Open Access

Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System

Boundary Value Problems20112011:475126

DOI: 10.1155/2011/475126

Received: 14 November 2010

Accepted: 24 February 2011

Published: 14 March 2011

Abstract

Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.

1. Introduction

In the past decades, fractional differential equations have been intensively studied. This is due to the rapid development of the theory of fractional differential equations itself and the applications of such construction in various sciences such as physics, mechanics, chemistry, and engineering [1, 2]. For the basic theory of fractional differential equations, we refer the readers to [37].

Recently, many researchers have devoted their attention to studying the existence of (positive) solutions of boundary value problems for differential equations with fractional order [823]. We mention that the fractional order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq1_HTML.gif involved is generally in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq2_HTML.gif with the exception that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq3_HTML.gif in [12, 23] and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq4_HTML.gif in [8, 17]. Though there have been extensive study on systems of fractional differential equations, not much has been done for boundary value problems for systems of fractional differential equations [1820].

On the other hand, we know that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays [2431].

As a result, in this paper, we consider the following nonlinear system of fractional order differential equations with delays,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq5_HTML.gif is the standard Riemann-Liouville fractional derivative of order   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq6_HTML.gif for some integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq8_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq10_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq11_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq12_HTML.gif is a nonlinear function from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq13_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq14_HTML.gif . The purpose is to establish sufficient conditions on the existence of positive solutions to (1.1) by using some fixed point theorems and some properties of the Green function. By a positive solution to (1.1) we mean a mapping with positive components on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq15_HTML.gif such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional differential equations when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq16_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq18_HTML.gif . Therefore, the obtained results generalize and include some existing ones.

The remaining part of this paper is organized as follows. In Section 2, we introduce some basics of fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4.

2. Preliminaries

We first introduce some basic definitions of fractional derivative for the readers' convenience.

Definition 2.1 (see [3, 32]).

The fractional integral of order   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq19_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq20_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ2_HTML.gif
(2.1)

provided that the integral exists on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq21_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq22_HTML.gif is the Gamma function.

Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq23_HTML.gif has the semigroup property, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ3_HTML.gif
(2.2)

Definition 2.2 (see [3, 32]).

The Riemann-Liouville derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq24_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq25_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ4_HTML.gif
(2.3)

provided that the right-hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq26_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq27_HTML.gif .

It is well known that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq28_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq29_HTML.gif . Furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq31_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq32_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq33_HTML.gif .

The following results on fractional integral and fractional derivative will be needed in establishing our main results.

Lemma 2.3 (see [10]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq34_HTML.gif . Then solutions to the fractional equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq35_HTML.gif can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ5_HTML.gif
(2.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq37_HTML.gif .

Lemma 2.4 (see [10]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq38_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ6_HTML.gif
(2.5)

for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq40_HTML.gif .

Now, we cite the fixed point theorems to be used in Section 3.

Lemma 2.5 (the Banach contraction mapping theorem [33]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq41_HTML.gif be a complete metric space and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq42_HTML.gif be a contraction mapping. Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq43_HTML.gif has a unique fixed point.

Lemma 2.6 (see [16, 34]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq44_HTML.gif be a closed and convex subset of a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq45_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq46_HTML.gif is a relatively open subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq47_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq49_HTML.gif is completely continuous. Then at least one of the following two properties holds:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq50_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq51_HTML.gif ;

(ii)there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq53_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq54_HTML.gif .

Lemma 2.7 (the Krasnosel'skii fixed point theorem [33, 35]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq55_HTML.gif be a cone in a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq56_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq58_HTML.gif are open subsets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq59_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq61_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq62_HTML.gif is a completely continuous operator such that either

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq63_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq65_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq66_HTML.gif

or

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq67_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq69_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq70_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq71_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq72_HTML.gif .

3. Existence of Positive Solutions

Throughout this paper, we let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq73_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq74_HTML.gif is a Banach space, where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ7_HTML.gif
(3.1)

In this section, we always assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq75_HTML.gif .

Lemma 3.1.

System (1.1) is equivalent to the following system of integral equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ8_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ9_HTML.gif
(3.3)

Proof.

