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 [1–15] 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 [17–23] 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 Letting , , then , for any , and is bounded on any closed subinterval of . Thus, condition (92) holds, but (91) does not hold; thus (93) does hold. The conclusion of Theorem 38 also does not hold, since attains its maximum in , but is not constant.

Example 43. Let , , , and . Then Let then These show that conditions (91) and (92) hold, but (93) does not hold on . The conclusion of Theorem 38 also does not hold on , since attains its maximum in , but is not constant.

Now, we establish a generalized maximum principle.

Theorem 44. Assume that the functions and satisfy (91), (92), and let and where . If at some point , then cannot attain its maximum at . Moreover, if , for each , then cannot attain its maximum in .

Proof. Assume that attains its nonnegative maximum at a point in and . If is left-scattered, then by Lemma 34, we have that and then This is a contradiction. If is left-dense, by Lemma 34 we have that and . Then and imply that . This is also a contradiction with Corollary 35. Thus, we have that cannot attain its maximum at . The proof is completed.

In Theorem 44, if we take , we have the following corollary which is an improvement for the result that appeared in [3].

Corollary 45. Let be functions, on and . If at some point , then cannot attain its maximum at . Moreover, if , for each , then cannot attain its maximum in .

In Theorem 44, 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 46. Assume that the functions , , and satisfy and on . Let . If for some then cannot attain its maximum at . If then cannot attain its maximum in .

Theorem 47. Assume that the functions and satisfy (91), (92), and (93), and let satisfy where and If attains a nonnegative maximum in , then .

Proof. Assume that attains its nonnegative maximum at a point in but does not identically equal . Thus, we can choose , such that , . If , we define a function by
Then It is similar to the proof of Theorem 38; we choose sufficiently larger such that holds on , where . Let us define a function by where is chosen so that Since , we have Moreover, the definition of yields that Finally, implies that . It implies that has a maximum in . However, holds on . This is 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 it is similar to the proof of Theorem 38; we have 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 38 we know that cannot attain its maximum in . Note that we get that is the maximum of on . This implies that ; however, we also have that This is a contradiction. The proof is completed.

Corollary 48. Assume that is not always equal to in Theorem 47; if attains its nonnegative maximum in , then the nonnegative maximum . Especially, if , , then for , unless .

In Theorem 47, if we take , we have the following corollary which is the result that appeared in [3].

Corollary 49. Assuming that the functions onin and on , if satisfies in , then cannot attain its maximum in , unless .

In Theorem 47, 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 50. Assume that the functions , , and satisfy and on . If satisfies then cannot attain its maximum in , unless .

All of the above results investigate the behavior of functions inside the considered interval. Now, we will discuss the behavior of functions by providing the information about the boundary points.

Theorem 51. Let , , and satisfy (91), (92), and . Assume that is not constant, such that for each , has unilateral derivative at points of , and (93), (130) hold on .(1)If attains its nonnegative maximum at a point of , then ;(2)If attains its nonnegative maximum at a point of , then .

Proof. We suppose that attains its nonnegative maximum at , that is, , and there exists a point , such that ; we define a function by where . It is similar to the proof of Theorem 38; we can choose a larger enough , such that Moreover, we define a function by where Thus and by using Theorem 47 to on , we get that attains its maximum at or . Note that , and thus attains its maximum at . Therefore, unilateral derivative of is not positive: However, and hence If , we can prove as the similar way above. The proof is completed.

In Theorem 51, if we take , we have the following corollary which is the result that appeared in [3].

Corollary 52. Assuming that the functions are bounded in and on , if satisfies in , and has unilateral derivative at points of .(1)If attains its nonnegative maximum at a point of , then ;(2)If attains its nonnegative maximum at a point of , then .

In Theorem 51, 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 53. Assume that the functions , , and satisfy and on , and assume that satisfies (1)If attains its nonnegative maximum at a point of , then ;(2)If attains its nonnegative maximum at a point of , then .

