Research Article | Open Access
Peiguang Wang, Ping Li, "Monotone Iterative Technique for Partial Dynamic Equations of First Order on Time Scales", Discrete Dynamics in Nature and Society, vol. 2008, Article ID 265609, 7 pages, 2008. https://doi.org/10.1155/2008/265609
Monotone Iterative Technique for Partial Dynamic Equations of First Order on Time Scales
This work is concerned with the monotone iterative technique for partial dynamic equations of first order on time scales and for this purpose, the existence, uniqueness, and comparison results are also established.
The study of dynamic equations on time scales is an area of research that has received a lot of attention. Many authors have been devoted to the qualitative research of ordinary differential equations on time scales. We can refer to monographs by Bohner and Peterson , Lakshmikantham et al. , and the references cited therein. With the development of technology, in recent years, some authors have explored the partial differential equations on time scales. Ahlbrandt and Morian  have introduced definitions of time scale derivatives and integrals for functions of two variables and obtained the results that include an Euler-Lagrange equation for double integral variational problems on time scales and a Picone identity which implies a Sturm-Picone comparison theorem for second-order elliptic partial differential equations on time scales. Jackson  has extended the existing ideas of the univariate case of the time scales calculus to the multivariate case.
Lakshmikantham et al.  investigated the monotone iterative technique for partial differential equations of first order in 1984. Nevertheless, up till now, we have not yet seen any research about monotone iterative technique for partial dynamic equations on time scales. In this paper, we will investigate the monotone iterative technique for partial dynamic equations of first order on time scales. Firstly, in Sections 2 and 3, we will introduce the univariable calculus and multivariable calculus on time scales that will be the preliminary work in order to define the partial dynamic equations on time scales. Secondly, in Sections 4 and 5, we will present existence, uniqueness, and comparison results and provide the monotone iterative technique for such partial dynamic equations.
2. Univariable Calculus on Time Scales
A time scale is a nonempty closed subset of the real number , and we denote it by the symbol . We define the forward and backward jump operators by (supplemented by ). A point is called right-scattered, right-dense, left-scattered, left-dense if hold, respectively. The set is defined to be if does not have a left-scattered maximum; otherwise it is without this left-scattered maximum. The graininess is defined by .
If is a function and , then the “delta-derivative” of at the point is defined to be the number (provided it exists) with the property that for each there is a neighborhood of such that
Lemma 2.1 (see ). Let be continuously differentiable and suppose that is delta differentiable. Then is delta differentiable and the formula holds.
3. Multivariable Calculus on Time Scales
In this section, we generalize existing ideas of the time scales calculus to the multivariate case.
Firstly, consider the product , where is a time scale for all , so is a time scale too. Then for any with , in which , , define the following: (i)the forward jump operator by where represents the forward jump operator of for all . Hereafter, the forward jump operator of will be denoted by ;(ii)the back jump operator by where represents the backward jump operator of for all . The backward jump operator of will be denoted by ;(iii)the graininess function by where represents the graininess function of on the time scale for all . Again, from this point on the graininess function of the time scale for will be denoted by ;(iv).
Secondly, we introduce the definitions of time-scale derivatives for the function , where and . The forward jump operator and are defined by and . The graininess is defined as . The graininess is defined as . We will use the notation and .
Because we will need notation for partial derivatives with respect to time-scale variables and , we employ lexigraphic ordering for consistency. Let denote the time-scale partial derivative with respect to , in order to distinguish for the sign of -derivative in Section 2, let denote the time-scale partial derivative with respect to .
The following definitions of these partial derivatives are now given.
Definition 3.1. Let be a
real-valued function on . At , we say that has a “ partial
derivative” , with respect to , if for each , there exists a neighborhood (open in the
relative topology of of ) such that
for all . Similarly, we say that has a “ partial
derivative” denoted by , with respect to , if for each there exists a
neighborhood of such that
for all .
Finally, we define the partial derivative of at with respect to the variable , where and . Having defined the multivariable calculus as earlier, we set
Definition 3.2. Let be a real-valued function and let . Then define to be the number (provided it exists) with the property that given any , there exists a neighborhood of , with for such that for all . is called the partial delta derivative of at with respect to variable .
4. Existence, Uniqueness, and Comparison Result
In this section, we consider the initial value problem (IVP) for partial dynamic equation of first order on time scales where , , , and .
