# 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## 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.

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.

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)]).

*ě*k [3] worked on the existence of two different solutions to the nonlinear differential equation with nonlinear boundary conditions

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

*H*1) Open image in new window is continuous, and there exist Open image in new window Open image in new window , such that

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.

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

Lemma 2.3.

Proof.

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

- (i)
- (ii)
- (iii)For Open image in new window , we have(2.15)Then(2.16)Furthermore,(2.17)which implies(2.18)In particular, it is not hard to obtain(2.19)

Similarly, we can obtain the following lemma.

Lemma 2.4.

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.

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.

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.

Case 2.

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.

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 ).

for Open image in new window .

Subsubcase 1.2.2 ( Open image in new window ).

Case 1.2.3 ( Open image in new window ).

for Open image in new window .

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

Remark 2.7.

Lemma 2.8.

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.

- (a)
- (b)
Open image in new window is a completely continuous operator;

- (c)
Open image in new window for Open image in new window .

First, we take Open image in new window , then

Thus Open image in new window is asserted.

Second, we prove Open image in new window .

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).

Case 1.

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

Case 2.

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.

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.

## 3. The Main Results

Theorem 3.1.

Proof.

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).

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.

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.

- (a)
Open image in new window is a completely continuous operator;

- (b)(3.11)

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

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

- (i)for all Open image in new window ,(3.16)
- (ii)For all Open image in new window ,(3.17)

Case 2.

- (i)
- (ii)
Similarly, we can prove Open image in new window for Open image in new window .

Combining Case 1 with Case 2, we get(3.23)

which contradicts with Open image in new window .

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

Theorem 3.2.

Proof.

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

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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