Next, we consider that , and may take positive value. Let , , and satisfy (91) and (92). Assume that we can find a function which satisfies Then there exists a function which is predifferentiable with region of differentiation such that and therefore, since .

We define a new function by and then satisfies

Lemma 54. Assume that (91), (92), and (93) hold, and a function satisfies (162). Then, the following inequalities hold in ;, , .
Moreover, if , , , and are bounded on , and there exist , such that then and are bounded on , where

Proof. (a) Since then If , then (b) and (c) are satisfied at , and so we suppose that .
(b) It is easy to see that Since , and hence then and, therefore,
(c) Finally, Since , and hence thus The boundness of and can be deduced by (166) and (173). The proof is completed.

From Theorem 47, Theorem 51, and Lemma 54 we obtain the following theorem.

Theorem 55. Assume that satisfies , and let , , , and be bounded on , such that (92) and (166) hold. Assume that there exists a function satisfying (162). If attains its nonnegative maximum in , then . If , then ; if , then .

In Theorem 55, if we take , we have the following corollary which is the result that appeared in [3].

Corollary 56. Assume that the functions are bounded in , and satisfies in . Assume that there exists a function such that If attains its nonnegative maximum in , then . If , then ; if , then .

In Theorem 55, 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 inequality.

Corollary 57. Assume that the functions , , , and satisfy , Let satisfy If attains its nonnegative maximum in , then . If , then ; if , then .

To show the value of Theorem 55, we need the following definition.

Definition 58. One says that is a change sign point of , if there exist and , such that has different sign on and , that is, either on and on or on and on .

Remark 59. Theorem 55 shows that a function which satisfies (151) cannot oscillate too rapidly. In fact, assuming that between two of its change sign points , , then must have a positive maximum between them. Hence, Theorem 55 will be violated. Thus, we have the following corollary.

Corollary 60. Assuming satisfies , then can have at most two change sign points (between which is negative) in any interval in which Theorem 55 holds.

By applying the same reasoning to both and , we can obtain the following corollary.

Corollary 61. If is a solution of equation , then can have at most one change sign point in any interval in which Theorem 55 holds.

Theorem 55 depends on the existence of the function , and now, we discuss the existence of the function .

Lemma 62. Assume that , , and satisfy the suppositions of Theorem 55, and there are positive numbers , such that the following properties hold.  .   .  .
Then there exists a function satisfying (162) and is bounded in .

Proof. We can choose and then Moreover, Since , Then ; we have and hence

Lemma 63. Let be a solution of equation where , , , and satisfy the conditions of Theorem 55. If is not identically zero and then cannot vanish in some right neighbourhood of .

Proof. If is right-scattered, then . Otherwise, we have that ; this shows that Then we can obtain . In fact, according to Theorem 55, cannot attain its maximum nor minimum at . If attains its maximum in , then since . If attains its maximum at , hence attains its maximum in . Next we apply Theorem 47 to and obtain that is constant; then since . Thus, in all cases we get that ; this implies that which is contradiction with the assumption.
If is right-dense, we obtain that cannot vanish in some right neighbourhood of . In fact, if it is not so, then there exists a sequence , and ; then . Again we obtain that by a similar proof of above, which is contradiction with the assumption. Thus, cannot vanish in some right neighbourhood of .

Remark 64. Under the conditions of Lemma 63, if has any change sign point at the right of , we denote the first one by and call it the conjugate change sign point of . Thus, does not change its sign in the interval . Without loss of the generality, we assume that Then function is positive in and is also a change sign point of . By the definition of change sign point, we have that . Hence, has a maximum in . Therefore by Theorem 55, cannot satisfy . That is, under these cases, there is no function satisfying the condition of Theorem 51.
On the other hand, if is any point in , a function can be found so that satisfies the maximum principle of Theorem 55. To see this, we observe first that is bounded from below by a positive number on any subinterval contained in . Consequently, for sufficiently small , the function is positive on . If is selected so that in , then is a function for which Theorem 55 holds. Thus, we get the following result.

