Abstract and Applied Analysis

Volume 2012 (2012), Article ID 391301, 11 pages

http://dx.doi.org/10.1155/2012/391301

## Duality Fixed Point and Zero Point Theorems and Applications

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 22 May 2012; Accepted 29 June 2012

Academic Editor: Rudong Chen

Copyright © 2012 Qingqing Cheng et al. 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.

#### Abstract

The following main results have been given. (1) Let be a -uniformly convex Banach space and let be a -Lipschitz mapping with condition . Then has a unique generalized duality fixed point and (2) let be a -uniformly convex Banach space and let be a inverse strongly monotone mapping with conditions , . Then has a unique generalized duality fixed point . (3) Let be a -uniformly smooth and uniformly convex Banach space with uniformly convex constant and uniformly smooth constant and let be a -lipschitz mapping with condition . Then has a unique zero point . These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.

#### 1. Introduction and Preliminaries

Let be a real Banach space with the dual and let be an operator from into . Firstly, for , we consider the variational inequality problem of finding an element such that Taking , the problem (1.1) becomes the following variational inequality problem of finding an element such that Secondly, for , we consider the optimal problem of finding an element such that Taking , the problem (1.3) becomes the following optimal problem of finding an element such that Thirdly, for , we consider the operator equation problem of finding an element such that Taking , the problem (1.5) becomes the following operator equation problem of finding an element such that Finally, we consider the operator equation problem of finding an element such that

Let be a real Banach space with the dual . Let be a given real number with . The generalized duality mapping from into is defined by where denotes the generalized duality pairing. In particular, is called the normalized duality mapping and for all . If is a Hilbert space, then , where is the identity mapping. The duality mapping has the following properties: (i)if is smooth, then is single-valued;(ii)if is strictly convex, then is one-to-one;(iii)if is reflexive, then is surjective;(iv)if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of ;(v)if is uniformly convex, then is uniformly continuous on each bounded subsets of and is single-valued and also one-to-one.For more details, see [1].

In this paper, we firstly present the definition of duality fixed point for a mapping from into its dual as follows.

Let be a Banach space with a single-valued generalized duality mapping . Let . An element is said to be a generalized duality fixed point of if . An element is said to be a duality fixed point of if .

*Example 1.1. * Let be a smooth Banach space with the dual , and let be an operator, then an element is a zero point of if and only if is a duality fixed point of for any . Namely, the is a duality fixed point of for any if and only if is a fixed point of (if is maximal monotone, then is, namely, the resolvent of ).

*Example 1.2. *In Hilbert space, the fixed point of an operator is always duality fixed point.

*Example 1.3. *Let be a -uniformly convex Banach space with the dual , then any element of must be the generalized duality fixed point of the generalized normalized duality mapping .

*Conclusion 1. *If is a generalized duality fixed point of , then must be a solution of variational inequality problem (1.1).

* Proof. *Suppose is a generalized duality fixed point of , then
Obverse that
for all .

Taking , we have the following result.

*Conclusion 2. *If is a duality fixed point of , then must be a solution of variational inequality problem (1.2).

*Conclusion 3. *If is a generalized duality fixed point of , then must be a solution of the optimal problem (1.3). Therefore, is also a solution of operator equation problem (1.5).

* Proof. *If is a generalized duality fixed point of , then , so that
All conclusions are obvious.

Take , we have the following result.

*Conclusion 4. *If is a duality fixed point of , then must be a solution of the optimal problem (1.4). Therefore, is also a solution of operator equation problem (1.6).

Let . A Banach space is said to be strictly convex if for any , implies . It is also said to be uniformly convex if for each , there exists such that for any , implies . It is well known that a uniformly convex Banach space is reflexive and strictly convex. And we define a function called the modulus of convexity of as follows:
It is well known that is uniformly convex if and only if for all . Let be a fixed real number with . Then is said to be -uniformly convex if there exists a constant such that for all . For example, see [2, 3] for more details. The constant is said to be uniformly convexity constant of .

A Banach space is said to be smooth if the limit
exists for all . It is also said to be uniformly smooth if the above limit is attained uniformly for . One should note that no Banach space is -uniformly convex for ; see [4] for more details. It is well known that the Hilbert and the Lebesgue spaces are -uniformly convex and uniformly smooth. Let be a Banach space and let be the Lebesgue-Bochner space on an arbitrary measure space . Let and let . Then is -uniformly convex if and only if is -uniformly convex; see [3].

