On some integral inequalities with iterated integrals

ON SOME INTEGRAL INEQUALITIES WITH ITERATED INTEGRALS

Yeol Je Cho, Sever S. Dragomir, and Young-Ho Kim

Abstract. The main aim of the present paper is to establish some new Gronwall type inequalities involving iterated integrals and give some applications of the main results.

1. Introduction

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.

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 lin- ear and nonlinear differential equations may be found in Bellman [3].

Some applications to the existence and uniqueness theory of differential equations may be found in Nemyckii-Stepanov [14], Bihari [4], and Lan- genhop [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 vari- ables. Of course, such results have application in the theory of partial differential equations and Volterra integral equations.

In [5] (see also [1, p.98]), Bykov and Salpagarov proved the following interesting integral inequality:

Received January 25, 2005. Revised November 17, 2005.

2000 Mathematics Subject Classification: 26D15, 35A05.

Key words and phrases: Gronwall-Bellman inequalities, integral inequality, iter- ated integrals, nondecreasing function.

The first author was supported from the Korea Research Foundation Grant (KRF-

2003-005-C00013).

Lemma 1.1. Let u(t), b(t), k(t, s) and h(t, s, σ) be nonnegative con- tinuous functions for α ≤ τ ≤ s ≤ t ≤ β. Suppose that

(1.1)

u(t) ≤ a + Z _t

α

b(s)u(s) ds + Z _t

α

Z _s

α

k(s, τ )u(τ ) dτ ds +

Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ)u(σ) dσ dτ ds

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

u(t) ≤ a exp µZ _t

α

b(s) ds + Z _t

α

Z _s

α

k(s, τ ) dτ ds +

Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ) dσ dτ ds

¶ .

In this paper, we consider simple inequalities involving iterated in- tegrals in the inequality (1.1) for functions when the function u in the right-hand side of the inequality (1.1) is replaced by the function u log u and the constant a is replaced by a nonnegative, nondecreasing func- tion a(t). We also provide some integral inequalities involving iterated integrals and some applications for the main results.

2. Some inequalities for Bykov type

Throughout the paper, all the functions which appear in the inequal- ities are assumed to be real-valued.

First, we state and prove some new interesting integral inequalities involving iterated integrals.

Theorem 2.1. Let u(t), b(t), k(t, s) and h(t, s, τ ) be nonnegative continuous functions for α ≤ τ ≤ s ≤ t ≤ β. Suppose that

u(t) ≤ a + Z _t

α

b(s)u(s) log u(s) ds +

Z _t

α

Z _s

α

k(s, τ )u(τ ) log u(τ ) dτ ds (2.1)

+ Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ)u(σ) log u(σ) dσ dτ ds

for any t ∈ [α, β], where a > 1 is a constant. Then, for any t ∈ [α, β],

(2.2) u(t) ≤ a ^exp(A(t)) ,

where

A(t) = Z _t

α

b(s) ds + Z _t

α

Z _s

α

k(s, τ ) dτ ds +

Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ) dσ dτ ds.

(2.3)

Proof. We denote the right-hand side of (2.1) by v(t). Then v(α) = a, the function v(t) is nondecreasing in t ∈ [α, β], u(t) ≤ v(t) and

v ⁰ (t) = b(t)u(t) log u(t) + Z _t

α

k(t, τ )u(τ ) log u(τ ) dτ +

Z _t

α

Z _τ

α

h(t, τ, σ)u(σ) log u(σ) dσ dτ

= b(t)v(t) log v(t) + Z _t

α

k(t, τ )v(τ ) log v(τ ) dτ +

Z _t

α

Z _τ

α

h(t, τ, σ)v(σ) log v(σ) dσ dτ

≤ B(t)v(t), where

B(t) = b(t) log v(t) + Z _t

α

k(t, τ ) log v(τ ) dτ +

Z _t

α

Z _τ

α

h(t, τ, σ) log v(σ) dσ dτ.

That is,

(2.4) v ⁰ (t)

v(t) ≤ B(t).

