This paper investigates the eigenvalue problem for a class of singular elastic beam
equations where one end is simply supported and the other end is clamped by sliding
clamps. Firstly, we establish 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 nonlinearity 
may be singular at 
and/or 
.
1. Introduction
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:
(11)where 
is a parameter and 
satisfies the following hypothesis:
(
)
, and there exist constants 
, 
, 
, 
,
such that for any 
, 
, 
satisfies
(12)Typical functions that satisfy the above sublinear hypothesis (
) are those taking the form
(13)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:
(14)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:
(15)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:
(1)
, 
;
, 
.
(2)
, 
;
, 
.
Then, 
has a fixed point in 
.
Lemma 1.2 (see [32]).
Let 
be a metric space and 
. Let 
and 
satisfy
(16)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
(17)where 
there exists a sequence 
such that 
.
2. Some Preliminaries and Lemmas
Let 
, 
, then 
is a Banach space, where 
. Define
(21)It is easy to conclude that 
is a cone of 
. Denote
(22)Let
(23)Then 
is the Green function of homogeneous boundary value problem
(24)Lemma 2.1.
, 
, and 
have the following properties:
(1)
, 
, 
, for all 
.
(2)
, 
, 
(or 
), for all 
.
(3)
, 
, 
, for all 
.
(4)
, 
, 
, for all 
.
Proof.
From (2.4), it is easy to obtain the property (2.18).
We now prove that property (2) is true. For 
, by (2.4), we have
(25)For 
, by (2.4), we have
(26)Consequently, property (2) holds.
From property (2), it is easy to obtain property (3).
We next show that property (4) is true. From (2.4), we know that property (4) holds
for 
.
For 
, if 
, then
(27)if 
, then
(28)Therefore, property (4) holds.
Lemma 2.2.
Assume that 
, then 
and
(29)
(210)Proof.
Assume that 
, then 
, 
, 
, so
(211)Therefore, (2.9) holds. From (2.9), we get
(212)By (2.9) and the definition of 
, we can obtain that
(213)Thus, (2.10) holds.
For any fixed 
, define an operator 
by
(214)Then, it is easy to know that
(215)
(216)Lemma 2.3.
Suppose that (
) and
(217)hold. Then 
.
Proof.
From (
), for any 
, 
, 
, we easily obtain the following inequalities:
(218)For every 
, 
, choose positive numbers 
, 
, 
. It follows from (
), (2.10), Lemma 2.1, and (2.17) that
(219)Similar to (2.19), from (
), (2.10), Lemma 2.1, and (2.17), for every 
, 
, we have
(220)Thus, 
is well defined on 
.
From (2.4) and (2.14)–(2.16), it is easy to know that
(221)Therefore, 
follows from (2.21).
Obviously, 
is a positive solution of BVP (1.1) if and only if 
is a positive fixed point of the integral operator 
in 
.
Lemma 2.4.
Suppose that (
) and (2.17) hold. Then for any 
, 
is completely continuous.
Proof.
First of all, notice that 
maps 
into 
by Lemma 2.3.
Next, we show that 
is bounded. In fact, for any 
, by (2.10) we can get
(222)Choose positive numbers 
, 
, 
. This, together with (
), (2.22), (2.16), and Lemma 2.1 yields that
(223)Thus, 
is bounded on 
.
Now we show that 
is a compact operator on 
. By (2.23) and Ascoli-Arzela theorem, it suffices to show that 
is equicontinuous for arbitrary bounded subset 
.
Since for each 
, (2.22) holds, we may choose still positive numbers 
, 
, 
. Then
(224)where 
. Notice that
(225)Thus for any given 
with 
and for any 
, we get
(226)From (2.25), (2.26), and the absolute continuity of integral function, it follows
that 
is equicontinuous.
Therefore, 
is relatively compact, that is, 
is a compact operator on 
.
Finally, we show that 
is continuous on 
. Suppose 
, 
and 
. Then 
,
and 
as 
uniformly, with respect to 
. From 
, choose still positive numbers 
, 
, 
. Then
(227)By (2.17), we know that 
is integrable on 
. Thus, from the Lebesgue dominated convergence theorem, it follows that
(228)Thus, 
is continuous on 
. Therefore, 
is completely continuous.
3. A Necessary and Sufficient Condition for Existence of Positive Solutions
In this section, by using the fixed point theorem of cone, we establish the following necessary and sufficient condition for the existence of positive solutions for BVP (1.1).
Theorem 3.1.
Suppose (
) holds, then BVP (1.1) has at least one positive solution for any 
if and only if the integral inequality (2.17) holds.
Proof.