Let be the modulus of smoothness of defined by
A Banach space is said to be uniformly smooth if as . Let . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that . It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable. If is -uniformly smooth, then and is uniformly smooth, and hence the norm of is uniformly Fréchet differentiable, in particular, the norm of is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for every .

Lemma 1.4 (see [5, 6]). *Let be a -uniformly convex Banach space with . Then, for all , and ,
**
where is the generalized duality mapping from into and is the -uniformly convexity constant of .*

Lemma 1.5. *Let be a -uniformly convex Banach space with . Then is one-to-one from onto and for all ,
**
where is the generalized duality mapping from into with range , and is the -uniformly convexity constant of .*

* Proof. *Let be a -uniformly convex Banach space with , then is one-to-one from onto . Since , then is single valued. From (1.5) we have
which implies that
That is
Hence
Then (1.6) has been proved. Therefore, from (1.6) we can see, for any , that implies that .

#### 2. Duality Contraction Mapping Principle and Applications

Let be a Banach space with the dual . An operator is said to be --Lipschitz, if where are two constants. If , the operator is said to be -Lipschitz.

Theorem 2.1 (generalized duality contraction mapping principle). *Let be a -uniformly convex Banach space and let be a --Lipschitz mapping with condition . Then has a unique generalized duality fixed point and for any given guess , the iterative sequence converges strongly to this generalized duality fixed point .*

* Proof. *Let , then is a mapping from into itself. By using Lemma 1.5, we have
for all , where . By using Banach's contraction mapping principle, there exists a unique element such that . That is, , so is a generalized unique duality fixed point of . Further, the Picard iterative sequence () converges strongly to this generalized duality fixed point .

Taking , we have the following result.

Theorem 2.2 (duality contraction mapping principle). *Let be a -uniformly convex Banach space and let be a -Lipschitz mapping with condition . Then has a unique duality fixed point and for any given guess , the iterative sequence converges strongly to this duality fixed point .*

From Conclusions 1–4 and Theorem 2.1, we have the following result for solving the variational inequality problems (1.1) and (1.2), the optimal problems (1.3) and (1.4), and the operator equation problems (1.5) and (1.6).

Theorem 2.3. *Let be a -uniformly convex Banach space and let be a --Lipschitz mapping with condition . Then the variational inequality problem (1.1) (the optimal problem (1.2) and operator equation problem (1.3)) has solutions and for any given guess , the iterative sequence converges strongly to a solution of the variational inequality problem (1.1) (the optimal problem (1.3) and the operator equation problem (1.5)).*

Taking , we have the following result.

Theorem 2.4. *Let be a -uniformly convex Banach space, let be a --Lipschitz mapping with condition . Then the variational inequality problem (1.2) (the optimal problem (1.4) and operator equation problem (1.6)) has solutions and for any given guess , the iterative sequence converges strongly to a solution of the variational inequality problem (1.2) (the optimal problem (1.4) and the operator equation problem (1.6)).*

Theorem 2.5 (generalized duality Mann weak convergence theorem). * Let be a p-uniformly convex Banach space which satisfying Opial's condition, let be a --Lipschitz mapping with nonempty generalized duality fixed point set. Assume , and the real sequence satisfies the condition . Then for any given guess , the generalized Mann iterative sequence
**
converges weakly to a generalized duality fixed point of .*

*Proof. *Letting , by using Lemma 1.4, we have
for all . Hence is a nonexpansive mapping from into itself. In addition, we have
By using the well-known result, we know that the sequence converges weakly to a fixed point of . This point is also a duality fixed point of .

Take , we have the following result.

