Fixed Point Theorems on Nonlinear Binary Operator Equations with

Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 241942, 4 pages http://dx.doi.org/10.1155/2014/241942

Research Article Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications Baomin Qiao Department of Mathematics, Shangqiu Normal College, Shangqiu 476000, China Correspondence should be addressed to Baomin Qiao; [email protected] Received 18 April 2014; Revised 11 June 2014; Accepted 11 June 2014; Published 19 June 2014 Academic Editor: Krzysztof Ciepli´nski Copyright © 2014 Baomin Qiao. 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. The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.

1. Introduction In recent years, more and more scholars have studied binary operator equations and have obtained many conclusions; see [1–6]. In this paper, we will discuss solutions for these equations which associated with an ordinal symmetric contraction operator and obtain some results which generalized and improved those of [3–6]. Finally, we apply our conclusions to two-point boundary value problem with two-degree superlinear ordinary differential equations. In the following, let 𝐸 always be a real Banach space which is partially ordered by a cone 𝑃, let 𝑃 be a normal cone of 𝐸, 𝑁 is normal constant of 𝑃, partial order ≤ is determined by 𝑃 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 operator equations, and so forth see [1, 5]. Definition 1. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be a binary operator. 𝐴 is said to be 𝐿-ordering symmetric contraction operator if there exists 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 a contraction operator of 𝐴.

2. Main Results Theorem 2. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetric contraction operator, and there exists a 𝛼 ∈ [0, 1) such that 𝐴 (𝑥2 , 𝑦2 ) − 𝐴 (𝑥1 , 𝑦1 ) ≥ −𝛼 (𝑥2 − 𝑥1 ) , 𝑢 ≤ 𝑥1 ≤ 𝑥2 ≤ V, 𝑢 ≤ 𝑦2 ≤ 𝑦1 ≤ V.

(1)

If condition (H1 ) 𝑢 ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V − 𝛼(V − 𝑢) or (H2 ) 𝑢 + 𝛼(V − 𝑢) ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V holds, then the following statements 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 𝑥0 , 𝑦0 ∈ 𝐷, we construct symmetric iterative sequences:

𝑥𝑛 =

1 [𝐴 (𝑥𝑛−1 , 𝑦𝑛−1 ) + 𝛼𝑥𝑛−1 ] , 𝛼+1

𝑦𝑛 =

1 [𝐴 (𝑦𝑛−1 , 𝑥𝑛−1 ) + 𝛼𝑦𝑛−1 ] , 𝛼+1 𝑛 = 1, 2, 3, . . . .

(2)

2

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Then 𝑥𝑛 → 𝑥∗ , 𝑦𝑛 → 𝑥∗ (𝑛 → ∞), and for any 𝛽 ∈ (𝑟(𝐿), 1), there exists a natural number 𝑚; and if 𝑛 ≥ 𝑚, we get error estimates for iterative sequences (2):

Hence, 󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩𝑥𝑛 (𝑦𝑛 ) − 𝑢𝑛 󵄩󵄩󵄩 ≤ 𝑁 󵄩󵄩󵄩V𝑛 − 𝑢𝑛 󵄩󵄩󵄩 , 󵄩 󵄩󵄩 ∗ 󵄩 󵄩 󵄩󵄩𝑥 − 𝑢𝑛 󵄩󵄩󵄩 ≤ 𝑁 󵄩󵄩󵄩V𝑛 − 𝑢𝑛 󵄩󵄩󵄩 ,

𝑛

𝛼+𝛽 󵄩󵄩 ∗󵄩 ) ‖𝑢 − V‖ . 󵄩󵄩𝑥𝑛 (𝑦𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 2𝑁( 𝛼+1

(3)

Proof. Set 𝐵(𝑥, 𝑦) = (1/(𝛼+1))[𝐴(𝑥, 𝑦)+𝛼𝑥], and if condition (H1 ) or (H2 ) holds, then it is obvious that 𝑢 ≤ 𝐵 (𝑢, V) ,

𝐵 (V, 𝑢) ≤ V.

