Abstract
This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique.
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1. Introduction
In this paper, we are interested in the existence, uniqueness, and iteration of positive solutions for the following nonlinear discrete fourth-order beam equation under Lidstone boundary conditions with explicit parameter 
given by
where 
is the usual forward difference operator given by 
, 
, 
, and 
is a real parameter.
In recent years, the theory of nonlinear difference equations has been widely applied to many fields such as economics, neural network, ecology, and cybernetics, for details, see [1–7] and references therein. Especially, there was much attention focused on the existence and multiplicity of positive solutions of fourth-order problem, for example, [8–10], and in particular the discrete problem with Lidstone boundary conditions [11–17]. However, very little work has been done on the uniqueness and iteration of positive solutions of discrete fourth-order equation under Lidstone boundary conditions. We would like to mention some results of Anderson and Minhós [11] and He and Su [12], which motivated us to consider the BVP (1.1) and (1.2).
In [11], Anderson and Minhós studied the following nonlinear discrete fourth-order equation with explicit parameters 
and 
given by
with Lidstone boundary conditions (1.2), where 
and 
are real parameters. The authors obtained the following result.
Theorem 1.1 (see [11]).
Assume that the following condition is satisfied
, where 
with 
, 
is continuous and nondecreasing, and there exists 
such that 
for 
and 
then, for any 
, the BVP (1.3) and (1.2) has a unique positive solution 
. Furthermore, such a solution 
satisfies the following properties:
-
(i)
and 
; -
(ii)
is nondecreasing in 
; -
(iii)
is continuous in 
, that is, if 
, then 
.
and 
;
is nondecreasing in 
;
is continuous in 
, that is, if 
, then 
.Very recently, in [12], He and Su investigated the existence, multiplicity, and nonexistence of nontrivial solutions to the following discrete nonlinear fourth-order boundary value problem
where 
denotes the forward difference operator defined by 
, 
, 
is the discrete interval given by 
with 
and 
(
) integers, 
are real parameters and satisfy
For the function 
, the authors imposed the following assumption:
(B1)
, where 
with 
, 
is continuous and nondecreasing, and there exists 
such that 
for 
and 
.
Their main result is the following theorem.
Theorem 1.2 (see [12]).
Assume that 
holds. Then for any 
, the BVP (1.4) has a unique positive solution 
. Furthermore, such a solution 
satisfies the properties (i)–(iii) stated in Theorem 1.1.
The aim of this work is to relax the assumptions 
and 
on the nonlinear term, without demanding the existence of upper and lower solutions, we present conditions for the BVP (1.1) and (1.2) to have a unique solution and then study the convergence of the iterative sequence. The ideas come from Zhai et al. [18, 19] and Liang [20].
Let 
denote the Banach space of real-valued functions on 
, with the supremum norm
Throughout this paper, we need the following hypotheses:
(H1)
are continuous and 
for 
(
);
(H2)
with 
;
(H3)
is nondecreasing, 
is nonincreasing, and there exist 
on interval 
with 
, for all
, there exists 
such that 
, and 
for all 
which satisfy
2. Two Lemmas
To prove the main results in this paper, we will employ two lemmas. These lemmas are based on the linear discrete fourth-order equation
with Lidstone boundary conditions (1.2).
Lemma 2.1 (see [11]).
Let 
be a function. Then the nonhomogeneous discrete fourth-order Lidstone boundary value problem (2.1), (1.2) has solution
where 
given by
with 
for 
, is the Green's function for the second-order discrete boundary value problem
and 
given by
is the Green's function for the second-order discrete boundary value problem
Lemma 2.2 (see [11]).
Let
Then, for 
, one has
3. Main Results
Theorem 3.1.
Assume that 
hold. Then, the BVP (1.1) and (1.2) has a unique solution 
in 
, where
Moreover, for any 
, constructing successively the sequences
One has 
converge uniformly to 
in 
.
Proof.
First, we show that the BVP (1.1) and (1.2) has a solution.
It is easy to see that the BVP (1.1) and (1.2) has a solution 
if and only if 
is a fixed point of the operator equation
In view of 
and (3.3), 
is nondecreasing in 
and nonincreasing in 
. Moreover, for any 
, we have
for 
and 
.
Let
condition 
implies 
. Since 
for 
(
), by Lemma 2.2, we have
for 
in (2.1) and 
in (3.5).
Moreover, we obtain
for 
in (2.1).
Thus
Therefore, we can choose a sufficiently small number 
such that
which together with 
implies that there exists 
such that 
, so
Since 
, we can take a sufficiently large positive integer 
such that
It is clear that
We define
Evidently, for 
, 
. Take any 
, then 
and 
.
By the mixed monotonicity of 
, we have 
. In addition, combining 
with (3.10) and (3.11), we get
From 
, we have
and hence
Thus, we have
In accordance with (3.12), we can see that
Construct successively the sequences
By the mixed monotonicity of 
, we have 
. By induction, we obtain 
. It follows from (3.14), (3.18), and the mixed monotonicity of 
that
Note that 
, so we can get 
. Let
Thus, we have
and then
Therefore, 
that is, 
is increasing with 
. Set 
. We can show that 
. In fact, if 
, by 
, there exists 
such that 
. Consider the following two cases.
