Open Access

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

Boundary Value Problems20102011:591219

DOI: 10.1155/2011/591219

Received: 29 May 2010

Accepted: 6 August 2010

Published: 10 August 2010

Abstract

The purpose of this article is to establish the existence of multiple positive solutions of the dynamic equation on time scales https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq1_HTML.gif , subject to the multi-point boundary condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq2_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq4_HTML.gif is an increasing homeomorphism and satisfies the relation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq5_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq6_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq7_HTML.gif is not necessarily odd.

1. Introduction

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 [26].

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq8_HTML.gif be a time scale (an arbitrary nonempty closed subset of the real numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq9_HTML.gif ). For each interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq10_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq11_HTML.gif , we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq12_HTML.gif . For more details on time scales, one can refer to [13, 5]. In this paper we are concerned with the existence of at least triple positive solutions to the following https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq13_HTML.gif -point boundary value problems on time scales
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq14_HTML.gif is an increasing homeomorphism and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq15_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq16_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq17_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq18_HTML.gif denotes the displacement of the bridge from the unloaded position, and we emphasize the position of the bridge at supporting points near https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq19_HTML.gif , 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, 1014] and the references therein.

Recently, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq20_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq21_HTML.gif -Laplacian operator, that is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq22_HTML.gif , 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, 1522] in the continuous case, see [15, 2325] 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq23_HTML.gif -Laplacian operator is key to the proof. However in this paper the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq24_HTML.gif is not necessary odd, so it improves and generalizes the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq25_HTML.gif -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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ3_HTML.gif
(1.3)

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

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq26_HTML.gif and the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq27_HTML.gif is not involved with the first-order derivative https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq28_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq29_HTML.gif -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] ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq30_HTML.gif ), Sun and Li [27] ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq31_HTML.gif ). We also emphasize that in this paper the nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq32_HTML.gif is involved with the first-order delta derivative https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq33_HTML.gif , the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq34_HTML.gif 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) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq39_HTML.gif

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq40_HTML.gif exists and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq41_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq43_HTML.gif is continuous.

2. Preliminaries

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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq44_HTML.gif is a cone in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq45_HTML.gif . The map https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq46_HTML.gif is said to be a nonnegative continuous concave (convex) functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq47_HTML.gif provided that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq48_HTML.gif is continuous and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ4_HTML.gif
(2.1)
Let the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq49_HTML.gif be endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq50_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ5_HTML.gif
(2.2)
and choose the cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq51_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ6_HTML.gif
(2.3)
Now we define the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq52_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ7_HTML.gif
(2.4)
From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq53_HTML.gif and the assumptions of (H1), (H2), we can easily obtain that for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq54_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq55_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq56_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq57_HTML.gif . From the fact that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ8_HTML.gif
(2.5)

we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq58_HTML.gif is concave in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq59_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq61_HTML.gif is the maximum value of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq62_HTML.gif . In addition, by direct calculation, we get that each fixed point of the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq63_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq64_HTML.gif is a positive solution of (1.1) and (1.2). Similar as the proof of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq65_HTML.gif in [27], it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq66_HTML.gif is completely continuous.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq68_HTML.gif are two nonnegative continuous convex functionals satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ9_HTML.gif
(2.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq69_HTML.gif is a positive constant, and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ10_HTML.gif
(2.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq71_HTML.gif be given, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq72_HTML.gif nonnegative continuous convex functionals on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq73_HTML.gif satisfying the relation (2.6) and (2.7), and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq74_HTML.gif a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq75_HTML.gif . We define the following convex sets:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ11_HTML.gif
(2.8)

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

Lemma 2.2 (see [16]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq76_HTML.gif be Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq77_HTML.gif a cone, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq78_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq79_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq81_HTML.gif are nonnegative continuous convex functionals satisfying (2.6) and (2.7), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq82_HTML.gif is a nonnegative continuous concave functional on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq83_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq84_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq85_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq86_HTML.gif is a completely continuous operator. Suppose

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq88_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq89_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq90_HTML.gif

(C2) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq93_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq94_HTML.gif ;

(C3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq97_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq98_HTML.gif

Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq99_HTML.gif has at least three fixed points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq100_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ12_HTML.gif
(2.9)

3. Main Results

In this section, we impose some growth conditions on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq101_HTML.gif which allow us to apply Lemma 2.2 to the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq102_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq104_HTML.gif , the solution of (1.1) and (1.2) is nonnegative and concave on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq105_HTML.gif .

First in view of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq106_HTML.gif in [27], we know that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq107_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq108_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq109_HTML.gif So we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ13_HTML.gif
(3.1)
Let the nonnegative continuous convex functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq110_HTML.gif and the nonnegative continuous concave functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq111_HTML.gif be defined on the cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq112_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ14_HTML.gif
(3.2)

Then, it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq113_HTML.gif and (2.6), (2.7) hold.