It is easy to see that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq76_HTML.gif satisfies (3.2) then it also satisfies (3.2). So, assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq77_HTML.gif is a solution to (1.1). Integrating both sides of the first equation of (1.1) of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq78_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq79_HTML.gif gives us
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ10_HTML.gif
(3.4)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq81_HTML.gif . It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ11_HTML.gif
(3.5)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq83_HTML.gif . This, combined with the boundary conditions in (1.1), yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ12_HTML.gif
(3.6)
Similarly, one can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ13_HTML.gif
(3.7)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ14_HTML.gif
(3.8)
   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq84_HTML.gif . Then it follows from (3.8) and the boundary condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq85_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ15_HTML.gif
(3.9)
Therefore, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq86_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ16_HTML.gif
(3.10)

This completes the proof.

The following two results give some properties of the Green functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq87_HTML.gif .

Lemma 3.2.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq88_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq90_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq91_HTML.gif .

Proof.

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq92_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq93_HTML.gif . It remains to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq94_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq95_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq97_HTML.gif . We only need to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq98_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq99_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq100_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ17_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ18_HTML.gif
(3.12)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ19_HTML.gif
(3.13)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq102_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq103_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq104_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq105_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq106_HTML.gif .

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq107_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq108_HTML.gif and the proof is complete.

Lemma 3.3.
  1. (i)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq109_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq110_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq111_HTML.gif .

     
  2. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq112_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq114_HTML.gif .

     
Proof.
  1. (i)
    Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq115_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq116_HTML.gif . Now, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq117_HTML.gif , we have
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ20_HTML.gif
    (3.14)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq118_HTML.gif is the function defined by (3.11). It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq119_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq120_HTML.gif . In summary, we have proved (i).
  1. (ii)
    Again, one can easily see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq121_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq122_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq123_HTML.gif , we have in this case that
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ21_HTML.gif
    (3.15)
     

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq124_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq125_HTML.gif . To summarize, we have proved (ii) and this completes the proof.

Now, we are ready to present the main results.

Theorem 3.4.

Suppose that there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq127_HTML.gif , 2, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq128_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ22_HTML.gif
(3.16)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq130_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq131_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ23_HTML.gif
(3.17)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ24_HTML.gif
(3.18)

then (1.1) has a unique positive solution.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ25_HTML.gif
(3.19)
It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq132_HTML.gif is a complete metric space. Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq133_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq134_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ26_HTML.gif
(3.20)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq135_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ27_HTML.gif
(3.21)
Because of the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq137_HTML.gif , it follows easily from Lemma 3.2 that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq138_HTML.gif maps https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq139_HTML.gif into itself. To finish the proof, we only need to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq140_HTML.gif is a contraction. Indeed, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq141_HTML.gif , by (3.16) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ28_HTML.gif
(3.22)

This, combined with Lemma 3.3 and (3.17) and (3.18), immediately implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq142_HTML.gif is a contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5.

The following result can be proved in the same spirit as that for Theorem 3.4.

Theorem 3.5.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq143_HTML.gif , suppose that there exist nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq144_HTML.gif and nonnegative constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq145_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq146_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ29_HTML.gif
(3.23)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq148_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ30_HTML.gif
(3.24)

then (1.1) has a unique positive solution.

Theorem 3.6.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq149_HTML.gif , suppose that there exist nonnegative real-valued functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq150_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ31_HTML.gif
(3.25)
for almost every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq151_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq152_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ32_HTML.gif
(3.26)

then (1.1) has at least one positive solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq154_HTML.gif be defined by (3.19) and (3.20), respectively. We first show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq155_HTML.gif is completely continuous through the following three steps.

Step 1.

Show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq156_HTML.gif is continuous. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq157_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq158_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq159_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq160_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq161_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq162_HTML.gif is continuous, it is uniformly continuous on any compact set. In particular, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq163_HTML.gif , there exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq164_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ33_HTML.gif
(3.27)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq166_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq167_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ34_HTML.gif
(3.28)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq169_HTML.gif . Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ35_HTML.gif
(3.29)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq170_HTML.gif is continuous.

Step 2.

Show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq171_HTML.gif maps bounded sets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq172_HTML.gif into bounded sets. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq173_HTML.gif be a bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq174_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq175_HTML.gif is bounded. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq176_HTML.gif is continuous, there exists an https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq177_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ36_HTML.gif
(3.30)
It follows that, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq179_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ37_HTML.gif
(3.31)

Immediately, we can easily see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq180_HTML.gif is a bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq181_HTML.gif .

