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Some new generalized retarded nonlinear integral inequalities with iterated integrals and their applications
Journal of Inequalities and Applications volume 2012, Article number: 236 (2012)
Abstract
Some new generalized retarded nonlinear integral inequalities with iterated integrals are discussed and upper bound estimations of unknown functions are given by analysis technique. These estimations can be used as tools in the study of differential-integral equations with the initial conditions.
MSC:26D10, 26D15, 26D20, 34A12, 34A40.
1 Introduction
Gronwall-Bellman inequalities [1, 2] can be used as important tools in the study of existence, uniqueness, boundedness, stability and other qualitative properties of solutions of differential equations and integral equations. There can be found a lot of generalizations of Gronwall-Bellman inequalities in various cases from literature (e.g., [3–15]).
Agarwal et al. [5] studied the inequality
Agarwal et al. [6] obtained the explicit bound to the unknown function of the following retarded integral inequality:
In 2011, Abdeldaim and Yakout [4] studied the following integral inequalities:
However, the bound given on such an inequality in [4] is not directly applicable in the study of certain retarded nonlinear differential and integral equations. It is desirable to establish new inequalities of the above type, which can be used more effectively in the study of certain classes of retarded nonlinear differential and integral equations.
In this paper, we discuss some new retarded nonlinear integral inequalities with iterated integrals
where is a positive constant, and give upper bound estimation of the unknown function by integral inequality technique. Furthermore, we apply our result to differential-integral equations for estimation.
2 Main result
In this section, we discuss some retarded integral inequalities with iterated integrals. Throughout this paper, let .
Lemma 1 (Abdeldaim and Yakout [4])
We assume that , and are nonnegative real-valued continuous functions defined on and satisfy the inequality
for all , where and p are positive constants. Then
where
such that for all .
Lemma 2 (Abdeldaim and Yakout [4])
We assume that , and are nonnegative real-valued continuous functions defined on and satisfy the inequality
for all , where is a positive constant. Then
where , and
where , and is the maximal solution of the differential equation
such that .
Lemma 3 Suppose that is a positive and increasing function on I with , , ; is a nonnegative real-valued continuous function defined on I with and satisfies the inequality
then has the following estimation:
where
and is the largest number such that
Remark 1 when .
Proof Integrating both sides of (2.3) from 0 to t,
where is a positive constant chosen arbitrarily, is defined by (2.6). Let
then is a nonnegative and nondecreasing function on I with . Then (2.7) is equivalent to
Differentiating with respect to t, from (2.8) and (2.9), we have
Since , from (2.10) we have
By taking in (2.11) and integrating it from 0 to t, we get
where Φ is defined by (2.5). From (2.9) we have
Letting , from (2.13) we get
Because is chosen arbitrarily, this proves (2.4). □
Lemma 4 Let , ; we assume that is a nonnegative real-valued continuous function defined on I with and satisfies the inequality
then has the following estimation:
for all , where is the largest number such that
Proof Integrating both sides of (2.14) from 0 to t, we get
where is a positive constant chosen arbitrarily, is defined by (2.16). Let
then is a nonnegative and nondecreasing function on I with . Then (2.17) is equivalent to
Differentiating with respect to t, from (2.18) and (2.19), we have
Since , from (2.20) we have
Let , then , from (2.21) we obtain
Consider the ordinary differential equation
The solution of equation (2.23) is
for all . Letting in (2.24), from (2.19), (2.22), (2.23) and (2.24), we obtain
Because is chosen arbitrarily, this proves (2.15). □
Lemma 5 Suppose that , are positive and increasing functions on I, , ; is a nonnegative real-valued continuous function defined on I with and satisfies the inequality
then has the following estimation:
for all , where
and is the largest number such that
Proof Integrating both sides of (2.26) from 0 to t, we get
for all , where is a positive constant chosen arbitrarily, is defined by (2.29). Let denote the function on the right-hand side of (2.30), which is a positive and nondecreasing function on with
Then (2.26) is equivalent to
Differentiating with respect to t and using (2.32), we have
From (2.33) we get
Integrating both sides of (2.34) from 0 to t and using (2.31), we get
here we use the monotonicity of and . Let
for all , then is a positive and nondecreasing function on with
(2.36) is equivalent to
Differentiating with respect to t and using (2.38), we have
From (2.39) we get
Integrating both sides of (2.40) from 0 to t,
for all , where is defined by (2.28). From (2.32), (2.38) and (2.41), we have
for all . Letting , from (2.42) we get
Because is chosen arbitrarily, this gives the estimation (2.27) of the unknown function in the inequality (2.26). □
Theorem 1 Suppose that is an increasing function with , , ; , , and are nonnegative real-valued continuous functions defined on and satisfy the inequality (1.1). Then
for all , where
is the largest number such that
Remark 2 Theorem 1 gives the explicit estimation (2.43) for the inequality (1.1) which is just the inequality (2.1) when . Lemma 1 gives the implicit estimation (2.2) for the inequality (2.1).
