Monotone Iterative Technique for First-Order Nonlinear Periodic Boundary Value Problems on Time Scales

Abstract

We investigate the following nonlinear first-order periodic boundary value problem on time scales: , , . Some new existence criteria of positive solutions are established by using the monotone iterative technique.

1. Introduction

Recently, periodic boundary value problems (PBVPs for short) for dynamic equations on time scales have been studied by several authors by using the method of lower and upper solutions, fixed point theorems, and the theory of fixed point index. We refer the reader to [1–10] for some recent results.

In this paper we are interested in the existence of positive solutions for the following first-order PBVP on time scales:

(1.1)

where will be defined in Section 2, is a time scale, is fixed and . For each interval of we denote by By applying the monotone iterative technique, we obtain not only the existence of positive solution for the PBVP (1.1), but also give an iterative scheme, which approximates the solution. It is worth mentioning that the initial term of our iterative scheme is a constant function, which implies that the iterative scheme is significant and feasible. For abstract monotone iterative technique, see [11] and the references therein.

2. Some Results on Time Scales

Let us recall some basic definitions and relevant results of calculus on time scales [12–15].

Definition 2.1.

For we define the forward jump operator by

(2.1)

while the backward jump operator is defined by

(2.2)

In this definition we put and where denotes the empty set If we say that is right scattered, while if we say that is left scattered. Also, if and then is called right dense, and if and then is called left dense. We also need below the set which is derived from the time scale as follows. If has a left-scattered maximum then Otherwise,

Definition 2.2.

Assume that is a function and let Then is called differentiable at if there exists a such that, for any given there is an open neighborhood of such that

(2.3)

In this case, is called the delta derivative of at and we denote it by If then we define the integral by

(2.4)

Definition 2.3.

A function is called rd-continuous provided that it is continuous at right-dense points in and its left-sided limits exist at left-dense points in The set of rd-continuous functions will be denoted by

Lemma 2.4 (see [13]).

If and then

(2.5)

where is the graininess function.

Lemma 2.5 (see [13]).

If then is increasing.

Definition 2.6.

For we define the Hilger complex numbers as

(2.6)

and for let

Definition 2.7.

For let be the strip

(2.7)

and for let

Definition 2.8.

For we define the cylinder transformation by

(2.8)

where is the principal logarithm function. For we define for all

Definition 2.9.

A function is regressive provided that

(2.9)

The set of all regressive and rd-continuous functions will be denoted by

Definition 2.10.

We define the set of all positively regressive elements of by

(2.10)

Definition 2.11.

If , then the generalized exponential function is given by

(2.11)

where the cylinder transformation is defined as in Definition 2.8.

Lemma 2.12 (see [13]).

If , then

• (i) ,

• (ii) ,

• (iii) ,

• (iv) for and

Lemma 2.13 (see [13]).

If and , then

(2.12)

3. Main Results

For the forthcoming analysis, we assume that the following two conditions are satisfied.

(H1) is rd-continuous, which implies that .

(H2) is continuous and is nondecreasing on .

If we denote that

(3.1)

then we may claim that , which implies that

In fact, in view of (H1) and Lemmas 2.12 and 2.13, we have

(3.2)

which together with Lemma 2.5 shows that is increasing on And so,

(3.3)

This indicates that .

Let

(3.4)

be equipped with the norm Then is a Banach space.

First, we define two cones and in as follows:

(3.5)

and then we define an operator

(3.6)

It is obvious that fixed points of are solutions of the PBVP (1.1).

Since is nondecreasing on , we have the following lemma.

Lemma 3.1.

is nondecreasing.

Lemma 3.2.

is completely continuous.

Proof.

Suppose that Then

(3.7)

so,

(3.8)

Therefore,

(3.9)

This shows that . Furthermore, with similar arguments as in [7], we can prove that is completely continuous by Arzela-Ascoli theorem.

Theorem 3.3.

Assume that there exist two positive numbers such that

(3.10)

Then the PBVP (1.1) has positive solutions and , which may coincide with

(3.11)

where and for .

Proof.

First, we define

(3.12)

Then we may assert that

(3.13)

In fact, if , then

(3.14)

which together with (H2) and (3.10) implies that

(3.15)

which shows that

(3.16)

Now, if we denote that for , then . Let

(3.17)

In view of , we have Since the set is bounded and the operator is compact, we know that the set is relatively compact, which implies that there exists a subsequence such that

(3.18)

Moreover, since

(3.19)

it follows from Lemma 3.1 that ; that is, . By induction, it is easy to know that

(3.20)

which together with (3.18) implies that

(3.21)

Since is continuous, it follows from (3.17) and (3.21) that

(3.22)

which shows that is a solution of the PBVP (1.1). Furthermore, we get from that

(3.23)

On the other hand, if we denote that for and that then we can obtain similarly that and there exists a subsequence such that

(3.24)

Moreover, since

(3.25)

it is also easy to know that

(3.26)

With the similar arguments as above, we can prove that is a solution of the PBVP (1.1) and satisfies

(3.27)

Corollary 3.4.

If the following conditions are fulfilled:

(3.28)

then there exist two positive numbers such that (3.10) is satisfied, which implies that the PBVP (1.1) has positive solutions and , which may coincide with

(3.29)

where and for .

Example 3.5.

Let We consider the following PBVP on :

(3.30)

Since and we can obtain that

(3.31)

Thus, if we choose and , then all the conditions of Theorem 3.3 are fulfilled. So, the PBVP (3.30) has positive solutions and , which may coincide with

(3.32)

where and for .