Generalized integral inequalities for discontinuous functions with two independent variables and their applications
Journal of Inequalities and Applications volume 2014, Article number: 524 (2014)
This paper investigates integral inequalities for discontinuous functions with two independent variables involving two nonlinear terms. We do not require that is in the class ℘ or the class ȷ in Gallo and Piccirllo’s paper (Nonlinear Stud. 19:115-126, 2012). My main results can be applied to generalize Borysenko and Iovane’s results (Nonlinear Anal., Theory Methods Appl. 66:2190-2230, 2007) and to give results similar to Gallo-Piccirllo’s. Examples to show the bounds of solutions of a partial differential equation with impulsive terms are also given, which cannot be estimated by Gallo and Piccirllo’s results.
Integral inequalities and their various linear and nonlinear generalizations involving continuous or discontinuous functions play very important roles in investigating different qualitative characteristics of solutions for differential equations, partial differential equations and impulsive differential equations such as existence, uniqueness, continuation, boundedness, continuous dependence of parameters, stability, and attraction. The literature on inequalities for continuous functions and their applications is vast (see [1–11]). In the one-dimensional case, all the main results in the theory of integral inequalities for continuous functions are almost based on the solvability of Chaplygin’s problem  for the integral inequality
Recently, more attention has been paid to generalizations of Gronwall-Bellman’s results for discontinuous functions and their applications (see [12–27]). One of the important things is that Samoilenko and Perestyuk  studied the following inequality:
for the nonnegative piecewise continuous function , where c, are nonnegative constants, is a positive function, and are the first kind discontinuity points of the function . Then Borysenko  investigated integral inequalities with two independent variables,
Here is an unknown nonnegative continuous function with the exception of the points where there is a finite jump: , .
In 2007, Borysenko and Iovane  considered the following inequalities:
Later, Gallo and Piccirllo  studied the following inequalities:
In this paper, motivated by the work above, we will establish the following much more general integral inequality:
with two independent variables involving two nonlinear terms and where we do not restrict and to the class ℘ or the class ȷ. Moreover, () has a more general form. We also show that many integral inequalities for discontinuous functions such as (1.3)-(1.6) can be reduced to the form of (1.8). Finally, our main result is applied to an estimation of the bounds of the solutions of a partial differential equation with impulsive terms.
2 Main results
for , and , and let denote the first-order partial derivative of with respect to x and .
Consider (1.8) and assume that
(H1) is defined on Ω and ; is a nonnegative constant for any positive integer i;
(H2) () are continuous and nonnegative functions on and satisfy a certain condition: () if , for arbitrary ;
(H3) and are continuous and nonnegative functions on and are positive on such that is nondecreasing;
(H4) is continuous and nonnegative on Ω;
(H5) is nonnegative and continuous on Ω with the exception of the points where there is a finite jump: , . Here if , , , and , ;
(H6) and () are continuously differentiable and nondecreasing such that on and on .
Let for and where is a given positive constant. Clearly, is strictly increasing so its inverse is well defined, continuous, and increasing in its corresponding domain.
Theorem 2.1 Suppose that (H k ) () hold and satisfies (1.8) for a positive constant m. If we let for , then the estimate of is recursively given, for , , by
The proof is given in Section 3.
Remark 2.1 If satisfies for , then i in Theorem 2.1 can be any nonzero integer. Reference  pointed out that different choices of in do not affect our results for . If , then define and (2.1) is still true.
Remark 2.2 If is nondecreasing, Theorem 2.1 generalizes many known results. For example:
If we take , , , , , and , then (1.8) reduces to (1.3). It is easy to check that and . From Theorem 2.1, we know that for
After recursive calculations, we have
which is the same as the expression in ;
If we take , , , , , and , then (1.8) reduces to (1.4) and Theorem 2.1 becomes Theorem 2.1 in ;
If we take , , , , , and , then (1.8) reduces to (1.5) and Theorem 2.1 becomes Theorem 2.2 in ;
If on where , then (1.6) can be rewritten as(2.4)
If we let , , and , the above inequality is the same as (1.8).
Consider the inequality
which looks much more complicated than (1.8).
Corollary 2.1 Suppose that (H k ) () hold, is positive on , is positive and strictly increasing on and satisfies (2.5). If we let for , then the estimate of is recursively given, for , , by
where , and are given in Theorem 2.1, is defined as follows:
Proof Let . Since the function φ is strictly increasing on , its inverse is well defined. Equation (2.5) becomes
Let and . Equation (2.7) becomes
It is easy to see that , and are continuous and nonnegative functions on , and is nondecreasing on . Even though is much more general, in the same way as in Theorem 2.1, for , , we can obtain the estimate of ,
This completes the proof of Corollary 2.1. □
If where is a constant, we can study the inequality
According to Corollary 2.1, we have the following result.
Corollary 2.2 Suppose that (H k ) () hold, , and satisfies (2.10). If we let for , then the estimate of is recursively given, for , , by
where , , and are given in Corollary 2.1.
3 Proof of Theorem 2.1
Obviously, for any , is positive and nondecreasing with respect to x and y, () is nonnegative and nondecreasing with respect to x and y for each fixed s and t. They satisfy and .
We first consider and have from (1.8)
Take any fixed , , and for arbitrary , we have
and . Hence, . Clearly, is a nonnegative, nondecreasing and differentiable function for and . Moreover, (or ) is differentiable and nondecreasing in (or ) for . Thus, (or ) for (or ). Since and , we have
Integrating both sides of the above inequality from to x, we obtain
for and , where , or equivalently
It is easy to check that , and is differentiable, positive and nondecreasing on and . Since is nondecreasing from the assumption (H3), we have
Integrating both sides of the inequality (3.6) from to x, we obtain
Since the above inequality is true for any , , we obtain
Replacing by x and by y yields
This means that (2.1) is true for and if replace with .
For and , (1.8) becomes
where the definition of is given in (2.2). Note that the estimate of is known. Clearly, (3.9) is the same as (3.1) if replace and by and . Thus, by (3.8) we have
This implies that (2.1) is true for and if replace by .
Assume that (2.1) is true for , i.e.,
For , (1.8) becomes
where we use the fact that the estimate of is already known for (). Again, (3.11) is the same as (3.1) if replace and by and . Thus, by (3.8) we have
This shows that (2.1) is true for if replace by . By induction, we know that (2.1) holds for for any nonnegative integer i. This completes the proof of Theorem 2.1.
Consider the following partial differential equation with an impulsive term:
where , , , and .
(C1) where , are nonnegative and continuous on Ω, , for , , ;
(C2) where and m are nonnegative constants.
Corollary 4.1 Suppose that (C1) and (C2) hold. If we let for , then the solution of system (4.1) has an estimate for
Proof The solution of (4.1) with an initial value is given by
Thus, (4.4) is the same as (1.8). It is easy to see that for any positive constants and
Therefore, for any nonnegative i and
Remark 4.1 From (4.4), we know . Clearly, does not hold for large . Thus, does not belong to the class ℘ in . Again does not hold for large so does not belong to the class ȷ in . Hence, the results in  cannot be applied to the inequality (4.1).
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The author is grateful to Dr. Shengfu Deng for many helpful conversations during the course of this work. This research was supported by National Natural Science Foundation of China (No. 11371314), Guangdong Natural Science Foundation (No. S2013010015957), and the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074).
The author declares that she has no competing interests.
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Mi, Y. Generalized integral inequalities for discontinuous functions with two independent variables and their applications. J Inequal Appl 2014, 524 (2014). https://doi.org/10.1186/1029-242X-2014-524