Theorem 2.6 (duality Mann weak convergence theorem). *Let be a 2-uniformly convex Banach space which satisfy Opial's condition and let be a -Lipschitz mapping with nonempty duality fixed point set. Assume , and the real sequence satisfies the condition . Then for any given guess , the generalized Mann iterative sequence
**
converges weakly to a duality fixed point of .*

Theorem 2.7 (duality Halpern strong convergence theorem). *Let be a p-uniformly convex Banach space which satisfying Opial's condition, let be a --Lipschitz mapping with nonempty generalized duality fixed point set. Assume , and the real sequence satisfies the condition:* *(C _{1}): ;*

*(C*

_{2}): ;*(C*

_{3}): or .*Let be given, then iterative sequence*

*converges strongly to a generalized duality fixed point of .*

* Proof. *Let , then is a mapping from into itself. By using Lemma 1.5, we have
for all , where . Hence is a nonexpansive mapping from into itself. In addition, we have
By using the well-known result of Xu [7, Theorem 2.3], we know that the iterative sequence converges strongly to a fixed point of nonexpansive mapping . Hence the sequence converges strongly to a generalized duality fixed point of .

Theorem 2.8. *Letting be a Hilbert space, then one has its uniformly convexity constant , that is .*

* Proof. *If . For any , by using Lemma 1.4, we have
This is a contradiction.

#### 3. Fixed Point Theorem of Inverse Strongly Monotone Mappings

*Definition 3.1. *Letting be a Banach space, the mapping is called --inverse strongly monotone, if

Lemma 3.2. *Let be a Banach space and let be a --inverse strongly monotone mapping. Then is -Lipschitz.*

* Proof. *Let be a --inverse strongly monotone mapping, that is,
It follows from the above inequality that
which leads to
Further
and hence
Then is -Lipschitz.

Theorem 3.3 (fixed point theorem of inverse strongly monotone mappings). *Let be a -uniformly convex Banach space and let be a --inverse strongly monotone mapping with conditions , . Then has a unique generalized duality fixed point and for any given guess , the iterative sequence converges strongly to this generalized duality fixed point .*

* Proof. *Letting , then is a mapping from into itself. By using Lemma 1.5 and Lemma 3.2, we have
for all . It follows from the condition that . By using Banach's contraction mapping principle, there exists a unique element such that . That is, , so is a generalized unique duality fixed point of . Further, the Picard iterative sequence () converges strongly to this generalized duality fixed point .

Taking , we have the following results.

Lemma 3.4. *Let be a Banach space and let be a --inverse strongly monotone mapping. Then is -Lipschitz.*

Theorem 3.5 (fixed point theorem of inverse strongly monotone mappings). *Let be a -uniformly convex Banach space, let be a --inverse strongly monotone mapping with condition . Then has a unique duality fixed point and for any given guess , the iterative sequence converges strongly to this generalized duality fixed point .*

#### 4. Application for Zero Point of Operators

Lemma 4.1 (see [8]). *Let be a -uniformly smooth Banach space with uniformly smooth constant . Then
*

Theorem 4.2. *Let be a -uniformly smooth and uniformly convex Banach space with uniformly convex constant and uniformly smooth constant and let be a -lipschitz mapping with condition . Then has a unique zero point and for any given guess , the iterative sequence converges strongly to this zero point .*

* Proof. *Let , then is a mapping from into itself. By using Lemma 1.5 and Lemma 4.1, we have
for all . Observing the condition , it follows that, there exists a positive number such that . By using Banach's contraction mapping principle, there exists a unique element such that . That is, which implies , so is a zero point of . Further, the Picard iterative sequence () converges strongly to this zero point .

*Remark 4.3. *Under the conditions of Theorem 4.2, we know that the operator equation has a unique solution which can be computed by the iterative scheme () starting any given guess .

#### Acknowledgment

This project is supported by the National Natural Science Foundation of China under Grant (11071279).

#### References

- T. Ibaraki and W. Takahashi, “A new projection and convergence theorems for the projections in Banach spaces,”
*Journal of Approximation Theory*, vol. 149, no. 1, pp. 1–14, 2007. View at Publisher · View at Google Scholar - D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,”
*Journal of Applied Analysis*, vol. 7, no. 2, pp. 151–174, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,”
*Numerical Functional Analysis and Optimization*, vol. 24, no. 5-6, pp. 489–508, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Su and X. Qin, “General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings,”
*Applied Mathematics and Computation*, vol. 186, no. 1, pp. 271–278, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,”
*Nonlinear Analysis*, vol. 64, no. 11, pp. 2400–2411, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,”
*Journal of Approximation Theory*, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,”
*Bulletin of the Australian Mathematical Society*, vol. 65, no. 1, pp. 109–113, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-S. Chang, Y. J. Cho, and H. Zhou,
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*, Nova Science, Huntington, NY, USA, 2002.