Theorem 65. If is the conjugate change sign point of , letting , , and be bounded on , such that (92) and (166) hold, then there exists a such that Theorem 55 holds on the interval if and only if . If (the solution of (186) which satisfies ) has no change sign point at the right of , one sets , and Theorem 55 holds on every interval .

In Theorem 65, if we take , we have the following corollary which is the result that appeared in [3].

Corollary 66. Assume that is the conjugate change sign point of , and the functions are bounded in ; then there exists a function such that Corollary 52 holds on the interval if and only if . If (the solution of , which satisfies ) has no change sign point at the right of , one sets , and Corollary 52 holds on every interval .

In Theorem 65, 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 inequality.

Corollary 67. Assuming that the functions , and satisfy then there exists a function , , such that Corollary 53 holds on the interval if and only if . If (the solution of the equation which satisfies ) has no change sign point at the right of , one sets , and Corollary 53 holds on every interval .

4. Applications to Initial Value Problems

In this section, as an application of the maximum principles established in section three, firstly, we will prove some uniqueness theorem of the solution for initial value problem: in . Secondly, we will discuss the existence of the lower and upper solutions of (192). Thirdly, we will give a general scheme for obtaining upper and lower solutions.

Theorem 68. Assume that , , and satisfy (91) and (92), and , (93), (130) hold on . If and are solutions of the initial value problem (192), then .

Proof. We define a function by Since both and satisfy (192), the function satisfies According to Theorem 51, cannot attain its maximum nor minimum at . If attains its maximum at an interior point of since . If attains its maximum at , hence attains its maximum at an interior point of . Next we apply Theorem 47 to and obtain that is constant; then since . The proof is completed.

It follows from Theorem 65; we get Theorem 69.

Theorem 69. Let , , and be bounded on , such that (92) and (166) hold. Assuming that and are solutions of the initial value problem (192), if , where is the conjugate change sign point of , then .

More generally, we can prove the following theorem which shows that the conclusion of Theorem 69 holds on any interval .

Theorem 70. Let , , and be bounded on , such that (92) and (166) hold. Assuming that and are solutions of the initial value problem (192), then .

Proof. We define a function by Since both and satisfy (192), the function satisfies (194). We give our proof by two steps.
(1) If is right-scattered, it follows from (194) that , , and then we have that . On the other hand, this implies that Note that (166); we know that . This shows that . If is right-dense, by using Lemma 62, there is a enough small and a function on . Let ; then According to Theorem 55, cannot attain its maximum nor minimum at . If attains its maximum in , then since . If attains its maximum at , hence attains its maximum in . Next we apply Theorem 55 to and obtain that is constant; then since . Thus, in all cases we get that ; this implies that on . Thus, we can get that on . If is left-dense, by using the continuous , we have that . If is left-scattered, then , and is right-scattered; then similar to the above proof of , we have that . Synthesizing the above proof, we have proven that there exists (if is right-scattered, ), such that and on .
(2) Let If , then the conclusion of Theorem 70 will be proved. If , then and on . If is left-dense, by using the continuous , we have that and . If is left-scattered, then , and is right-scattered; then similar to the above proof of , we can prove that . By using the conclusion of step (1), we have that there exists such that and on . Then we get that and on . This is a contradiction with the definition of . Thus, ; this shows that on . The proof is completed.

Remark 71. Theorems 68 and 69 show that , at most, has one solution satisfying , . On the other hand, in many cases, it is difficult to find a solution of the initial value problem directly, and therefore, it becomes important to find a lower and upper solution.

Assume that , , and are bounded on , on and satisfy (91), (92), (93), and (130) for each . If we can find a function satisfying we define a function by where is the solution of (192). Thus, Since , has a nonnegative maximum at any interval , and using Theorem 47, we know that the maximum point must be or . However, , and from Theorem 51 maximum point cannot be unless constant. Thus, we obtain Since is arbitrary, we can deduce that Using to take the place of , inequality (204) implies and inequality (205) implies Since inequality (206) implies Similarly, assume that we can find a function satisfying The same as the above statement, define and we obtain Therefore, we have established the following theorem, which gives a sufficient condition for the lower and upper solutions.