𝜃 ≤ 𝐵 (𝑦, 𝑥) − 𝐵 (𝑥, 𝑦) ≤ 𝐻 (𝑦 − 𝑥) ,

(5)

where 𝐻 = (1/(𝛼+1))(𝐿+𝛼𝐼) is a bounded linear and positive operator and 𝐼, is identical operator. By the mathematical induction, we easily prove that 𝜃 ≤ 𝐵𝑛 (𝑦, 𝑥) − 𝐵𝑛 (𝑥, 𝑦) ≤ 𝐻𝑛 (𝑦 − 𝑥) ,

𝑛 = 1, 2, 3, . . . . Moreover, if 𝑛 ≥ 𝑚, we get 󵄩󵄩 󵄩 󵄩 ∗󵄩 󵄩󵄩𝑥𝑛 (𝑦𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 2𝑁 󵄩󵄩󵄩V𝑛 − 𝑢𝑛 󵄩󵄩󵄩

(4)

By (1), we easily prove that 𝐵 : 𝐷×𝐷 → 𝐸 is mixed monotone operator, and for any 𝑥, 𝑦 ∈ 𝐷, 𝑢 ≤ 𝑥 ≤ 𝑦 ≤ V,

𝑢 ≤ 𝑥 ≤ 𝑦 ≤ V, (6)

𝑛

𝛼+𝛽 󵄩 󵄩 ≤ 2𝑁 󵄩󵄩󵄩𝐻𝑛 󵄩󵄩󵄩 ‖V − 𝑢‖ ≤ 2𝑁( ) ‖𝑢 − V‖ . 𝛼+1

Remark 3. When 𝛼 = 0, Theorem 1 in [4] is a special case of this paper Theorem 2 under condition (H1 ) or (H2 ). Corollary 4. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetric contraction operator; if there exists a 𝛼 ∈ [0, 1) such that 𝐴 satisfies condition of Theorem 2, the following statement holds. (C3 ) For any 𝛽 ∈ (𝑟(𝐿), 1) and 𝛼 + 𝛽 < 1, we make iterative sequences:

𝑢𝑛 = 𝐵 (𝑢𝑛−1 , V𝑛−1 ) ,

V𝑛 = 𝐵 (V𝑛−1 , 𝑢𝑛−1 ) ,

(8)

𝑦𝑛 = 𝐵 (𝑦𝑛−1 , 𝑥𝑛−1 ) ,

𝜃 ≤ V𝑛 − 𝑢𝑛 ≤ 𝐻𝑛 (V − 𝑢) ,

(9)

by the mathematical induction and characterization of mixed monotone of 𝐵; then 𝑢𝑛 ≤ 𝑥∗ ≤ V𝑛 ,

𝑢𝑛 ≤ 𝑥𝑛 ≤ V𝑛 ,

V𝑛 = 𝐴 (V𝑛−1 , 𝑢𝑛−1 ) + 𝛼 (V𝑛−1 − 𝑢𝑛−1 ) ,

𝑢𝑛 ≤ 𝑦𝑛 ≤ V𝑛 .

(10)

(13)

𝑛 = 1, 2, 3, . . . , or 𝑢𝑛 = 𝐴 (𝑢𝑛−1 , V𝑛−1 ) − 𝛼 (V𝑛−1 − 𝑢𝑛−1 ) , V𝑛 = 𝐴 (V𝑛−1 , 𝑢𝑛−1 ) ,

(14) 𝑛 = 1, 2, 3, . . . ,

where 𝑢0 = 𝑢, V0 = V. Thus, 𝑢𝑛 → 𝑥∗ , V𝑛 → 𝑥∗ (𝑛 → ∞), and there exists a natural number 𝑚, and if 𝑛 ≥ 𝑚, we have error estimates for iterative sequences (13) or (14): 𝑛 󵄩󵄩 ∗󵄩 (15) 󵄩󵄩𝑢𝑛 (V𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 𝑁(𝛼 + 𝛽) ‖𝑢 − V‖ . Proof. By the character of mixed monotone of 𝐴, then (1) and (C1 ), (C2 ) [in (1), (C2 ) 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) ≤ V1 .

