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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 165429, 28 pages

http://dx.doi.org/10.1155/2014/165429
Research Article

Delta-Nabla Type Maximum Principles for Second-Order Dynamic Equations on Time Scales and Applications

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received 6 February 2014; Accepted 6 April 2014; Published 11 May 2014

Academic Editor: Dragoş-Pătru Covei

Copyright © 2014 Jiang Zhu and Dongmei Liu. 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.

Abstract

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

1. Introduction

Maximum principles are a well known tool for studying differential equations, which can be used to receive prior information about solutions of differential inequalities and to obtain lower and upper solutions of differential equations and so on. Maximum principles include continuous maximum principles and discrete maximum principles. It is well known that there are many results and applications for continuous and discrete maximum principles. For example, about these theories and applications, we can refer to [115] and the references therein. On the other hand, Hilger [16] established the theory of time scales calculus to unify the continuous and discrete calculus in 1990. After that, ordinary dynamic equations and partial dynamic equations on time scales have been extensively studied by some authors. For example, about these, we can refer to [1723] and the references therein. However, the study on the maximum principles on time scales is very little, about these, we can refer to Stehik and Thompson’s recent works [24, 25].

Inspired by the above works, we will be devoted to study delta-nabla type maximum principles for second-order dynamic equations on one-dimensional time scales and the applications of these maximum principles.

This paper is organized as follows. In Section 2, we state and prove some basic notations and results on time scales. In Section 3, we will first prove some delta-nabla type maximum principles for second-order dynamic equations on time scales; then, by using these maximum principles, we get some maximum principles for second-order mixed forward and backward difference dynamic system and discuss the oscillation of second-order mixed delta-nabla differential equations. In Section 4, we apply the maximum principles proved in Section 3 to obtain uniqueness of the solutions, the approximating techniques of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear initial value problems. In Section 5, we apply the maximum principles proved in Section 3 to obtain uniqueness of the solutions, the approximating techniques of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear boundary value problems. Finally, in Section 6, we extended the results of linear operator established in Sections 4 and 5 to nonlinear operators.

2. Preliminaries

Definition 1 (see [22]). A time scale is a nonempty closed subset of the real numbers. Throughout this paper, denotes a time scale.

Definition 2 (see [22]). Let be a time scale. For one defines the forward jump operator by , while the backward jump operator is defined by . If , one says that is right-scattered, while if , we say that is left-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if and , then is called right-dense, and if and , then is called left-dense. Finally, the graininess function is defined by

Definition 3 (see [22]). If has a left-scattered maximum , then one defines ; otherwise . Assume is a function and let . Then one defines to be the number (provided it exists) with the property that, given any , there is a neighborhood of (i.e., for some ) such that We call the delta derivative of at .

Definition 4 (see [22]). If has a right-scattered minimum , then one defines ; otherwise . The backwards graininess function is defined by Assume is a function and let . Then we define to be the number (provided it exists) with the property that, given any , there is a neighborhood of (i.e., for some ) such that We call the nabla derivative of at . Define the second derivative by .

Definition 5 (see [21]). Let . Define and denote as right-dense continuous if for each

Definition 6 (see[21]). Let . Define and denote as left-dense continuous if for each

Theorem 7 (see [21]). Assume that and let .(i)If is differentiable at then is continuous at .(ii)If is continuous at and is right-scattered then is differentiable at with (iii)If is right-dense, then is differentiable at if and only if the limit exists as a finite number. In this case (iv)If is differentiable at , then

Theorem 8 (see [22]). Assume that and let .(i)If is nabla differentiable at then is continuous at .(ii)If is continuous at and is left-scattered then is nabla differentiable at with (iii)If is left-dense, then is nabla differentiable at if and only if the limit exists as a finite number. In this case (iv)If is nabla differentiable at , then

Theorem 9 (see [22]). If is differentiable and is right-dense continuous on , then is differentiable, and If is differentiable and is left-dense continuous on , then is differentiable, and where

Corollary 10 (see [22]). If is differentiable and is continuous on , is differentiable, and is continuous on , then

Theorem 11 (see [21]). Assume are differentiable at . Then

the sum is differentiable at with

for any constant : is differentiable at with

the product is differentiable at with

if , then is differentiable at with

Theorem 12 (see [22]). Assume are nabla differentiable at . Then

the sum : is nabla differentiable at with

for any constant : is nabla differentiable at with

   the product is nabla differentiable at with

if , then is nabla differentiable at with

Theorem 13 (see [22]). If , , and are continuous, then ; ; ; .