Theorem 72. Assume that , , and are bounded on , on and satisfy (91), (92), (93), and (130) for each . Let be a solution of (192). Let and satisfy (200), (201) and (210), (211). Then , .

In the following, we will discuss the existence of the lower and upper solutions.

Theorem 73. Assume that , , and are bounded on , are bounded on , and satisfy (166), (92), (93), and (130).(1)If is continuous on , then there exist functions and which satisfy (200), (201) and (210), (211), respectively.(2)Moreover, if on , then , .

Proof. It follows from (166) that so we can select large enough, such that , where is defined by Let we show that, under the stated assumptions, the function satisfies (200) and (201). To see that (200) is satisfied, we note that To see that (201) is satisfied, we note that Similarly, we can choose where To see that (210) is satisfied, we note that To see that (211) is satisfied note that Thus, conclusion (1) holds. Conclusion (2) can be deduced from Theorem 72.
The proof is completed.

As we all know, the accuracy of the approximation will depend on how well we can choose the functions and . So we next search for the following general scheme for obtaining upper and lower bounds. Suppose we divide the interval into subintervals On each subinterval, we will select as the following form: and choose the coefficients so that , , and . Also, will be selected so that inequality (200) holds in each subinterval . We set The constants , and the number of subintervals will be chosen so that all the required conditions are satisfied. We proceed in a step by step manner starting with the interval . The initial conditions require that and . Next, we divide our proof into three parts.

If is right-scattered and is also right-scattered, we let , and then we only have one point in , and hence, in this point, the inequality becomes If and , and are bounded, then can be properly selected so that (229) is an equality. Thus, in this case, is a solution of (192) in .

If is right-scattered and is right-dense, we let , and then the inequality becomes If , we have that Thus, if , , , and are bounded, then can be selected so close to , and can be taken so large that (231) holds for . Moreover, when is sufficiently close to , we can properly select , such that (231) is close to an equality; then is also close to the solution of (192) in .

If is right-dense, the inequality becomes If , , and are bounded, then can be selected so close to that where is a positive constant. If, in addition, is bounded, then can be taken so large that (234) holds for all in . Moreover, when is sufficiently close to , we can properly select , such that (234) is close to an equality; then is also close to the solution of (192) in .

Following all of the above proof, we have proved that there exists an and a large enough , such that (200) holds for all in for

We now turn to the interval , with being defined by

To insure the continuity of , , and at , we choose In fact, by computing we get that Thus, are continuous at , and is left-dense continuous at . In the interval , we apply the same reasoning of to and get that there exists an and a large enough , such that (200) holds for all in .

Proceeding in this fashion, we determine each so that and are continuous everywhere; is left-dense continuous everywhere, and if is a left-dense point, we always take interval so small, such that the coefficient of satisfies: where is a positive constant. Also, we take the constant to be large enough, so that holds on . In fact, the quantities are determined by the recursion formulas In an actual computation to determine the , it is convenient to replace by its maximum in the th subinterval and to replace , , and by either their maximum or minimum, whichever may be appropriate for making throughout.

In a similar manner we may construct lower bounds. The constants , are selected in exactly the same way, and the quantities are taken so large that holds everywhere.

If , , , and are continuous, by the above process, it can be shown that, as the maximum length of the subintervals, the upper and lower bounds both tend to the solution . The above discussion leads to the following theorem.

Theorem 74. Assume that , , and are bounded and continuous on and satisfy (166), (92), (93), and (130). If is continuous on and on , then there exist the upper solution sequence and lower solution ; they both tend to the solution of (192).

Thus far in this section, we have assumed that . We now take up the problem of approximating the solution of the equation with initial conditions when the function may be positive. Under these circumstances we employ the generalized maximum principle (Theorem 51). To do so, we suppose that there is a function which is positive on and which has property that For example, we can take the function defined in Lemma 62.