By taking t = s in (2.4) and then integrating it from α to any t ∈ [α, β], we have

(2.5) log v(t) ≤ log a +

Z _t

α

B(s) ds.

Now, by a suitable application of Lemma 1.1 to (2.5), we get (2.6) log v(t) ≤ (log a) exp

µZ _t

α

B(s) ds

¶

= log a ^exp(A(t)) , where A(t) is defined by (2.3). From (2.6), we observe that

(2.7) v(t) ≤ a ^exp(A(t)) .

Now, by using (2.7) in u(t) ≤ v(t), we get the required inequality in

(2.2). This completes the proof. ¤

Remark 2.1. If, in (2.1) and (2.2), the constant a is replaced by a nonnegative, nondecreasing function a(t) ≥ 1, then the inequality (2.1) implies that, for all α ≤ t ≤ T ≤ β,

u(t) ≤ a(T ) + Z _t

α

b(s)u(s) log u(s) ds + Z _t

α

Z _s

α

k(s, τ )u(τ ) log u(τ ) dτ ds +

Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ)u(σ) log u(σ) dσ dτ ds.

Therefore, Theorem 2.1 implies u(t) ≤ (a(T )) ^exp(A(t)) and, for T = t, we obtain u(t) ≤ (a(t)) ^exp(A(t)) .

Lemma 2.2. Let u(t), a(t), b(t), k(t, s) and h(t, s, τ ) be nonnegative continuous functions for α ≤ τ ≤ s ≤ t ≤ β. Suppose that

(2.8)

u(t) ≤ a(t) + Z _t

α

b(s)u(s) ds + Z _t

α

Z _s

α

k(s, τ )u(τ ) dτ ds +

Z _t

α

Z _s

α

Z _τ

α

h(s, τ, σ)u(σ) dσ dτ ds for any t ∈ [α, β]. Then, for any t ∈ [α, β],

(2.9) u(t) ≤ a(t) + Z _t

α

f (s) exp µZ _t

s

B(δ) dδ

¶ ds, where

(2.10) f (t) = a(t)b(t) + Z _t

α

k(t, τ )a(τ ) dτ + Z _t

α

Z _τ

α

h(t, τ, σ)a(σ) dσdτ,

(2.11) B(t) = b(t) + Z _t

α

k(t, τ ) dτ + Z _t

α

Z _τ

α

h(t, τ, σ) dσdτ.

Proof. We denote the right-hand side of (2.8) by a(t) + v(t). Then v(α) = 0, the function v(t) is nondecreasing in t ∈ [α, β], u(t) ≤ a(t) + v(t) and

v ⁰ (t) = b(t)u(t) + Z _t

α

k(t, τ )u(τ ) dτ +

Z _t

α

Z _τ

α

h(t, τ, σ)u(σ) dσ dτ (2.12)

≤ f (t) + B(t)v(t),

where f (t) and B(t) are defined by (2.10) and (2.11), respectively. The condition (2.12) implies that

[v ⁰ (s) − B(s)v(s)] exp µZ _t

s

B(δ) dδ

¶

≤ f (s) exp µZ _t

s

B(δ) dδ

¶

for α ≤ s or d ds

·

v(s) exp µZ _t

s

B(δ) dδ

¶¸

≤ f (s) exp µZ _t

s

B(δ) dδ

¶ . Integrating over s from α to any t ∈ [α, β], we have

(2.13) v(t) ≤

Z _t

α

f (s) exp µZ _t

s

B(δ) dδ

¶ ds.

