Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations

In recent years, boundary value problems (BVPs) of differential and difference equations have been studied widely and there are many excellent results (see Gai et al. [1], Guo and Tian [2], Henderson and Peterson [3], and Yang et al. [4]). By using the critical point theory, Deng and Shi [5] studied the existence and multiplicity of the boundary value problems to a class of second-order functional difference equations

(1.1)

with boundary value conditions

(1.2)

where the operator is the Jacobi operator

(1.3)

Ntouyas et al. [6] and Wong [7] investigated the existence of solutions of a BVP for functional differential equations

(1.4)

where is a continuous function, , and .

Weng and Guo [8] considered the following two-point BVP for a nonlinear functional difference equation with -Laplacian operator

(1.5)

where , , , , , , is continuous, .

Yang et al. [9] considered two-point BVP of the following functional difference equation with -Laplacian operator:

(1.6)

where , , and , , , and are nonnegative real constants.

For and , let

(1.7)

Then and are both Banach spaces endowed with the max-norm

(1.8)

For any real function defined on the interval and any with , we denote by an element of defined by , .

In this paper, we consider the following second-order four-point BVP of a nonlinear functional difference equation:

(1.9)

where and , , , , is a continuous function, and for , , , and are nonnegative real constants, and for .

At this point, it is necessary to make some remarks on the first boundary condition in (1.9). This condition is a generalization of the classical condition

(1.10)

from ordinary difference equations. Here this condition connects the history with the single . This is suggested by the well-posedness of BVP (1.9), since the function depends on the term (i.e., past values of ).

As usual, a sequence is said to be a positive solution of BVP (1.9) if it satisfies BVP (1.9) and for with for .

First we consider the nonexistence of positive solutions of (1.9). We have the following result.

Lemma 2.1.

Assume that

(2.1)

or

(2.2)

Then (1.9) has no positive solution.

Proof.

From , we know that is convex for .

Assume that is a positive solution of (1.9) and (2.1) holds.

1. (1)

   Consider that .

    

If , then . It follows that

(2.3)

which is a contradiction to the convexity of .

If , then . If , then we have

(2.4)

Hence

(2.5)

which is a contradiction to the convexity of . If for , then is a trivial solution. So there exists a such that .

We assume that . Then

(2.6)

Hence

(2.7)

which is a contradiction to the convexity of .

If , similar to the above proof, we can also get a contradiction.

1. (2)

   Consider that .

    

Now we have

(2.8)

which is a contradiction to the convexity of .

Assume that is a positive solution of (1.9) and (2.2) holds.

1. (1)

   Consider that .

    

If , then we obtain

(2.9)

which is a contradiction to the convexity of .

If , similar to the above proof, we can also get a contradiction.

If , and so , then there exists a such that . Otherwise, is a trivial solution. Assume that , then

(2.10)

which implies that

(2.11)

A contradiction to the convexity of follows.

If , we can also get a contradiction.

1. (2)

   Consider that .

    

Now we obtain

(2.12)

which is a contradiction to the convexity of .

Next, we consider the existence of the Green's function of equation

(2.13)

We always assume that

( ) , and .

Motivated by Zhao [10], we have the following conclusions.

Theorem 2.2.

The Green's function for second-order four-point linear BVP (2.13) is given by

(2.14)

where

(2.15)

Proof.

Consider the second-order two-point BVP

(2.16)

It is easy to find that the solution of BVP (2.16) is given by

(2.17)

(2.18)

The three-point BVP

(2.19)

can be obtained from replacing by in (2.16). Thus we suppose that the solution of (2.19) can be expressed by

(2.20)

where and are constants that will be determined.

From (2.18) and (2.20), we have

(2.21)

Putting the above equations into (2.19) yields

(2.22)

By (H ), we obtain and by solving the above equation:

(2.23)

By (2.19) and (2.20), we have

(2.24)

The four-point BVP (2.13) can be obtained from replacing by in (2.19). Thus we suppose that the solution of (2.13) can be expressed by

(2.25)

where and are constants that will be determined.

