Nonlinear Systems of Second-Order ODEs

Abstract

We study existence of positive solutions of the nonlinear system in ; in ; , where and . Here, it is assumed that , are nonnegative continuous functions, , are positive continuous functions, , , and that the nonlinearities satisfy superlinear hypotheses at zero and . 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 and .

1. Introduction

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

(1.1)

where , are nonnegatives constants, the functions are continuous, the functions are continuous, and We will suppose the following four hypotheses.

(H₁) We have

(1.2)

uniformly for all

(H₂) There exist , , and for such that

(1.3)

(H₃) The functions are continuous and

(1.4)

In addition, we suppose that there exists an such that , are nondecreasing for all . Here, , are nondecreasing, meaning that

(1.5)

whenever where the inequality is understood inside every component.

(H₄) We have

(1.6)

where .

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 . About hypothesis (H₃), the fact that , are unbounded leads us to use the strategy of considering a truncation system. Note that if , 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 and , 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 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 and In order to overcome these difficulties, we introduce a truncation of system (1.1) depending on 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 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 so that implies

(2.1)

for all . Thus, we can define for every , such that , the functions

(2.2)

for , 2.

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

(2.3)

For this purpose we need to establish a priori bounds for solutions of a family of systems parameterized by In fact, for every , consider the family

(2.4)

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

(2.5)

Here, , are Green's functions given by

(2.6)

where denotes

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

(2.7)

where with

Proof.

A simple computation shows that every solution satisfies

(2.8)

where

Since

(2.9)

we have that (2.7) is proved .

Lemma 2.2.

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

(2.10)

where

(2.11)

Theorem 2.3.

Assume hypotheses (H₂) and (H₃). Then there is a positive constant which does not depend on , such that for every solution of system (2.4), we have

(2.12)

where with .

Proof.

By Lemmas 2.1 and 2.2, every solution of system (2.4) satisfies

(2.13)

where

Thus,

(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 sufficiently large.

Proof.

Let be a solution of system (2.4), in other words,

(3.1)

Then,

(3.2)

By Theorem 2.3, we know that thus

(3.3)

which proves Theorem 3.1.

4. Fixed Point Operators

Consider the following Banach space:

(4.1)

endowed with the norm where Define the cone by

(4.2)

and the operator by

(4.3)

where

(4.4)

Note that a simple calculation shows us that the fixed points of the operator are the positive solutions of system (2.4).

Lemma 4.1.

The operator is compact, and the cone is invariant under .

Proof.

The compactness of follows from the well-known Arzelá-Ascoli theorem. The invariance of the cone 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 be a cone in a Banach space, and let be a compact operator such that . Suppose there exists an verifying

(a) for all and suppose further that there exist a compact homotopy and an such that

(b) for all ;

(c) for all and ;

(d) for all

Then has a fixed point verifying

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 the cone defined in Section 4 and define the homotopy by

(5.1)

where is a sufficiently large parameter, and where

(5.2)

Note that is a compact homotopy and that which verifies (b).

On the other hand, we have

(5.3)

Taking with sufficiently small, from hypothesis, we have

(5.4)

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

Finally, choosing sufficiently large in the homotopy 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 such that every solution of system (2.4) with satisfies

(6.1)

Proof.

For otherwise, there would exist a sequence of solutions of system (2.4) such that , for all with . Using the same argument as in Theorem 2.3, we would obtain the estimate

(6.2)

We have and with . Moreover, there exists a constant such that . Then

(6.3)

which is impossible, since and by hypothesis (H₄).

7. Remarks

• (i) We note that the solutions of nonlinear system (1.1) are of functions in and almost every where, in . Note also that when , are continuous functions, the solutions of system (1.1) are classic.

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

(7.1)

where , satisfy (H₂). In addition, we must assume that there exist continuous functions satisfying (H₁) and (H₂), and nonnegative functions so that for all

(7.2)