Research ArticleFixed Point Theorems on Nonlinear Binary OperatorEquations with Applications

Baomin Qiao

Department of Mathematics, Shangqiu Normal College, Shangqiu 476000, China

Correspondence should be addressed to Baomin Qiao; bmqiao@126.com

Received 18 April 2014; Revised 11 June 2014; Accepted 11 June 2014; Published 19 June 2014

Academic Editor: Krzysztof Ciepliński

Copyright © 2014 Baomin Qiao.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone andpartial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some correspondingresults are obtained. Finally, the applications of our results are given.

1. Introduction

In recent years, more and more scholars have studied binaryoperator equations and have obtained many conclusions; see[1–6]. In this paper, we will discuss solutions for these equa-tionswhich associatedwith an ordinal symmetric contractionoperator and obtain some results which generalized andimproved those of [3–6]. Finally, we apply our conclusions totwo-point boundary value problem with two-degree super-linear ordinary differential equations.

In the following, let𝐸 always be a realBanach spacewhichis partially ordered by a cone 𝑃, let 𝑃 be a normal cone of 𝐸,𝑁 is normal constant of 𝑃, partial order ≤ is determined by𝑃 and 𝜃 denotes zero element of 𝐸. Let 𝑢, V ∈ 𝐸, 𝑢 < V, 𝐷 =[𝑢, V] = {𝑥 ∈ 𝐸 : 𝑢 ≤ 𝑥 ≤ V} denote an ordering interval of 𝐸.

For the concepts of normal cone and partially order,mixed monotone operator, coupled solutions of operatorequations, and so forth see [1, 5].

Definition 1. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be a binary operator. 𝐴 issaid to be 𝐿-ordering symmetric contraction operator if thereexists a bounded linear and positive operator 𝐿 : 𝐸 → 𝐸,where spectral radius 𝑟(𝐿) < 1 such that 𝐴(𝑦, 𝑥) − 𝐴(𝑥, 𝑦) ≤𝐿(𝑦 − 𝑥) for any 𝑥, 𝑦 ∈ 𝐷, 𝑥 ≤ 𝑦, where 𝐿 is called acontraction operator of 𝐴.

2. Main Results

Theorem 2. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetriccontraction operator, and there exists a 𝛼 ∈ [0, 1) such that

𝐴 (𝑥2, 𝑦2) − 𝐴 (𝑥

1, 𝑦1) ≥ −𝛼 (𝑥

2− 𝑥1) ,

𝑢 ≤ 𝑥1≤ 𝑥2≤ V, 𝑢 ≤ 𝑦

2≤ 𝑦1≤ V.

(1)

If condition (H1) 𝑢 ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V − 𝛼(V − 𝑢) or (H

2)

𝑢 + 𝛼(V − 𝑢) ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V holds, then the followingstatements hold.

(C1) 𝐴(𝑥, 𝑥) = 𝑥 has a unique solution 𝑥∗ ∈ 𝐷, and for

any coupled solutions 𝑥, 𝑦 ∈ 𝐷, 𝑥 = 𝑦 = 𝑥∗.(C2) For any 𝑥

0, 𝑦0∈ 𝐷, we construct symmetric iterative

sequences:

𝑥𝑛=

1

𝛼 + 1[𝐴 (𝑥𝑛−1

, 𝑦𝑛−1

) + 𝛼𝑥𝑛−1

] ,

𝑦𝑛=

1

𝛼 + 1[𝐴 (𝑦𝑛−1

, 𝑥𝑛−1

) + 𝛼𝑦𝑛−1

] ,

𝑛 = 1, 2, 3, . . . .

(2)

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 241942, 4 pageshttp://dx.doi.org/10.1155/2014/241942

2 Discrete Dynamics in Nature and Society

Then 𝑥𝑛

→ 𝑥∗, 𝑦𝑛

→ 𝑥∗(𝑛 → ∞), and for any 𝛽 ∈

(𝑟(𝐿), 1), there exists a natural number 𝑚; and if 𝑛 ≥ 𝑚, weget error estimates for iterative sequences (2):

𝑥𝑛 (𝑦𝑛) − 𝑥∗ ≤ 2𝑁(

𝛼 + 𝛽

𝛼 + 1)

𝑛

‖𝑢 − V‖ . (3)