We saw in Section 3 that satisfies an equation of the form with , , . Now, we define the comparison functions and , so that and provide the bounds for . First, we take and , such that the inequalities hold. Then, at , Moreover, it is easily seen by computing that Hence, if the conditions of Lemma 54 hold, by using Lemma 54 and Theorem 72, we know that, for , The first of these sets of inequalities gives the bounds

The second set yields Since is positive on , we find If , we may substitute the upper bound of as given in (251) into the left side of (253) and we may substitute the lower bound of into the right side of (253). If , we use the lower bound of on the left and the upper bound of on the right. We thus find that Inequalities (251) and (254) give the bounds for and which are more precise when and are smaller.

It is always possible to find a positive function which satisfies on a sufficiently small interval, but in general, there is no such function if the interval is too large. Once more we resort to breaking up the interval and piecing together functions defined on subintervals. Let and on an interval , and let be another positive function which satisfies on an interval . We wish to find bounds for the solution of the initial value problem (192), on the whole interval .

Let and satisfy the conditions on the interval , and Then From these, we get the bounds for and . In addition, if a function which satisfies on an interval is given, we can then find bounds for and , as before. Let the functions and be defined on and they satisfy Then we find, as we did previously, that While we do not know and at , but we know their bounds, therefore, we can give explicit conditions on the values of , , , and at which assure that the above inequalities are satisfied.

In fact, since at we should have that if , then we can give the conditions on the values of , , , and at which assure that inequality (258) holds as follows: If , we replace by in the coefficient of in the first row of the inequalities and replace by in the second row. If these conditions are satisfied, we have the bounds

We now consider as an extension of to the interval , as an extension to the interval , and as an extension of to the interval . Then these extended functions are, in general, discontinuous at . However, the above inequalities relating ,  ,  ,  ,  , and at establish the relation between the right and left limits at discontinuous points. It may, of course, be necessary or desirable to divide the interval into more than two subintervals.

The above discussion leads to the following theorem.

Theorem 75. Let , , and be piecewise continuous functions on the interval . Moreover, we assume that , and , are piecewise continuous on the interval , and are piecewise left-dense continuous on the interval , and , , and satisfy the conditions of Lemma 54. If the following properties hold:(a) on ;(b);(c);(d), hold at all points where the derivatives occurring in these formulas are continuous;(e)at each point of discontinuity the functions , , and have nonnegative jumps, the jump of is at least times as many as jump of , and the jump of is at most times as many as jump of ,then

5. Applications to Boundary Value Problems

In this section, by using the maximum principles proved in Section 3 to some general boundary value problems, the uniqueness of the solutions, the existence of the upper and lower solutions, and some necessary and sufficient conditions for the existence of the approximation solutions are discussed. First, we consider the following boundary value problems:

Theorem 76. Assume that , , and are bounded on such that (91), (92), (93), and (130) hold and at each . If are solutions of (264) and satisfy the boundary value problem (265), then .

Proof. We define a function by Since both and satisfy (264) and (265), the function satisfies It follows from Theorem 47 that , for each . Since satisfies the same boundary value problem, we have , for each , and thus , for each .

Next we study general boundary value problems of the form where are all constant, and . If , (269) becomes (265).

Theorem 77. Assume that , , and are bounded on such that (91), (92), (93), and (130) hold. If and are solutions of (268) and (269), then , except that is a constant when , .

Proof. We define a function by Since both and satisfy (268) and (269), the function satisfies It is clear that satisfies all the above conditions, if and only if . Then we assume first that at some point and is not constant. Using Theorem 47 we know that attains its maximum at or . Suppose that , and by using Theorem 51 we get , which do not satisfy (272). Suppose that , and by using Theorem 51 we get , which do not satisfy (273). Thus, we obtain . We can also prove that , and then , for each .

Similar to the initial value problems, in most cases it is impossible to find such a solution explicitly. But, it is frequently desirable to approximate a solution in such a way that an explicit bound for the error is known. Such an approximation is equivalent to the determination of both upper and lower bounds for the values of the solution. Thus, in the following, we will discuss the existence of the upper and lower solutions for boundary value problems.

