Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. Therefore, singular boundary value problems have been investigated extensively in recent years (see [1–4] and references therein).

This paper investigates the following fourth-order nonlinear singular eigenvalue problem:

where
is a parameter and
satisfies the following hypothesis:

()

, and there exist constants

,

,

,

,

such that for any

,

,

satisfies

Typical functions that satisfy the above sublinear hypothesis (
) are those taking the form

where
,
,
,
,
,
,
. The hypothesis (
) is similar to that in [5, 6].

Because of the extensive applications in mechanics and engineering, nonlinear fourth-order two-point boundary value problems have received wide attentions (see [7–12] and references therein). In mechanics, the boundary value problem (1.1) (BVP (1.1) for short) describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. The term
in
represents bending effect which is useful for the stability analysis of the beam. BVP (1.1) has two special features. The first one is that the nonlinearity
may depend on the first-order derivative of the unknown function
, and the second one is that the nonlinearity
may be singular at
and/or
.

In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP (1.1). Firstly, we construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from
. Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.

By singularity of
, we mean that the function
in (1.1) is allowed to be unbounded at the points
,
,
, and/or
. A function
is called a (positive) solution of the BVP (1.1) if it satisfies the BVP (1.1) (
for
and
for
). For some
, if the
(1.1) has a positive solution
, then
is called an eigenvalue and
is called corresponding eigenfunction of the BVP (1.1).

The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see [

7–

20] and the references therein. Yao [

15,

18] studied the following BVP:

where

is a closed subset and

,

. In [

15], he obtained a sufficient condition for the existence of positive solutions of

(1.4) by using the monotonically iterative technique. In [

13,

18], he applied Guo-Krasnosel'skii's fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP (1.4) and the following BVP:

These differ from our problem because
in (1.4) cannot be singular at
,
and the nonlinearity
in (1.5) does not depend on the derivatives of the unknown functions.

In this paper, we first establish a necessary and sufficient condition for the existence of positive solutions of BVP (1.1) for any
by using the following Lemma 1.1. Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs by the lower and upper solution method can be found, for example, in [5, 6, 21–23]. In [5, 6, 22, 23] they considered the case that
depends on even order derivatives of
. Although the nonlinearity
in [21] depends on the first-order derivative, where the nonlinearity
is increasing with respect to the unknown function
. Papers [24, 25] derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity
does not depend on the derivatives of the unknown functions, and
is decreasing with respect to
.

Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated (see [26–28] and references therein). Ma and An [26] and Ma and Xu [27] discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems (see [29, 30]). The terms
in [26] and
in [27] are not singular at
,
,
. Yao [14] obtained one or two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using Guo-Krasnosel'skii's fixed point theorem, but the global structure of positive solutions was not considered. Since the nonlinearity
in BVP (1.1) may be singular at
and/or
, the global bifurcation theorems in [29, 30] do not apply to our problem here. In Section 4, we also investigate the global structure of positive solutions for BVP (1.1) by applying the following Lemma 1.2.

The paper is organized as follows: in the rest of this section, two known results are stated. In Section 2, some lemmas are stated and proved. In Section 3, we establish a necessary and sufficient condition for the existence of positive solutions. In Section 4, we prove that the closure of positive solution set possesses an unbounded connected branch which comes from
.

Finally we state the following results which will be used in Sections 3 and 4, respectively.

Lemma 1.1 (see [31]).

Let
be a real Banach space, let
be a cone in
, and let
,
be bounded open sets of
,
. Suppose that
is completely continuous such that one of the following two conditions is satisfied:

Then,
has a fixed point in
.

Lemma 1.2 (see [32]).

Let

be a metric space and

. Let

and

satisfy

Suppose also that
is a family of connected subsets of
, satisfying the following conditions:

and
for each
.

(2)For any two given numbers
and
with
,
is a relatively compact set of
.

Then there exists a connected branch

of

such that

where
there exists a sequence
such that
.