From (2.24) and (2.25), we get

(2.26)

Putting the above equations into (2.13) yields

(2.27)

By (H ), we can easily obtain

(2.28)

Then by (2.17), (2.20), (2.23), (2.25), and (2.28), the solution of BVP (2.13) can be expressed by

(2.29)

where is defined in (2.14). That is, is the Green's function of BVP (2.13).

Remark 2.3.

By (H ), we can see that for . Let

(2.30)

Then .

Lemma 2.4.

Assume that ( ) holds. Then the second-order four-point BVP (2.13) has a unique solution which is given in (2.29).

Proof.

We need only to show the uniqueness.

Obviously, in (2.29) is a solution of BVP (2.13). Assume that is another solution of BVP (2.13). Let

(2.31)

Then by (2.13), we have

(2.32)

(2.33)

From (2.32) we have, for ,

(2.34)

which implies that

(2.35)

Combining (2.33) with (2.35), we obtain

(2.36)

Condition (H ) implies that (2.36) has a unique solution . Therefore for . This completes the proof of the uniqueness of the solution.

In this section, we discuss the BVP (1.9).

Assume that , .

We rewrite BVP (1.9) as

(3.1)

with .

Suppose that is a solution of the BVP (3.1). Then it can be expressed as

(3.2)

Lemma 3.1 (see Guo et al. [11]).

Assume that is a Banach space and is a cone in . Let . Furthermore, assume that is a completely continuous operator and for . Thus, one has the following conclusions:

1. (1)

   if  for , then ;

    
2. (2)

   if  for , then .

    

Assume that . Then (3.1) may be rewritten as

(3.3)

Let be a solution of (3.3). Then by (3.2) and , it can be expressed as

(3.4)

Let be a solution of BVP (3.1) and . Then for we have and

(3.5)

Let

(3.6)

Then is a Banach space endowed with norm and is a cone in .

For , we have by (H ) and the definition of ,

(3.7)

For every , , by the definition of and (3.5), if , we have

(3.8)

If , we have, by (3.4),

(3.9)

hence by the definition of , we obtain for

(3.10)

Lemma 3.2.

For every , there is , such that

(3.11)

Proof.

For , and , by the definitions of and , we have

(3.12)

Obviously, there is a , such that (3.11) holds.

Define an operator by

(3.13)

Then we may transform our existence problem of positive solutions of BVP (3.1) into a fixed point problem of operator (3.13).

Lemma 3.3.

Consider that .

Proof.

If and , and , respectively. Thus, (H ) yields

(3.14)

It follows from the definition of that

(3.15)

which implies that .

Lemma 3.4.

Suppose that ( ) holds. Then is completely continuous.

We assume that

(H )

(H ) .

We have the following main results.

Theorem 3.5.

Assume that ( )–( ) hold. Then BVP (3.1) has at least one positive solution if the following conditions are satisfied:

(H ) there exists a such that, for , if , then ;

(H ) there exists a such that, for , if , then

or

(H ) ;

(H ) there exists a such that, for , if , then ;

(H ) there exists an , such that, for , if , then

where

(3.16)

Proof.

Assume that ( ) and ( ) hold. For every , we have , thus

(3.17)

which implies by Lemma 3.1 that

(3.18)

For every , by (3.8)–(3.10) and Lemma 3.2, we have, for , . Then by (3.13) and ( ), we have

(3.19)

which implies by Lemma 3.1 that

(3.20)

So by (3.18) and (3.20), there exists one positive fixed point of operator with .

Assume that ( )–( ) hold, for every and , , by ( ), we have

(3.21)

Thus we have from Lemma 3.1 that

(3.22)

For every , by (3.8)–(3.10), we have ,

(3.23)

Thus we have from Lemma 3.1 that

(3.24)

So by (3.22) and (3.24), there exists one positive fixed point of operator with .

