- Open Access
Nonlinear integral inequalities with delay for discontinuous functions and their applications
© Mi et al.; licensee Springer. 2013
Received: 16 May 2013
Accepted: 26 August 2013
Published: 11 September 2013
This paper investigates integral inequalities with delay for discontinuous functions involving two nonlinear terms. We do not require the classes ℘ and ȷ in Gallo and Piccirillo’s paper (Bound. Value Probl. 2009:808124, 2009). Our main results can be applied to generalize Gallo and Piccirillo’s results and Iovane’s results (Nonlinear Anal., Theory Methods Appl. 66:498-508, 2007). Examples to show the bounds of solutions of an impulsive differential equation are also given, which can not be estimated by Gallo and Piccirillo’s results.
with a general nonlinear term of u. They assumed that or , where the class ℘ consists of all nonnegative, nondecreasing and continuous functions on such that and for all and , and the class ȷ consists of all positive, nondecreasing and continuous functions on such that and for all and . The classes ℘ and ȷ allow a reduction of to the case of a constant by dividing if is a positive and nondecreasing function. Actually, when we study behaviors of solutions of impulsive differential equations, may not be a nondecreasing function, and w may not satisfy the condition or . For example, does not belong to the class ℘ and ȷ for any and large . Thus, it is interesting to avoid such conditions.
with two nonlinear terms and of u, where we do not restrict and to the class ℘ or the class ȷ. We also show that many integral inequalities for discontinuous functions such as (1.3), (1.4) and (1.6) can be reduced to the form of (1.7). Our main result is applied to estimate the bounds of solutions of an impulsive ordinary differential equation.
2 Main results
Consider (1.7), and assume that
(C1) and are continuous and nondecreasing functions on and are positive on such that is nondecreasing;
(C2) is defined on and ; is a nonnegative constant for any positive integer i;
(C3) and are continuous and nonnegative functions on ;
(C4) and are continuously differentiable and nondecreasing such that and on ;
(C5) For , is nonnegative and piecewise-continuous with the first kind of discontinuities at the points , where i is a nonnegative integer and .
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.
- (2)Take , , , , and . Hence, (1.7) becomes (1.1). It is easy to check that and . From Theorem 2.1, we know that for ,
- (3)Clearly, (1.2) and (1.3) are special cases of (1.7). If on , then (1.6) can be rewritten as
Let and , the inequality above is same as (1.7). Similarly, (1.5) can also be reduced to (1.7).
which looks more complicated than (1.7).
where and its related functions are defined as in Theorem 2.1 by replacing with , .
Remark 2.2 Using the same way, we can change inequality (1.4) into the form of (1.7) with , , , and .
3 Proof of Theorem 2.1
Obviously, is positive and nondecreasing in x, and is nonnegative and nondecreasing in x for each fixed s and . 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 yields 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 , , (), , , for all .
, where , are nonnegative and continuous on ;
, where and m are nonnegative constants.
Remark 4.1 From (4.3), we know that . Clearly, does not hold for large . Thus, does not belong to the class ℘. Again does not hold for large , so does not belong to the class ȷ. Hence, the results in  can not be applied to inequality (4.3).
This work was supported by the Project of Department of Education of Guangdong Province, China (No. 2012KJCX0074), the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province, China (No. S2011040000464), the China Postdoctoral Science Foundation-Special Project (No. 201104077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. (2012)940), the Natural Fund of Zhanjiang Normal University (No. LZL1101), and the Doctoral Project of Zhanjiang Normal University (No. ZL1109).
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