Now, by using (2.13) in u(t) ≤ a(t) + v(t), we get the required inequality

in (2.9). This completes the proof. ¤

Theorem 2.3. Let u(t), b(t), k(t, s) and σ(t) be nonnegative contin- uous functions for α ≤ s ≤ t ≤ β. Suppose σ(t) ≥ 1 and

u(t) ≤ σ(t)

½ a +

Z _t

α

b(s)u(s) log u(s) ds +

Z _t

α

Z _s

α

k(s, τ )u(τ ) log u(s) dτ ds

¾

for any t ∈ [α, β], where a ≥ 1 is a constant. Then (2.14) u(t) ≤ σ(t) exp

µ

B ₁ (t) + Z _t

α

f ₁ (s) exp µZ _t

s

f ₂ (δ) dδ

¶ ds

¶

for any t ∈ [α, β], where

(2.15)

B ₁ (t) = a + Z _t

α

b(s)σ(s) log σ(s) ds +

Z _t

α

Z _s

α

k(s, τ )σ(τ ) log σ(τ ) dτ ds,

(2.16) f ₁ (t) = b(t)σ(t)B ₁ (t) + Z _t

α

k(t, τ )σ(τ )B ₁ (τ ) dτ,

(2.17) f ₂ (t) = b(t)σ(t) + Z _t

α

k(t, τ )σ(τ ) dτ.

Proof. We deduce from the hypothesis on u(t) that u(t) ≤ σ(t)v(t), where the function v(t) is defined by

v(t) = a + Z _t

α

b(s)u(s) log u(s) ds + Z _t

α

Z _s

α

k(s, τ )u(τ ) log u(s) dτ ds.

The function v(t) is nondecreasing in t ∈ [α, β], v(α) = a and v ⁰ (t) = b(t)u(t) log u(t) +

Z _t

α

k(t, τ )u(τ ) log u(τ ) dτ

≤ µ

b(t)σ(t) log(σ(t)v(t)) + Z _t

α

k(t, τ )σ(τ ) log(σ(τ )v(τ )) dτ

¶ v(t).

This implies that v ⁰ (s)

v(s) ≤ b(s)σ(s) log(σ(s)v(s)) + Z _s

α

k(s, τ )σ(τ ) log(σ(τ )v(τ )) dτ.

Integrating over s from α to any t ∈ [α, β], we have

(2.18)

log v(t) ≤ B 1 (t) + Z _t

α

b(s)σ(s) log v(s) ds +

Z _t

α

Z _s

α

k(s, τ )σ(τ ) log v(τ ) dτ ds,

where B 1 (t) is defined by (2.15). Applying Lemma 2.2 in (2.17), we have (2.19) v(t) ≤ exp

µ

B ₁ (t) + Z _t

α

f ₁ (s) exp µZ _t

s

f ₂ (δ) dδ

¶ ds

¶ , where f 1 (t), f 2 (t) are defined in (2.16) and (2.17), respectively. Now, by using (2.19) in u(t) ≤ σ(t)v(t), we get the required inequality in (2.14).

This completes the proof. ¤

In the same manner, we can prove the following theorem:

Theorem 2.4. Let u(t) be nonnegative, continuous functions for α ≤ t ≤ β. Suppose that

(2.20)

u(t) ≤ a + Z _t

α

k(t, s)u(s) log u(s) ds +

Z _t

α

Z _s

α

h(t, s, σ)u(σ) log u(σ) dσds,

where a ≥ 1 is a constant and k(t, s) and h(t, s, σ) are nonnegative, continuous functions for α ≤ σ ≤ s ≤ t ≤ β. Suppose the partial derivatives ^∂k _∂t (t, s), ^∂h _∂t (t, s, σ) exist and are nonnegative, continuous for α ≤ σ ≤ s ≤ t ≤ β. Then, for any t ∈ [α, β],

(2.21) u(t) ≤ a ^exp(B

^(t)) , where

B ₂ (t) = Z _t

α

k(s, s) ds + Z _t

α

Z _s

α

[k _s (s, σ) + h(s, s, σ)] dσds +

Z _t

α

Z _s

α

Z _σ

α

h _s (s, σ, τ ) dτ dσ ds.

