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Advances in Difference Equations

, 2008:586020 | Cite as

Existence and Multiple Solutions for Nonlinear Second-Order Discrete Problems with Minimum and Maximum

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Abstract

Consider the multiplicity of solutions to the nonlinear second-order discrete problems with minimum and maximum: Open image in new window , Open image in new window , Open image in new window , Open image in new window , where Open image in new window are fixed numbers satisfying Open image in new window are satisfying Open image in new window , Open image in new window , Open image in new window .

Keywords

Differential Equation Continuous Function Functional Equation Difference Equation Bounded Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

It is clear that the above are norms on Open image in new window and Open image in new window , respectively, and that the finite dimensionality of these spaces makes them Banach spaces.

In this paper, we discuss the nonlinear second-order discrete problems with minimum and maximum:

where Open image in new window is a continuous function, Open image in new window are fixed numbers satisfying Open image in new window and Open image in new window satisfying   Open image in new window .

Functional boundary value problem has been studied by several authors [1, 2, 3, 4, 5, 6, 7]. But most of the papers studied the differential equations functional boundary value problem [1, 2, 3, 4, 5, 6]. As we know, the study of difference equations represents a very important field in mathematical research [8, 9, 10, 11, 12], so it is necessary to investigate the corresponding difference equations with nonlinear boundary conditions.

Our ideas arise from [1, 3]. In 1993, Brykalov [1] discussed the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions

where Open image in new window is a bounded function, that is, there exists a constant Open image in new window , such that Open image in new window . The proofs in [1] are based on the technique of monotone boundary conditions developed in [2]. From [1, 2], it is clear that the results of [1] are valid for functional differential equations in general form and for some cases of unbounded right-hand side of the equation (see [1, Remark 3 and (5)], [2, Remark 2 and (8)]).

In 1998, Staně k [3] worked on the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions
where Open image in new window satisfies the condition that there exists a nondecreasing function Open image in new window satisfying Open image in new window Open image in new window , such that

It is not difficult to see that when we take Open image in new window , (1.8) is to be (1.7), and Open image in new window may not be bounded.

But as far as we know, there have been no discussions about the discrete problems with minimum and maximum in literature. So, we use the Borsuk theorem [13] to discuss the existence of two different solutions to the second-order difference equation boundary value problem (1.5), (1.6) when Open image in new window satisfies

where Open image in new window .

In our paper, we assume Open image in new window , if Open image in new window .

2. Preliminaries

Definition 2.1.

Remark 2.2.

then boundary condition (1.6) is equal to

So, in the rest part of this paper, we only deal with BVP (1.5), (2.4).

Lemma 2.3.

Furthermore, one has

Proof.

Without loss of generality, we suppose Open image in new window .

  1. (i)
     
  2. (ii)
     
  3. (iii)
    Furthermore,
    which implies
    In particular, it is not hard to obtain
     

Similarly, we can obtain the following lemma.

Lemma 2.4.

In particular, one has

Lemma 2.5.

then there exist Open image in new window , such that Open image in new window .

Proof.

We only prove that there exists Open image in new window , such that Open image in new window , and the other can be proved similarly.

Suppose Open image in new window for Open image in new window . Then Open image in new window . Furthermore, Open image in new window , which contradicts with Open image in new window .

Lemma 2.6.

Suppose Open image in new window is a solution of (1.5) and Open image in new window . Then

Proof.

and Open image in new window be the number of elements in Open image in new window the number of elements in Open image in new window .

If Open image in new window , then Open image in new window ; if Open image in new window , then Open image in new window . Equation (2.24) is obvious.

Now, suppose Open image in new window and Open image in new window . It is easy to see that
At first, we prove the inequality

Since Open image in new window , by Lemma 2.5, there exist Open image in new window , such that Open image in new window . Without loss of generality, we suppose Open image in new window .

For any Open image in new window , there exits Open image in new window satisfying one of the following cases:

Case 1.

Open image in new window ,

Case 2.

Open image in new window .

We only prove that (2.27) holds when Case 1 occurs, (if Case 2 occurs, it can be similarly proved).

If Case 1 holds, we divide the proof into two cases.

Subcase 1.1.

If Open image in new window , without loss of generality, we suppose Open image in new window , then by Lemma 2.3, we have
Combining this with

Subcase 1.2 ( Open image in new window ).

Without loss of generality, we suppose Open image in new window . Then Open image in new window will be discussed in different situations.

Subsubcase 1.2.1 ( Open image in new window ).

By Lemma 2.3 (we take Open image in new window Open image in new window ), it is not difficult to see that
So, we get
At the same time, for Open image in new window ,
Combining this with Open image in new window , we have

for Open image in new window .

Similarly, we get
By (2.39) and (2.41), for Open image in new window ,

Subsubcase 1.2.2 ( Open image in new window ).

By Lemma 2.3 (we take Open image in new window ), it is easy to obtain that
At the same time, for Open image in new window ,
Together with Open image in new window , we have

Case 1.2.3 ( Open image in new window ).

Without loss of generality, we suppose Open image in new window (when Open image in new window , by Lemma 2.4, it can be proved similarly). Then from Lemma 2.3 (we take Open image in new window ), it is not difficult to see that

for Open image in new window .