Consequently, or is a positive solution of BVP (3.1).

Theorem 3.6.

Assume that ( )–( ) hold. Then BVP (3.1) has at least one positive solution if ( ) and ( ) or ( ) and ( ) hold.

Theorem 3.7.

Assume that ( )–( ) hold. Then BVP (3.1) has at least two positive solutions if (H ), (H ), and (H )  or (H ), (H ), and (H ) hold.

Theorem 3.8.

Assume that (H )–(H ) hold. Then BVP (3.1) has at least three positive solutions if (H )–(H ) hold.

Assume that , , and

( )

Define as follows:

(3.25)

which satisfies

H .

Obviously, exists.

Assume that is a solution of (1.9). Let

(3.26)

where

(3.27)

By (1.9), (3.26), (3.27), (H ), (H ), and the definition of , we have

(3.28)

(3.29)

and, for ,

(3.30)

Let

(3.31)

Then by (3.27), (H ), (H ), and the definition of , we have for Thus, the BVP (1.9) can be changed into the following BVP:

(3.32)

with and .

Similar to the above proof, we can show that (1.9) has at least one positive solution. Consequently, (1.9) has at least one positive solution.

Example 3.9.

Consider the following BVP:

(3.33)

That is,

(3.34)

Then we obtain

(3.35)

Let

(3.36)

where

By calculation, we can see that (H )–(H ) hold, then by Theorem 3.8, the BVP (3.33) has at least three positive solutions.

In this section, we consider the following BVP with parameter :

(4.1)

with .

The BVP (4.1) is equivalent to the equation

(4.2)

Let be the solution of (3.3), . Then we have

(4.3)

Let and be defined as the above. Define by

(4.4)

Then solving the BVP (4.1) is equivalent to finding fixed points in . Obviously is completely continuous and keeps the invariant for .

Define

(4.5)

respectively. We have the following results.

Theorem 4.1.

Assume that (H ), (H ), (H ),

(H )   ,

(H )   

hold, where , then BVP (4.1) has at least one positive solution, where is a positive constant.

Proof.

Assume that condition (H ) holds. If and , there exists an sufficiently small, such that

(4.6)

By the definition of , there is an , such that for ,

(4.7)

It follows that, for and ,

(4.8)

For every and , by (3.9), we have

(4.9)

Therefore by (3.13) and Lemma 3.2, we have

(4.10)

If then for a sufficiently small , we have Similar to the above, for every , we obtain by (3.10)

(4.11)

If , choose sufficiently large, such that

(4.12)

By the definition of , there is an such that, for and

(4.13)

For every , by (3.8)–(3.10) and (3.13), we have

(4.14)

which implies that

(4.15)

Finally, we consider the assumption . By the definition of , there is

, such that, for and ,

(4.16)

We now show that there is    , such that, for , In fact, for   and every , hence in a similar way, we have

(4.17)

which implies that

(4.18)

Theorem 4.2.

Assume that (H ),(H ), and (H ) hold. If or , then there is a such that for , BVP (4.1) has at least one positive solution.

Proof.

Let be given. Define

(4.19)

Then , where .

For every , we know that . By the definition of operator , we obtain

(4.20)

It follows that we can take such that, for all and all ,

(4.21)

Fix . If , for , we obtain a sufficiently large such that, for ,

(4.22)

It follows that, for and ,

(4.23)

For every , by the definition of , and the definition of Lemma 3.2, there exists a such that and , thus . Hence

(4.24)

If there is , such that, for and ,

(4.25)

where .

For every , by(3.8)–(3.10) and Lemma 3.2,

(4.26)

which by combining with (4.21) completes the proof.

Example 4.3.

Consider the BVP(3.33) in Example 3.9 with

(4.27)

where , is some positive constant, .

By calculation, , , and ; let . Then by Theorem(4.1), for , the above equation has at least one positive solution.