Proof. Set𝐵(𝑥, 𝑦) = (1/(𝛼+1))[𝐴(𝑥, 𝑦)+𝛼𝑥], and if condition(H1) or (H

2) holds, then it is obvious that

𝑢 ≤ 𝐵 (𝑢, V) , 𝐵 (V, 𝑢) ≤ V. (4)

By (1), we easily prove that𝐵 : 𝐷×𝐷 → 𝐸 ismixedmonotoneoperator, and for any 𝑥, 𝑦 ∈ 𝐷, 𝑢 ≤ 𝑥 ≤ 𝑦 ≤ V,

𝜃 ≤ 𝐵 (𝑦, 𝑥) − 𝐵 (𝑥, 𝑦) ≤ 𝐻 (𝑦 − 𝑥) , (5)

where𝐻 = (1/(𝛼+1))(𝐿+𝛼𝐼) is a bounded linear and positiveoperator and 𝐼, is identical operator.

By the mathematical induction, we easily prove that

𝜃 ≤ 𝐵𝑛(𝑦, 𝑥) − 𝐵

𝑛(𝑥, 𝑦) ≤ 𝐻

𝑛(𝑦 − 𝑥) , 𝑢 ≤ 𝑥 ≤ 𝑦 ≤ V,

(6)

where 𝐵𝑛(𝑥, 𝑦) = 𝐵(𝐵𝑛−1(𝑥, 𝑦), 𝐵𝑛−1(𝑦, 𝑥)), 𝑥, 𝑦 ∈ 𝐷, 𝑛 ≥ 2.By the character of normal cone 𝑃, it is shown that𝐵𝑛(𝑦, 𝑥) − 𝐵

𝑛(𝑥, 𝑦)

≤ 𝑁𝐻𝑛

𝑦 − 𝑥 , 𝑢 ≤ 𝑥 ≤ 𝑦 ≤ V.

(7)

For any 𝛽 ∈ (𝑟(𝐿), 1), since lim𝑛→∞

||𝐻𝑛||1/𝑛

= 𝑟(𝐻) ≤

(𝛼 + 𝑟(𝐿))/(𝛼 + 1) < (𝛼 +𝛽)/(𝛼 + 1) < 1, there exists a naturalnumber𝑚, and if 𝑛 ≥ 𝑚, we have ||𝐻𝑛|| < ((𝛼 + 𝛽)/(𝛼 + 1))𝑛,and𝑁||𝐻𝑚|| < 1. Considering mixed monotone operator 𝐵𝑚and constant𝑁||𝐻𝑚||, 𝐵𝑚(𝑥, 𝑥) = 𝑥 has a unique solution 𝑥∗and for any coupled solution 𝑥, 𝑦 ∈ 𝐷, such that 𝑥 = 𝑦 = 𝑥∗byTheorem 3 in [3].

From 𝐵𝑚(𝐵(𝑥∗, 𝑥∗), 𝐵(𝑥∗, 𝑥∗)) = 𝐵(𝐵𝑚(𝑥∗, 𝑥∗), 𝐵𝑚(𝑥∗,𝑥∗)) = 𝐵(𝑥

∗, 𝑥∗), and the uniqueness of solution with

𝐵𝑚(𝑥, 𝑥) = 𝑥, then we have 𝐵(𝑥∗, 𝑥∗) = 𝑥∗ and 𝐴(𝑥∗, 𝑥∗) =

𝑥∗.We take note of that 𝐴(𝑥, 𝑥) = 𝑥 and 𝐵(𝑥, 𝑥) = 𝑥 have

the same coupled solution; therefore, a coupled solution for𝐵(𝑥, 𝑥) = 𝑥 must be a coupled solution for 𝐵𝑚(𝑥, 𝑥) = 𝑥;consequently, (C

1) has been proved.