Now, for convenience we introduce the following notations. Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ15_HTML.gif
(3.3)

Theorem 3.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq114_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq115_HTML.gif . If there are positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq116_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq117_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq118_HTML.gif , such that the following conditions are satisfied

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq119_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq120_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq121_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq122_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq123_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq124_HTML.gif

then the problem (1.1), (1.2) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq125_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq126_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq127_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ16_HTML.gif
(3.4)

Proof.

By the definition of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq128_HTML.gif and its properties, it suffices to show that the conditions of Lemma 2.2 hold with respect to the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq129_HTML.gif

We first show that if the condition (i) is satisfied, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ17_HTML.gif
(3.5)
In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq130_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ18_HTML.gif
(3.6)
so assumption (i) implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ19_HTML.gif
(3.7)
On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq131_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq132_HTML.gif ; then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq133_HTML.gif is concave in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq134_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq135_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq136_HTML.gif , so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ20_HTML.gif
(3.8)

Therefore, (3.5) holds.

In the same way, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq137_HTML.gif , then condition (iii) implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ21_HTML.gif
(3.9)

As in the argument above, we can get that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq138_HTML.gif Thus, condition ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq139_HTML.gif ) of Lemma 2.2 holds.

Next we show that condition ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq140_HTML.gif ) in Lemma 2.2 holds. We choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq141_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq142_HTML.gif It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ22_HTML.gif
(3.10)
and consequently
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ23_HTML.gif
(3.11)
Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq143_HTML.gif there are
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ24_HTML.gif
(3.12)
Hence in view of hypothesis (ii), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ25_HTML.gif
(3.13)
So by the definition of the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq144_HTML.gif , we see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ26_HTML.gif
(3.14)

Therefore, we get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq145_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq146_HTML.gif , and condition ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq147_HTML.gif ) in Lemma 2.2 is fulfilled.

We finally prove that ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq148_HTML.gif ) in Lemma 2.2 holds. In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq149_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq150_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ27_HTML.gif
(3.15)
Thus from Lemma 2.2 and the assumption that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq151_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq152_HTML.gif , the BVP (1.1) and (1.2) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq153_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq154_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq155_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq156_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ28_HTML.gif
(3.16)
The fact that the functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq158_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq159_HTML.gif satisfy an additional relation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq160_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq161_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ29_HTML.gif
(3.17)

The proof is complete.

From Theorem 3.1, we see that, when assumptions as (i), (ii), and (iii) are imposed appropriately on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq162_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ30_HTML.gif
(3.18)
with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ31_HTML.gif
(3.19)

such that the following conditions hold:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq163_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq164_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq165_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq166_HTML.gif

Then, BVP (1.1) and (1.2) has at least https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq167_HTML.gif positive solutions.

Proof.

When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq168_HTML.gif it is immediate from condition (i) that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq169_HTML.gif which means that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq170_HTML.gif has at least one fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq171_HTML.gif by the Schauder fixed point theorem. When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq172_HTML.gif it is clear that the hypothesis of Theorem 3.1 holds. Then we can obtain at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq173_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq175_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq176_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq177_HTML.gif denote nonnegative integer numbers set. If we choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq179_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq180_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq181_HTML.gif and consider the following BVP on time scale https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq182_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ32_HTML.gif
(3.20)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ33_HTML.gif
(3.21)
obviously the hypotheses (H1), (H2) hold and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq183_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq184_HTML.gif . By simple calculations, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ34_HTML.gif
(3.22)
Observe that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ35_HTML.gif
(3.23)
If we choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq185_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq186_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq187_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ36_HTML.gif
(3.24)
So all conditions of Theorem 3.1 hold. Thus by Theorem 3.1, the problem (3.20) has at least three positive solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq188_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ37_HTML.gif
(3.25)

Declarations

Acknowledgments

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

Authors’ Affiliations

(1)
Department of Applied Mathematics, Lanzhou University of Technology
(2)
School of Mathematics and Statistics, Lanzhou University