Remark 2.2. If, in (2.20) and (2.21), the constant a is replaced by a nonnegative, nondecreasing function a(t) ≥ 1, then the inequality (2.20) implies that, for all α ≤ t ≤ T ≤ β,

u(t) ≤ a(T ) + Z _t

α

k(t, s)u(s) log u(s) ds +

Z _t

α

Z _s

α

h(t, s, σ)u(σ) log u(σ) dσds.

Therefore, Theorem 2.4 implies u(t) ≤ (a(T )) ^exp(B

^(t)) and, for T = t, we obtain u(t) ≤ (a(t)) ^exp(B

^(t)) .

3. Some further inequalities

In this section, we present a generalization of the result obtained

in Theorem 2.4 and some new integral inequalities involving iterated

integrals. 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.

Theorem 3.1. Let u(t) and b(t) be nonnegative continuous functions in J = [α, β] and a ≥ 1 be a constant. Suppose that

(3.1) u(t)

≤ a + Z _t

α

k ₁ (t, t ₁ )u(t ₁ ) log u(t ₁ ) dt ₁ + · · · +

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k _n (t, t ₁ , . . . , t _n )u(t _n ) log u(t _n ) dt _n

¶

· · ·

¶ dt ₁

for any t ∈ J, where k _i (t, t ₁ , . . . , t _i ) are nonnegative, continuous func- tions in J i+1 for i = 1, 2, . . . , n.

Suppose the partial derivative ^∂k _∂t

(t, t ₁ , . . . , t _i ) exists and are nonneg- ative, continuous in J i+1 for i = 1, 2, . . . , n, which are nondecreasing in t ∈ J for any fixed (t ₁ , . . . , t _i ) ∈ J _i for i = 1, . . . , n. Then, for any t ∈ J,

(3.2) u(t) ≤ a ^exp(B

^(t)) ,

where B ₃ (t) = R _t

α (R[1](τ ) + Q[1](τ )) dτ, and

R[1](t) = k 1 (t, t) + Z _t

α

k 2 (t, t, t 2 ))dt 2

+ X n i=3

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k _i (t, t, t ₂ , . . . , t _i ) dt _i

¶

· · ·

¶ dt ₂ ,

Q[1](t) = Z _t

α

∂k 1

∂t (t, t ₁ ) dt ₁ +

X n i=2

Z _t

α

µZ _t

α

· · ·

µZ _t

α

∂k _i

∂t (t, t ₁ , . . . , t _i ) dt _i

¶

· · ·

¶ dt ₁ .

Proof. We denote the right-hand side of (3.1) by v(t). Then, for α ≤ t ≤ β, (3.1) implies v(α) = a, the function v(t) is nondecreasing continuous, u(t) ≤ v(t) and we have

(3.3) v ⁰ (t) = R[u log u](t) + Q[u log u](t),

where

R[u log u](t)

= k ₁ (t, t)u(t) log u(t) + Z _t

α

k ₂ (t, t, t ₂ )u(t ₂ ) log u(t ₂ )dt ₂ +

X n i=3

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k _i (t, t, t ₂ , . . . , t _i )u(t _i ) log u(t _i ) dt _i

¶

· · ·

¶ dt ₂ , Q[u log u](t)

= Z _t

α

∂k ₁

∂t (t, t 1 )u(t 1 ) log u(t 1 ) dt 1

+ X n i=2

Z _t

α

µZ _t

α

· · ·

µZ _t

α

∂k _i

∂t (t, t ₁ , . . . , t _i )u(t _i ) log u(t _i ) dt _i

¶

· · ·

¶ dt ₁ . We note that R[u log u] and Q[u log u] are linear functionals and

R[u ₁ log u ₁ ] ≤ R[u ₂ log u ₂ ], Q[u ₁ log u ₁ ] ≤ Q[u ₂ log u ₂ ] if u ₁ log u ₁ ≤ u ₂ log u ₂ for any t ∈ J, and

R[u log u] ≤ R[log u]u, Q[u log u] ≤ Q[log u]u

if log u(t) is nonnegative in J and u(t) is nondecreasing in J. The equality (3.3) implies the estimate

(3.4) v ⁰ (t) ≤ {R[log v](t) + Q[log v](t)}v(t).

