Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
A Note on Gronwall’s Inequality on Time Scales
This paper gives a new version of Gronwall’s inequality on time scales. The method used in the proof is much different from that in the literature. Finally, an application is presented to show the feasibility of the obtained Gronwall’s inequality.
1. Introduction and Motivation
Recently, an interesting field of research is to study the dynamic equations on time scales, which have been extensively studied. For example, one can see [1–17] and references cited therein. A time scale is an arbitrary nonempty closed subset of the real numbers . The forward and backward jump operators are defined by . A point , , is said to be left dense if and right dense if and . The mapping defined by is called graininess. A function is said to be rd-continuous provided is continuous at right-dense points. The set of all such rd-continuous functions is denoted by . A function is regressive provided for . Denote .
One of important topics is the differential inequalities on time scales. A nonlinear version of Gronwall’s inequality is presented in [2, Theorem 6.4, pp 256]. This version is stated as follows.
Theorem A. Let , , and . Then implies
Taking , a classical version of Gronwall’s inequality follows (see [2, Corollary 6.7, pp 257]).
Theorem B. Let , , , and . Then implies
This paper presents a new version of Gronwall’s inequality as follows.
Theorem 1. Let and . Suppose that , , and . Then implies
Remark 2. Note that, for , inequality (5) reduces to
which is different from inequality (3) in Theorem B. Since Theorem B requires , we see that Theorem B cannot be applied to (7). Moreover, the method used to prove Theorem A cannot be used to prove Theorem 1. To explain this, recall the proof of Theorem A in . Let . Then and
By comparing theorem and variation of constants formula,
and hence Theorem A follows in view of .
Now we try to adopt the same idea used in  to estimate inequality (7). Let . Then and By comparing theorem and variation of constants formula, we have which implies If we were to use the same idea as in , we should combine (12) with However, on one side, ; on the other side, . These two inequalities cannot lead us anywhere.
Therefore, some novel proof is employed to prove Theorem 1. One can see the detailed proof in the next section.
2. Proof of Main Result
Before our proof of Theorem 1, we need some lemmas.
Lemma 3 (chain rule ). Assume is delta differentiable on . Assume further that is continuously differentiable. Then is delta differentiable and satisfies
Lemma 4. Suppose that is positive delta differentiable on and is regressive. Then is a preantiderivative of function , where and is the principal logarithm function.
Proof of Theorem 1. To prove Theorem 1, we divide it into two cases.
Case 1. For , in this case, we have Hence, it is easy to conclude that for .
Case 2. For , let . For any , we have Noting that , we have . Thus, we have Multiplied by on both sides of the above inequality, it follows that or Since , . Using the fact that is nondecreasing with respect to for , we have An integration of the above inequality over leads to It follows from Lemma 4 that or which leads to Therefore, for . This completes the proof of Theorem 1.
3. An Application
Inequality (5) has many potential applications. For instance, it can be used to study the property of the solutions to the dynamic systems. Consider the following linear system: Let and be two solutions of (26) satisfying the initial conditions and , respectively.
Theorem 5. Suppose that is bounded on . Then one has where
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was supported by JB12254.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.