- Open Access
Viscosity approximation methods for two nonexpansive semigroups in CAT(0) spaces
© Tang and Chang; licensee Springer. 2014
- Received: 6 November 2013
- Accepted: 10 July 2014
- Published: 15 August 2014
The purpose of this paper is by using the viscosity approximation method to study the strong convergence problem for two one-parameter continuous semigroups of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of two one-parameter continuous semigroups of nonexpansive mappings are proved, which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.
MSC:47J05, 47H09, 49J25.
- viscosity approximation method
- nonexpansive semigroup
- implicit and explicit iterative scheme
- CAT(0) space
- fixed point
Throughout this paper, we assume that X is a CAT(0) space, ℕ is the set of positive integers, ℝ is the set of real numbers, is the set of nonnegative real numbers and C is a nonempty closed and convex subset of a complete CAT(0) space X.
- (i)for each , is a nonexpansive mapping on C, i.e.,
for all ;
for each , the mapping from into C is continuous.
for all .
where is an arbitrary fixed element. In the case of T having a fixed point, Browder  proved that converged strongly to a fixed point of T that is nearest to u in the framework of Hilbert spaces. Reich  extended Browder’s result to the setting of a uniformly smooth Banach space and proved that converged strongly to a fixed point of T.
Under suitable conditions, they proved strong convergence of to a member of ℱ.
They proved that defined by (1.5) and (1.6) both converged to the same point of ℱ in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm.
They proved that defined by (1.7) and defined by (1.8) converged strongly to a fixed point of T in the framework of CAT(0) spaces.
Motivated and inspired by the research going on in this direction, especially inspired by Wangkeeree and Preechasilp , in this paper we study the strong convergence theorems of Moudafi’s viscosity approximation methods for two one-parameter continuous semigroups of nonexpansive mappings in CAT(0) spaces. We prove that the implicit and explicit iteration algorithms both converge strongly to the same point such that , which is the unique solution of the variational inequality (1.9) where ℱ is the set of common fixed points of the two semigroups of nonexpansive mappings.
Lemma 2.1 
Lemma 2.2 
This implies that (2.5) holds.
Next, we prove that (2.6) holds.
Indeed, it is obvious that (2.6) holds for . Suppose that (2.6) holds for some . Next we prove that (2.6) is also true for .
This completes the proof of (2.6). □
The concept of Δ-convergence introduced by Lim  in 1976 was shown by Kirk and Panyanak  in CAT(0) spaces to be very similar to the weak convergence in the Banach space setting (see also ). Now, we give the concept of Δ-convergence.
It is known from Proposition 7 of  that in a complete CAT(0) space, consists of exactly one point. A sequence is said to Δ-converge to if for every subsequence of .
Lemma 2.4 
Every bounded sequence in a complete CAT(0) space always has a Δ-convergent subsequence.
for all .
Recently, Dehghan and Rooin  presented a characterization of metric projection in CAT(0) spaces as follows.
Lemma 2.6 
Let X be a complete CAT(0) space, be a sequence in X and . Then Δ-converges to x if and only if for all .
Lemma 2.7 
Then converges to zero as .
In this section, we present the strong convergence theorems of Moudafi’s viscosity approximation implicit and explicit iteration algorithms for two one-parameter continuous semigroups of nonexpansive mappings and in CAT(0) spaces.
Before proving main results, we need the following two vital lemmas.
which is the desired result. □
Now we are in a position to state and prove our main results.
for any bounded subset B of C, .
Proof We shall divide the proof of Theorem 3.3 into five steps.
Step 1. The sequence defined by (3.2) is well defined for all .
This implies that is a contraction mapping. Hence, the sequence is well defined for all .
Step 2. The sequence is bounded.
Hence is bounded, so are , and .
Step 3. For any , and .
Step 4. The sequence contains a subsequence converging strongly to such that , which is equivalent to (3.3).
Since is bounded, by Lemma 2.4, there exists a subsequence of (without loss of generality, we denote it by ) which Δ-converges to a point .
This is a contraction, and hence . Similarly, we can obtain that . So we have .
It follows from (3.5) and that converges strongly to .
That is, solves inequality (3.3).
Step 5. The sequence converges strongly to .
Since , we have that , and so . Hence the sequence converges strongly to , which is the unique solution to the variational inequality (3.3).
This completes the proof. □
, and ;
for all , ;
for any bounded subset B of C, .
Then converges strongly to such that , which is equivalent to the variational inequality (3.3).
for all . Hence is bounded, so are , and .
It follows from Theorem 3.3 that converges strongly to a fixed point , which solves the variational inequality (3.3).
Applying Lemma 2.7 and , we can conclude that as . This completes the proof. □
The authors would like to express their thanks to the editors and referees for their helpful comments and advice. This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199) and the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and the National Natural Science Foundation of China (Grant No. 11361070).
- Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleGoogle Scholar
- Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleGoogle Scholar
- Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar
- Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semi-groups in Hilbert spaces. Nonlinear Anal. 1998, 34: 87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleGoogle Scholar
- Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 2003, 131: 2133–2136. 10.1090/S0002-9939-02-06844-2View ArticleGoogle Scholar
- Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. J. Math. Anal. Appl. 2008, 338: 152–161. 10.1016/j.jmaa.2007.05.021MathSciNetView ArticleGoogle Scholar
- Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011,24(2):224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
- Dhompongsa S, Fupinwong W, Kaewkhao A: Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. Nonlinear Anal. 2009, 70: 4268–4273. 10.1016/j.na.2008.09.012MathSciNetView ArticleGoogle Scholar
- Shi LY, Chen RD: Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces. J. Appl. Math. 2012., 2012: Article ID 421050 10.1155/2012/421050Google Scholar
- Wangkeeree R, Preechasilp P: Viscosity approximation methods for nonexpansive semigroups in CAT(0) spaces. Fixed Point Theory Appl. 2013., 3013: Article ID 160 10.1186/1687-1812-2013-160Google Scholar
- Dhompongsa S, Panyanak B: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036MathSciNetView ArticleGoogle Scholar
- Bridson M, Haefliger A: Metric Spaces of Nonpositive Curvature. Springer, Berlin; 1999.View ArticleGoogle Scholar
- Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-XView ArticleGoogle Scholar
- Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleGoogle Scholar
- Ahmadi Kakavandi B: Weak topologies in complete CAT(0) metric spaces. Proc. Am. Math. Soc. 2013, 141: 1029–1039. S 0002–9939(2012)11743–5MathSciNetView ArticleGoogle Scholar
- Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006, 65: 762–772. 10.1016/j.na.2005.09.044MathSciNetView ArticleGoogle Scholar
- Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl. 2008,68(12):3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleGoogle Scholar
- Berg ID, Nikolaev IG: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 2008, 133: 195–218. 10.1007/s10711-008-9243-3MathSciNetView ArticleGoogle Scholar
- Dehghan H, Rooin J: A characterization of metric projection in CAT(0) spaces. In International Conference on Functional Equation, Geometric Functions and Applications. Payame Noor University, Tabriz; 2012:41–43. (ICFGA 2012) 10–12th, May 2012Google Scholar
- Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
- Liu XD, Chang SS: Viscosity approximation methods for hierarchical optimization problems in CAT(0) spaces. J. Inequal. Appl. 2013., 2013: Article ID 47Google Scholar
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