Abstract

We study the existence of positive solutions for a class of -point boundary value problems on time scales. Our approach is based on the monotone iterative technique and the cone expansion and compression fixed point theorem of norm type. Without the assumption of the existence of lower and upper solutions, we do not only obtain the existence of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

1. Introduction

The purpose of this paper is to consider the existence of positive solutions and establish the corresponding iterative schemes for the following -point boundary value problems (BVP) on time scales:

The study of dynamic equations on time scales goes back to its founder Hilger [1], and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics. Some preliminary definitions and theorems on time scales can be found in [25] which are good references for the calculus of time scales.

In recent years, by applying fixed point theorems, the method of lower and upper solutions and critical point theory, many authors have studied the existence of positive solutions for two-point and multipoint boundary value problems on time scales, for details, see [2, 3, 618] and references therein. However, to the best of our knowledge, there are few papers which are concerned with the computational methods of the multipoint boundary value problems on time scales. We would like to mention some results of Sun and Li [16], Aykut Hamel and Yoruk [12], Anderson and Wong [10], Wang et al. [18], and Jankowski [13], which motivated us to consider the BVP (1.1).

In [16], Sun and Li considered the existence of positive solutions of the following dynamic equations on time scales:

where , . They obtained the existence of single and multiple positive solutions of (1.2) by using a fixed point theorem and the Leggett-Williams fixed point theorem, respectively.

Very recently, in [12], Aykut Hamel and Yoruk discussed the following dynamic equation on time scales:

They obtained some results for the existence of at least two and three positive solutions to the BVP (1.3) by using fixed point theorems in a cone and the associated Green's function.

In related paper, in [10], Anderson and Wong studied the second-order time scale semipositone boundary value problem with Sturm-Liouville boundary conditions or multipoint conditions as in

On the other hand, the method of lower and upper solutions has been effectively used for proving the existence results for dynamic equations on time scales. In [18], Wang et al. considered a method of generalized quasilinearization, with even-order convergence, for the BVP

The main contribution in [18] relaxed the monotone conditions on including a more general concept of upper and lower solution in mathematical biology, so that the high-order convergence of the iterations was ensured for a larger class of nonlinear functions on time scales.

In [13], Jankowski investigated second-order differential equations with deviating arguments on time scales of the form

They formulated sufficient conditions, under which such problems had a minimal and a maximal solution in a corresponding region bounded by upper-lower solutions.

We would also like to mention the result of Yao [19]. In [19], Yao considered the positive solutions to the following two classes of nonlinear second-order three-point boundary value problems:

where both and are given constants satisfying . By improving the classical monotone iterative technique of Amann [20], two successive iterative schemes were established for the BVP (1.7). It was worth stating that the first terms of the iterative schemes were constant functions or simple functions. We note that Ma et al. [21] and Sun et al. [22, 23] have also applied the similar methods to -laplacian boundary value problems with .

In this paper, we will investigate the iterative and existence of positive solutions for the BVP (1.1), by considering the “heights” of the nonlinear term on some bounded sets and applying monotone iterative techniques on a Banach space, we do not only obtain the existence of positive solutions for the BVP (1.1), but also give the iterative schemes for approximating the solutions. We should point out that the monotone condition imposed on the nonlinear term will play crucial role in obtaining the iterative schemes for approximating the solutions. In essence, we combine the method of lower and upper solutions with the cone expansion and compression fixed point theorem of norm type. The idea of this paper comes from Yao [19, 24, 25].

Let be a time scale which has the subspace topology inherited from the standard topology on . For each interval of , we define .

For the remainder of this article, we denote the set of continuous functions from to by . Let be endowed with the ordering if for all , and is defined as usual by maximum norm. The is a Banach space.

Throughout this paper, we will assume that the following assumptions are satisfied:

() for with ;() with for and ; () is continuous.

