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Some new integral inequalities on time scales containing integration on infinite intervals
Journal of Inequalities and Applications volume 2013, Article number: 245 (2013)
Abstract
In this paper, some new Gronwall-Bellman type integral inequalities on time scales containing integration on infinite intervals are established. They provide new bounds for unknown functions concerned and can be used as a handy tool in the research of qualitative and quantitative properties of solutions of certain dynamic equations on time scales.
MSC:26E70, 26D15, 26D10.
1 Introduction
The development of the theory of time scales was initiated by Hilger [1] as a theory capable to contain both difference and differential calculus in a consistent way. Since then many authors have expounded on various aspects of the theory of dynamic equations on time scales (for example, see [2–9], and the references therein). In these investigations, integral inequalities on time scales have been paid much attention by many authors, and a lot of integral inequalities on time scales have been established (for example, see [5–16] and the references therein), which are mainly designed to unify continuous and discrete analysis and play an important role in the research of boundedness, uniqueness, stability of solutions of dynamic equations on time scales. Among these inequalities, Gronwall-Bellman type integral inequalities are of particular importance as such inequalities provide explicit bounds for unknown functions. For related results, we refer to [10–16]. But to our knowledge, Gronwall-Bellman type integral inequalities on time scales containing integration on infinite intervals have been paid little attention so far in the literature.
In this paper, we establish some new Gronwall-Bellman type integral inequalities on time scales containing integration on infinite intervals. New explicit bounds for unknown functions concerned are obtained due to the presented inequalities. Also we present some applications for the established results.
2 Some preliminaries
Throughout this paper, ℝ denotes the set of real numbers and , while ℤ denotes the set of integers. For two given sets G, H, we denote the set of maps from G to H by .
A time scale is an arbitrary nonempty closed subset of real numbers. In this paper, denotes an arbitrary time scale. On we define the forward and backward jump operators and such that , .
Definition 2.1 The graininess is defined by .
Definition 2.2 The cylinder transformation is defined by
where Log is the principal logarithm function.
Definition 2.3 For , the exponential function is defined by
Definition 2.4 If , , we define
Remark 2.1 If , then we have
If , then we have
The following two theorems include some known properties on the exponential function.
Theorem 2.1 [[17], Theorem 5.2]
If , then the following conclusions hold:
-
(i)
and ,
-
(ii)
,
-
(iii)
if , then , ,
-
(iv)
if , then ,
-
(v)
,
where .
Remark 2.2 If , then Theorem 2.1(iii), (v) still holds.
Theorem 2.2 [[17], Theorem 5.1]
If , and fix , then the exponential function is the unique solution of the following initial value problem:
For more details about the calculus of time scales, we refer to [18, 19].
The following lemma is important for proving our results.
Lemma 2.1 Suppose that , , , and u is delta differential at , then
implies
Proof Since , then from Theorem 2.1(iv) we have , and furthermore, from Theorem 2.1(iii) we obtain , .
So,
On the other hand, from Theorem 2.2 we have
So, combining (2.3), (2.4) and Theorem 2.1, it follows that
Substituting t with s and an integration for (2.5) with respect to s from α to ∞ yield
Since , from (2.1) and (2.6) we have
which is followed by
Since is arbitrary, after substituting α with t, we obtain the desired inequality. □
Lemma 2.2 [20]
Assume that , , and , then for any
3 Main results
Theorem 3.1 Suppose that , , and a is decreasing. If satisfies the following inequality:
then
provided that , where
Proof Fix , and let . Since is decreasing on , from (3.1) we have
Denote the right-hand side of (3.4) by . Then it follows that
and furthermore,
where is defined in (3.3).
Under the condition , in fact we have . So, a suitable application of Lemma 2.1 yields
Since , then (3.7) can be rewritten as
Combining (3.8) with (3.5), we have
Setting in (3.9), since T is selected from arbitrarily, after substituting T with t, we obtain the desired inequality (3.2). □
Since is an arbitrary time scale, if we take for some peculiar cases, then we have the following corollaries.
Corollary 3.1 Suppose that . , and a is decreasing on ℝ. If satisfies the following inequality:
then
where .
Proof When , we have , and then . Obviously, . Furthermore, from Remark 2.1 we have
Combining (3.12) and (3.2), we obtain the desired result. □
Corollary 3.2 Suppose that . , and a is decreasing on ℤ. If satisfies the following inequality:
then
provided that , where .
Proof When , we have , , and then . Furthermore, from Remark 2.1 we have
Combining (3.15) and (3.2), we obtain the desired result. □
Theorem 3.2 Suppose that , u, a, f, g are defined as in Theorem 3.1. p is a positive number with . If satisfies the following inequality:
then
provided , where
Proof Fix . Let , and
Then
and furthermore,
On the other hand, from Lemma 2.2 we have
So, combining (3.21) and (3.22), we obtain
where and are defined in (3.18).
