Open Access

Positive Solutions for Third-Order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq1_HTML.gif -Laplacian Functional Dynamic Equations on Time Scales

Boundary Value Problems20102011:279752

DOI: 10.1155/2011/279752

Received: 31 March 2010

Accepted: 9 December 2010

Published: 15 December 2010

Abstract

The authors study the boundary value problems for a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq2_HTML.gif -Laplacian functional dynamic equation on a time scale, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq8_HTML.gif . By using the twin fixed-point theorem, sufficient conditions are established for the existence of twin positive solutions.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq9_HTML.gif be a closed nonempty subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq10_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq11_HTML.gif have the subspace topology inherited from the Euclidean topology on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq12_HTML.gif . In some of the current literature, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq13_HTML.gif is called a time scale (or measure chain). For notation, we shall use the convention that, for each interval of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq14_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq16_HTML.gif will denote time scales interval, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq17_HTML.gif .

In this paper, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq18_HTML.gif be a time scale such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq19_HTML.gif , 0, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq20_HTML.gif . We are concerned with the existence of positive solutions of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq21_HTML.gif -Laplacian dynamic equation on a time scale
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq22_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq23_HTML.gif -Laplacian operator, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq26_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq27_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq28_HTML.gif and

the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq30_HTML.gif is continuous,

the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq32_HTML.gif is left dense continuous (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq33_HTML.gif and does not vanish identically on any closed subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq34_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq35_HTML.gif denotes the set of all left dense continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq36_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq37_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq39_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq40_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq42_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq43_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq44_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq46_HTML.gif is continuous and satisfies that there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq47_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ2_HTML.gif
(1.2)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq48_HTML.gif -Laplacian problems with two-, three-, m-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example see [14] and references therein. However, there are not many concerning the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq49_HTML.gif -Laplacian problems on time scales, especially for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq50_HTML.gif -Laplacian functional dynamic equations on time scales.

The motivations for the present work stems from many recent investigations in [58] and references therein. Especially, Kaufmann and Raffoul [8] considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions. In this paper, we apply the twin fixed-point theorem to obtain at least two positive solutions of boundary value problem (BVP for short) (1.1) when growth conditions are imposed on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq51_HTML.gif . Finally, we present two corollaries, which show that under the assumptions that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq52_HTML.gif is superlinear or sublinear, BVP (1.1) has at least two positive solutions.

Given a nonnegative continuous functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq53_HTML.gif on a cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq54_HTML.gif of a real Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq55_HTML.gif , we define for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq56_HTML.gif the sets
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ3_HTML.gif
(1.3)

The following twin fixed-point lemma due to [9] will play an important role in the proof of our results.

Lemma 1.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq57_HTML.gif be a real Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq58_HTML.gif a cone of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq61_HTML.gif two nonnegative increasing continuous functionals, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq62_HTML.gif a nonnegative continuous functional, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq63_HTML.gif . Suppose that there are two positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq65_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq66_HTML.gif is completely continuous. There are positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq67_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ5_HTML.gif
(1.5)

and

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq68_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq69_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq70_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq71_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq73_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq74_HTML.gif .

Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq75_HTML.gif has at least two fixed points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq77_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ6_HTML.gif
(1.6)

2. Positive Solutions

We note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq78_HTML.gif is a solution of (1.1) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ7_HTML.gif
(2.1)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq79_HTML.gif be endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq81_HTML.gif is concave and nonnegative valued on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq82_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq83_HTML.gif .

Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq84_HTML.gif is a Banach space with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq85_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq86_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq87_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq88_HTML.gif , extend https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq89_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq90_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq91_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq92_HTML.gif .

Define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq93_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ8_HTML.gif
(2.2)
We seek a fixed point, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq94_HTML.gif , of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq95_HTML.gif in the cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq96_HTML.gif . Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ9_HTML.gif
(2.3)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq97_HTML.gif denotes a positive solution of BVP (1.1).

It follows from (2.2) that

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq98_HTML.gif be defined by (2.2). If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq99_HTML.gif , then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq100_HTML.gif .

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq101_HTML.gif is completely continuous.

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq103_HTML.gif .

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq104_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq105_HTML.gif .

The proof is similar to the proofs of Lemma  2.3 and Theorem  3.1 in [7], and is omitted.

Fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq106_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq107_HTML.gif , and set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ10_HTML.gif
(2.4)

Throughout this paper, we assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq109_HTML.gif .

Now, we define the nonnegative, increasing, continuous functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq110_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq111_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq112_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq113_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ11_HTML.gif
(2.5)
We have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ12_HTML.gif
(2.6)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ13_HTML.gif
(2.7)
We also see that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ14_HTML.gif
(2.8)
For the notational convenience, we denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq114_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq117_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ15_HTML.gif
(2.9)

Theorem 2.2.

Suppose that there are positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq118_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ16_HTML.gif
(2.10)

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq119_HTML.gif satisfies the following conditions:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq121_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq122_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq123_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq125_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq126_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq127_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ17_HTML.gif
(2.11)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq130_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq131_HTML.gif .

Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ18_HTML.gif
(2.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq132_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq134_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq135_HTML.gif .

Proof.

By the definition of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq136_HTML.gif and its properties, it suffices to show that the conditions of Lemma 1.1 hold with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq137_HTML.gif .