We will assume that the functions , , , , are bounded and in . Under these circumstances it is possible to use the maximum principle in Theorem 55 to obtain a bound for a solution without any actual knowledge of itself.

Suppose we can find a function satisfying Then the function satisfies The maximum principles as given in Theorem 47 in Section 3 may be applied to , and we conclude that on . That is, The function is an upper bound for .

Similarly, a lower bound for may be obtained by finding a function with the properties By using the maximum principle (Theorem 51) to , we can get that

Functions , with the desired properties are easily constructed. For example, we may set with and try to select and so that (274) and (275) are satisfied. In fact, if (166) holds, we can choose to be so large that for . Let for , and For the selections of and as just described, the function satisfies (274) and (275), where .

To determine a lower bound, we choose with , and try to select and such that (279) and (280) are satisfied. In fact, we choose to be so large that (283) holds. Let With the selections of and as just described, the function satisfies (279) and (280). Then In particular, we have

If is a solution of (264) and (265) and is a solution of the related problem then the difference satisfies Inequality (289) shows that Therefore, if the quantities are all small, then is small for all in the interval . Under these circumstances, we say that the solution of the problem (264), (265) depends continuously on and the boundary values ,.

Combining the above discussions, we get the following result.

Theorem 78. If , , and are bounded on such that (92), (93), (130), and (166) hold and in , then the following conclusions hold:(1)there exist functions satisfying (274), (275) and (279), (280), respectively;(2)the solution of the problem (264), (265) satisfies , ;(3)the solution of the problem (264), (265) depends continuously on and the boundary values , .

Next, we consider the question of approximations of solutions of (268) and (269).

If we can find a function satisfying we define a function by Thus, It follows from Theorem 51, (297) we know that cannot attain its maximum at or . If at some point , then by using Theorem 47, we have that is positive constant which implies that and . Otherwise, if , , are not all hold, then we have that , that is . Similarly, we assume that we can find a function satisfying: The same as the above statement, we define and we obtain Therefore, we establish an approximation theorem as in the following.

Theorem 79. Assume that , , and are bounded on and satisfy (91), (92), (93), and (130) at each . If is a solution of (268) and (269), where , , . Let and satisfy (294), (295), (298), and (299). If , , are not all hold, then .

Now, we consider deleting the conditions ,, in Theorem 79. Without loss of generality, we can assume that , .

In order to use the generalized maximum principle established in Section 3, we suppose that there is a positive function which satisfies the inequalities We set where is a solution of (268) and (269). Then must satisfy with , , , and We can rewrite (306) to be where By (303) we may choose and in the range , .

Suppose that there is a function , which is positive on and satisfies conditions (302) and (303). If and satisfy (294), (295) and (298), (299), respectively, then the functions satisfy the analogous conditions with respect to (305) with boundary (307); that is, the following inequalities hold: Hence, by Theorem 79, we have the inequalities unless and .

If satisfies (302) and (303) with equality rather than inequality, we may add a multiple of to a solution of (268) and (269) to obtain another solution. That is, the solution is not unique. Of course, there may be no solution at all, but if there is at least one, then there are many. Therefore, if there is a positive function that satisfies (302) and (303) but such that not all the inequalities are equations, we obtain the bounds as before.

If inequality (312) holds for the solution of (268) and (269), then, particularly, the solution of which satisfies the boundary conditions must be nonnegative.

In fact, if we select , then with respect to (313) and (314), satisfies (298) and (299) for , . Then we have that , that is, . Moreover, if in , then is a maximum of in ; by using Lemma 34 to we obtain If is left-scattered, since we have and hence by (166), (92), and (315), we obtain (if , we can also have that ). If is left-dense, by Lemma 34, we get that So we have proved that if . The uniqueness theorem for the initial value problem implies that , which violates the boundary conditions (314). So cannot vanish in . If vanishes at endpoint, say at , then the first condition in (314) becomes , which is a contradiction. Therefore and, similarly, . Hence on .