2. Preliminaries and Several Lemmas

To prove the main results in this paper, we will employ several lemmas. These lemmas are based on the linear boundary value problem

Lemma 2.1 (see [12]). It holds ; then the Green's function for the BVP is given by Here for the sake of convenience, one writes for .

Lemma 2.2 (see [12]). Assume that conditions are satisfied. Then(i) for ; (ii)there exist a number and a continuous function such that where

Let . It is easy to see that the BVP (1.1) has a solution if and only if is a fixed point of the operator equation:

Denote

where is the same as in Lemma 2.2. By [12, Lemma  3.1], we can obtain that and is completely continuous.

3. Successive Iteration and One Positive Solution for (1.1)

For notational convenience, we denote

Constants are not easy to compute explicitly. For convenience, we can replace by , by , where

Obviously, .

Theorem 3.1. Assume hold, and there exist two positive numbers with such that(), ;() for any . Then the BVP (1.1) has at least one positive solution such that and , that is, converges uniformly to in , where .

Remark 3.2. The iterative scheme in Theorem 3.1 is It starts off with constant function .

Proof of Theorem 3.1. Denote . If , then By Assumptions and , we have It follows that Thus, we assert that .
Let , then . Let , then . Denote Since , we have Since is completely continuous, we assert that has a convergent subsequence and there exists , such that .
Now, since , we have
By Assumption , By the induction, then Hence, . Applying the continuity of and , we get . Since and is a nonnegative concave function, we conclude that is a positive solution of the BVP (1.1).

Corollary 3.3. Assume that hold, and the following conditions are satisfied:() (particularly, );() for any . Then the BVP (1.1) has at least one positive solution and there exists a positive number such that , that is, where .

Theorem 3.4. Assume hold, and the following conditions are satisfied:( there exists such that is nondecreasing for any and ; (), for any . Then the BVP (1.1) has one positive solution such that and , that is, converges uniformly to in . Furthermore, if there exists such that Then .

Proof. Denote . Similarly to the proof of Theorem 3.1, we can know that . Let , then . Denote Copying the corresponding proof of Theorem 3.1, we can prove that
Since is completely continuous, we can get that there exists such that . Applying the continuity of and , we can obtain that . We note that , it implies that the zero function is not the solution of the problem (1.1). Therefore, is a positive solution of (1.1).
Now, since
If and , , then Hence, we can deduce that It implies that The proof is complete.

4. Existence of positive solutions

Theorem 4.1. Assume hold, and there exist positive numbers with such that(), ; () for any . Then the BVP (1.1) has positive solutions such that and , that is, where .

Corollary 4.2. Assume that and hold, and the following condition is satisfied ( there exist positive numbers such that Then the BVP (1.1) has positive solutions , and there exists a positive number with such that , where .

5. Examples

Example 5.1. Let . Considering the following BVP: where , it is easy to check that , for any .
Further calculations give us
Choose , it is easy to check that is nondecreasing for any and Let , for we have By Theorem 3.4, the BVP (5.1) has one positive solution such that and . On the other hand, for any , we have Then, The first and second terms of this scheme are as follows: Now, we compute the third term of this scheme.
For ,
For ,

Example 5.2. Let . Considering the BVP on By direct computation, we can get Choose , it is easy to see that the nonlinear term possesses the following properties(a) is continuous; (b) for any and ; (c), . By Theorem 3.1, the BVP (5.10) has one positive solution such that and , where , . Let . For we have

Remark 5.3. By Theorems  3.1, 3.2, and  3.3 in [12, 16, 17], the existence of positive solutions for the BVP (5.1) can be obtained, however, we cannot give a way to find the solutions which will be useful from an application viewpoint. Therefore, Theorem 3.1 improves and extends the main results of [12, 16, 17]. On the other hand, in Example 5.2, since , we cannot obtain the above mentioned results by use of Theorem 3.4, thus, Theorems 3.1 and 3.4 do not contain each other.

Acknowledgment

This project supported by the Youth Science Foundation of Shanxi Province (2009021001-2).