First, we verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq138_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq139_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq140_HTML.gif , one gets https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq141_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq142_HTML.gif . Recalling that (2.7), we know https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq143_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq144_HTML.gif . Then, we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ19_HTML.gif
(2.13)

Secondly, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq145_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq146_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq147_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq148_HTML.gif , it holds that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq149_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq150_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq151_HTML.gif implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ20_HTML.gif
(2.14)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ21_HTML.gif
(2.15)
So, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ22_HTML.gif
(2.16)
Finally, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ23_HTML.gif
(2.17)
It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq152_HTML.gif . On the other hand, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq153_HTML.gif and (2.7) imply
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ24_HTML.gif
(2.18)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ25_HTML.gif
(2.19)
By Lemma 1.1, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq154_HTML.gif has at least two different fixed points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq156_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ26_HTML.gif
(2.20)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ27_HTML.gif
(2.21)

which are twin positive solutions of BVP (1.1). The proof is complete.

In analogy to Theorem 2.2, we have the following result.

Theorem 2.3.

Suppose that there are positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq157_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ28_HTML.gif
(2.22)

Assume https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq158_HTML.gif satisfies the following conditions: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq159_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq161_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq162_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq163_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ29_HTML.gif
(2.23)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq165_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq166_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq167_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq169_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq170_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq171_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ30_HTML.gif
(2.24)
Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ31_HTML.gif
(2.25)

Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ32_HTML.gif
(2.26)
and choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq174_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ33_HTML.gif
(2.27)

From above, we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq175_HTML.gif .

Theorem 2.4.

If the following conditions are satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq177_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq178_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq179_HTML.gif ,

there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq181_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq182_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ34_HTML.gif
(2.28)
Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ35_HTML.gif
(2.29)

Proof.

First, choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq183_HTML.gif , one gets
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ36_HTML.gif
(2.30)
Secondly, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq184_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq185_HTML.gif sufficiently small such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ37_HTML.gif
(2.31)
Without loss of generality, suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq186_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq187_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq188_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq189_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq190_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq191_HTML.gif . Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ38_HTML.gif
(2.32)
Thirdly, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq192_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq193_HTML.gif sufficiently large such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ39_HTML.gif
(2.33)
Without loss of generality, suppose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq194_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq195_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ40_HTML.gif
(2.34)

We get now https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq196_HTML.gif , and then the conditions in Theorem 2.2 are all satisfied. By Theorem 2.2, BVP (1.1) has at least two positive solutions. The proof is complete.

Theorem 2.5.

If the following conditions are satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq198_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq199_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq200_HTML.gif ,

there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq202_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq203_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ41_HTML.gif
(2.35)
Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ42_HTML.gif
(2.36)

The proof is similar to that of Theorem 2.4 and we omitted it.

The following Corollaries are obvious.

Corollary 2.6.

If the following conditions are satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq206_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq207_HTML.gif ,

there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq209_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq210_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ43_HTML.gif
(2.37)
Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ44_HTML.gif
(2.38)

Corollary 2.7.

If the following conditions are satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq212_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq214_HTML.gif ;

there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq216_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq217_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ45_HTML.gif
(2.39)
Then, BVP (1.1) has at least two positive solutions of the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ46_HTML.gif
(2.40)

3. Example

Example 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq218_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq223_HTML.gif .

We consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ47_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq225_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq226_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq227_HTML.gif .

Choosing https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq229_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq231_HTML.gif , direct calculation shows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ48_HTML.gif
(3.2)

Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq233_HTML.gif satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq235_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq236_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq237_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ49_HTML.gif
(3.3)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq239_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq240_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq241_HTML.gif ,

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq243_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq244_HTML.gif , uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_IEq245_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F279752/MediaObjects/13661_2010_Article_33_Equ50_HTML.gif
(3.4)

Then all conditions of Theorem 2.3 hold. Thus, with Theorem 2.3, the BVP (3.1) has at least two positive solutions.

Declarations

Acknowledgment

This paper is supported by Grants nos. (10871052) and (10901060) from the NNSF of China, and by Grant (no. 10151009001000032) from the NSF of Guangdong.

Authors’ Affiliations

(1)
School of Applied Mathematics, Guangdong University of Technology

References

  1. Avery R, Henderson J: Existence of three positive pseudo-symmetric solutions for a one-dimensional p -Laplacian. Journal of Mathematical Analysis and Applications 2003, 277(2):395-404. 10.1016/S0022-247X(02)00308-6View ArticleMathSciNetMATH
  2. 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 ArticleMathSciNetMATH
  3. Cabada A: Extremal solutions for the difference ϕ -Laplacian problem with nonlinear functional boundary conditions. Computers & Mathematics with Applications 2001, 42(3–5):593-601.View ArticleMathSciNetMATH
  4. Wong F-H: Existence of positive solutions for m -Laplacian boundary value problems. Applied Mathematics Letters 1999, 12(3):11-17. 10.1016/S0893-9659(98)00164-5View ArticleMathSciNetMATH
  5. Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003, 82: 1-11.MathSciNet
  6. He Z: Double positive solutions of three-point boundary value problems for p -Laplacian dynamic equations on time scales. Journal of Computational and Applied Mathematics 2005, 182(2):304-315. 10.1016/j.cam.2004.12.012View ArticleMathSciNetMATH
  7. Bian L, He X, Sun H: Multiple positive solutions of m -point BVPs for third-order p -Laplacian dynamic equaitons on time scales. Advance in Difference Equations 2009, 2009:-12.
  8. Kaufmann ER, Raffoul YN: Positive solutions for a nonlinear functional dynamic equation on a time scale. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(7):1267-1276. 10.1016/j.na.2005.04.031View ArticleMathSciNetMATH
  9. 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.View ArticleMathSciNetMATH

Copyright

© C. Song and X. Gao. 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.