Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter

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.

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

(1.1)

(1.2)

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

(1.3)

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 fourth-order boundary value problem

(1.4)

where denotes the forward difference operator defined by , , is the discrete interval given by with and () integers, are real parameters and satisfy

(1.5)

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

(1.6)

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

(1.7)

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

(2.1)

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

(2.2)

where given by

(2.3)

with for , is the Green's function for the second-order discrete boundary value problem

(2.4)

and given by

(2.5)

is the Green's function for the second-order discrete boundary value problem

(2.6)

Lemma 2.2 (see [11]).

Let

(2.7)

Then, for , one has

(2.8)

3. Main Results

Theorem 3.1.

Assume that hold. Then, the BVP (1.1) and (1.2) has a unique solution in , where

(3.1)

Moreover, for any , constructing successively the sequences

(3.2)

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

(3.3)

In view of and (3.3), is nondecreasing in and nonincreasing in . Moreover, for any , we have

(3.4)

for and .

Let

(3.5)

condition implies . Since for (), by Lemma 2.2, we have

(3.6)

for in (2.1) and in (3.5).

Moreover, we obtain

(3.7)

for in (2.1).

Thus

(3.8)

Therefore, we can choose a sufficiently small number such that

(3.9)

which together with implies that there exists such that , so

(3.10)

Since , we can take a sufficiently large positive integer such that

(3.11)

It is clear that

(3.12)

We define

(3.13)

Evidently, for , . Take any , then and .

By the mixed monotonicity of , we have . In addition, combining with (3.10) and (3.11), we get

(3.14)

From , we have

(3.15)

and hence

(3.16)

Thus, we have

(3.17)

In accordance with (3.12), we can see that

(3.18)

Construct successively the sequences

(3.19)

By the mixed monotonicity of , we have . By induction, we obtain . It follows from (3.14), (3.18), and the mixed monotonicity of that

(3.20)

Note that , so we can get . Let

(3.21)

Thus, we have

(3.22)

and then

(3.23)

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

(3.27)

Let , we have and this is also a contradiction. Hence, .

Thus, combining (3.20) with (3.22), we have

(3.28)

for , where is a nonnegative integer. Thus,

(3.29)

Therefore, there exists a function such that

(3.30)

By the mixed monotonicity of and (3.20), we have

(3.31)

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

(3.32)

Then and for .

We now show that . In fact, if , then, in view of , there exists such that . Furthermore, we have

(3.33)

(3.34)

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

(3.35)

It follows from that there exists such that , and hence

(3.36)

Thus, we can choose a sufficiently large positive integer such that

(3.37)

Define

(3.38)

Obviously, . Let

(3.39)

for . By induction, we get , , .

Similarly to the above proof, it follows that there exists such that

(3.40)

By the uniqueness of fixed points in , we get . Therefore, we have

(3.41)

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

(3.42)

Therefore .

Theorem 3.3.

Assume that holds, and the following conditions are satisfied:

is continuous and for ;

is nondecreasing;

(3.43)

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

(3.47)

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

(3.48)

Since , we can find a sufficiently large positive integer such that

(3.49)

For , we still define

(3.50)

By the proof of Theorem 3.1, it is sufficient to show that

(3.51)

Obviously, and .

In this case, it follows from conditions , , and (3.49) that

(3.52)

In accordance with (3.16), we have

(3.53)

which together with condition and (3.48) implies that

(3.54)

(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

(3.55)

Since , we can take a sufficiently large positive integer such that

(3.56)

For , we still define

(3.57)

We continue to prove that

(3.58)

By (3.52), combining (3.55) with the monotonicity of , we have

(3.59)

In accordance with (3.54), combining the monotonicity of and (3.55), we get

(3.60)

An application of (3.56) yields

(3.61)

Therefore, we obtain

(3.62)

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:

(4.1)

We claim that the BVP (4.1) and (1.2) has a unique solution in , where

(4.2)

Moreover, for any , constructing successively the sequences

(4.3)

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

(4.4)

. It is easy to see that

(4.5)

The conclusion then follows from Theorem 3.1.