}^{s} ∫_{α}^{τ} h(s, τ, σ) dσ dτ) ds]^{\frac{1}{q}}

and, for t = T, we obtain (3.4). This completes the proof.

Let α < β, and set

J_{i} = {(t_{1}, t_{2}, …, t_{i}) ∈ R^{i} : α ≤ t_{i} ≤ ⋯ ≤ t_{1} ≤ β}

for i = 1, ⋯, n. By a similar reasoning to the proof of Theorem 3.2, we also can prove the following result:

Theorem 3.4. Let u(t), and b(t) be nonnegative continuous functions in J = [α, β] and suppose that

\begin{aligned}u(t) & ≤ b(t)[a + ∫_{α}^{t} k_{1}(t, t_{1})u^{p}(t_{1}) dt_{1} + ⋯ \\ & + ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{n−1}} k_{n}(t, t_{1}, ⋯, t_{n})u^{p}(t_{n}) dt_{n}) ⋯)dt_{1}] \end{aligned}

for any t ∈ J, where a > 0 and p ≥ 0 (p ≠ 1) is a constant, k_{i}(t, t_{1}, ⋯, t_{i}) are nonnegative continuous functions in J_{i+1} for i = 1, 2, ⋯, n. Suppose that the partial derivatives \frac{∂k_{i}}{∂t} (t, t_{1}, ⋯, t_{i}) exist and are nonnegative and continuous in J_{i+1} for i = 1, 2, ⋯, n. Then, for any t ∈ J,

(3.6)  u(t) ≤ b(t)[a^{q} + q ∫_{α}^{t}(R[b^{p}](s) + Q[b^{p}](s)) ds]^{1/q}

for any t ∈ [α, β_{1}), where q = 1 − p, β_{1} is chosen so that the expression between [⋯] is positive in the subinterval [α, β_{1}),

R[w](t) = k_{1}(t, t)w(t) + ∫_{α}^{t} k_{2}(t, t, t_{2})w(t_{2})dt_{2}

+ ∑_{i=3}^{n} ∫_{α}^{t}(∫_{α}^{t_{2}} ⋯ (∫_{α}^{t_{i−1}} k_{i}(t, t, t_{2}, ⋯, t_{i})w(t_{i}) dt_{i}) ⋯) dt_{2},

Q[w](t) = ∫_{α}^{t} \frac{∂k_{1}}{∂t} (t, t_{1})w(t_{1}) dt_{1}

+ ∑_{i=2}^{n} ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{i−1}} \frac{∂k_{i}}{∂t} (t, t_{1}, ⋯, t_{i})w(t_{i}) dt_{i}) ⋯) dt_{1}

for each continuous function w(t) in J.

Proof. We set

\begin{aligned}v(t) & = a + ∫_{α}^{t} k_{1}(t, t_{1})u^{p}(t_{1}) dt_{1} + ⋯ \\ & + ∫_{α}^{t}(∫_{α}^{t_{1}} ⋯ (∫_{α}^{t_{n−1}} k_{n}(t, t_{1}, ⋯, t_{n})u^{p}(t_{n}) dt_{n}) ⋯)dt_{1}. \end{aligned}

Since v(α) = a, u(t) ≤ b(t)v(t) and v(t) is nondecreasing and continuous in J, we have

\begin{aligned}v′(t) & = R[u^{p}](t) + Q[u^{p}](t) ≤ R[b^{p}u^{p}](t) + Q[b^{p}u^{p}](t) \\ & ≤ (R[b^{p}](t) + Q[b^{p}](t))v^{p}(t), \end{aligned}

from which, by the same method as in the proof of Theorem 3.2, we find the inequality (3.6). This completes the proof.

Corollary 3.5. Let u(t) be nonnegative continuous function for α ≤ t ≤ βand suppose that

u(t) ≤ a + ∫_{α}^{t} k_{1}(t, s)u^{p}(s) ds + ∫_{α}^{t}(∫_{α}^{s} h(t, s, σ)u^{p}(σ) dσ)ds,

where a > 0 and p ≥ 0 (p ≠ 1) is a constant, k(t, s) and h(t, s, σ) are nonnegative continuous functions for α ≤ σ ≤ s ≤ t ≤ β. Suppose that the partial derivatives \frac{∂k}{∂t} (t, s) and \frac{∂h}{∂t}(t, s, σ) exist and are nonnegative and continuous for α ≤ σ ≤ s ≤ t ≤ β. Then, for any t ∈ J,

u(t) ≤ [a^{q} + q ∫_{α}^{t}(R(s) + Q(s)) ds]^{1/q}, t ∈ [α, β_{1}),

where q = 1 − p, β_{1} is chosen so that the expression between [⋯] is positive in the subinterval [α, β_{1}),

R(t) = k(t, t) + ∫_{α}^{t} h(t, t, σ)dσ

and

Q(t) = ∫_{α}^{t} \frac{∂k}{∂t} (t, σ) dσ + ∫_{α}^{t}(∫_{α}^{s} \frac{∂h}{∂t}(t, s, σ) dσ) ds.

Acknowledgement

Y. J. Cho and S. S. Dragomir greatly acknowledge the financial support from the Brain Pool Program (2002) of the Korean Federation of Science and Technology Societies. The research was performed under the “Memorandum of Understanding” between Victoria University and Gyeongsang National University.

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