where 𝑢0 = 𝑢, V0 = V, it is obvious that 𝑥𝑛 = 𝐵 (𝑥𝑛−1 , 𝑦𝑛−1 ) ,

𝑢𝑛 = 𝐴 (𝑢𝑛−1 , V𝑛−1 ) ,

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 natural number 𝑚, and if 𝑛 ≥ 𝑚, we have ||𝐻𝑛 || < ((𝛼 + 𝛽)/(𝛼 + 1))𝑛 , and 𝑁||𝐻𝑚 || < 1. Considering mixed monotone operator 𝐵𝑚 and constant 𝑁||𝐻𝑚 ||, 𝐵𝑚 (𝑥, 𝑥) = 𝑥 has a unique solution 𝑥∗ and for any coupled solution 𝑥, 𝑦 ∈ 𝐷, such that 𝑥 = 𝑦 = 𝑥∗ by Theorem 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 for 𝐵(𝑥, 𝑥) = 𝑥 must be a coupled solution for 𝐵𝑚 (𝑥, 𝑥) = 𝑥; consequently, (C1 ) has been proved. Considering iterative sequence (2), we construct iterative sequences:

(12)

Consequently, 𝑥𝑛 → 𝑥∗ , 𝑦𝑛 → 𝑥∗ (𝑛 → ∞).

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 that 󵄩 󵄩 𝑛󵄩 󵄩 󵄩 󵄩󵄩 𝑛 𝑛 󵄩󵄩𝐵 (𝑦, 𝑥) − 𝐵 (𝑥, 𝑦)󵄩󵄩󵄩 ≤ 𝑁 󵄩󵄩󵄩𝐻 󵄩󵄩󵄩 󵄩󵄩󵄩𝑦 − 𝑥󵄩󵄩󵄩 ,

(11)

(16)

By the mathematical induction, we easily prove 𝑢𝑛 ≤ 𝑥∗ ≤ V𝑛 , 𝑛 ≥ 1, hence 𝜃 ≤ 𝑥∗ − 𝑢𝑛 ≤ V𝑛 − 𝑢𝑛 ,

𝜃 ≤ V𝑛 − 𝑥∗ ≤ V𝑛 − 𝑢𝑛 .

(17)

It is clear that 𝜃 ≤ V𝑛 − 𝑢𝑛 ≤ (𝐿 + 𝛼𝐼) (V𝑛−1 − 𝑢𝑛−1 ) = (𝐿 + 𝛼𝐼)𝑛 (V − 𝑢) ,

𝑛 ≥ 1.

(18)


3

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

(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 𝑥0 , 𝑦0 ∈ 𝐷, we make symmetric iterative sequence:

󵄩 󵄩1/𝑛 lim 󵄩󵄩(𝐿 + 𝛼𝐼)𝑛 󵄩󵄩󵄩 = 𝑟 (𝐿 + 𝛼𝐼) ≤ 𝑟 (𝐿) + 𝛼 < 𝛼 + 𝛽 < 1, (19)

𝑛 → ∞󵄩

there exists a natural number 𝑚, if 𝑛 ≥ 𝑚, such that 𝑛 󵄩󵄩 𝑛󵄩 󵄩󵄩(𝐿 + 𝛼𝐼) 󵄩󵄩󵄩 < (𝛼 + 𝛽) .

(20)

𝑥𝑛 =

1 𝐴 (𝑥𝑛−1 , 𝑦𝑛−1 ) , 1+𝛼

𝑦𝑛 =

1 𝐴 (𝑦𝑛−1 , 𝑥𝑛−1 ) , 1+𝛼

Moreover, 󵄩 󵄩󵄩 ∗󵄩 𝑛󵄩 󵄩󵄩𝑢𝑛 (V𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 𝑁 󵄩󵄩󵄩(𝐿 + 𝛼𝐼) 󵄩󵄩󵄩 ‖𝑢 − V‖ 𝑛

≤ 𝑁(𝛼 + 𝛽) ‖𝑢 − V‖ ,

(𝑛 ≥ 𝑚) .

