Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter
© Yanbin Sang et al. 2010
Received: 9 January 2010
Accepted: 26 March 2010
Published: 30 March 2010
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.
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  and He and Su , which motivated us to consider the BVP (1.1) and (1.2).
In , 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 ).
Assume that the following condition is satisfied
, where with , is continuous and nondecreasing, and there exists such that for and
is nondecreasing in ;
is continuous in , that is, if , then .
Very recently, in , 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 ).
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 .
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 ;
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 ).
Lemma 2.2 (see ).
3. Main Results
One has converge uniformly to in .
First, we show that the BVP (1.1) and (1.2) has a solution.
for and .
for in (2.1) and in (3.5).
for in (2.1).
Evidently, for , . Take any , then and .
- (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 thatBy the definition of , we have(3.24)(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)
Let , we have and this is also a contradiction. Hence, .
Let and we get , . That is, is a nontrivial solution of the BVP (1.1) and (1.2).
Then and for .
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.
for . By induction, we get , , .
This completes the proof of the theorem.
Assume that holds, and the following conditions are satisfied:
is continuous and for ;
for all , where , for all , there exists such that , and , with for all ;
- (i)is nonincreasing with respect to , and there exists such that(3.44)
- (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 .
In the following, we consider the following two cases.
(i) For fixed , is nonincreasing with respect to .
Obviously, and .
(ii) For fixed , is nondecreasing with respect to .
For , the proof is similar and hence omitted. This completes the proof of the theorem.
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
we have converge uniformly to in .
The conclusion then follows from Theorem 3.1.
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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