Proof Let denote the function on the right-hand side of (1.1), which is a positive and nondecreasing function on I with . Then (1.1) is equivalent to
Differentiating with respect to t and using (2.45), we have
where
Then is a positive and nondecreasing function on I with and
Differentiating with respect to t and using (2.46), (2.47) and (2.48), we get
for all . Applying Lemma 4 to (2.49), we obtain
From (2.46) and (2.50), we get
Applying Lemma 3 to (2.51) and using Remark 1, we obtain
From (2.45), the estimation (2.43) of the unknown function in the inequality (1.1) is obtained. □
Theorem 2 Suppose and are increasing functions with , , . We assume that , , and are nonnegative real-valued continuous functions defined on I and satisfy the inequality (1.2). Then
where Φ is defined by (2.5),
is defined by (2.28), is the largest number such that
Proof Let denote the function on the right-hand side of (1.2), which is a positive and nondecreasing function on I with . Then (1.2) is equivalent to
Differentiating with respect to t and using (2.55), we have
where
Then is a positive and nondecreasing function on I with and
Differentiating with respect to t and using (2.56), (2.57) and (2.58), we get
for all . Applying Lemma 5 to (2.59), we obtain
where and are defined by (2.53) and (2.54) respectively. From (2.56) and (2.60), we get
Applying Lemma 3 to (2.61), we obtain
where Φ is defined by (2.5). From (2.55), the estimation (2.52) of the unknown function in the inequality (1.2) is obtained. □
Theorem 3 Suppose are increasing functions with , , . We assume that , and are nonnegative real-valued continuous functions defined on I and satisfy the inequality (1.3). Then
where
such that for all .
Remark 3 If , then Theorem 3 reduces Lemma 1.
Proof Let denote the function on the right-hand side of (1.3), which is a positive and nondecreasing function on I with . Then (1.3) is equivalent to
Differentiating with respect to t and using (2.64), we have
where
Then is a positive and nondecreasing function on I with and
Differentiating with respect to t and using (2.65), (2.66) and (2.67), we get
From (2.68) we have
Let , then . From (2.69) we obtain
Consider the ordinary differential equation
The solution of equation (2.71) is
for all . By (2.70), (2.71) and (2.72), we obtain
where is as defined in (2.63). From (2.65) and (2.73), we have
By taking in the above inequality and integrating it from 0 to t, from (2.64) we get
The estimation (2.62) of the unknown function in the inequality (1.3) is obtained. □
Theorem 4 Suppose are increasing functions with , , , , . We assume that , and are nonnegative real-valued continuous functions defined on and satisfy the inequality (1.4). Then
where
and is the largest number such that
Proof Let denote the function on the right-hand side of (1.4), which is a positive and nondecreasing function on I with . Then (1.4) is equivalent to
Differentiating with respect to t and using (2.77), we have
where
Then is a positive and nondecreasing function on I with and
Differentiating with respect to t and using (2.78), (2.79) and (2.80), we get
Since , , from (2.81) we have
By taking in the above inequality and integrating it from 0 to t, we get
for all , where is defined by (2.75). From (2.82) we have
for all , where is chosen arbitrarily. Let denote the function on the right-hand side of (2.83), which is a positive and nondecreasing function on I with and
Differentiating with respect to t and using the hypothesis on , from (2.84) we have
By the definition of in (2.76), from (2.85) we obtain
Let , from (2.86) we have
Since is chosen arbitrarily, from (2.77), (2.80), (2.84) and (2.87), we have
This proves (2.74). □
3 Application
In this section we apply our Theorem 4 to the following differential-integral equation:
where , , is a constant, satisfy the following conditions:
where f, g are nonnegative real-valued continuous functions defined on I.
Corollary 1 Consider the nonlinear system (3.88) and suppose that K, H satisfy the conditions (3.89) and (3.90), and are increasing functions with , , , , . Then all the solutions of equation (3.88) exist on I and satisfy the following estimation:
for all , where
and is the largest number such that
Proof Integrating both sides of equation (3.88) from 0 to t, we get
Using the conditions (3.89) and (3.90), from (3.92) we obtain
for all . Applying Theorem 4 to (3.93), we get the estimation (3.91). This completes the proof of Corollary 1. □
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Acknowledgements
The author is very grateful to the editor and the referees for their helpful comments and valuable suggestions. This research was supported by the National Natural Science Foundation of China (Project No. 11161018), Guangxi Natural Science Foundation (Project No. 0991265 and 2012GXNSFAA053009), Scientific Research Foundation of the Education Department of Guangxi Province of China (Project No. 201106LX599), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).
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Wang, WS. Some new generalized retarded nonlinear integral inequalities with iterated integrals and their applications. J Inequal Appl 2012, 236 (2012). https://doi.org/10.1186/1029-242X-2012-236
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DOI: https://doi.org/10.1186/1029-242X-2012-236