References

  1. Hilger S: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results in Mathematics 1990, 18(1-2):18-56.View ArticleMathSciNetMATHGoogle Scholar
  2. Bohner M, Peterson A: Dynamic Equations on Time Scales: an Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar
  3. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar
  4. Jones MA, Song B, Thomas DM: Controlling wound healing through debridement. Mathematical and Computer Modelling 2004, 40(9-10):1057-1064. 10.1016/j.mcm.2003.09.041View ArticleMathSciNetMATHGoogle Scholar
  5. Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic systems on measure chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Boston, Mass, USA; 1996:x+285.View ArticleGoogle Scholar
  6. Spedding V: Taming nature's numbers. New Scientist 2003, 7(19):28-32.Google Scholar
  7. Adem J, Moshinsky M: Self-adjointness of a certain type of vectorial boundary value problems. Boletín de la Sociedad Matemática Mexicana 1950, 7: 1-17.MathSciNetGoogle Scholar
  8. Zou Y, Hu Q, Zhang R: On numerical studies of multi-point boundary value problem and its fold bifurcation. Applied Mathematics and Computation 2007, 185(1):527-537. 10.1016/j.amc.2006.07.064View ArticleMathSciNetMATHGoogle Scholar
  9. Ilin VA, Moiseev EI: Nonlocal boundary value problem of the second kind for a Sturm-CLiouville operator. Difference Equation 1987, 23: 979-987.Google Scholar
  10. Agarwal RP, O'Regan D, Wong PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Boston, Mass, USA; 1999:xii+417.View ArticleMATHGoogle Scholar
  11. Anderson DR, Avery R, Henderson J: Existence of solutions for a one dimensional p -Laplacian on time-scales. Journal of Difference Equations and Applications 2004, 10(10):889-896. 10.1080/10236190410001731416View ArticleMathSciNetMATHGoogle Scholar
  12. Bai Z, Ge W, Wang Y: Multiplicity results for some second-order four-point boundary-value problems. Nonlinear Analysis 2005, 60(3):491-500.MathSciNetMATHGoogle Scholar
  13. Li WT, Sun HR: Positive solutions for second-order m -point boundary value problems on time scales. Acta Mathematica Sinica 2006, 22(6):1797-1804. 10.1007/s10114-005-0748-5View ArticleMathSciNetMATHGoogle Scholar
  14. Su Y-H, Li W-T, Sun H-R: Triple positive pseudo-symmetric solutions of three-point BVPs for p -Laplacian dynamic equations on time scales. Nonlinear Analysis 2008, 68(6):1442-1452.View ArticleMathSciNetMATHGoogle Scholar
  15. Avery RI, Chyan CJ, Henderson J: Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. Computers & Mathematics with Applications 2001, 42(3-5):695-704. 10.1016/S0898-1221(01)00188-2View ArticleMathSciNetMATHGoogle Scholar
  16. Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems. Computers & Mathematics with Applications 2004, 48(5-6):699-707. 10.1016/j.camwa.2004.03.002View ArticleMathSciNetMATHGoogle Scholar
  17. Bereanu C, Mawhin J:Existence and multiplicity results for some nonlinear problems with singular https://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq189_HTML.gif -Laplacian. Journal of Differential Equations 2007, 243(2):536-557. 10.1016/j.jde.2007.05.014View ArticleMathSciNetMATHGoogle Scholar
  18. Liu B: Positive solutions of a nonlinear three-point boundary value problem. Applied Mathematics and Computation 2002, 132(1):11-28. 10.1016/S0096-3003(02)00341-7View ArticleMathSciNetMATHGoogle Scholar
  19. Lü H, O'Regan D, Zhong C: Multiple positive solutions for the one-dimensional singular p -Laplacian. Applied Mathematics and Computation 2002, 133(2-3):407-422. 10.1016/S0096-3003(01)00240-5View ArticleMathSciNetMATHGoogle Scholar
  20. Wang J: The existence of positive solutions for the one-dimensional p -Laplacian. Proceedings of the American Mathematical Society 1997, 125(8):2275-2283. 10.1090/S0002-9939-97-04148-8View ArticleMathSciNetMATHGoogle Scholar
  21. Webb JRL: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Analysis 2001, 47(7):4319-4332. 10.1016/S0362-546X(01)00547-8View ArticleMathSciNetMATHGoogle Scholar
  22. Zeidler E: Nonlinear Analysis and Its Applications I: Fixed-Point Theorems. Springer, New York, NY, USA; 1993.MATHGoogle Scholar
  23. He Z: Double positive solutions of three-point boundary value problems for p -Laplacian difference equations. Zeitschrift für Analysis und ihre Anwendungen 2005, 24(2):305-315.View ArticleMATHGoogle Scholar
  24. Li Y, Lu L: Existence of positive solutions of p -Laplacian difference equations. Applied Mathematics Letters 2006, 19(10):1019-1023. 10.1016/j.aml.2005.10.020View ArticleMathSciNetMATHGoogle Scholar
  25. Liu Y, Ge W: Twin positive solutions of boundary value problems for finite difference equations with p -Laplacian operator. Journal of Mathematical Analysis and Applications 2003, 278(2):551-561. 10.1016/S0022-247X(03)00018-0View ArticleMathSciNetMATHGoogle Scholar
  26. Sun H-R, Li W-T: Existence theory for positive solutions to one-dimensional p -Laplacian boundary value problems on time scales. Journal of Differential Equations 2007, 240(2):217-248. 10.1016/j.jde.2007.06.004View ArticleMathSciNetMATHGoogle Scholar
  27. Sun H-R, Li W-T: Positive solutions of p -Laplacian m -point boundary value problems on time scales. Taiwanese Journal of Mathematics 2008, 12(1):105-127.MathSciNetMATHGoogle Scholar

Copyright

© J. Liu and H.-R. Sun. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.