Step 3.

Show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq182_HTML.gif maps bounded sets of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq183_HTML.gif into equicontinuous sets. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq184_HTML.gif be a bounded subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq185_HTML.gif . Similarly as in Step 2, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq186_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ38_HTML.gif
(3.32)
Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq189_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ39_HTML.gif
(3.33)

Now the equicontituity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq190_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq191_HTML.gif follows easily from the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq192_HTML.gif is continuous and hence uniformly continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq193_HTML.gif .

Now we have shown that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq194_HTML.gif is completely continuous. To apply Lemma 2.6, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ40_HTML.gif
(3.34)
Fix https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq195_HTML.gif and define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ41_HTML.gif
(3.35)
We claim that there is no https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq196_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq197_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq198_HTML.gif . Otherwise, assume that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq200_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq201_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ42_HTML.gif
(3.36)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq202_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ43_HTML.gif
(3.37)

Similarly, we can have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq203_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq204_HTML.gif . To summarize, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq205_HTML.gif , a contradiction to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq206_HTML.gif . This proves the claim. Applying Lemma 2.6, we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq207_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq208_HTML.gif , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete.

As a consequence of Theorem 3.6, we have the following.

Corollary 3.7.

If all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq210_HTML.gif , are bounded, then (1.1) has at least one positive solution.

To state the last result of this section, we introduce
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ44_HTML.gif
(3.38)

Theorem 3.8.

Suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq211_HTML.gif and positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq212_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq213_HTML.gif such that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq214_HTML.gif   for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq215_HTML.gif

and

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq216_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq217_HTML.gif ,

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq218_HTML.gif . Then (1.1) has at least a positive solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq219_HTML.gif be defined by (3.19) and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq220_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq221_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq222_HTML.gif . From the proof of Theorem 3.6, we know that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq223_HTML.gif defined by (3.20) is completely continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq224_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq225_HTML.gif , it follows from Lemma 3.3 and condition (ii) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ45_HTML.gif
(3.39)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ46_HTML.gif
(3.40)
On the other hand, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq226_HTML.gif , it follows from Lemma 3.3 and condition (i) that, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq227_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ47_HTML.gif
(3.41)
if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq228_HTML.gif , whereas
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ48_HTML.gif
(3.42)
if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq229_HTML.gif . In summary,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ49_HTML.gif
(3.43)

Therefore, we have verified condition (ii) of Lemma 2.7. It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq230_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq231_HTML.gif , which is a positive solution to (1.1). This completes the proof.

4. Examples

In this section, we demonstrate the feasibility of some of the results obtained in Section 3.

Example 4.1.

Consider
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ50_HTML.gif
(4.1)
Here
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ51_HTML.gif
(4.2)
One can easily see that (3.16) is satisfied with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ52_HTML.gif
(4.3)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ53_HTML.gif
(4.4)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ54_HTML.gif
(4.5)

It follows from Theorem 3.4 that (4.1) has a unique positive solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq232_HTML.gif .

Example 4.2.

Consider
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ55_HTML.gif
(4.6)
Here
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ56_HTML.gif
(4.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ57_HTML.gif
(4.8)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq234_HTML.gif satisfy (3.25). Moreover, simple calculations give us
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ58_HTML.gif
(4.9)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq235_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ59_HTML.gif
(4.10)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq237_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ60_HTML.gif
(4.11)
Then, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq239_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ61_HTML.gif
(4.12)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq240_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq241_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_Equ62_HTML.gif
(4.13)

By now we have verified all the assumptions of Theorem 3.8. Therefore, (4.6) has at least one positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq242_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F475126/MediaObjects/13661_2010_Article_41_IEq243_HTML.gif .

Declarations

Acknowledgment

Supported partially by the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, the Foundation of Science and Technology Department of Hunan Province under Grant no. 2009RS3019 and the Subject Lead Foundation of University of South China no. 2007XQD13. Research was partially supported by the Natural Science and Engineering Re-search Council of Canada (NSERC) and the Early Researcher Award (ERA) Pro-gram of Ontario.

Authors’ Affiliations

(1)
School of Mathematics and Physics, School of Nuclear Science and Technology, University of South China
(2)
Department of Mathematics, Wilfrid Laurier University
(3)
School of Nuclear Science and Technology, University of South China

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© Zigen Ouyang et al. 2011

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