We define the set , .
Now, we begin to prove the following comparison result.
Theorem 4.1. Assume that
(A0), and for , where .(A1) is
quasimonotone nonincreasing in for each and , .(A2) whenever for some .
Then on .
Proof. Let us prove the theorem for strict inequalities firstly. For example, we suppose that and on , and prove that on . If this conclusion is not true, then consider the set Let be the projection of on the -axis and let . Clearly, and there exists an such that It then follows that and if , we also have In this case, we get the following contradiction: If, on the other hand, for some , and , and , then we have , and , then we have . Hence using the assumption (A1), we obtain and . Consequently, we get the inequality which is a contradiction as before. This proves that on . If one of the inequalities in (A0) is not strict, and on one hand, if , we set and note that . Then using (A0) and (A2), we have and on . On the other hand, if , we set and note that . Then using (A0), (A2), and Lemma 2.1, we have and on . Thus the foregoing arguments imply that on . Taking limit as , we then get on , and the proof is complete.
Theorem 4.2. Assume that (A1) and (A2) hold. Suppose further that (A3) for each , there exists a unique solution of
on , is continuously differentiable with respect to . Assume (A4) for each and , there exists a unique solution of
on , where is the unique
solution of (4.8) and is continuously
differentiable with respect to .
Then there exists a unique solution for the problem (4.1) on .
Proof. By (A3) and (A4), are unique
solutions of (4.8) and (4.9), respectively, on . Choose and note that
if , then because of uniqueness, . Hence, is a unique
solution of (4.9). Then we have
that is, satisfies (4.1)
and consequently, is a solution
of the problem (4.1).
To show uniqueness of solution of (4.1), suppose that are two solutions of (4.1) on . Then setting , and applying Theorem 4.1, we get on . Similarly we can prove that on . Hence the proof is complete.
5. Monotone Iterative Technique
we are now in a position to describe the monotone iterative technique which yields monotone sequences on time scales. We prove the following result specifically.
Theorem 5.1. Assume that (A0), (A1), and (A3) hold with on . Suppose further (A5) for some , whenever on . Assume (A6) for each and , there exists a unique solution of
Then there exist monotone sequences , and the functions , such that if is any solution of (4.1), then
Proof. Consider the linear IVP
where and is such that on .
By (A5), we have so it follows that . Similarly, we obtain that . Hence (A0) holds. Also, if , and therefore (A2) is satisfied for . Furthermore, by (A6), (A4) is satisfied relative to As a result, by Theorem 4.2, there exists a unique solution of (5.3) on for every such that on .
Defining a mapping by , where is the unique solution of (5.3) corresponding to . Concerning this mapping , we will show that (i) , , and (ii) is monotone on the sector , namely, if , then .
Let and let , where is the unique solution of (5.3). Then we have , and on . By Theorem 4.1, we see that . Similarly, we can show that .
To prove (ii), let be such that and let , , where , are the unique solutions of (5.3) corresponding to , , respectively. Then, and . Also, . Hence by Theorem 4.1, we have on . This proves .
Consider the sequence and note that on , where is the unique solution of (5.6) such that . Thus and . Since is monotone sequence, it is easy to conclude that converges uniformly and monotonically as . Suppose that on . Then it is clear that , . Consequently, we can now define on . Similar arguments hold relative to the sequence and one defines on .
Finally, we show that on , where is any solution of (4.1) such that on . For this, it is enough to show that on and this we do by induction. Suppose that for some on . Then we have By Theorem 4.1, we have on . We can show similarly that . Hence it follows that on for all , which prove the claim.
Remark 5.2. We note that if (A4) holds, then and are actually solutions of (4.1).
This work is supported by the Key Project of Chinese Ministry of Education (207014) and the Natural Science Foundation of Hebei Province of China (A2006000941).
- M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.
- V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, vol. 370 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.
- C. D. Ahlbrandt and C. Morian, “Partial differential equations on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 35–55, 2002.
- B. Jackson, “Partial dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 186, no. 2, pp. 391–415, 2006.
- V. Lakshmikantham, M. N. Oguztöreli, and A. S. Vatsala, “Monotone iterative technique for partial differential equations of first order,” Journal of Mathematical Analysis and Applications, vol. 102, no. 2, pp. 393–398, 1984.
Copyright © 2008 Peiguang Wang and Ping Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.