𝑛 = 1, 2, 3, . . . . (21)

Consequently, 𝑢𝑛 → 𝑥∗ , V𝑛 → 𝑥∗ , (𝑛 → ∞). Similarly, we can prove (14). Theorem 5. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be a 𝐿-ordering symmetric contraction 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 statements hold. (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 𝑥0 , 𝑦0 , 𝑤0 , 𝑧0 ∈ 𝐷, we make symmetric iterative sequences 𝑥𝑛 =

1 𝐴 (𝑥𝑛−1 , 𝑦𝑛−1 ) , 1−𝛼

𝑦𝑛 =

1 𝐴 (𝑦𝑛−1 , 𝑥𝑛−1 ) , 1−𝛼

(22)

𝑧𝑛 = 𝐴 (𝑧𝑛−1 , 𝑤𝑛−1 ) + 𝛼𝑧𝑛−1 ,

(23)

𝑛 = 1, 2, 3, . . . . ∗

𝛽 𝑛 󵄩󵄩 ∗󵄩 ) ‖𝑢 − V‖ , 󵄩󵄩𝑥𝑛 (𝑦𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 2𝑁( 𝛼+1

(26)

(C8 ) For any 𝛽 ∈ (𝑟(𝐿), 1)(𝛼 + 𝛽 < 1), 𝑤0 , 𝑧0 ∈ 𝐷, we make symmetry iterative sequence 𝑤𝑛 = 𝐴(𝑤𝑛−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 of this paper Theorems 2–6.

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

3. Application

𝑤𝑛 = 𝐴 (𝑤𝑛−1 , 𝑧𝑛−1 ) + 𝛼𝑤𝑛−1 ,

∗

Then 𝑥𝑛 → 𝑥∗ , 𝑦𝑛 → 𝑥∗ (𝑛 → ∞); moreover, 𝛽 ∈ (𝑟(𝐿), 1), and there exists natural number 𝑚, and if 𝑛 ≥ 𝑚, then we have error estimates for iterative sequence (25):

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

𝑛 = 1, 2, 3, . . . ,

∗

(25)

We consider that two-point boundary value problem for twodegree super linear ordinary differential equations: 𝑥󸀠󸀠 + 𝑎 (𝑡) 𝑥𝑚 +

∗

Then 𝑥𝑛 → 𝑥 , 𝑦𝑛 → 𝑥 , 𝑤𝑛 → 𝑥 , 𝑧𝑛 → 𝑥 (𝑛 → ∞), and for any 𝛽 ∈ (𝑟(𝐿), 1), 𝛼 + 𝛽 < 1, there exists a natural number 𝑚, and if 𝑛 ≥ 𝑚, then we have error estimates for iterative sequences (22) and (23), respectively, 𝛽 𝑛 󵄩󵄩 ∗󵄩 ) ‖𝑢 − V‖ , 󵄩󵄩𝑥𝑛 (𝑦𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 2𝑁( 1−𝛼 𝑛 󵄩󵄩 ∗󵄩 󵄩󵄩𝑤𝑛 (𝑧𝑛 ) − 𝑥 󵄩󵄩󵄩 ≤ 2𝑁(𝛼 + 𝛽) ‖𝑢 − V‖ .

Similarly, we can prove the following theorems. Theorem 6. Let 𝐴 : 𝐷 × 𝐷 → 𝐸 be 𝐿-ordering symmetric contraction operator; if there exists a 𝛼 ∈ [0, 1) such that 𝑢 + 𝛼V ≤ 𝐴(𝑢, V), 𝐴(V, 𝑢) ≤ V + 𝛼𝑢, then the following statements hold.

𝑡 ∈ [0, 1] , (𝑚 ≥ 2) (27)

󸀠

𝑥 (0) = 𝑥 (1) = 0. Let 𝑘(𝑡, 𝑠) be Green function with boundary value problem (23); that is, 𝑡, 𝑘 (𝑡, 𝑠) = min {𝑡, 𝑠} = { 𝑠,

(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 proof of Theorem 2.

1 = 0, 1 + 𝑏 (𝑡) 𝑥

𝑡≤𝑠 𝑠 < 𝑡.