Considering iterative sequence (2), we construct iterativesequences:

𝑢𝑛= 𝐵 (𝑢

𝑛−1, V𝑛−1

) , V𝑛= 𝐵 (V

𝑛−1, 𝑢𝑛−1

) , (8)

where 𝑢0= 𝑢, V

0= V, it is obvious that

𝑥𝑛= 𝐵 (𝑥

𝑛−1, 𝑦𝑛−1

) , 𝑦𝑛= 𝐵 (𝑦

𝑛−1, 𝑥𝑛−1

) ,

𝜃 ≤ V𝑛− 𝑢𝑛≤ 𝐻𝑛(V − 𝑢) ,

(9)

by themathematical induction and characterization ofmixedmonotone of 𝐵; then

𝑢𝑛≤ 𝑥∗≤ V𝑛, 𝑢

𝑛≤ 𝑥𝑛≤ V𝑛, 𝑢

𝑛≤ 𝑦𝑛≤ V𝑛. (10)

Hence,𝑥𝑛 (𝑦𝑛) − 𝑢𝑛

≤ 𝑁V𝑛 − 𝑢𝑛

,

𝑥∗− 𝑢𝑛

≤ 𝑁V𝑛 − 𝑢𝑛

,

𝑛 = 1, 2, 3, . . . .

(11)

Moreover, if 𝑛 ≥ 𝑚, we get𝑥𝑛 (𝑦𝑛) − 𝑥

∗ ≤ 2𝑁V𝑛 − 𝑢𝑛

≤ 2𝑁𝐻𝑛 ‖V − 𝑢‖ ≤ 2𝑁(

𝛼 + 𝛽

𝛼 + 1)

𝑛

‖𝑢 − V‖ .(12)

Consequently, 𝑥𝑛→ 𝑥∗, 𝑦𝑛→ 𝑥∗(𝑛 → ∞).

Remark 3. When 𝛼 = 0, Theorem 1 in [4] is a special case ofthis paper Theorem 2 under condition (H

1) or (H

2).

Corollary 4. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetriccontraction operator; if there exists a 𝛼 ∈ [0, 1) such that 𝐴satisfies condition of Theorem 2, the following statement holds.

(C3) For any 𝛽 ∈ (𝑟(𝐿), 1) and 𝛼+𝛽 < 1, we make iterative

sequences:

𝑢𝑛= 𝐴 (𝑢

𝑛−1, V𝑛−1

) ,

V𝑛= 𝐴 (V

𝑛−1, 𝑢𝑛−1

) + 𝛼 (V𝑛−1

− 𝑢𝑛−1

) ,

𝑛 = 1, 2, 3, . . . ,

(13)

or𝑢𝑛= 𝐴 (𝑢

𝑛−1, V𝑛−1

) − 𝛼 (V𝑛−1

− 𝑢𝑛−1

) ,

V𝑛= 𝐴 (V

𝑛−1, 𝑢𝑛−1

) ,

𝑛 = 1, 2, 3, . . . ,

(14)

where 𝑢0= 𝑢, V

0= V.

Thus, 𝑢𝑛

→ 𝑥∗, V𝑛

→ 𝑥∗(𝑛 → ∞), and there exists a

natural number 𝑚, and if 𝑛 ≥ 𝑚, we have error estimates foriterative sequences (13) or (14):

𝑢𝑛 (V𝑛) − 𝑥∗ ≤ 𝑁(𝛼 + 𝛽)

𝑛

‖𝑢 − V‖ . (15)

Proof. By the character of mixedmonotone of𝐴, then (1) and(C1), (C2) [in (1), (C

2) where 𝛼 = 0] hold.

In the following, we will prove (C3).

Consider iterative sequence (13); since 𝑢 ≤ 𝑥∗ ≤ V, we get

𝑢1= 𝐴 (𝑢, V) ≤ 𝐴 (𝑥∗, 𝑥∗) = 𝑥∗ ≤ 𝐴 (V, 𝑢)

= V1− 𝛼 (V − 𝑢) ≤ V

1.

(16)

By themathematical induction, we easily prove 𝑢𝑛≤ 𝑥∗≤ V𝑛,

𝑛 ≥ 1, hence

𝜃 ≤ 𝑥∗− 𝑢𝑛≤ V𝑛− 𝑢𝑛, 𝜃 ≤ V

𝑛− 𝑥∗≤ V𝑛− 𝑢𝑛. (17)

It is clear that𝜃 ≤ V𝑛− 𝑢𝑛≤ (𝐿 + 𝛼𝐼) (V

𝑛−1− 𝑢𝑛−1

)

= (𝐿 + 𝛼𝐼)𝑛(V − 𝑢) , 𝑛 ≥ 1.