Definition 14 (see [21]). One says that a function is regressive provided for all holds. The set of all regressive and rd-continuous functions will be denoted by .

Definition 15 (see [21]). One defines , where . If , then one defines the exponential function by If , then the first-order linear dynamic equation is called regressive.

Theorem 16 (see [21]). Suppose (28) is regressive and fix . Then is a solution of the initial value problem on .

Theorem 17 (see [21]). Suppose (28) is regressive; then the only solution of (29) is given by .

Theorem 18 (see [21]). If , then and ; ; ; .

According to the above theorems and definitions, we can obtain the following corollary.

Corollary 19. Suppose (28) is regressive and fix , and if one chooses , where is a positive constant, then the following equality holds on . . .

Proof. (a) Since we have and thus (b) Obviously,

Definition 20 (see [22]). One defines , where . If , then one defines the exponential function by If , then the first-order linear dynamic equation is called regressive.

Theorem 21 (see [22]). Suppose (35) is regressive and fix . Then is a solution of the initial value problem on .

Theorem 22 (see [22]). Suppose (35) is regressive; then the only solution of (36) is given by .

Theorem 23 (see [22]). If , then(i) and ;(ii) ; (iii) ; (iv) ; (v) .

Definition 24 (see [22]). One defines the set of all positively regressive elements of by

Corollary 25 (see [22]). If and , then .

According to the above theorems and definitions, we can obtain the following corollary.

Corollary 26. Suppose (35) is regressive and fix , and if one chooses , where is a negative constant, then the following equality holds on . . . . .

Proof. (a) It is easy to see that and we have which can obtain and thus

(b) Obviously, and we have which can obtain and therefore, we get

(c) We have and then

(d) Obviously, And hence, we get

Theorem 27 (see [22]). Let be a continuous function on , that is, delta differentiable on . Then is increasing, decreasing, nondecreasing, and nonincreasing on if for all , respectively.

Definition 28. One says that a function is left-increasing at provided(i)if is left-scattered, then ;(ii)if is left-dense, then there is a neighbourhood of such that for all with .

Similarly, we say that is left-decreasing if in the above (i) and in (ii) .

Theorem 29. Suppose is nabla differentiable at . If , then is left-increasing. If , then is left-decreasing.

Proof. We only show as the second statement can be shown similarly. If is left-scattered, then and hence ; that is, is left-increasing. Let now be left-dense. Then and therefore for there is a neighbourhood of such that for all with . Hence Therefore, for all with . Combining what we have proved, we can get that if , then is left-increasing.

Definition 30. We say that a function assumes its local left-minimum at provided(i)if is left-scattered, then ;(ii)if is left-dense, then there is a neighbourhood of such that for all with .

Similarly, we say that assumes its local left-maximum if in the above (i) and in (ii) .

Theorem 31. Suppose is nabla differentiable at . If attains its local left-minimum at , then . If attains its local left-maximum at , then .

Proof. Suppose that attains its local left-minimum at . To show that , we assume the opposite, that is, . Then is left-increasing by Theorem 29, contrary to the assumption that attains its local left-minimum at . Thus, we must have . The second statement can be shown similarly.

Theorem 32. Let be a continuous function on , that is, nabla differentiable on (the differentiability at is understood as left-sided) and satisfies . Then, there exists such that

Proof. Since is continuous function on , attains its minimum and its maximum . Therefore, here exists such that . Since , we may assume that . Clearly attains its local left-minimum at and its local left-maximum at . Then, by Theorem 31 we have and .

Theorem 33. Let be a continuous function on , that is, nabla differentiable on . Then is increasing, decreasing, nondecreasing, and nonincreasing on if , , , for all , respectively.

Proof. Suppose the function defined on by Clearly is continuous on and nabla differentiable on . Also , and so for some by Theorem 32. Hence, taking into account that then we have for some .

If , , , for all , then , , , , respectively. Considering the arbitrary of , we arrive at the statement of the theorem.

3. Delta-Nabla Type Maximum Principles

In this paper, we denote as an interval on time scales. We study those functions defined on which belong to , where is the set of all functions , such that is continuous on , is continuous on , and exists in .