(28)

Then the solution with boundary value problem (23) and solution for nonlinear integral equation with type of Hammerstein 1

𝑥 (𝑡) = ∫ 𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥 (𝑠)]𝑚 + 0

1 } 𝑑𝑠 (29) 1 + 𝑏 (𝑠) 𝑥 (𝑠)

1

are equivalent, where max𝑡∈[0,1] ∫0 𝑘(𝑡, 𝑠)𝑑𝑠 = 1/2. Theorem 10. Let 𝑎(𝑡), 𝑏(𝑡) be nonnegative continuous function in [0, 1], 𝑝 = max𝑡∈[0,1] 𝑎(𝑡), 𝑞 = max𝑡∈[0,1] 𝑏(𝑡). If 𝑝 < 1, 𝑚𝑝 +

4


𝑞 < 2, then boundary value problem (23) has a unique solution 𝑥∗ (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 ≤ 𝑦0 (𝑡) ≤ 1

(𝑡 ∈ [0, 1]) ,

(30)

we make iterative sequence: 1

𝑚

𝑥𝑛 (𝑡) = ∫ 𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥𝑛−1 (𝑠)] + 0

1

𝑚

𝑦𝑛 (𝑡) = ∫ 𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑦𝑛−1 (𝑠)] + 0

1 } 𝑑𝑠, 1 + 𝑏 (𝑠) 𝑦𝑛−1 (𝑠) 1 } 𝑑𝑠, 1 + 𝑏 (𝑠) 𝑥𝑛−1 (𝑠) 𝑛 = 1, 2, 3, . . . . (31)

Then 𝑥𝑛 (𝑡) and 𝑦𝑛 (𝑡) are all uniformly converge to 𝑥∗ (𝑡) on [0, 1], and we have error estimates: 𝑚𝑝 + 𝑞 𝑛 󵄨 󵄨󵄨 ∗ ) , 󵄨󵄨𝑥𝑛 (𝑡) (𝑦𝑛 (𝑡)) − 𝑥 (𝑡)󵄨󵄨󵄨 ≤ 2( 2

(32)

𝑡 ∈ [0, 1] , 𝑛 = 1, 2, 3, . . . . Proof. Let 𝐸 = 𝐶[0, 1], 𝑃 = {𝑥 ∈ 𝐸 | 𝑥(𝑡) ≥ 0, 𝑡 ∈ [0, 1]}, ‖𝑥‖ = max𝑡∈[0,1] |𝑥(𝑡)| denote norm of; then 𝐸 has become 𝐵𝑎𝑛𝑎𝑐ℎ space, 𝑃 is normal cone of 𝐸, and its normal constant 𝑁 = 1. It is obvious that integral Equation (24) transforms to operator equation 𝐴(𝑥, 𝑥) = 𝑥, where 𝐴 (𝑥, 𝑦) (𝑡) 1

= ∫ 𝑘 (𝑡, 𝑠) {𝑎 (𝑠) [𝑥 (𝑠)]𝑚 + 0

1 } 𝑑𝑠, 1 + 𝑏 (𝑠) 𝑦 (𝑠)

𝑡 ∈ [0, 1] . (33)

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

𝐿𝑥 (𝑡) = ∫ 𝑘 (𝑠, 𝑡) [𝑚𝑎 (𝑠) + 𝑏 (𝑠)] 𝑥 (𝑠) 𝑑𝑠, 0

𝑡 ∈ [0, 1] . (34)

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

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

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

References [1] D. J. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear 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 operators with applications,” Journal of Mathematical Analysis and Applications, vol. 156, no. 1, pp. 240–252, 1991. [3] Q. Zhang, “Contraction mapping principle of mixed monotone mapping and applications,” Henan Science, vol. 18, no. 2, pp. 121– 125, 2000. [4] J. X. Sun and L. S. Liu, “An iterative solution method for nonlinear operator equations and its applications,” Acta Mathematica Scientia A, vol. 13, no. 2, pp. 141–145, 1993. [5] Q. Zhang, “Iterative solutions of ordering symmetric contraction operator with applications,” Journal of Engineering Mathematics, vol. 17, no. 2, pp. 131–134, 2000. [6] X. L. Yan, “Fixed-point theorems for mixed monotone operators and their applications,” Mathematica Applicata, vol. 4, no. 4, pp. 107–114, 1991.

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