(18)

Discrete Dynamics in Nature and Society 3

For any 𝛽 ∈ (𝑟(𝐿), 1), 𝛼 + 𝛽 < 1, since

lim𝑛→∞

(𝐿 + 𝛼𝐼)𝑛

1/𝑛

= 𝑟 (𝐿 + 𝛼𝐼) ≤ 𝑟 (𝐿) + 𝛼 < 𝛼 + 𝛽 < 1,

(19)

there exists a natural number𝑚, if 𝑛 ≥ 𝑚, such that(𝐿 + 𝛼𝐼)

𝑛 < (𝛼 + 𝛽)𝑛

. (20)

Moreover,𝑢𝑛 (V𝑛) − 𝑥

∗ ≤ 𝑁(𝐿 + 𝛼𝐼)

𝑛 ‖𝑢 − V‖

≤ 𝑁(𝛼 + 𝛽)𝑛

‖𝑢 − V‖ , (𝑛 ≥ 𝑚) .(21)

Consequently, 𝑢𝑛→ 𝑥∗, V𝑛→ 𝑥∗, (𝑛 → ∞).

Similarly, we can prove (14).

Theorem 5. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be a 𝐿-ordering symmetriccontraction operator; if there exists a 𝛼 ∈ [0, 1) such that (1 −𝛼)𝑢 ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ (1−𝛼)V, then the following statementshold.

(C4) Operator equation 𝐴(𝑥, 𝑥) = (1 − 𝛼)𝑥 has a unique

solution 𝑥∗ ∈ 𝐷, and for any coupled solutions 𝑥, 𝑦 ∈ 𝐷, 𝑥 =𝑦 = 𝑥

∗.(C5) For any 𝑥

0, 𝑦0, 𝑤0, 𝑧0

∈ 𝐷, we make symmetriciterative sequences

𝑥𝑛=

1

1 − 𝛼𝐴 (𝑥𝑛−1

, 𝑦𝑛−1

) ,

𝑦𝑛=

1

1 − 𝛼𝐴 (𝑦𝑛−1

, 𝑥𝑛−1

) ,

𝑛 = 1, 2, 3, . . . ,

(22)

𝑤𝑛= 𝐴 (𝑤

𝑛−1, 𝑧𝑛−1

) + 𝛼𝑤𝑛−1

,

𝑧𝑛= 𝐴 (𝑧

𝑛−1, 𝑤𝑛−1

) + 𝛼𝑧𝑛−1

,

𝑛 = 1, 2, 3, . . . .

(23)

Then 𝑥𝑛

→ 𝑥∗, 𝑦𝑛

→ 𝑥∗, 𝑤𝑛

→ 𝑥∗, 𝑧𝑛

→ 𝑥∗(𝑛 → ∞),

and for any 𝛽 ∈ (𝑟(𝐿), 1), 𝛼 + 𝛽 < 1, there exists a naturalnumber 𝑚, and if 𝑛 ≥ 𝑚, then we have error estimates foriterative sequences (22) and (23), respectively,

𝑥𝑛 (𝑦𝑛) − 𝑥∗ ≤ 2𝑁(

𝛽

1 − 𝛼)

𝑛

‖𝑢 − V‖ ,

𝑤𝑛 (𝑧𝑛) − 𝑥∗ ≤ 2𝑁(𝛼 + 𝛽)

𝑛

‖𝑢 − V‖ .

(24)

Proof. Set 𝐵(𝑥, 𝑦) = (1/(1 − 𝛼))𝐴(𝑥, 𝑦) or 𝐶(𝑥, 𝑦) =𝐴(𝑥, 𝑦) + 𝛼𝑥; we can prove that this theorem imitates proofof Theorem 2.

Similarly, we can prove the following theorems.

Theorem 6. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetriccontraction operator; if there exists a 𝛼 ∈ [0, 1) such that 𝑢 +𝛼V ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V + 𝛼𝑢, then the following statementshold.

