Abstract
In this paper, we prove a strong convergence theorem for two different hybrid methods by using CQ method for a finite family of G-nonexpansive mappings in a Hilbert space. We give an example and numerical results for supporting our main results and compare the rate of convergence of the two iterative methods.
1 Introduction
Let be a Hilbert space with the inner product , norm and be a nonempty subset of . A nonlinear mapping is said to be
1. contraction if there exists such that for all ;
2. nonexpansive if for all .
The fixed point set of is denoted by , that is, .
Since 1922, fixed point theorems and the existence of fixed points of a single-valued nonlinear mapping have been intensively studied and considered by many authors (see, for examples [1–7Citation[1]Citation[2]Citation[3]Citation[4]Citation[5]Citation[6]Citation[7]]).
In 1953, Mann [Citation8] introduced the famous iteration procedure as follows: where and the set of all positive integers. Many researchers have used Mann’s iteration for obtaining weak convergence theorem (see for example [Citation9–Citation11]).
In 2003, Nakajo and Takahashi [Citation12] introduced a modification of the Mann iteration and called it CQ method. They generated the sequence by (1.1) (1.1) for , where for some . They showed that converges strongly to .
In 2015, Khan et al. [Citation13] used the following definition defined by Berinde [Citation14] to compare the convergence rate:
Let and be two sequences of real numbers with limits and , respectively. Assume that there exists . If , then we say that converges faster to than to .
Let be a nonempty subset of a real Banach space . Let denote the diagonal of the cartesian product . Consider a directed graph such that the set of its vertices coincides with , and the set of its edges with . We assume has no parallel edge. So we can identify the graph with the pair . A mapping is said to be
1. G-contraction if satisfies the conditions:
(i) preserves edges of , i.e.,
(ii) decreases weights of edges of in the following way: there exists such that
2. G-nonexpansive if satisfies the conditions:
(i) preserves edges of , i.e.,
(ii) non-increases weights of edges of in the following way:
In 2008, Jachymski [Citation15] proved some generalizations of the Banach’s contraction principle in complete metric spaces endowed with a graph. To be more precise, Jachymski proved the following result.
Theorem 1.1
[Citation15]
Let be a complete metric space, and a triple has the following property:
for any sequence if and for and there is the subsequence of with for .
Let be a G-contraction, and . Then if and only if .
When is a sequence in , we denote the strong convergence of to by and the weak convergence by .
In 2015, Tiammee et al. [Citation16] and Alfuraidan [Citation17] employed the above theorem to establish the existence and the convergence results for G-nonexpansive mappings with graphs.
Motivated by Nakajo and Takahashi [Citation12] and Tiammee et al. [Citation16], we introduce the modified CQ method for proving a strong convergence theorem for G-nonexpansive mappings in a Hilbert space endowed with a directed graph. Moreover, we provide some numerical examples to support our main theorem.
2 Preliminaries and lemmas
Let be a nonempty, closed and convex subset of a Hilbert space . The nearest point projection of onto is denoted by , that is, for all and . Such is called the metric projection of onto . We know that the metric projection is firmly nonexpansive, i.e., for all . Furthermore, holds for all and ; see [Citation18].
A mapping is called -inverse strongly monotone if there exists a positive real number such that We know that if is nonexpansive, then is -inverse strongly monotone; see [Citation19] for more details. We consider the following variational inequality problem: find a such that The set of solutions of the variational inequality (2.1) is denoted by .
We need the following known results.
Lemma 2.1
[Citation18]
Let be a real Hilbert space. Then for all and .
Lemma 2.2
[Citation20]
Let be a nonempty, closed and convex subset of a real Hilbert space . Given that and , the set is convex and closed.
Lemma 2.3
[Citation12]
Let be a nonempty, closed and convex subset of a real Hilbert space and be the metric projection from onto . Then the following inequality holds:
Lemma 2.4
[Citation21]
Let be a real Hilbert space and let . For such that , the following identity holds:
Lemma 2.5
Let be a nonempty subset of a Hilbert space and a directed graph such that . Let be a G-nonexpansive mapping. Then, for any , there exists a positive number such that for all , whenever with , and .
Proof
Let , for some and .
We consider the following two cases.
Case I. If , then If , then we have
Case II. If , then for any nonnegative number , we have If and , then we have
We next prove the demiclosedness principle of a G-nonexpansive mapping.
Lemma 2.6
Let be a nonempty, closed and convex subset of a Hilbert space and a directed graph such that . Let be a G-nonexpansive mapping and be a sequence in such that for some . If, there exists a subsequence of such that for all and for some , then .
Proof
Let be a sequence in such that and for some . Set , . Then . We may assume, without loss of generality, that . By the assumption, there exists a subsequence of such that . We set . Let . Since as , there exists such that Lemma 2.5 gives that, for each , . By the weak compactness of , it contains the weak limit of . This shows that . Hence , that is, , since is arbitrary. □
3 Main results
In this section, by using the CQ method, we obtain two different strong convergence theorems for finding the same common fixed point of a finite family of G-nonexpansive mappings in real Hilbert spaces.
