It is well known that beam is one of the basic structures in architecture. It is greatly used in the designing of bridge and construction. Recently, scientists bring forward the theory of combined beams. That is to say, we can bind up some stratified structure copings into one global combined beam with rock bolts. The deformations of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by following equation of deflection curve:

where
is Yang's modulus constant,
is moment of inertia with respect to
axes, determined completely by the beam's shape cross-section. Specially,
if the cross-section is a rectangle with a height of
and a width of
Also,
is loading at
. If the loading of beam considered is in relation to deflection and rate of change of deflection, we need to research the more general equation

According to the forms of supporting, various boundary conditions should be considered. Solving corresponding boundary value problems, one can obtain the expression of deflection curve. It is the key in design of constants of beams and rock bolts.

Owing to its importance in physics and engineering, the existence of solutions to this problem has been studied by many authors, see [1–10]. However, in practice, only its positive solution is significant. In [1, 9, 11, 12], Aftabizadeh, Del Pino and Manásevich, Gupta, and Pao showed the existence of positive solution for

under some growth conditions of
and a nonresonance condition involving a two-parameter linear eigenvalue problem. All of these results are based on the Leray-Schauder continuation method and topological degree.

The lower and upper solution method has been studied for the fourth-order problem by several authors [2, 3, 7, 8, 13, 14]. However, all of these authors consider only an equation of the form

with diverse kind of boundary conditions. In [10], Ehme et al. gave some sufficient conditions for the existence of a solution of

with some quite general nonlinear boundary conditions by using the lower and upper solution method. The conditions assume the existence of a strong upper and lower solution pair.

Recently, Krasnosel'skii's fixed point theorem in a cone has much application in studying the existence and multiplicity of positive solutions for differential equation boundary value problems, see [3, 6]. With this fixed point theorem, Bai and Wang [6] discussed the existence, uniqueness, multiplicity, and infinitely many positive solutions for the equation of the form

where
is a constant.

In this paper, via a new fixed point theorem in a cone and concavity of function, we show the existence of positive solutions for the following problem:

where
is continuous.

We point out that positive solutions of (1.7) are concave and this concavity provides lower bounds on positive concave functions of their maximum, which can be used in defining a cone on which a positive operator is defined, to which a new fixed point theorem in a cone due to Bai and Ge [5] can be applied to obtain positive solutions.