(C6) Equation 𝐴(𝑥, 𝑥) = (1 + 𝛼)𝑥 has a unique solution

𝑥∗∈ 𝐷, and for any coupled solutions 𝑥, 𝑦 ∈ 𝐷𝑥 = 𝑦 = 𝑥∗.(C7) For any 𝑥

0, 𝑦0

∈ 𝐷, we make symmetric iterativesequence:

𝑥𝑛=

1

1 + 𝛼𝐴 (𝑥𝑛−1

, 𝑦𝑛−1

) ,

𝑦𝑛=

1

1 + 𝛼𝐴 (𝑦𝑛−1

, 𝑥𝑛−1

) ,

𝑛 = 1, 2, 3, . . . .

(25)

Then𝑥𝑛→ 𝑥∗,𝑦𝑛→ 𝑥∗(𝑛 → ∞); moreover,𝛽 ∈ (𝑟(𝐿), 1),

and there exists natural number𝑚, and if 𝑛 ≥ 𝑚, then we haveerror estimates for iterative sequence (25):

𝑥𝑛 (𝑦𝑛) − 𝑥∗ ≤ 2𝑁(

𝛽

𝛼 + 1)

𝑛

‖𝑢 − V‖ , (26)

(C8) For any 𝛽 ∈ (𝑟(𝐿), 1)(𝛼+𝛽 < 1), 𝑤

0, 𝑧0∈ 𝐷, we make

symmetry iterative sequence 𝑤𝑛= 𝐴(𝑤

𝑛−1, 𝑧𝑛−1

) − 𝛼𝑧𝑛−1

, 𝑧𝑛=

𝐴(𝑧𝑛−1

, 𝑤𝑛−1

) − 𝛼𝑤𝑛−1

, 𝑛 ≥ 1; then 𝑤𝑛

→ 𝑥∗, 𝑧𝑛

→

𝑥∗(𝑛 → ∞), and there exists a natural number 𝑚, and if

𝑛 ≥ 𝑚, we have error estimates for iterative sequence (24).

Remark 7. When 𝛼 = 0, Corollary 2 in [4] is a special case ofthis paper Theorems 2–6.

Remark 8. The contraction constant of operator in [5] isexpand into the contraction operator of this paper.

Remark 9. Operator 𝐴 of this paper does not need characterof mixed monotone as operator in [6].

3. Application

We consider that two-point boundary value problem for two-degree super linear ordinary differential equations:

𝑥+ 𝑎 (𝑡) 𝑥

𝑚+

1

1 + 𝑏 (𝑡) 𝑥= 0, 𝑡 ∈ [0, 1] , (𝑚 ≥ 2)

𝑥 (0) = 𝑥(1) = 0.

(27)

Let 𝑘(𝑡, 𝑠) be Green function with boundary value prob-lem (23); that is,

𝑘 (𝑡, 𝑠) = min {𝑡, 𝑠} = {𝑡, 𝑡 ≤ 𝑠

𝑠, 𝑠 < 𝑡.(28)

Then the solution with boundary value problem (23) andsolution for nonlinear integral equation with type of Ham-merstein

𝑥 (𝑡) = ∫

1

0

𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥 (𝑠)]𝑚+

1

1 + 𝑏 (𝑠) 𝑥 (𝑠)} 𝑑𝑠 (29)

are equivalent, where max𝑡∈[0,1]

∫1

0𝑘(𝑡, 𝑠)𝑑𝑠 = 1/2.

Theorem10. Let 𝑎(𝑡), 𝑏(𝑡) be nonnegative continuous functionin [0, 1], 𝑝 = max

𝑡∈[0,1]𝑎(𝑡), 𝑞 = max

𝑡∈[0,1]𝑏(𝑡). If 𝑝 < 1,𝑚𝑝 +

4 Discrete Dynamics in Nature and Society

𝑞 < 2, then boundary value problem (23) has a unique solution𝑥∗(𝑡) such that 0 ≤ 𝑥∗(𝑡) ≤ 1 (𝑡 ∈ [0, 1]). Moreover, for any

initial function 𝑥0(𝑡), 𝑦0(𝑡), such that

0 ≤ 𝑥0(𝑡) ≤ 1, 0 ≤ 𝑦

0(𝑡) ≤ 1 (𝑡 ∈ [0, 1]) , (30)

we make iterative sequence:

𝑥𝑛(𝑡) = ∫

1

0

𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥𝑛−1

(𝑠)]𝑚

+1

1 + 𝑏 (𝑠) 𝑦𝑛−1

(𝑠)} 𝑑𝑠,

𝑦𝑛(𝑡) = ∫

1

0

𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑦𝑛−1

(𝑠)]𝑚

+1

1 + 𝑏 (𝑠) 𝑥𝑛−1

(𝑠)} 𝑑𝑠,

𝑛 = 1, 2, 3, . . . .

