The Iterative Method of Generalized -Concave Operators

Abstract

We define the concept of the generalized -concave operators, which generalize the definition of the -concave operators. By using the iterative method and the partial ordering method, we prove the existence and uniqueness of fixed points of this class of the operators. As an example of the application of our results, we show the existence and uniqueness of solutions to a class of the Hammerstein integral equations.

1. Introduction and Preliminary

In [1, 2], Collatz divided the typical problems in computation mathematics into five classes, and the first class is how to solve the operator equation

(1.1)

by the iterative method, that is, construct successively the sequence

(1.2)

for some initial to solve (1.1).

Let be a cone in real Banach space and the partial ordering ≤ defined by , that is, if and only if . The concept and properties of the cone can be found in [3–5]. People studied how to solve (1.1) by using the iterative method and the partial ordering method (see [1–11]).

In [7], Krasnosel'skiĭ gave the concept of concave operators and studied the existence and uniqueness of the fixed point for the operator by the iterative method. The concept of concave operators was defined by Krasnosel'skiĭ as follows.

Let operator and . Suppose that

(i)for any , there exist and , such that

(1.3)

(ii)for any satisfying  , ) and any , there exists , such that

(1.4)

Then is called an concave operator.

In many papers, the authors studied concave operators and obtained some results (see [3–5, 8–15]). In this paper, we generalize the concept of concave operators, give a concept of the generalized concave operators, and study the existence and uniqueness of fixed points for this class of operators by the iterative method. Our results generalize the results in [3, 4, 7, 15].

2. Main Result

In this paper, we always let be a cone in real Banach space and the partial ordering ≤ defined by . Given , let .

Definition 2.1.

Let operator and . Suppose that

(i)for any , there exist and , such that

(2.1)

(ii)for any satisfying (, ) and any , there exists , such that

(2.2)

Then is called a generalized concave operator.

Remark 2.2.

In Definition 2.1, let , we get the definition of the concave operator.

Theorem 2.3.

Let operator be generalized -concave and increasing (i.e., ), then has at most one fixed point in .

Proof.

Let ,  be two different fixed points of , that is, , , and . By Definition 2.1, there exist real numbers , , , , such that

(2.3)

Hence .

Let , , we get that , that is, . Let

(2.4)

hence , that is, , then .

Next we will show that . Assume that ; by (2.2) and (2.4), there exists , such that

(2.5)

By (2.2), there exists , such that

(2.6)

hence,

(2.7)

Therefore,

(2.8)

Obviously, by (2.5) and (2.8), we get

(2.9)

Let , we have

(2.10)

in contradiction to the definition of . Therefore, .

By (2.4), . The proof is completed.

To prove the following Theorem 2.4, we will use the definition of the -norm as follows.

Given , set

(2.11)

It is easy to see that becomes a normed linear space under the norm . is called the norm of the element (see [3, 4]).

Theorem 2.4.

Let operator be increasing and generalized concave. Suppose that has a fixed point in , then, constructing successively the sequence for any initial , we have .

Proof.

Suppose that is generated from . Take , such that . Let , and constructing successively the sequences , . Since is a generalized concave operator, we know that there exists , such that

(2.12)

hence, , then

(2.13)

By (2.2), we can easily get . So it is easy to show that

(2.14)

Let

(2.15)

then,

(2.16)

which implies that the limit of exists. Let , then .

Next we will show that . Suppose that . Since is a generalized concave operator, then there exists , such that

(2.17)

Moreover,

(2.18)

Therefore,

(2.19)

By (2.17) and (2.19), for any , there exists , such that

(2.20)

Particularly, for any , we have

(2.21)

where .

Hence,

(2.22)

By (2.15), and (2.22), we get therefore, , in contradiction to . Hence,

(2.23)

Since is a generalized -concave operator, thus there exist real numbers , , such that , and , we have

(2.24)

Moreover

(2.25)

Hence,

(2.26)

Consequently, by (2.23), we get .

To prove the following Theorem 2.5, we will use the definition of the normal cone as follows.

Let be a cone in . Suppose that there exist constants , such that

(2.27)

then is said to be normal, and the smallest is called the normal constant of (see [3–5]).

Theorem 2.5.

Let be a normal cone of . If operator is increasing and generalized -concave, and is irrelevant to in (2.2), then has the only one fixed point . Moreover, constructing successively the sequence for any initial , we have .

Proof.

Since is a generalized concave operator, hence there exist real numbers , such that . Take small enough, then .

Therefore, , that is, is an increasing sequence and , hence, the limit of exists. Set , then .

Let , where which is irrelevant to is in (2.2), and is increasing, so . By , we can choose a natural number big enough, such that

(2.28)

Let

(2.29)

Obviously, . Since is increasing, we have

(2.30)

Since , we get . Hence

(2.31)

then . It is easy to see

(2.32)

Let

(2.33)

Obviously, . So .

Therefore, , that is, is an increasing sequence and , hence, the limit of exists. Set , then .

Next we will show that . Suppose that , we have the following.

• (i) If for any natural number n, , then

  

  (2.34)

hence,

(2.35)

Taking limits, we have , a contradiction.

• (ii) Suppose that there exists a natural number , such that .

When , so we have

(2.36)

then , a contradiction.

Therefore, .

For any natural numbers , we have

(2.37)

Similarly, . By the normality of and , we get

(2.38)

where is the normal constant of . Hence the limits of and exist. Let , then , hence,

(2.39)

That is, . Let , then .

Taking limits, we get . Hence , that is, is a fixed point of . By Theorem 2.4, the conclusions of Theorem 2.5 hold. The proof is completed.

3. Examples

Example 3.1.

Let , let is continuous}, let, let , , then is a real Banach space and is a normal and solid cone in ( is called solid if it contains interior points, i.e., ). Take , let , , , and .

Considering the Hammerstein integral equation

(3.1)

where is continuous, is increasing for . Suppose that

(1)there exist real numbers , such that , , and ,,

(2)for any and , there exists , such that

(3.2)

Then (3.1) has the only one solution . Moreover, constructing successively the sequence:

(3.3)

for any initial , we have .

Proof.

Considering the operator

(3.4)

Obviously, is increasing. Therefore, (i) of Definition 2.1 is satisfied. For any , by (3.2), we have

(3.5)

Therefore, (ii) of Definition 2.1 is satisfied. Hence the operator is generalized -concave. Consequently, operator satisfies all conditions of Theorem 2.5, thus the conclusion of Example 3.1 holds.

Example 3.2.

Let be a real numbers set, and let , , then is a real Banach space and is a normal and solid cone in . Let . Considering the equation: . Obviously, is a generalized -concave operator and satisfies all the conditions of Theorem 2.5. Hence has the only one fixed point . Moreover, we know by computing.

In Example 3.2, we know that operator doesn't satisfy the definition of -concave operators. Therefore, we can't obtain the fixed point of by the fixed point theorem of -concave operators. The -concave operators' fixed points are all positive, but here 's fixed point is negative.