Generalized integral inequalities for discontinuous functions with two independent variables and their applications
© Mi; licensee Springer 2014
Received: 29 October 2013
Accepted: 8 May 2014
Published: 15 December 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.
Here is an unknown nonnegative continuous function with the exception of the points where there is a finite jump: , .
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.
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.
- (1)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
If we let , , and , the above inequality is the same as (1.8).
which looks much more complicated than (1.8).
This completes the proof of Corollary 2.1. □
According to Corollary 2.1, we have the following result.
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 .
This means that (2.1) is true for and if replace with .
This implies that (2.1) is true for and if replace by .
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.
where , , , and .
(C1) where , are nonnegative and continuous on Ω, , for , , ;
(C2) where and m are nonnegative constants.
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).
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).
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