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Existence and Uniqueness of Positive Solutions for Discrete FourthOrder Lidstone Problem with a Parameter
Advances in Difference Equations volume 2010, Article number: 971540 (2010)
Abstract
This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourthorder 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.
1. Introduction
In this paper, we are interested in the existence, uniqueness, and iteration of positive solutions for the following nonlinear discrete fourthorder 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 fourthorder 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 fourthorder 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 fourthorder 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 .
Very recently, in [12], He and Su investigated the existence, multiplicity, and nonexistence of nontrivial solutions to the following discrete nonlinear fourthorder 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:
(B_{1}), 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 realvalued functions on , with the supremum norm
Throughout this paper, we need the following hypotheses:
(H_{1}) are continuous and for ();
(H_{2}) with ;
(H_{3}) 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 fourthorder equation
with Lidstone boundary conditions (1.2).
Lemma 2.1 (see [11]).
Let be a function. Then the nonhomogeneous discrete fourthorder Lidstone boundary value problem (2.1), (1.2) has solution
where given by
with for , is the Green's function for the secondorder discrete boundary value problem
and given by
is the Green's function for the secondorder 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)
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 "moreoverpart 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 .
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 fourthorder 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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Acknowledgments
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 (20090210012).
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Sang, Y., Wei, Z. & Dong, W. Existence and Uniqueness of Positive Solutions for Discrete FourthOrder Lidstone Problem with a Parameter. Adv Differ Equ 2010, 971540 (2010). https://doi.org/10.1155/2010/971540
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Keywords
 Unique Solution
 Nonlinear Term
 Nonnegative Integer
 Nontrivial Solution
 Operator Equation