From (3.4), we have

(3.5) v ⁰ (t)

v(t) ≤ R[log v](t) + Q[log v](t).

By taking t = s in (3.5) and then integrating it from 0 to any t ∈ J, we obtain

(3.6) log v(t) ≤ log a + Z _t

α

(R[log v](s) + Q[log v](s)) ds.

Now, we denotes the right-hand side of (3.6) by v 1 (t). Then v 1 (α) = log a, log v(t) ≤ v ₁ (t), the function v ₁ (t) is nondecreasing in t ∈ [α, β]

and

(3.7) v ₁ ⁰ (t) ≤ {R[1](t) + Q[1](t)}v ₁ (t).

The condition (3.7) implies that µ

v ₁ ⁰ (s) − {R[1](s) + Q[1](s)}v ₁ (s)

¶ exp

µZ _t

s

{R[1](τ ) + Q[1](τ )} dτ

¶

≤ 0 for α ≤ s or

d ds

·

v ₁ (s) exp µZ _t

s

{R[1](τ ) + Q[1](τ )} dτ

¶¸

≤ 0.

Integrating over s from α to any t ∈ [α, β], we have (3.8) v ₁ (t) ≤ log a exp(B ₃ (t)) = log a ^exp(B

^(t)) , where B ₃ (t) = R _t

α {R[1](τ ) + Q[1](τ )} dτ. The desired inequality in (3.2) now follows by using log v(t) ≤ v ₁ (t) in (3.8) and then u(t) ≤ v(t). This

completes the proof. ¤

Theorem 3.2. Let u(t) and b(t) be nonnegative continuous functions in J = [α, β] and a ≥ 1 be a constant. Suppose that

(3.9) u(t)

≤ a + Z _t

α

k 1 (t, t 1 )u(t 1 ) log u(t 1 ) dt 1 + · · ·

+ Z _t

α

à Z _t

α

· · ·

µZ _t

α

k n (t, t 1 , . . . , t n )u(t n ) log u(t n ) dt n

¶

· · ·

! dt 1

for any t ∈ J, where k _i (t, t ₁ , . . . , t _i ) are nonnegative, continuous func- tions in J _i+1 for i = 1, 2, . . . , n, which are nondecreasing in t ∈ J for all fixed (t 1 , . . . , t i ) ∈ J i for i = 1, . . . , n. Then, for any t ∈ J,

(3.10) u(t) ≤ a ^exp(B

^(t)) , where B ₄ (t) = R _t

α R[1](t, τ ) dτ, and b R[1](t, τ ) = k b ₁ (t, τ ) +

Z _t

α

k ₂ (t, τ, t ₂ ))dt ₂ +

X n i=3

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k i (t, τ, t 2 , . . . , t i ) dt i

¶

· · ·

¶

dt 2 .

Proof. For any fixed T ∈ (α, β] and α ≤ t ≤ T , we define a function w(t) by

w(t)

= a + Z _t

α

k ₁ (T, t ₁ )u(t ₁ ) log u(t ₁ ) dt ₁ + · · · +

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k n (T, t 1 , . . . , t n )u(t n ) log u(t n ) dt n

¶

· · ·

¶ dt 1 .

Then w(α) = a, the function w(t) is nondecreasing continuous and u(t) ≤ w(t). Since ^∂k _∂t

(T, t ₁ , . . . , t _i ) = 0 for all i = 1, . . . , n and t ∈ J = [α, β], we have

(3.11) w ⁰ (t) = b R[u log u](T, t), where

R[u log u](T, t) b

= k 1 (T, t)u(t) log u(t) + Z _t

α

k 2 (T, t, t 2 )u(t 2 ) log u(t 2 )dt 2

+ X n i=3

Z _t

α

µZ _t

α

· · ·

µZ _t

α

k _i (T, t, t ₂ , . . . , t _i )u(t _i ) log u(t _i ) dt _i

¶

· · ·

¶ dt ₂ .

