Abstract
In this work, some new generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions are established, which can be used as a handy tool in the quantitative and qualitative analysis of solutions of certain integral equations. The established results generalize the main results of Gao and Meng in (Math. Pract. Theory 39(22):198-203, 2009).
MSC:26D10, 26D15.
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1 Introduction
In recent years, the research of various mathematical inequalities has been paid much attention to by many authors, and many integral inequalities have been established, which provide handy tools for investigating the quantitative and qualitative properties of solutions of integral and differential equations. Among these inequalities, Gronwall-Bellman type integral inequalities are of particular importance as such inequalities provide explicit bounds for unknown functions, and a lot of generalizations of Gronwall-Bellman type integral inequalities have been established (for example, see [1–16] and the references therein). But to our knowledge, most of the known integral inequalities are concerned with continuous functions, while very few authors undertake research in integral inequalities for discontinuous functions [14–16]. Furthermore, delay integral inequalities containing integration on infinite intervals for discontinuous functions have not been reported in the literature so far.
In [1], Gao and Meng established a generalized Gronwall-Bellman type integral inequality, named Mate-Nevai type nonlinear integral inequality for continuous functions, which is one case of inequalities containing integration on infinite intervals for continuous functions. The related theorem reads as follows.
Theorem A Suppose that , , and , are decreasing in t for every fixed s. is strictly increasing, and . is increasing. is strictly increasing, and is submultiplicative. If for and , satisfies
then for , we have
where , , is the inverse of G, and is chosen so that
The inequality above appears to be useful in deriving bounds of solutions of certain integral equations, but it is inadequate to deal with boundedness for solutions of both integral equations for discontinuous functions and delay integral equations.
In the present paper, motivated by the work above, we establish some new generalized Mate-Nevai type nonlinear delay integral inequalities for discontinuous functions, which extend the main results in [1]. The establishment for the desired inequalities will be discussed in 1-D and 2-D cases respectively.
2 Main results
In the rest of the paper, we denote the set of real numbers as ℝ, and .
Theorem 2.1 Suppose that is a nonnegative continuous function defined on with the first kind of discontinuities at the points , , and . , , and , are decreasing in t for every fixed s. is a constant. with , and for , . , and ω is nondecreasing with for . Furthermore, ω is submultiplicative, that is, for . C, are constants, and , , . If for , satisfies the following inequality:
then
where
Proof Denote the right-hand side of (2.1) by . Then
Define
Case 1: If (in fact, ), then . From the definition of τ, we have for . So, , and from (2.4) we have
Then
and
where , are defined in (2.3), and obviously . Furthermore, we have
Multiplying on both sides of (2.8) yields
Setting in (2.9), by , an integration for (2.9) with respect to s from t to ∞ yields
which implies
Define . Then
Since , and ω is submultiplicative, we have
that is,
Setting in (2.13), an integration for (2.13) with respect to s from t to ∞ yields
By , , and is increasing, we obtain
Combining (2.12) and (2.14), we obtain
Especially,
where is defined in (2.3). At the same time, we have
Case 2: If , then from (2.1), (2.16) and (2.17), we have
Then, following in a similar manner as in Case 1, we obtain
Especially,
where is defined in (2.3).
Case 3: If for , , the following inequalities hold:
where is defined in (2.3). Then for , from (2.1) we obtain
Then, following in a similar manner as in Case 1, we obtain
and the proof is complete. □
Remark 2.1 If we take , , , and furthermore is continuous on , then Theorem 2.1 reduces to Theorem A (i.e., [[1], Theorem 1]).
Remark 2.2 Under the assumptions of Remark 2.1, furthermore, if we take , then Theorem 2.1 reduces to [[1], Corollary 1].
Remark 2.3 If we take in Theorem 2.1, Remark 2.1 and Remark 2.2, then we can obtain three corollaries, which are omitted here.
Based on Theorem 2.1, we will establish another Mate-Nevai type inequality for discontinuous functions in the 2-D case.
Theorem 2.2 Suppose is a nonnegative continuous function on , with the exception at the points , , where there are finite jumps, and , . . with , and for , . with , and for , . ω is defined the same as in Theorem 2.1. Furthermore, , for , . is a constant. If for , satisfies the following inequality:
then
where
Proof Denote the right-hand side of (2.23) by . Then
Define
Case 1: If , then . As for and for , so , and from (2.26) we have
Then
and
where , are defined in (2.25), and obviously .
Furthermore, we have
Multiplying on both sides of (2.30) yields
Setting in (2.31), by , an integration for (2.31) with respect to s from x to ∞ yields
which implies
Denote by . Then
By , we have
that is,
Setting in (2.35), and an integration for (2.35) with respect to s from x to ∞ yields
Since , , and is increasing, it follows that
Combining (2.34) and (2.36), we obtain
Especially,
where is defined in (2.25). At the same time, we have
Case 2: If , then from (2.23), (2.38) and (2.39), under the given conditions for , , we have
Then, following in a similar manner as in Case 1, we obtain
Especially,
where is defined in (2.25).
Case 3: If for , , the following inequalities hold:
where is defined in (2.25), then for , by (2.23), we obtain
Then, following in a similar manner as in Case 1, we obtain
and the proof is complete. □
Remark 2.4 If we take and, furthermore, , or take in Theorem 2.2, then we can obtain corresponding corollaries, which are omitted here.
3 Application
In this section, we present one application to illustrate the validity of our results in deriving explicit bounds for the discontinuous solutions of certain integral equations.
Consider an integral equation of the form
where u is a continuous function defined on with the first kind of discontinuities at the points , , and . with , and for , . , , and .
Theorem 3.1 Assume that is a solution of Eq. (3.1), and
where q, C are constants with , , , , and , are decreasing in t for every fixed s. Then we have
where , , , , are defined the same as in (2.3), and .
Proof From (3.1) we have
Then a suitable application of Theorem 2.1 yields the desired result. □
Theorem 3.2 Under the conditions of Theorem 3.1, furthermore, we have
Proof As long as we notice , and
combining (3.5) and Theorem 3.1, we can easily deduce the desired result. □
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The authors would like to thank the referees very much for their careful comments and valuable suggestions on this paper.
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Zheng, B. Some generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions. J Inequal Appl 2013, 297 (2013). https://doi.org/10.1186/1029-242X-2013-297
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DOI: https://doi.org/10.1186/1029-242X-2013-297