Since , and , we have , and a suitable application of Lemma 2.1 yields
Combining (3.24) and (3.20), we have
Setting in (3.25), and considering T is selected from arbitrarily, after substituting T with t, we obtain the desired inequality (3.17). □
Corollary 3.3 Suppose that . , and a is decreasing on ℝ. p is defined as in Theorem 3.2. If satisfies the following inequality:
then
where , are the same as in Theorem 3.2.
Corollary 3.4 Suppose that . , and a is decreasing on ℤ. p is defined as in Theorem 3.2. If satisfies the following inequality:
then
where , are the same as in Theorem 3.2.
The proofs for Corollaries 3.3 and 3.4 are similar to those for Corollaries 3.1 and 3.2, and we omit them here.
Theorem 3.3 Suppose that , u, a, f, g, h are defined as in Theorem 3.1, and . If satisfies the following inequality:
then
provided that , , where
Proof Let
Then
Since , we have , and furthermore . Treating as a variable, by Corollary 3.5 we obtain
Now fix . Let , and
Then
Since , we have . From (3.35) and Lemma 2.2, we obtain
where , are defined in (3.32).
Since , we have , and a suitable application of Lemma 2.1 yields
Since , we obtain
Combining (3.35), (3.37) and (3.40), we obtain
Setting in (3.41) yields
Since T is selected from arbitrarily, substituting T with t in (3.42), we obtain the desired inequality (3.31). □
Corollary 3.5 Suppose that . , and a is decreasing on ℝ. p is defined as in Theorem 3.3. If satisfies the following inequality:
then
where , and is defined as in Theorem 3.3.
Corollary 3.6 Suppose that . , and a is decreasing on ℤ. p is the same as in Theorem 3.3. If satisfies the following inequality:
then
where , and is defined as in Theorem 3.3.
Theorem 3.4 Suppose that , u, a, f are defined as in Theorem 3.1, and . , and for , where . If satisfies the following inequality:
then
provided that , , where
Proof Let
Then
Similar to the process of (3.34) to (3.35), we obtain
Now fix . Let , and
Then
Since , we have , and furthermore . From (3.52), (3.54) and Lemma 2.2, we obtain
where , are defined in (3.49).
We note that the structure of (3.55) is just similar to (3.38). So, following in a similar manner as the process of (3.38)-(3.40), we deduce
Combining (3.52), (3.54) and (3.56), we obtain
Setting in (3.57), since T is selected from arbitrarily, after substituting T with t, we obtain the desired inequality (3.48). □
Remark 3.1 If we take or in Theorem 3.4, then we obtain another two corollaries, which are similar to Corollaries 3.1-3.6.
4 Applications
In this section, we give some applications for the results presented above. In Examples 1 and 3, new explicit bounds for solutions of certain dynamic equations on time scales are derived by the results established, while in Example 2, we deal with the quantitative properties of solutions of a kind of a dynamical equation on time scales.
Example 1 Consider the following dynamic integral equation on time scales:
with the condition , where , p is a positive number with .
Assume that , where , then
provided , where , are defined the same as in Theorem 3.2.
In fact, from (4.1) we have
Then a suitable application of Theorem 3.2 yields the desired result.
Example 2 Consider the following dynamic integral equation on time scales:
where u is the same as in Example 1.
Assume that , where , then Eq. (4.3) has at most one solution.
In fact, if and are two solutions to Eq. (4.3), then
Then a suitable application of Theorem 3.1 yields , which implies .
Example 3 Consider the following dynamic differential equation on time scales:
with the condition , where , and p is a constant with .
Assume that , where f, L are defined the same as in Theorem 3.4, then
provided that , , where , are the same as in Theorem 3.4.
In fact, considering , then the equivalent form of (4.5) is denoted by
Furthermore,
Then a suitable application of Theorem 3.4 yields the desired inequality (4.6).
5 Conclusions
We have established some new Gronwall-Bellman type integral inequalities on time scales containing integration on infinite intervals, which can be used in the research of boundedness and other qualitative properties as well as quantitative properties of solutions of certain dynamic equations on time scales. From the corollaries in Section 3, one can see the established inequalities unify continuous and discrete analysis. Finally, we note that the process of Theorems 3.1-3.4 can be applied to establish delay inequalities with two independent variables on time scales.
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The authors would like to thank the referees very much for their valuable suggestions on improving this paper.
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ZM and CW carried out the main part of this article. All authors read and approved the final manuscript.
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Meng, Z., Zheng, B. & Wen, C. Some new integral inequalities on time scales containing integration on infinite intervals. J Inequal Appl 2013, 245 (2013). https://doi.org/10.1186/1029-242X-2013-245
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DOI: https://doi.org/10.1186/1029-242X-2013-245