The equality (3.11) implies the estimate

(3.12) w ⁰ (t) ≤ { b R[log w](T, t)}w(t).

From (3.12), we have

(3.13) w ⁰ (t)

w(t) ≤ b R[log w](T, t).

By taking t = s in (3.13) and then integrating it from α to any t ∈ J, we obtain

(3.14) log w(t) ≤ log a + Z _t

α

R[log w](T, s) ds. b

Now, we denotes the right-hand side of (3.14) by w 1 (t). Then w 1 (α) = log a, log v(t) ≤ w ₁ (t), the function w ₁ (t) is nondecreasing in t ∈ [α, β]

and

(3.15) w ⁰ ₁ (t) ≤ { b R[1](T, t)}w ₁ (t).

The inequality (3.15) implies the estimate µ

w ⁰ ₁ (s) − { b R[1](T, t)}w 1 (s)

¶ exp

µZ _t

s

{ b R[1](T, τ )} dτ

¶

≤ 0 for α ≤ s or

d ds

·

w ₁ (s) exp µZ _t

s

{ b R[1](T, τ )} dτ

¶¸

≤ 0.

Integrating over s from α to any t ∈ J = [α, β], we have (3.16) w ₁ (t) ≤ log a exp(B ₄ (t)) = log a ^exp(B

^(t)) , where B 4 (t) = R _t

α { b R[1](T, τ )} dτ. In particular, for T = t, we find the desired inequality in (3.10) now follows by using log w(t) ≤ w 1 (t) in (3.8) and then u(t) ≤ w(t). This completes the proof. ¤ Theorem 3.3. Let u, f ₁ , . . . , f _n be nonnegative, continuous func- tions in J = [α, β]. Suppose that

(3.17) u(t)

≤ a + Z _t

α

f ₁ (t ₁ )u(t ₁ ) log u(t ₁ ) dt ₁ + · · · +

Z _t

α

f 1 (t 1 ) µZ _t

α

f 2 (t 2 ) · · ·

µZ _t

α

f n (t n )u(t n ) log u(t n ) dt n

¶

· · ·

¶ dt 1

for any t ∈ J, where a ≥ 1 is a constant. Then

(3.18) u(t) ≤ R 1 (t)

for any t ∈ J, where R n (t) =

· 1 a −

Z _t

α

(2 ⁿ⁻¹ f 1 (s) + 2 ⁿ⁻² f 2 (s) + · · · + f n (s)) ds

¸ ₋₁

for any t ∈ [α, β ₁ ), where β ₁ is chosen so that the expression between [· · · ] is positive in the subinterval [α, β 1 ), and

R i (t) = a + Z _t

α

R i+1 (s) µ X _i

k=1

2 ^i−k f k (s)

¶

ds

for any t ∈ J and i = n − 1, . . . , 1.

Proof. We set

u ₁ (t) = a + L ₁ [u log u](t), u _j+1 (t) = u ² _j + L _j+1 [u log u](t) for any t ∈ J and j = 1, . . . , n − 1, where

L _k [u log u](t)

= Z _t

α

f k (t k )u(t k ) log u(t k ) dt k + · · · +

Z _t

α

f _k (t _k ) µZ _t

α

f _k+1 (t _k+1 ) · · · µZ _t

α

f _n (t _n )u(t _n ) log u(t _n ) dt _n

¶

· · ·

¶ dt _k for any t ∈ J and k = 1, . . . , n. Now, (3.17) implies

(3.19) u(t) ≤ u 1 (t).