Suppose first that 
be a positive solution of BVP (1.1) for any fixed 
. Then there exist constants 
(
) with 
, 
such that
(31)In fact, it follows from 
, 
and 
, that 
for 
and 
, 
for 
. By the concavity of 
and 
, we have
(32)On the other hand,
(33)Let 
let
and let
, then (3.1) holds.
Choose positive numbers 
, 
, 
. This, together with (
), (1.2), and (2.18) yields that
(34)where 
. Hence, integrating (3.4) from 
to 1, we obtain
(35)Since 
increases on 
, we get
(36)that is,
(37)Notice that 
, integrating (3.7) from 0 to 1, we have
(38)That is,
(39)Thus,
(310)By an argument similar to the one used in deriving (3.5), we can obtain
(311)where 
. So,
(312)Integrating (3.12) from 0 to 1, we have
(313)That is,
(314)So,
(315)This and (3.10) imply that (2.17) holds.
Now assume that (2.17) holds, we will show that BVP (1.1) has at least one positive
solution for any 
. By (2.17), there exists a sufficient small 
such that
(316)For any fixed 
, first of all, we prove
(317)where 
.
Let 
, then
(318)From Lemma 2.1, (3.18), and (
), we get
(319)Thus, (3.17) holds.
Next, we claim that
(320)where 
.
Let 
, then for 
, we get
(321)Therefore, by Lemma 2.1 and (
), it follows that
(322)This implies that (3.20) holds.
By Lemmas 1.1 and 2.4, (3.17), and (3.20), we obtain that 
has a fixed point in 
. Therefore, BVP (1.1) has a positive solution in 
for any 
.
4. Unbounded Connected Branch of Positive Solutions
In this section, we study the global continua results under the hypotheses (
) and (2.17). Let
(41)then, by Theorem 3.1, 
for any 
.
Theorem 4.1.
Suppose (
) and (2.17) hold, then the closure 
of positive solution set possesses an unbounded connected branch 
which comes from 
such that
(i)for any 
, and
(ii)
Proof.
We now prove our conclusion by the following several steps.
First, we prove that for arbitrarily given 
is bounded. In fact, let
(42)then for 
and 
, we get
(43)Therefore, by Lemma 2.1 and (
), it follows that
(44)Let
(45)where 
is given by (3.16). Then for 
and 
, we get
(46)Therefore, by Lemma 2.1 and (
), it follows that
(47)Therefore, 
has no positive solution in 
. As a consequence, 
is bounded.
By the complete continuity of 
, 
is compact.
Second, we choose sequences 
and 
satisfy
(48)We are to prove that for any positive integer 
, there exists a connected branch 
of 
satisfying
(49)Let 
be fixed, suppose that for any 
, the connected branch 
of 
, passing through 
, leads to 
. Since 
is compact, there exists a bounded open subset 
of 
such that 
, 
, and 
, where 
and later 
denote the closure and boundary of 
with respect to 
. If 
, then 
and 
are two disjoint closed subsets of 
. Since 
is a compact metric space, there are two disjoint compact subsets 
and 
of 
such that 
, 
, and 
. Evidently, 
. Denoting by 
the 
-neighborhood of 
and letting 
, then it follows that
(410)If 
, then taking 
.
It is obvious that in 
, the family of 
makes up an open covering of 
. Since 
is a compact set, there exists a finite subfamily 
which also covers 
. Let 
, then
(411)Hence, by the homotopy invariance of the fixed point index, we obtain
(412)By the first step of this proof, the construction of 
, (4.4), and (4.7), it follows easily that there exist 
such that
(413)
(414)
(415)However, by the excision property and additivity of the fixed point index, we have
from (4.12) and (4.14) that 
, which contradicts (4.15). Hence, there exists some 
such that the connected branch 
of 
containing 
satisfies that 
. Let 
be the connected branch of 
including 
, then this 
satisfies (4.9).
By Lemma 1.2, there exists a connected branch 
of 
such that 
for any 
. Noticing 
, we have 
. Let 
be the connected branch of 
including 
, then 
for any 
. Similar to (4.4) and (4.7), for any 
, 
, we have, by (
), (4.2), (4.3), (4.5), (4.6), and Lemma 2.1,
(416)
(417)where 
is given by (3.16). Let 
in (4.16) and 
in (4.17), we have
(418)Therefore, Theorem 4.1 holds and the proof is complete.
Acknowledgments
This work is carried out while the author is visiting the University of New England. The author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities. The author also thanks the anonymous referees for their carefully reading of the first draft of the manuscript and making many valuable suggestions. Research is supported by the NSFC (10871120) and HESTPSP (J09LA08).
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