Open Access

Nonlinear Systems of Second-Order ODEs

Boundary Value Problems20072008:236386

DOI: 10.1155/2008/236386

Received: 2 February 2007

Accepted: 16 November 2007

Published: 10 December 2007

Abstract

We study existence of positive solutions of the nonlinear system https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq2_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq4_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq5_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq7_HTML.gif . Here, it is assumed that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq9_HTML.gif are nonnegative continuous functions, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq11_HTML.gif are positive continuous functions, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq13_HTML.gif , and that the nonlinearities https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq14_HTML.gif satisfy superlinear hypotheses at zero and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq15_HTML.gif . The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq17_HTML.gif .

1. Introduction

We study existence of positive solutions for the following nonlinear system of second-order ordinary differential equations:

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq19_HTML.gif are nonnegatives constants, the functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq20_HTML.gif are continuous, the functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq21_HTML.gif are continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq22_HTML.gif We will suppose the following four hypotheses.

(H1) We have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ2_HTML.gif
(1.2)

uniformly for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq23_HTML.gif

(H2) There exist https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq25_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq26_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq27_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ3_HTML.gif
(1.3)

(H3) The functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq28_HTML.gif are continuous and

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ4_HTML.gif
(1.4)

In addition, we suppose that there exists an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq29_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq30_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq31_HTML.gif are nondecreasing for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq32_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq34_HTML.gif are nondecreasing, meaning that

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ5_HTML.gif
(1.5)

whenever https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq35_HTML.gif where the inequality is understood inside every component.

(H4) We have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq36_HTML.gif .

Here are some comments on the above hypotheses. Hypothesis (H1) is a superlinear condition at 0 and Hypothesis (H2) is a local superlinear condition at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq37_HTML.gif . About hypothesis (H3), the fact that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq38_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq39_HTML.gif are unbounded leads us to use the strategy of considering a truncation system. Note that if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq41_HTML.gif are bounded, we would not need to use that system. Hypothesis (H4) allows us to have a control on the nonlinear operator in system 1.1.

We remark that, the case when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq43_HTML.gif , systems of type (1.1) have been extensively studied in the literature under different sets of conditions on the nonlinearities. For instance, assuming superlinear hypothesis, many authors have obtained multiplicity of solutions with applications to elliptic systems in annular domains. For homogeneous Dirichlet boundary conditions, see de Figueiredo and Ubilla [1], Conti et al. [2], Dunninger and Wang [3, 4] and Wang [5]. For nonhomogeneous Dirichlet boundary conditions, see Lee [6] and Marcos do Ó et al. [7]. Our main goal is to study systems of type (1.1) by considering local superlinear assumptions at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq44_HTML.gif and global superlinear at zero.

The main result is the following.

Theorem 1.1.

Assume hypotheses (H1) through (H4). Then system (1.1) has at least one positive solution.

One of the main difficulties here lies in the facts that the coefficients of the differential operators of System (1.1) are nonlinear and that they may not necessarily be bounded from below by a positive bound which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq46_HTML.gif In order to overcome these difficulties, we introduce a truncation of system (1.1) depending on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq47_HTML.gif so that the new coefficient of the truncation system becomes bounded from below by a uniformly positive constant. (See (2.2).) This allows us to use a fixed point argument for the truncation system. Finally, we show the main result proving that, for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq48_HTML.gif sufficiently large, the solutions of the truncation system are solutions of system (1.1). Observe that, in general, this system has a nonvariational structure.

The paper is organized as follows. In Section 2, we obtain the a priori bounds for the truncation system. In Section 3, we show that the a priori bounds imply a nonexistence result for system (2.4). In Section 4, we introduce a operator of fixed point in cones. In Section 5, we show the existence of positive solutions of the truncation system. In Section 6, we prove the main result, that is, we show the existence of a solution of system (1.1). Finally, in Section 7 we give some remarks.

2. A Priori Bounds for a Truncation System

In this section, we establish a priori bounds for the truncation system. The hypothesis (H3) allows us to find a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq49_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq50_HTML.gif implies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ7_HTML.gif
(2.1)

for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq51_HTML.gif . Thus, we can define for every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq52_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq53_HTML.gif , the functions

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ8_HTML.gif
(2.2)

for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq54_HTML.gif , 2.

In the next section, we will prove the existence of a positive solution for the following truncation system:

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ9_HTML.gif
(2.3)

For this purpose we need to establish a priori bounds for solutions of a family of systems parameterized by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq55_HTML.gif In fact, for every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq56_HTML.gif , consider the family

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ10_HTML.gif
(2.4)

It is not difficult to prove that every solution of system (2.4) satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ11_HTML.gif
(2.5)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq58_HTML.gif are Green's functions given by

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ12_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq59_HTML.gif denotes https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq60_HTML.gif

In order to establish the a priori bound result we need the following two lemmas

Lemma 2.1.

Assume hypotheses (H2) and (H3). Then every solution of system (2.4) satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ13_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq61_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq62_HTML.gif

Proof.

A simple computation shows that every solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq63_HTML.gif satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ14_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq64_HTML.gif

Since

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ15_HTML.gif
(2.9)

we have that (2.7) is proved .

Lemma 2.2.

Assume hypotheses (H2) and (H3). Then Green's functions satisfy

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ16_HTML.gif
(2.10)

where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ17_HTML.gif
(2.11)

Theorem 2.3.

