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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq2_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq3_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq5_HTML.gif for http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq9_HTML.gif ). For each interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq10_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq11_HTML.gif , we define http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ1_HTML.gif
(1.1)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq14_HTML.gif is an increasing homeomorphism and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq15_HTML.gif for http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq20_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq21_HTML.gif -Laplacian operator, that is http://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 http://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 http://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 http://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:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq26_HTML.gif and the nonlinearity http://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 http://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 http://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] ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq30_HTML.gif ), Sun and Li [27] ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq33_HTML.gif , the operator http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq37_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq39_HTML.gif

(H2) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq40_HTML.gif exists and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq41_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq42_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq45_HTML.gif . The map http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq47_HTML.gif provided that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq48_HTML.gif is continuous and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ4_HTML.gif
(2.1)
Let the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq49_HTML.gif be endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq50_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ5_HTML.gif
(2.2)
and choose the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq51_HTML.gif as
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq52_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ7_HTML.gif
(2.4)
From the definition of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq54_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq55_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq57_HTML.gif . From the fact that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ8_HTML.gif
(2.5)

we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq58_HTML.gif is concave in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq59_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq61_HTML.gif is the maximum value of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq63_HTML.gif in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq66_HTML.gif is completely continuous.

Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq67_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ9_HTML.gif
(2.6)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq69_HTML.gif is a positive constant, and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ10_HTML.gif
(2.7)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq71_HTML.gif be given, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq72_HTML.gif nonnegative continuous convex functionals on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq74_HTML.gif a nonnegative continuous concave functional on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq75_HTML.gif . We define the following convex sets:
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq76_HTML.gif be Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq77_HTML.gif a cone, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq78_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq79_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq80_HTML.gif and http://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), http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq83_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq84_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq85_HTML.gif , and http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq88_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq89_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq90_HTML.gif

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

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

Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq99_HTML.gif has at least three fixed points http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq100_HTML.gif with
http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq103_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq105_HTML.gif .

First in view of Lemma http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq106_HTML.gif in [27], we know that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq107_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq108_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq109_HTML.gif So we get
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq110_HTML.gif and the nonnegative continuous concave functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq111_HTML.gif be defined on the cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq112_HTML.gif by
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ15_HTML.gif
(3.3)

Theorem 3.1.

Assume http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq114_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq115_HTML.gif . If there are positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq116_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq117_HTML.gif with http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq119_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq120_HTML.gif

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

(iii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq123_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq125_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq126_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq127_HTML.gif satisfying
http://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 http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ17_HTML.gif
(3.5)
In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq130_HTML.gif then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ18_HTML.gif
(3.6)
so assumption (i) implies
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq131_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq132_HTML.gif ; then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq133_HTML.gif is concave in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq134_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq135_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq136_HTML.gif , so
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq137_HTML.gif , then condition (iii) implies
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq138_HTML.gif Thus, condition ( http://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 ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq141_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq142_HTML.gif It is easy to see that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ22_HTML.gif
(3.10)
and consequently
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ23_HTML.gif
(3.11)
Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq143_HTML.gif there are
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq144_HTML.gif , we see that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ26_HTML.gif
(3.14)

Therefore, we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq145_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq146_HTML.gif , and condition ( http://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 ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq149_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq150_HTML.gif we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq151_HTML.gif on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq153_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq154_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq155_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq156_HTML.gif with
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq158_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq159_HTML.gif satisfy an additional relation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq160_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq161_HTML.gif implies that
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ30_HTML.gif
(3.18)
with
http://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) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq163_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq164_HTML.gif

(ii) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq165_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq167_HTML.gif positive solutions.

Proof.

When http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq169_HTML.gif which means that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq170_HTML.gif has at least one fixed point http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq173_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq174_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq176_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq179_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq180_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq182_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ32_HTML.gif
(3.20)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq183_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq184_HTML.gif . By simple calculations, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ34_HTML.gif
(3.22)
Observe that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_Equ35_HTML.gif
(3.23)
If we choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq186_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq187_HTML.gif satisfies
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F591219/MediaObjects/13661_2010_Article_48_IEq188_HTML.gif such that
http://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

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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.