Multiple Positive Solutions for m-Point Boundary Value Problem on Time Scales

The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales , subject to the multi-point boundary condition , where is an increasing homeomorphism and satisfies the relation for , which generalizes the usually p-Laplacian operator. An example applying the result is also presented. The main tool of this paper is a generalization of Leggett-Williams fixed point theorem, and the interesting points are that the nonlinearity f contains the first-order derivative explicitly and the operator is not necessarily odd.

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. On one hand, the time scales approach not only unifies calculus and difference equations, but also solves other problems that have a mix of stop-start and continuous behavior. On the other hand, the time scales calculus has tremendous potential for application in biological, phytoremediation of metals, wound healing, stock market and epidemic models [2–6].

Let be a time scale (an arbitrary nonempty closed subset of the real numbers ). For each interval of , we define . For more details on time scales, one can refer to [1–3, 5]. In this paper we are concerned with the existence of at least triple positive solutions to the following -point boundary value problems on time scales

where is an increasing homeomorphism and for .

Multipoint boundary value problem (BVP) arise in a variety of different areas of applied mathematics and physics, such as the vibrations of a guy wire of a uniform cross section and composed of parts of different densities can be set up as a multipoint boundary value problem [7]. Small size bridges are often designed with two supported points, which leads to a standard two-point boundary value condition. And large size bridges are sometimes contrived with multipoint supports, which corresponds to a multipoint boundary value condition [8]. Especially, if we let denotes the displacement of the bridge from the unloaded position, and we emphasize the position of the bridge at supporting points near , we can obtain the multipoint boundary condition (1.2). The study of multipoint BVPs for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [9], since then many authors studied more general nonlinear multipoint boundary value problems. We refer readers to [8, 10–14] and the references therein.

Recently, when is -Laplacian operator, that is , and the nonlinear term does not depend on the first-order derivative, the existence problems of positive solutions of boundary value problems have attracted much attention, see [10, 12, 15–22] in the continuous case, see [15, 23–25] in the discrete case and [11, 13, 14, 26, 27] in the general time scale setting. From the process of proving main results in the above references, one can notice that the oddness of the -Laplacian operator is key to the proof. However in this paper the operator is not necessary odd, so it improves and generalizes the -Laplacian operator. One may note this from Example 3.3 in Section 3. In addition, Bai and Ge [16] generalized the Leggett-Williams fixed point theorems by using fixed point index theory. An application of the theorem is given to prove the existence of three positive solutions to the following second-order BVP:

with Dirichlet boundary condition. They also extended the results to four-point BVP in [12].

When and the nonlinearity is not involved with the first-order derivative , in [27], Sun and Li discussed the existence and multiplicity of positive solutions for problems (1.1) and (1.2). The main tools used are fixed point theorems in cones.

Thanks to the above-mentioned research articles [16, 27], in this paper we consider the existence of multiple positive solutions for the more general dynamic equation on time scales (1.1) with -point boundary condition (1.2). An example is also given to illustrate the main results. The obtained results are even new for the special cases of difference equations and differential equations, as well as in the general time scale setting. The main result extends and generalizes the corresponding results of Liu [18] and Webb [21] (), Sun and Li [27] (). We also emphasize that in this paper the nonlinear term is involved with the first-order delta derivative , the operator is not necessary odd and have the more generalized form, and the tool is a generalized Leggett-Williams fixed point theorem [16].

The rest of the paper is organized as follows: in Section 2, we give some preliminaries which are needed later. Section 3 is due to develop existence criteria for at least three and arbitrary odd number positive solution of the boundary value problem (1.1) and (1.2). In the final part of this section, we present an example to illustrate the application of the obtained result.

Throughout this paper, the following hypotheses hold:

(H1), , for and

(H2) exists and such that and is continuous.

In this section, we first present some basic definition, then we define an appropriate Banach space, cone, and integral operator, and finally we list the fixed-point theorem which is needed later.

Definition 2.1.