-
(i)
There exists an integer
such that 
. In this case, we have 
for all 
holds. Hence, for 
, it follows from (3.4) and the mixed monotonicity of 
that 
(3.24)By the definition of
, we have
(3.25)This is a contradiction.
-
(ii)
For all integer
, 
. In this case, we have 
. In accordance with 
, there exists 
such that 
. Hence, combining (3.4) with the mixed monotonicity of 
, we have 
(3.26)
such that 
. In this case, we have 
for all 
holds. Hence, for 
, it follows from (3.4) and the mixed monotonicity of 
that 
, we have
, 
. In this case, we have 
. In accordance with 
, there exists 
such that 
. Hence, combining (3.4) with the mixed monotonicity of 
, we have 
By the definition of 
, we have
Let 
, we have 
and this is also a contradiction. Hence, 
.
Thus, combining (3.20) with (3.22), we have
for 
, where 
is a nonnegative integer. Thus,
Therefore, there exists a function 
such that
By the mixed monotonicity of 
and (3.20), we have
Let 
and we get 
, 
. That is, 
is a nontrivial solution of the BVP (1.1) and (1.2).
Next, we show the uniqueness of solutions of the BVP (1.1) and (1.2). Assume, to the contrary, that there exist two nontrivial solutions 
and 
of the BVP (1.1) and (1.2) such that 
and 
for 
. According to (3.9), we can know that there exists 
such that 
for 
. Let
Then 
and 
for 
.
We now show that 
. In fact, if 
, then, in view of 
, there exists 
such that 
. Furthermore, we have
In (3.34), we used the relation formula (3.16). Since 
, this contradicts the definition of 
. Hence 
. Therefore, the BVP (1.1) and (1.2) has a unique solution.
Finally, we show that "moreover
part of the theorem. For any initial 
, in accordance with (3.9), we can choose a sufficiently small number 
such that
It follows from 
that there exists 
such that 
, and hence
Thus, we can choose a sufficiently large positive integer 
such that
Define
Obviously, 
. Let
for 
. By induction, we get 
, 
, 
.
Similarly to the above proof, it follows that there exists 
such that
By the uniqueness of fixed points 
in 
, we get 
. Therefore, we have
This completes the proof of the theorem.
Remark 3.2.
From the proof of Theorem 3.1, we easily know that assume 
, 
, thus, let 
, 
, we have
Therefore 
.
Theorem 3.3.
Assume that 
holds, and the following conditions are satisfied:
is continuous and 
for 
;
is nondecreasing;
for all 
, where 
, for all 
, there exists 
such that 
, and 
, with 
for all 
;
for fixed 
, one has
-
(i)
is nonincreasing with respect to 
, and there exists 
such that
(3.44)or
-
(ii)
is nondecreasing with respect to 
, and there exists 
such that
(3.45)where
are defined in (2.1), 
is defined in (3.5). Then, the BVP
(3.46)has a unique solution
.
is nonincreasing with respect to 
, and there exists 
such that
is nondecreasing with respect to 
, and there exists 
such that
are defined in (2.1), 
is defined in (3.5). Then, the BVP
.Proof.
For convenience, we still define the operator equation 
by
In the following, we consider the following two cases.
(i) For fixed 
, 
is nonincreasing with respect to 
.
According to condition 
and Lemma 2.2, we can know that there exists 
such that
Since 
, we can find a sufficiently large positive integer 
such that
For 
, we still define
By the proof of Theorem 3.1, it is sufficient to show that
Obviously, 
and 
.
In this case, it follows from conditions 
, 
, and (3.49) that
In accordance with (3.16), we have
which together with condition 
and (3.48) implies that
(ii) For fixed 
, 
is nondecreasing with respect to 
.
In this case, by condition 
and Lemma 2.2, we can know that there exists 
such that
Since 
, we can take a sufficiently large positive integer 
such that
For 
, we still define
We continue to prove that
By (3.52), combining (3.55) with the monotonicity of 
, we have
In accordance with (3.54), combining the monotonicity of 
and (3.55), we get
An application of (3.56) yields
Therefore, we obtain
For 
, the proof is similar and hence omitted. This completes the proof of the theorem.
Remark 3.4.
In Theorem 3.1, the more general conditions are imposed on the nonlinear term than Theorem 1.1. In particular, in Theorem 3.3, 
contains the variable 
; therefore, the more comprehensive functions can be incorporated.
4. An Example
Example 4.1.
Consider the following discrete fourth-order Lidstone problem:
We claim that the BVP (4.1) and (1.2) has a unique solution 
in 
, where
Moreover, for any 
, constructing successively the sequences
we have 
converge uniformly to 
in 
.
In fact, we choose 
, 
, 
, thus 
for 
, 
. It is easy to check that 
is nondecreasing on 
, 
is nonincreasing on 
. In addition, we set
. It is easy to see that
The conclusion then follows from Theorem 3.1.
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The authors were supported financially by the National Natural Science Foundation of China (10971046), the Natural Science Foundation of Shandong Province (ZR2009AM004), and the Youth Science Foundation of Shanxi Province (2009021001-2).
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Sang, Y., Wei, Z. & Dong, W. Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter. Adv Differ Equ 2010, 971540 (2010). https://doi.org/10.1155/2010/971540
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DOI: https://doi.org/10.1155/2010/971540