Positive Solution for a Singular Fourth-Order Differential System ()
1. Introduction
It is well known that the bending of an elastic beam can be described with fourth-order boundary value problems. An elastic beam with its two ends simply supported, can be described by the fourth-order boundary value problem
(1)
(2)
Existence of solutions for problem (1) was established for example by Gupta [1] [2], Liu [3], Ma [4], Ma et al. [5], Ma and Wang [6], Aftabizadeh [7], Yang [8], Del Pino and Manasevich [9], RP Agarwal et al. [10] [11] [12] (see also the references therein). All of those results are based on the Leray-Schauder continuation method, topological degree and the method of lower and upper solutions.
Recently, Wang and An [13] studied the existence of positive solutions for the second-order differential system by using the fixed point theorem of cone expansion and compression.
By applying the cone compression and expansion fixed point theorem, Cui and Zou [14] showed that a fourth-order singular boundary problem has a unique positive solution.
By constructing a new type of cone and using fixed point index theory, López-Somoza and Minhós [15] investigated existence of solutions for the Hammerstein equations.
In [16], the authors use a mixed monotone operator method to investigate the existence of positive solution to a fourth-order boundary value problem which describes the deflection of an elastic beam.
In this paper we shall discuss the existence of positive solutions for the fourth-order boundary value problem
(3)
where
is a positive parameter and
is continuous. In fact as we will see below one could consider in Section 2 and 3
with
and
is continuous provided
here K is as defined in Section 2. Moreover, our hypotheses allow but do not require
to be singular at
.
2. Preliminaries
Let
and
It is well known that Y is a Banach space equipped with the norm
. We denote the norm
by
It is easy to show that
is complete with the norm
and
.
Suppose that
is the Green function associated with
(4)
which is explicitly expressed by
We need the following lemmas.
Lemma 1.
has the following properties:
1)
;
2)
;
3)
;
4)
, for all
Lemma 2. ( [17] ) Let E be a real Banach space and let
be a cone in E. Assume
are open subset of E with
,
, and let
be a completely continuous operator such that either
1)
and
; or
2)
and
.
Then Q has a fixed point in
.
The boundary value problem
can be solved by using the Green’s function, namely,
(5)
Thus inserting (5) into the first equation of (3), we have
(6)
Now we consider the existence of a positive solution of (6). The function
is a positive solution of (6), if
,
, and
.
Then the solution of (6) can be expressed as
(7)
and the second-order derivative
can be expressed by
(8)
Set
Note P is a cone in
. For
, write
. We now define a mapping
by
(9)
It is easy to see that if
than
(10)
where
.
Lemma 3. Let
. Then the following relations hold:
1)
for
, and
2)
for
.
Proof. For simplicity we denote
and
From Lemma 1 it is easy to see that
(11)
(12)
Using (11-12), we have
hence
(H1) Throughout this paper, we assume additionally that the function
satisfies
where
and
is continuous provided
Moreover the function
satisfies.
(H2) There exists an
such that
is non-decreasing in
for each fixed
and
, i.e. if
then
.
(H3) For each fixed
Let us introduce the following notations
Lemma 4. Let (H1), (H2) and (H3) hold. Then for all
where
the following hold
and
where
Proof. It is easy to see that
and
. Let
, then by Lemma 6,
and by Corollary 7,
. Thus
. Also since
we have
,
.
By Lemma 1. and (H1)-(H3) we have
and similarly we also have
Lemma 5.
and
is completely continuous.
Proof. Let
, then we define mapping
by (9). Then for any
, it is clear that
(13)
By Lemma 3,
and
Hence
.
Let
be a bounded set. Then there exists a
, such that
.
First we prove
is bounded. Since
, and
we have
Let
. Now from Lemma 4 we have for any
and
that
(14)
We have a similar type inequality for
.
Therefore
is bounded.
Next we prove that
is equicontinuous. Now from Lemma 4, we have for any
and any
that
We have a similar type inequality for
.
Therefore
is equicontinuous.
Next we prove that T is continuous. Suppose
and
which implies that
uniformly on
. Similarly for
,
uniformly on
and
uniformly on
. The assertion follows from the estimate
and the similar estimate for
by an application of the standard theorem on the convergence of integrals.
The Ascoli-Arzela theorem guarantees that
is completely continuous.
Lemma 6. If
and
, then
, and so,
.
Proof. Since
, there is a
such that
, and so
,
. Hence
,
. Thus
. Since
, we have
,
, and so
. Thus
. Since
and
, we obtain that
.
Corollary 7. Let
and let
. Then
.
3. Main Results
Theorem 1. Let (H1), (H2) and (H3) hold. Assume that the following condition holds
(H4)
and
If
, then problem (3) has at least one positive solution.
Proof. Let us choose
. Then by (H4), there exist
such that
(15)
Let
, then by Corollary 7,
and
. Also since
we have
,
,
.
Let
.
We now show that
To see this, let
, then we have
Thus using by (15) we obtain
Thus, by Lemma 4, and (H1-H2) we have
Consequently,
(16)
Similarly we also have
Hence
Consequently,
(17)
Using (16) and (17) we have
(18)
Let us choose
. Then by condition (H4), there exists
such that
Let
. Let
, i.e.
. Thus by using (10) we have
Then, by Lemma 1, we have
so
Consequently,
Then due to Lemma 2, by (18) and the above inequality we see that the problem (3) has at least one positive solution.
Theorem 2. Let (H1), (H2) and (H3) hold. Assume that the following conditions hold
(H5)
and
(H6) there exists
such that
(19)
If
, then problem (3) has at least two positive solutions.
We note for the argument below that
and
.
Proof. By condition (H6) there exists
such that (19) is fulfilled. Let
, by Corollary 7,
,
. Also since
we have
,
,
.
Let
.
We now show that
To see this, let
, then we have
By condition (H6),
and
, we have
Consequently, we get
(20)
Similarly we also have
Hence
Consequently,
(21)
Using (20) and (21) we have
(22)
Let us choose
. The by condition (H5), there exists
such that
Let
, by Corollary 7,
,
. Also since
we have
Thus by using (10) we have
The estimate for
is similar to that in the proof of Theorem 1 i.e. from Lemma 1 and (H5) we have
Thus
Consequently,
Finally we show that for sufficiently large
, it holds
To see this, we choose
. Due to condition (H5), there exist
such that
Let
. Let
, by Corollary 7,
. Thus by using (10) we have
Then, by Lemma 1, (H1) and (H4), we have
so
Consequently,
Then by Lemma 2, we know that T has at least two fixed points in
and
, i.e. problem (3) has at least two positive solutions.
4. Conclusion
This paper investigates the existence of positive solutions for a singular fourth-order differential system using a fixed point theorem of cone expansion and compression type. The nonlinear terms may be singular with respect to both the time and space variables. The problem comes from the deformation analysis of an elastic beam in the equilibrium state, whose two ends are simply supported. The results obtained herein generalize and improve some known results including singular and non-singular cases.