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.

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 [

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

A time scale

Let

If

If

Let

Let

Assume that

If

If

If

exists as a finite number. In this case

If

Assume that

If

If

If

exists as a finite number. In this case

If

If

If

Assume

Assume

If

One says that a function

One defines

Suppose (

Suppose (

If

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

Suppose (

(a) Since

One defines

Suppose (

Suppose (

If

One defines the set

If

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

Suppose (

(a) It is easy to see that

(b) Obviously,

(c) We have

(d) Obviously,

Let

One says that a function

if

if

Similarly, we say that

Suppose

We only show

We say that a function

if

if

Similarly, we say that

Suppose

Suppose that

Let

Since

Let

Suppose the function

If

In this paper, we denote

First we give a necessary condition that

If

Let us divide our proof into three parts.

(i) If

(ii) If

(iii) If

According to Lemma

Assuming that

We give a variant of Corollary

Let

We suppose that the result is false. Then there are

However,

Then we have that

However,

As a natural extension of the above simple maximum principle, we consider the operator of the following type:

Assume that the functions

We suppose that

(i) If

However, it follows from Lemma

(ii) If

Therefore,

Next, we weaken the condition

Assume that the functions

Assume that

In Theorem

Assuming that the function

In Theorem

Assume that the functions

To show that conditions (

Let

Let

Then

Let

Now, we establish a generalized maximum principle.

Assume that the functions

Assume that

In Theorem

Let

In Theorem

Assume that the functions

Assume that the functions

Assume that

Then

Assume that

In Theorem

Assuming that the functions

In Theorem

Assume that the functions

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.

Let

If

If

We suppose that

In Theorem

Assuming that the functions

If

If

In Theorem

Assume that the functions

If

If

Next, we consider that

We define a new function

Assume that (

Moreover, if

(a) Since

(b) It is easy to see that

(c) Finally,

From Theorem

Assume that

In Theorem

Assume that the functions

In Theorem

Assume that the functions

To show the value of Theorem

One says that

Theorem

Assuming

By applying the same reasoning to both

If

Theorem

Assume that

Then there exists a function

We can choose

Let

If

If

Under the conditions of Lemma