Assume hypotheses (H2) and (H3). Then there is a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq65_HTML.gif which does not depend on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq66_HTML.gif , such that for every solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq67_HTML.gif of system (2.4), we have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ18_HTML.gif
(2.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq68_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq69_HTML.gif .

Proof.

By Lemmas 2.1 and 2.2, every solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq70_HTML.gif of system (2.4) satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ19_HTML.gif
(2.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq71_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq72_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq73_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq74_HTML.gif

Thus,

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ20_HTML.gif
(2.14)

which proves (2.12).

3. A Nonexistence Result

In this section, we see that the a priori bounds imply a nonexistence result for system (2.4).

Theorem 3.1.

System (2.4) has no solution for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq75_HTML.gif sufficiently large.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq76_HTML.gif be a solution of system (2.4), in other words,

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ21_HTML.gif
(3.1)

Then,

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ22_HTML.gif
(3.2)

By Theorem 2.3, we know that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq77_HTML.gif thus

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ23_HTML.gif
(3.3)

which proves Theorem 3.1.

4. Fixed Point Operators

Consider the following Banach space:

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ24_HTML.gif
(4.1)

endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq78_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq79_HTML.gif Define the cone https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq80_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ25_HTML.gif
(4.2)

and the operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq81_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ26_HTML.gif
(4.3)

where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ27_HTML.gif
(4.4)

Note that a simple calculation shows us that the fixed points of the operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq82_HTML.gif are the positive solutions of system (2.4).

Lemma 4.1.

The operator https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq83_HTML.gif is compact, and the cone https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq84_HTML.gif is invariant under https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq85_HTML.gif .

Proof.

The compactness of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq86_HTML.gif follows from the well-known Arzelá-Ascoli theorem. The invariance of the cone https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq87_HTML.gif is a consequence of the fact that the nonlinearities are nonnegative.

In Section 5, we will give an existence result of the truncation system (2.3). The proof will be based on the following well-known fixed point result due to Krasnoselskis, which we state without proof (compare [8, 9]).

Lemma 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq88_HTML.gif be a cone in a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq89_HTML.gif be a compact operator such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq90_HTML.gif . Suppose there exists an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq91_HTML.gif verifying

(a) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq92_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq94_HTML.gif suppose further that there exist a compact homotopy https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq95_HTML.gif and an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq96_HTML.gif such that

(b) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq97_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq98_HTML.gif ;

(c) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq99_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq101_HTML.gif ;

(d) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq102_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq103_HTML.gif

Then https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq104_HTML.gif has a fixed point https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq105_HTML.gif verifying https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq106_HTML.gif

5. Existence Result of Truncation System (2.3)

The following is an existence result of the truncation system.

Theorem 5.1.

Assume hipotheses (H1) through (H3). Then there exists a positive solution of system (2.3).

Proof.

We will verify the hypotheses of Lemma 4.2. Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq107_HTML.gif the cone defined in Section 4 and define the homotopy https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq108_HTML.gif by

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ28_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq109_HTML.gif is a sufficiently large parameter, and where

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ29_HTML.gif
(5.2)

Note that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq110_HTML.gif is a compact homotopy and that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq111_HTML.gif which verifies (b).

On the other hand, we have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ30_HTML.gif
(5.3)

Taking https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq112_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq113_HTML.gif sufficiently small, from hypothesis, we have

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ31_HTML.gif
(5.4)

which verifies (a) of Lemma 4.2. By Theorem 2.3, we clearly have (c).

Finally, choosing https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq114_HTML.gif sufficiently large in the homotopy https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq115_HTML.gif we see that condition (d) of Lemma 4.2 is satisfied by Theorem 3.1. The proof of Theorem 5.1 is now complete.

6. Proof of Main Result Theorem 1.1

The proof of Theorem 1.1 is direct consequence of the following.

Theorem 6.1.

Assume hypotheses (H1) through (H4). Then there exists an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq116_HTML.gif such that every solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq117_HTML.gif of system (2.4) with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq118_HTML.gif satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ32_HTML.gif
(6.1)

Proof.

For otherwise, there would exist a sequence of solutions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq119_HTML.gif of system (2.4) such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq120_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq121_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq122_HTML.gif . Using the same argument as in Theorem 2.3, we would obtain the estimate

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ33_HTML.gif
(6.2)

We have https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq124_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq125_HTML.gif . Moreover, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq126_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq127_HTML.gif . Then

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ34_HTML.gif
(6.3)

which is impossible, since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq129_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq130_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq131_HTML.gif by hypothesis (H4).

7. Remarks

  1. (i)

    We note that the solutions of nonlinear system (1.1) are of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq132_HTML.gif functions in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq133_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq134_HTML.gif almost every where, in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq135_HTML.gif . Note also that when https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq136_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq137_HTML.gif are continuous functions, the solutions of system (1.1) are classic.

     
  2. (ii)

    A little modification of our argument may be done to obtain an existence result of the following more general system:

     
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ35_HTML.gif
(7.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq139_HTML.gif satisfy (H2). In addition, we must assume that there exist continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq140_HTML.gif satisfying (H1) and (H2), and nonnegative functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq141_HTML.gif so that for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_Equ36_HTML.gif
(7.2)

Declarations

Acknowledgment

The authors are supported by FONDECYT, Grant no. 1040990.

Authors’ Affiliations

(1)
Departamento de Matemática y C. C., Universidad de Santiago de Chile

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Copyright

© P. Cerda and P. Ubilla. 2008

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.