(31)

Then 𝑥𝑛(𝑡) and 𝑦

𝑛(𝑡) are all uniformly converge to 𝑥∗(𝑡) on

[0, 1], and we have error estimates:

𝑥𝑛 (𝑡) (𝑦𝑛 (𝑡)) − 𝑥∗(𝑡)

≤ 2(𝑚𝑝 + 𝑞

2)

𝑛

,

𝑡 ∈ [0, 1] , 𝑛 = 1, 2, 3, . . . .

(32)

Proof. Let 𝐸 = 𝐶[0, 1], 𝑃 = {𝑥 ∈ 𝐸 | 𝑥(𝑡) ≥ 0, 𝑡 ∈[0, 1]}, ‖𝑥‖ = max

𝑡∈[0,1]|𝑥(𝑡)| denote norm of; then 𝐸 has

become 𝐵𝑎𝑛𝑎𝑐ℎ space, 𝑃 is normal cone of 𝐸, and its normalconstant 𝑁 = 1. It is obvious that integral Equation (24)transforms to operator equation 𝐴(𝑥, 𝑥) = 𝑥, where

𝐴 (𝑥, 𝑦) (𝑡)

= ∫

1

0

𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥 (𝑠)]𝑚+

1

1 + 𝑏 (𝑠) 𝑦 (𝑠)} 𝑑𝑠, 𝑡 ∈ [0, 1] .

(33)

Set 𝑢 = 𝑢(𝑡) ≡ 0, V = V(𝑡) ≡ 1; then 𝐷 = [0, 1] denoteordering interval of 𝐸, 𝐴 : 𝐷 × 𝐷 → 𝐸 is mixed monotoneoperator, and 0 ≤ 𝐴(0, 1), 𝐴(1, 0) ≤ (1 + 𝑝)/2 < 1.

Set

𝐿𝑥 (𝑡) = ∫

1

0

𝑘 (𝑠, 𝑡) [𝑚𝑎 (𝑠) + 𝑏 (𝑠)] 𝑥 (𝑠) 𝑑𝑠, 𝑡 ∈ [0, 1] .

(34)

Then 𝐿 : 𝐸 → 𝐸 is bounded linear operator, its spectralradius 𝑟(𝐿) ≤ (𝑚𝑝+𝑞)/2 < 1, and for any𝑥, 𝑦 ∈ 𝐸, 0 ≤ 𝑥(𝑡) ≤𝑦(𝑡) ≤ 1 such that 0 ≤ 𝐴(𝑦, 𝑥)(𝑡)−𝐴(𝑥, 𝑦)(𝑡) ≤ 𝐿(𝑦−𝑥)(𝑡),𝐴is 𝐿-ordering symmetric contraction operator, by Theorem 2(where 𝛼 = 0); thenTheorem 10 has been proved.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by the NSF of Henan EducationBureau (2000110019) and by the NSF of Shangqiu(200211125).

References

[1] D. J. Guo and V. Lakshmikantham, “Coupled fixed points ofnonlinear operators with applications,” Nonlinear Analysis:Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.

[2] Y. Sun, “A fixed point theorem for mixed monotone operatorswith applications,” Journal of Mathematical Analysis and Appli-cations, vol. 156, no. 1, pp. 240–252, 1991.

[3] Q. Zhang, “Contraction mapping principle of mixed monotonemapping and applications,”Henan Science, vol. 18, no. 2, pp. 121–125, 2000.

[4] J. X. Sun and L. S. Liu, “An iterative solutionmethod for nonlin-ear operator equations and its applications,” Acta MathematicaScientia A, vol. 13, no. 2, pp. 141–145, 1993.

[5] Q. Zhang, “Iterative solutions of ordering symmetric contrac-tion operator with applications,” Journal of Engineering Mathe-matics, vol. 17, no. 2, pp. 131–134, 2000.

[6] X. L. Yan, “Fixed-point theorems formixedmonotone operatorsand their applications,”Mathematica Applicata, vol. 4, no. 4, pp.107–114, 1991.

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