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

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

(H_{1}) We have

uniformly for all

(H_{2}) There exist
,
, and
for
such that

(H_{3}) The functions
are continuous and

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

whenever
where the inequality is understood inside every component.

(H_{4}) We have

where
.

Here are some comments on the above hypotheses. Hypothesis (H_{1}) is a superlinear condition at 0 and Hypothesis (H_{2}) is a local superlinear condition at
. About hypothesis (H_{3}), 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 (H_{4}) 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 (H_{1}) through (H_{4}). 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.