Some new generalized Gronwall-Bellman-type integral inequalities in two independent variables on time scales
© Wang et al.; licensee Springer. 2013
Received: 30 January 2013
Accepted: 23 April 2013
Published: 8 May 2013
In this paper, some new Gronwall-Bellman-type integral inequalities in two independent variables on time scales are established, which can be used as a handy tool in the research of qualitative and quantitative properties of solutions of dynamic equations on time scales. The inequalities established unify some of the integral inequalities for continuous functions in (Meng and Li in Appl. Math. Comput. 148:381-392, 2004) and their discrete analysis in (Meng and Li in J. Comput. Appl. Math. 158:407-417, 2003).
MSC:26E70, 26D15, 26D10.
It is well known that Gronwall-Bellman inequality [1, 2] plays an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. During the past decades, a lot of generations of Gronwall-Bellman inequality have been discovered; see, for example, [3–10]. On the other hand, in the 1980s, Hilger created the theory of time scales  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. See, for example, [12–14] 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, [15–20], which have been designed in order to unify continuous and discrete analysis. But to our knowledge, Gronwall-Bellman-type integral inequalities containing integration on infinite intervals on time scales have been paid little attention in the literature so far.
In this paper, we establish some new Gronwall-Bellman-type integral inequalities in two independent variables containing integration on infinite intervals on time scales, which unify some of the continuous inequalities in  and the corresponding discrete analysis in .
2 Some preliminaries
Throughout the paper, ℝ denotes the set of real numbers and . 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 .
Remark 2.1 Obviously, if , while if .
Definition 2.2 A point with is said to be left-dense if and right-dense if , left-scattered if and right-scattered if .
Definition 2.3 The set is defined to be if does not have a left-scattered maximum; otherwise, it is without the left-scattered maximum.
Definition 2.4 A function is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while f is called regressive if . denotes the set of rd-continuous functions, while ℜ denotes the set of all regressive and rd-continuous functions, and .
where , and is a neighborhood of t. The function is called delta differential on .
where , and is a neighborhood of x. The function is called partial delta differentiable with respect to x on .
Remark 2.2 If , then becomes the usual derivative , while if , which represents the forward difference.
The following two theorems include some important properties for partial delta derivative and Cauchy partial integral on time scales.
- (ii)If f, g are partial delta differentiable at x, then fg is also partial delta differentiable at x, and
if for all , then .
Remark 2.3 If , then all the conclusions of Theorem 2.2 still hold.
where Log is the principal logarithm function.
The following two theorems include some known properties on the exponential function.
if with respect to x, then , ,
if with respect to x, then ,
Remark 2.5 If , then Theorem 2.3(v) still holds.
Remark 2.7 For more details about time scales, we advise the reader to refer to .
3 Main results
First we give some important lemmas as follows.
Proof Fix , since , then from Theorem 2.3(iv) we have ; furthermore, from Theorem 2.3(iii) we obtain , .
which is the desired inequality. □
Lemma 3.2 
Then by Theorem 2.3(v) and after letting , we obtain the desired result. □
where , are defined in (3.13).
where , are defined in (3.13).
Combining (3.15) and (3.21), we can obtain the desired inequality (3.12). □
Since is an arbitrary time scale, then if we take for some peculiar cases, we can obtain some corollaries immediately. Especially, if we let or , we obtain the following two corollaries.
where , are the same as in Theorem 3.1.
which is the desired result. □
If we let or in Theorem 3.2, then we obtain the following two corollaries.
where , are defined the same as in (3.22).
provided that , , where , are defined the same as in (3.27).
where Lemma 3.3 is used in the last step.
where , are defined in (3.34).
Combining (3.39), (3.40), and (3.43), we get the desired inequality (3.33). □
where , are defined the same as in (3.34).
The proof of Theorem 3.4 is similar to that of Theorem 3.2, and we omit it here.
The proof for Corollary 3.5 is similar to that for Corollary 3.1.
Remark 3.2 Corollary 3.5 is equivalent to [, Theorem 4].
Remark 3.3 If we take in Theorem 3.3, or in Theorems 3.3 and 3.4, then we can obtain another three corollaries, which are omitted here. Especially, if we take in Theorem 3.4, then Theorem 3.4 reduces to [, Theorem 4] with a slight difference, which is one case of discrete inequality.
Remark 3.4 The inequalities with two independent variables established in (3.11) and (3.32) are essentially different from the cases with a single variable. As these inequalities can be used in deriving bounds for solutions of certain dynamic equations in two independent variables, which are shown in Section 4, while inequalities with a single variable can only be used to the boundedness analysis for solutions of certain dynamic equations in a single variable.
Remark 3.5 The inequalities established above are related to infinite intervals. The main difference between the results here and those of finite intervals lies in their applications. The results above are valid in the qualitative analysis of certain dynamic equations including integrals on infinite intervals, which are shown in the two examples in Section 4, while results of finite intervals are invalid in such cases.
4 Some applications
In this section, we present some applications for the results established above.
with the condition , where , , and p is a constant with .
Then a suitable application of Theorem 3.1 yields the desired inequality (4.2). □
In the last step of Theorem 4.1, if we apply Theorem 3.2 to (4.5), then we can obtain the following theorem.
where , are defined in (4.3).
Theorem 4.3 Suppose in (4.1) and , where f, g are defined the same as in Theorem 4.1, then under the condition , Eq. (4.1) has at most one solution.
that is, , and the proof is complete. □
where , , and is a constant.
Then using Theorem 3.3 in (4.15), we can obtain the desired result. □
We have established several new Gronwall-Bellman-type integral inequalities containing integration on infinite intervals on time scales, which unify some known continuous and discrete results in the literature. As one can see from the present applications, the inequalities established are useful in the investigation of qualitative and quantitative properties of solutions of certain dynamic equations on time scales.
The authors would like to thank the referees very much for their valuable suggestions on improving this paper.
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