Suppose is a cone in a Banach space . The map is said to be a nonnegative continuous concave (convex) functional on provided that is continuous and

Let the Banach space be endowed with the norm , where

and choose the cone as

Now we define the operator by

From the definition of and the assumptions of (H1), (H2), we can easily obtain that for each for and . From the fact that

we know that is concave in . Thus and is the maximum value of . In addition, by direct calculation, we get that each fixed point of the operator in is a positive solution of (1.1) and (1.2). Similar as the proof of Lemma in [27], it is easy to see that is completely continuous.

Suppose and are two nonnegative continuous convex functionals satisfying

where is a positive constant, and

Let , be given, nonnegative continuous convex functionals on satisfying the relation (2.6) and (2.7), and a nonnegative continuous concave functional on . We define the following convex sets:

In order to prove our main results, the following fixed point theorem is important in our argument.

Lemma 2.2 (see [16]).

Let be Banach space, a cone, and . Assume that and are nonnegative continuous convex functionals satisfying (2.6) and (2.7), is a nonnegative continuous concave functional on such that for all , and is a completely continuous operator. Suppose

(), for

(), for ;

() for with

Then has at least three fixed points with

In this section, we impose some growth conditions on which allow us to apply Lemma 2.2 to the operator defined in Section 2 to establish the existence of three positive solutions of (1.1) and (1.2). We note that, from the nonnegativity of and , the solution of (1.1) and (1.2) is nonnegative and concave on .

First in view of Lemma in [27], we know that for , there is for So we get

Let the nonnegative continuous convex functionals and the nonnegative continuous concave functional be defined on the cone by

Then, it is easy to see that and (2.6), (2.7) hold.

Now, for convenience we introduce the following notations. Let

Theorem 3.1.

Assume for . If there are positive numbers with , such that the following conditions are satisfied

(i) for

(ii) for

(iii) for

then the problem (1.1), (1.2) has at least three positive solutions satisfying

Proof.

By the definition of operator and its properties, it suffices to show that the conditions of Lemma 2.2 hold with respect to the operator

We first show that if the condition (i) is satisfied, then

In fact, if then

so assumption (i) implies

On the other hand, for , there is ; then is concave in , and for , so

Therefore, (3.5) holds.

In the same way, if , then condition (iii) implies

As in the argument above, we can get that Thus, condition () of Lemma 2.2 holds.

Next we show that condition () in Lemma 2.2 holds. We choose for It is easy to see that

and consequently

Therefore, for there are

Hence in view of hypothesis (ii), we have

So by the definition of the functional , we see that

Therefore, we get for , and condition () in Lemma 2.2 is fulfilled.

We finally prove that () in Lemma 2.2 holds. In fact, for with we have

Thus from Lemma 2.2 and the assumption that on , the BVP (1.1) and (1.2) has at least three positive solutions , and in with

The fact that the functionals and on satisfy an additional relation for implies that

The proof is complete.

From Theorem 3.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on we can establish the existence of an arbitrary odd number of positive solutions of (1.1) and (1.2).

Theorem 3.2.

Suppose that there exist constants

with

such that the following conditions hold:

(i) for

(ii) for

Then, BVP (1.1) and (1.2) has at least positive solutions.

Proof.

When it is immediate from condition (i) that which means that has at least one fixed point by the Schauder fixed point theorem. When it is clear that the hypothesis of Theorem 3.1 holds. Then we can obtain at least three positive solutions and . Following this way, we finish the proof by induction. The proof is complete.

In the final part of this section, we give an example to illustrate our results.

Example 3.3.

Let , where denote nonnegative integer numbers set. If we choose , , and and consider the following BVP on time scale :

where

obviously the hypotheses (H1), (H2) hold and on . By simple calculations, we have

Observe that

If we choose and , then satisfies

So all conditions of Theorem 3.1 hold. Thus by Theorem 3.1, the problem (3.20) has at least three positive solutions such that

This work was supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).

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