This being combined with Open image in new window , we get
From (2.50) and (2.52),
At last, from Case 1 and Case 2, we obtain
Then by the definition of Open image in new window and (2.54),
Similarly, we can prove

From (2.26), (2.55), and (2.56), the assertion is proved.

Remark 2.7.

It is easy to see that Open image in new window is continuous, and

Lemma 2.8.

Let Open image in new window be a positive constant as in (2.3), Open image in new window as in (2.3), Open image in new window as in (2.23). Set

where Open image in new window denotes Brouwer degree, and Open image in new window the identity operator on Open image in new window .

Proof.

Obviously, Open image in new window is a bounded open and symmetric with respect to Open image in new window subset of Banach space Open image in new window .

By Borsuk theorem, to prove Open image in new window , we only need to prove that the following hypothesis holds.

  1. (a)
     
  2. (b)

    Open image in new window is a completely continuous operator;

     
  3. (c)
     

Thus Open image in new window is asserted.

Second, we prove Open image in new window .

Let Open image in new window be a sequence. Then for each Open image in new window and the fact Open image in new window Open image in new window . The Bolzano-Weiestrass theorem and Open image in new window is finite dimensional show that, going if necessary to subsequences, we can assume Open image in new window Open image in new window . Then

Since Open image in new window and Open image in new window are continuous, Open image in new window is a continuous operator. Then Open image in new window is a completely continuous operator.

At last, we prove (c).

Assume, on the contrary, that
By (2.67) and Lemma 2.5 (take Open image in new window ), there exists Open image in new window , such that Open image in new window . Also from (2.67), we have Open image in new window , then we get

Case 1.

So, Open image in new window , which contradicts with Open image in new window .

Case 2.

If Open image in new window , then from (2.67), Open image in new window and the definition of Open image in new window , we have
Together with (2.69), we get Open image in new window , and
Furthermore, Open image in new window shows that Open image in new window is strictly increasing. From (2.68) and Lemma 2.5, there exist Open image in new window satisfying Open image in new window . Thus, Open image in new window . It is not difficult to see that
that is,

Similarly, Open image in new window , then we get Open image in new window and Open image in new window , which contradicts with Open image in new window .

Case 3.

By (2.69), we have

If Open image in new window , then Open image in new window . Furthermore, Open image in new window , which contradicts with Open image in new window .

If Open image in new window , then Open image in new window . Furthermore, Open image in new window , which contradicts with Open image in new window .

If Open image in new window , then Open image in new window a contradiction.

Then (c) is proved.

From the above discussion, the conditions of Borsuk theorem are satisfied. Then, we get
Similarly, we can prove

3. The Main Results

Theorem 3.1.

Suppose Open image in new window holds. Then (1.5) and (1.6) have at least two different solutions when Open image in new window and

Proof.

Let Open image in new window . Consider the boundary conditions
Suppose Open image in new window is a solution of (1.5). Then from Remark 2.7,
Now, if (1.5) and (3.2) have a solution Open image in new window , then Lemma 2.6 and (3.2) show that Open image in new window and

So, Open image in new window is a solution of (1.5) and (2.4), that is, Open image in new window is a solution of (1.5) and (1.6).

Similarly, if (1.5), (3.3) have a solution Open image in new window , then Open image in new window and

So, Open image in new window is a solution of (1.5) and (2.4).

Furthermore, since Open image in new window and Open image in new window .

Next, we need to prove BVPs (1.5), (3.2), and (1.5) and (3.3) have solutions, respectively.

Obviously,
Consider the parameter equation

Now, we prove (3.10) has a solution, when Open image in new window .

By Lemma 2.8, Open image in new window . Now we prove the following hypothesis.

  1. (a)

    Open image in new window is a completely continuous operator;

     
  2. (b)
     

Since Open image in new window is finite dimensional, Open image in new window is a completely continuous operator.

Suppose (b) is not true. Then,

From (3.13), Open image in new window is a solution of second-order difference equation Open image in new window . By Remark 2.7, Open image in new window . And from (3.14), there exist Open image in new window , such that Open image in new window . Now, we can prove it in two cases.

Case 1.

If there exists Open image in new window , such that Open image in new window , then

Case 2.

  1. (i)
     
  2. (ii)

    Similarly, we can prove Open image in new window for Open image in new window .

    Combining Case 1 with Case 2, we get
     

which contradicts with Open image in new window .

Similarly, consider the operator Open image in new window ,

we can obtain a solution of BVP (1.5) and (3.3).

Theorem 3.2.

Suppose Open image in new window holds. Then (1.5) and (1.6) have at least two different solutions when Open image in new window and

Proof.

Set Open image in new window . Then Open image in new window satisfies Open image in new window .

By Theorem 3.1,
have at least two difference solutions Open image in new window . Since Open image in new window is a solution of (3.30), if and only if Open image in new window is a solution of (1.5), we see that

are two different solutions of (1.5) and (2.4), then Open image in new window are the two different solutions of (1.5) and (1.6).

Notes

Acknowledgments

This work was supported by the NSFC (Grant no. 10671158), the NSF of Gansu Province (Grant no. 3ZS051-A25-016), NWNU-KJCXGC, the Spring-sun Program (no. Z2004-1-62033), SRFDP (Grant no. 20060736001), and the SRF for ROCS, SEM (2006[311]).

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Copyright information

© R. Ma and C. Gao. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceNorthwest Normal UniversityLanzhouChina

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