First we give a necessary condition that attains its maximum at some point .

Lemma 34. If attains a maximum at a point , then The strict inequality in the last two inequalities can occur only at left-scattered points.

Proof. Let us divide our proof into three parts.

(i) If is left-scattered, then the maximality of at implies that and and consequently

(ii) If is left-dense and right-scattered, in this case, we have . If there is no positive number sequence such that and , then there exists a such that for each ; by Theorem 27, a contraction with attains its maximum at interior point of . Thus, there exists such that and . This yields Furthermore, the continuity of the delta derivative implies that and consequently . Then by using Corollary 10 we have that

(iii) If is left-dense and right-dense, in this case the maximality of at and standard continuous necessary conditions imply that

According to Lemma 34, we can obtain the first simple maximum principle for the time scale.

Corollary 35. Assuming that , if at some point , then cannot attain its maximum at . Moreover, if in , then cannot attain its maximum in .

We give a variant of Corollary 35 where we weaken the condition .

Theorem 36. Let . If in , then cannot attain its maximum in , unless .

Proof. We suppose that the result is false. Then there are , such that and . Let us assume first that and let us define a function by where and is an exponential function on (see Section 2), and then Considering and the positivity of , we obtain Let us define a function by where is chosen so that Since , we have Furthermore, the definition of yields that Finally, derives It shows that attains its maximum in .

However, which contradicts the statement of Corollary 35. If ,

Then we have that Let us define a function by where is chosen so that Since , we have Furthermore, the definition of yields that Finally, since , we derive It shows that attains its maximum in .

However, which is a contradiction with Corollary 35. The proof is completed.

As a natural extension of the above simple maximum principle, we consider the operator of the following type: By the above results, we can obtain Theorem 37.

Theorem 37. Assume that the functions satisfy Letting , if at some point , then cannot attain its maximum at . Moreover, if , for each , then cannot attain its maximum in .

Proof. We suppose that at some point and attains its maximum at a point . We divide our proof into two parts.

(i) If is left-scattered, in this case, we have Multiplying by , we obtain

However, it follows from Lemma 34 and the conditions that , which is a contradiction.

(ii) If is left-dense, then by Lemma 34 we know that

Therefore, reduces to which is a contradiction with Lemma 34. Combining the proof of (i) and (ii), we get that cannot attain its maximum at . The proof is completed.

Next, we weaken the condition to

Theorem 38. Assume that the functions satisfy Let satisfy , for each . Then cannot attain its maximum in , unless .

Proof. Assume that attains its maximum at a point in but does not identically equal . That is, , and there exists such that . Let us assume first that and let us define a function by Therefore, we have Thus, by (93) we can take arbitrary , such that in , where if and if . Then we have in . Let us define a function by where is chosen so that If , since , we have that and Moreover, the definition of yields that Finally, implies that . It follows that has a maximum in . However, which is a contradiction with Theorem 37. If , then we have . It follows that has a maximum in . This is again a contradiction with Theorem 37. Thus, we have proved that if is a maximum point, then for any . Let From this, we obtain that and . Then we have that and for any . If is left-scattered, then Since , we multiply by and get that This is a contradiction. If is left-dense, let where , such that in . We choose closely enough to , such that , on , and where such that Therefore, we have Thus, on . By Theorem 37 we know that cannot attain its maximum in . Note that We get that is the maximum of on . Since for any and is increasing for , we have that ; however, we also have that This is a contradiction. The proof is completed.

In Theorem 38, if we take , we have the following corollary which is the result that appeared in [3].

Corollary 39. Assuming that the function is bounded on any closed subinterval of , if satisfies in , then cannot attain its maximum in , unless .

In Theorem 38, if we take , where is the set of all integral numbers, we can obtain the following new maximum principle for second-order mixed and difference dynamic system.

Corollary 40. Assume that the functions and satisfy and let ; if then cannot attain its maximum in , unless .

To show that conditions (91), (92), and (93) are necessary for the validity of our results, we give the following examples.

Example 41. Let , where is the set of all integral numbers and , and is defined by Then Letting , , , then

, for any , and is bounded on any closed subinterval of . Thus, conditions (91) and (93) hold, but (92) does not hold. The conclusion of Theorem 38 also does not hold, since attains its maximum in , but is not constant.

Example 42. Let , where is the set of all integral numbers and , and is defined by

Then