Theorem 3.1
Let be a nonempty closed and convex subset of a real Hilbert space and let be a directed graph such that and is convex. Let be a finite family of G-nonexpansive mappings of into itself. Assume that , is closed and for all . For an initial point with , let be a sequence generated by (3.1) (3.1) where for all such that for all . Assume that dominates for all and if there exists a subsequence of such that , then . Then the sequence converges strongly to
Proof
We split the proof into six steps.
Step 1. Show that is well-defined for each
As shown in Theorem 3.2 of Tiammee et al. [Citation16], is convex for all . It follows now from the assumption that is closed and convex. Hence, is well-defined.
Step 2. Show that is well-defined for each
From the definition of and , it is obvious that is closed and convex for all . It follows from Lemma 2.2 that for each , is closed and convex for all . This implies that is also closed and convex for all . Let and . Since dominates , we have
Step 3. Show that exists.
Since is a nonempty, closed and convex subset of , there exists a unique such that From and , , we get (3.3) (3.3) On the other hand, as , we obtain (3.4) (3.4) It follows from (Equation3.3)(3.3) (3.3) and Equation(3.4(3.4) (3.4) ) that the sequence is bounded and nondecreasing. Therefore exists.
Step 4. Show that as .
For , by the definition of , we see that . Noting that and , by Lemma 2.3, we get From Step 3, we obtain that is a Cauchy sequence. Hence, there exists such that as . In particular, we have (3.5) (3.5) Step 5. Show that .
Since , it follows from (Equation3.5(3.5) (3.5) ) that (3.6) (3.6) as for all . Since , we have It follows from for all and (Equation3.6(3.6) (3.6) ) that (3.7) (3.7) as for all . From and Lemma 2.6, we have .
Step 6. Show that .
Since , we have (3.8) (3.8) By taking the limit in (Equation3.8(3.8) (3.8) ), we obtain (3.9) (3.9) Since , so . This completes the proof. □
Theorem 3.2
Let be a nonempty closed and convex subset of a real Hilbert space and let be a directed graph such that and is convex. Let be a finite family of G-nonexpansive mappings of into itself. Assume that , is closed and for all . For an initial point with , let be a sequence generated by (3.10) (3.10) where for all such that and for all . Assume that dominates for all and if there exists a subsequence of such that , then . Then the sequence converges strongly to
Proof
From Step 1 in the proof of Theorem 3.1, we have that is well-defined for each . We know from Step 2 in the proof of Theorem 3.1 that is closed and convex for all . Let . Since dominates , we have
Remark 3.3
The sequences generated in Theorem 3.1 and 3.2 converge to under the different conditions on the sequence .
We know that every G-nonexpansive mapping is nonexpansive [Citation16], then we obtain the following results.
Corollary 3.4
Let be a nonempty closed and convex subset of a real Hilbert space . Let be a finite family of nonexpansive mappings of into itself. Assume that . For an initial point with , let be a sequence generated by (3.12) (3.12) where for all . Assume that for all Then the sequence converges strongly to
Corollary 3.5
Let be a nonempty closed and convex subset of a real Hilbert space . Let be a finite family of nonexpansive mappings of into itself. Assume that . For an initial point with , let be a sequence generated by (3.13) (3.13) where for all such that . Assume that for all Then the sequence converges strongly to
Putting where is an identity mapping in Theorem 3.1 –3.2, we obtain the following result.
Corollary 3.6
Let be a nonempty closed and convex subset of a real Hilbert space and let be a directed graph such that and is convex. Let be a G-nonexpansive mapping of into itself. Assume that , is closed and . For an initial point with , let be a sequence generated by (3.14) (3.14) where such that . Assume that dominates for all and if there exists a subsequence of such that , then . Then the sequence converges strongly to
Remark 3.7
The iterative scheme (Equation3.14(3.14) (3.14) ) extends the CQ method (Equation1.1(1.1) (1.1) ) from a nonexpansive mapping to a G-nonexpansive mapping.
If , we know that is nonexpansive; see [Citation19]. Then, we obtain the following results.
Corollary 3.8
Let be a nonempty closed and convex subset of a real Hilbert space . Let be a finite family of -inverse strongly monotone with . For an initial point with , let be a sequence generated by (3.15) (3.15) where for all and . Assume that for all Then the sequence converges strongly to
Corollary 3.9
Let be a nonempty closed and convex subset of a real Hilbert space . Let be a finite family of -inverse strongly monotone with . For an initial point with , let be a sequence generated by (3.16) (3.16) where for all such that and . Assume that for all Then the sequence converges strongly to
4 Examples and numerical results
In this section, we give examples and numerical results for supporting our main theorem.