Taking into account that u k (t) ≤ u k+1 (t), (L k [u log u]) ⁰ ≤ f k (u ² (t) + L _k+1 [u log u]) for k = 1, . . . , n − 1 and

(L _n [u log u]) ⁰ = f _n (t)u(t) log u(t), for k = 2, . . . , n − 1, we successively have

(3.20)

 

 



 

 

u ⁰ ₁ (t) = (L ₁ [u log u](t)) ⁰ = f ₁ [u ² (t) + L ₂ [u log u]]

≤ f ₁ [u ² ₁ (t) + L ₂ [u log u]] = f ₁ u ₂ , u ⁰ _k (t) ≤ (2 ^k−1 f ₁ + 2 ^k−2 f ₂ + · · · + f _k )u _k+1 , u ⁰ _n (t) ≤ (2 ⁿ⁻¹ f ₁ + 2 ⁿ⁻² f ₂ + · · · + f _n )u ² _n (t).

According to (3.20), the function z = −u ⁻¹ _n satisfies (3.21) z ⁰ (t) = u ⁻² _n u ⁰ _n ≤ 2 ⁿ⁻¹ f ₁ + 2 ⁿ⁻² f ₂ + · · · + f _n .

By taking t = s in (3.21) and then integrating it from α to any t ∈ [α, β ₁ ), we have

(3.22) u n (t) ≤

· 1 a −

Z _t

α

(2 ⁿ⁻¹ f 1 (s) + 2 ⁿ⁻² f 2 (s) + · · · + f n (s)) ds

¸ ₋₁ , where β ₁ is chosen so that the expression between [· · · ] is positive in the subinterval [α, β ₁ ). Define a function R _n (t) by the right side of (3.22).

Then (3.20) and (3.22) gives (3.23) u _i (t) ≤ a +

Z _t

α

R _i+1 µ X _i

k=1

2 ^i−k f _k (s)

¶ ds

for i = n − 1, . . . , 1. For i = 1, (3.19) and (3.23) imply (3.18). This

completes the proof. ¤

4. Applications

In this section, we show that our main results are useful in showing the global existence of solutions to certain integro-differential equations of the form

(4.1) x ⁰ (t) = F µ

t, s, x(t), Z _t

t

f (t, s, x(s)) ds

¶

for any t ∈ J = [α, β] with the given initial condition

(4.2) x(t 0 ) = x 0 ,

where f ∈ C(J ² × R, R), F ∈ C(J ² × R ² , R) and x 0 ≥ 1 is constant.

Assume that (4.3)

Z _t

t

|F (s, σ, u, v)| ds ≤ Z _t

t

(k(t, s)|u| log |u| + h ₁ (t)|v|) ds,

(4.4) |f (t, s, u)| ≤ h ₂ (t, s)|u| log |u|, (4.5) h ₁ (t)h ₂ (s, σ) ≤ h(t, s, σ),

where the functions k(t, s) and h(t, s, σ) are defined as in Theorem 2.4 for α ≤ σ ≤ s ≤ t ≤ β.

If x(t) is a solution of the equation (4.1) with (4.2), the solution x(t) can be written as

(4.6) x(t) = x 0 + Z _t

t

F µ

s, σ, x(σ), Z _s

t

f (s, σ, x(σ)) dσ

¶ ds

for any t ∈ J. Using (4.3)∼(4.5) in (4.6) and making the change of variables, we have

(4.7)

|x(t)| ≤ |x ₀ | + Z _t

t

k(t, s)|x(s)| log |x(s)| ds

+ Z _t

t

Z _s

t

h(t, s, σ)|x(σ)| log |x(σ)| dσ ds.

Now, a suitable application of Theorem 2.4 to (4.7) yields

log |x(t)| ≤ (log |x ₀ |) exp(B ₂ (t)),

which implies that x(t) is bounded.

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Frunze (1965), (In Russian).

Yeol Je Cho

Department of Mathematics Education and the RINS College of Education

Gyeongsang National University Chinju 660-701, Korea

E-mail: [email protected] Sever S. Dragomir

School of Computer Science and Mathematics Victoria University of Technology

P.O. Box 14428, MCMC, Melbourne Victoria 8001, Australia

E-mail: [email protected] Young-Ho Kim

Department of Applied Mathematics Changwon National University Changwon 641-773, Korea

E-mail: [email protected]