Nonlinear Systems of Second-Order ODEs

  • Patricio Cerda1 and

    Affiliated with

    • Pedro Ubilla1Email author

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq1_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq2_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq3_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq4_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq5_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq7_HTML.gif . Here, it is assumed that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq9_HTML.gif are nonnegative continuous functions, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq11_HTML.gif are positive continuous functions, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq12_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq13_HTML.gif , and that the nonlinearities http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq14_HTML.gif satisfy superlinear hypotheses at zero and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq16_HTML.gif and http://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:

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq19_HTML.gif are nonnegatives constants, the functions http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq20_HTML.gif are continuous, the functions http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq21_HTML.gif are continuous, and http://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

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

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

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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq29_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq31_HTML.gif are nondecreasing for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq32_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq34_HTML.gif are nondecreasing, meaning that

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

      whenever http://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

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

      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq37_HTML.gif . About hypothesis (H3), the fact that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq38_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq40_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq42_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq45_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq49_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq50_HTML.gif implies

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

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

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

      for http://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:

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq55_HTML.gif In fact, for every http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq56_HTML.gif , consider the family

      http://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

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

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

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

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

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq61_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq63_HTML.gif satisfies

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

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

      Since

      http://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

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

      where

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq65_HTML.gif which does not depend on http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq66_HTML.gif , such that for every solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq67_HTML.gif of system (2.4), we have

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq68_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq70_HTML.gif of system (2.4) satisfies

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

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

      Thus,

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq75_HTML.gif sufficiently large.

      Proof.

      Let http://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,

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

      Then,

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq77_HTML.gif thus

      http://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:

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

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

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

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

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

      where

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

      Proof.

      The compactness of http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq89_HTML.gif be a compact operator such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq90_HTML.gif . Suppose there exists an http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq91_HTML.gif verifying

      (a) http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq92_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq93_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq95_HTML.gif and an http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq96_HTML.gif such that

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

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

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

      Then http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq104_HTML.gif has a fixed point http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq105_HTML.gif verifying http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq108_HTML.gif by

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

      where http://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

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

      Note that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq110_HTML.gif is a compact homotopy and that http://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

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

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

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq114_HTML.gif sufficiently large in the homotopy http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq116_HTML.gif such that every solution http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq117_HTML.gif of system (2.4) with http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq118_HTML.gif satisfies

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq119_HTML.gif of system (2.4) such that http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq120_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq121_HTML.gif with http://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

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

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

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

      which is impossible, since http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq129_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq130_HTML.gif http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq132_HTML.gif functions in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq134_HTML.gif almost every where, in http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq135_HTML.gif . Note also that when http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq136_HTML.gif , http://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:

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq138_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq141_HTML.gif so that for all http://static-content.springer.com/image/art%3A10.1155%2F2008%2F236386/MediaObjects/13661_2007_Article_795_IEq142_HTML.gif

      http://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.