Example 4.1
Let and . Assume that if and only if or . Define two mappings by for any . Let and . It is easy to check that are G-nonexpansive. On the other hand, are not nonexpansive for and . This implies that and .
For generated by (Equation3.1(3.1) (3.1) ), we divide the process of our iteration into 3 Steps as follows:
Step 1. Find . Since and , we obtain that and We observe the following cases:
Case 1 : If , then ,
Case 2 : If , then . Thus, ,
Case 3 : If , then . Thus, ,
Similarly, we have ; where , ; where and ; where ,
Step 2. Find . We observe the following cases:
Case 1 : If , then ,
Case 2 : If , then . Thus, ,
Case 3 : If , then . Thus, , .
Step 3. Compute the numerical results of . Choose , we have
From , we see that is the solution of iteration (Equation3.1(3.1) (3.1) ).
We next show that our iteration generated by (Equation3.10(3.10) (3.10) ) also converges to 1. As the same above process, choose . (See .)
From , we see that the iteration (Equation3.10(3.10) (3.10) ) converges to . (See .)
Table 1 Numerical results of iteration (Equation3.1(3.1) (3.1) ).
Table 2 Numerical results of iteration (Equation3.10(3.10) (3.10) ).
In the same way of Khan ([Citation13], Example 3), we can conclude the following remark.
Remark 4.2
We see that the iteration (Equation3.1(3.1) (3.1) ) converges faster than the iteration (Equation3.10(3.10) (3.10) ) under the same conditions.
Notes
Peer review under responsibility of Kalasalingam University.
References
- Agarwal R.P. O’Regan D. Sahu D.R. Fixed Point Theory for Lipschitzian-Type Mappings with Applications 2009 Springer
- Banach S. Sur les oprations dans les ensembles abstraits et leur application aux quations intgrales Fund. Math. 3 1922 133 181
- Browder F.E. Nonexpansive nonlinear operators in a Banach space Proc. Natl. Acad. Sci. USA 54 1965 1041 1044
- Browder F.E. Convergence of approximants to fixed points of non-expansive maps in Banach spaces Arch. Ration. Mech. Anal. 24 1967 82 90
- Goebel K. Kirk W.A. Topics in Metric Fixed Point Theory 1990 Cambridge University Press Cambridge
- Goebel K. Reich S. Uniform Convexity Hyperbolic Geometry, and Nonexpansive Mappings vol. 73 (1984) Marcel Dekker. New York, NY, USA.
- Takahashi W. Yao J.C. Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces Taiwanese J. Math. 15 2011 457 472
- Mann W.R. Mean value methods in iteration Proc. Amer. Math. Soc. 4 1953 506 510
- Genal A. Lindenstrass J. An example concerning fixed points Israel J. Math. 22 1975 81 86
- Reich S. Weak convergence theorems for nonexpansive mappings in Banach spaces J. Math. Anal. Appl. 67 1979 274 276
- Schu J. Iterative construction of fixed points of asymptotically nonexpansive mapping J. Math Anal. Appl. 158 1991 407 413
- Nakajo K. Takahashi W. Strongly convergence theorems for nonexpansive mappings and nonexpansive semigroups J. Math. Anal. Appl. 279 2003 372 379
- Khan A.R. Gürsoy F. Karakaya V. Jungck–Khan iterative scheme and higher convergence rate Int. J. Comput. Math. 2015 10.1080/00207160.2015.1085030
- Berinde V. Picard iteration converges faster than the Mann iteration in the class of quasi-contractive operators Fixed Point Theory Appl. 2 2004 97 105
- Jachymski J. The contraction principle for mappings on a metric space with a graph Proc. Amer. Math. Soc. 136 4 2008 1359 1373
- Tiammee J. Kaewkhao A. Suantai S. On Browder’s convergence theorem and Halpern iteration process for G-nonexpansive mappings in Hilbert spaces endowed with graphs Fixed Point Theory Appl. 2015 2015 187 10.1186/s13663-015-0436-9
- Alfuraidan M.R. Fixed points of monotone nonexpansive mappings with a graph Fixed Point Theory Appl. 2015 2015 49 10.1186/s13663-015-0299-0
- Takahashi W. Nonlinear Functional Analysis 2000 Yokohama Publishers Yokohama
- Takahashi S. Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space Nonlinear Anal. 69 2008 1025 1033
- Kim T.H. Xu H.K. Strongly convergence of modified Mann iterations for with asymptotically nonexpansive mappings and semigroups Nonlinear Anal. 64 2006 1140 1152
- Chidume C.E. Ezeora J.N. Krasnoselkii-type algorithm for family of multi-valued strictly pseudo-contractive mappings Fixed Point Theory Appl. 2014 2014 111
- Marino G. Xu H.K. Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces J. Math. Anal. Appl. 329 2007 336 346