Thus far, under the hypothesis that the problem (313), (314) has a solution, we have established the following result.

Theorem 80. Let be a solution of (268), (269) with . Let and satisfy (294), (295) and (298), (299), respectively; , , and are bounded on and satisfy (91), (92). If problem (313), (314) has a solution, then the bounds hold in if and only if there exists a positive function on which satisfies inequalities (302) and (303) in such a way that not all the inequalities in (302) and (303) are equalities.

Remark 81. Moreover, if , , and , then the function satisfies conditions (302) and (303); thus Theorem 79 can be deduced from Theorem 80.

The function does not appear in the inequality (320). Therefore, it is of interest to obtain a theorem which eliminates entirely and which provides conditions on and guaranteeing that they form the upper and lower bounds of the solution of (268), (269). The next result gives a necessary and sufficient condition for this case.

Theorem 82. Let be a solution of (268), (269) with . Suppose and satisfy (294), (295) and (298), (299), respectively, and in all these inequalities, at least there is one which is a strict inequality. Let be bounded and satisfy , . Then the bounds hold if and only if for .

Proof. If (321) holds, then it is clear that . We now assume that is nonnegative, and we must show that (321) holds. If is strictly positive on , then we may select as the function of Theorem 80. All the requirements are fulfilled and Theorem 80 implies that (321) holds. Therefore, we need only to investigate the possibility that has a zero point on .
According to (294), (295), (298), and (299), satisfies the inequalities and, in view of the hypotheses, at least there is one which is a strict inequality in (323), (324).
First suppose , . Then has a minimum at , which shows that has a maximum at . So by Lemma 34 we obtain If is left-scattered, since we have and hence Therefore If is left-dense by Lemma 34, we have that And we conclude from Theorem 68 that . Then equality holds in all the conditions (323), (324) contrary to our hypotheses.
The only remaining possibility is that in but that at an endpoint, say at . Then according to Theorem 55, . But then the first inequality in (324) is violated unless , which is a contradiction with . Similarly, if vanishes at , then , which is a contradiction with . Therefore, we have proved that is positive on , and it can be used as an auxiliary function in Theorem 80. The proof is completed.

6. Applications to Nonlinear Operator

In this section, we discuss nonlinear equations. We can extend the results of linear operator in Sections 4 and 5 to nonlinear operator.

Suppose that is a solution of Assume that are all continuous functions in their domain and satisfy where is defined on . Besides, we suppose that for each , or equivalently

Remark 83. Note that if then (331) is the linear equation considered in the previous sections.

Assume that satisfies We define a function by Subtracting (331) from (337), we derive that Applying mean value theorem to , we obtain where , , and are calculated at the point of , , , and . Obviously, the function satisfies a linear equation. We can use Theorems 47 and 72 to get the following results.

Theorem 84. Assume that , satisfy where , ,  ,  , are all continuous functions such that (333) and (335) hold and are bounded on . If attains its maximum in , then .

Theorem 85. Assume that is a solution of where , , , and are all continuous functions such that (333) and (335) hold and are bounded on . If a function satisfies and a function satisfies then we have

Remark 86. Theorem 85 implies that the solution of (343), (344) is unique. In fact, if and are solutions of (343), (344), we just need to let , and then we can obtain .

By using Theorem 79, we have the following result.

Theorem 87. Assume that is a solution of where , , and are not all equal to 0, , , , and are all continuous functions such that (333) and (335) hold, and are bounded on . If a function satisfies and a function satisfies then we have

Remark 88. Theorem 87 implies that the solution of (349), (350) is unique. In fact, if and are solutions of (349), (350), we just need to let , and then we obtain .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the paper and the valuable comments and suggestions, which greatly improved it. This work was supported by the National Natural Science Foundation of China (no. 11171286) and by Jiangsu Province Colleges and Universities Graduate Scientific Research Innovative Program (no. CXZZ12-0974).