Abstract
Let K be a compact convex subset of a real Hilbert space H and , , be a family of continuous hemicontractive mappings. Let be such that and satisfying for some , . For arbitrary , define the sequence by (1.9) see below, then converges strongly to a common fixed point in .
MSC:05C38, 15A15, 05A15, 15A18.
Similar content being viewed by others
1 Introduction
Let H be a Hilbert space. A mapping is said to be pseudocontractive (see [1, 2]) if
for all and T is said to be strongly pseudocontractive if there exists such that
for all .
Let and K be a nonempty subset of H. A mapping is said to be hemicontractive if and
for all and . It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractive mappings.
The following example, due to Rhoades [3], shows that the inclusion is proper. For any , define a mapping by . It is shown in [4] that T is not Lipschitz and so T cannot be nonexpansive. A straightforward computation (see [5]) shows that T is pseudocontractive. For the importance of fixed points of pseudocontractive mappings, the reader may refer to [1].
In the last ten years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive (and, correspondingly, Lipschitz strongly accretive) mappings using the Mann iteration process (see, for example, [6]). The results which were known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see [3–5, 7–33]) and the references cited therein).
In 1974, Ishikawa [34] introduced an iteration process which, in some sense, is more general than Mann iteration and which converges, under this setting, to a fixed point of T. He proved the following theorem.
Theorem 1.1 If K is a compact convex subset of a Hilbert space H, is a Lipschitzian pseudocontractive mapping and is any point in K, then the sequence converges strongly to a fixed point of T, where is defined iteratively by
for each , where , are the sequences of positive numbers satisfying the following conditions:
-
(a)
;
-
(b)
;
-
(c)
.
In [35], Qihou extended Theorem 1.1 to a slightly more general class of Lipschitz hemicontractive mappings and, in [25], Reich proved, under the setting of Theorem 1.1, the convergence of the recursion formula (1.3) to a fixed point of T, when T is a continuous hemicontractive mapping, under an additional hypothesis that the number of fixed points of T is finite. The iteration process (1.3) is generally referred to as the Ishikawa iteration process in light of Ishikawa [34]. Another iteration process which has been studied extensively in connection with fixed points of pseudocontractive mappings is the following.
Let K be a nonempty convex subset of E and be a mapping.
The sequence is defined iteratively by
for each , where is a real sequence satisfying the following conditions:
-
(d)
;
-
(e)
;
-
(f)
.
The iteration process (1.4) is generally referred to as the Mann iteration process in light of [36].
In 1995, Liu [37] introduced the iteration process with errors as follows.
(I-a) The sequence defined by
for each , where , are the sequences in satisfying appropriate conditions and , , is called the Ishikawa iteration process with errors.
(I-b) The sequence defined by
for each , where is a sequence in satisfying appropriate conditions and , is called the Mann iteration process with errors.
While it is known that the consideration of error terms in the iterative processes (1.5), (1.6) is an important part of the theory, it is also clear that the iterative processes with errors introduced by Liu in (I-a) and (I-b) are unsatisfactory. The occurrence of errors is random so the conditions imposed on the error terms in (I-a) and (I-b), which imply, in particular, that they tend to zero as n tends to infinity, are unreasonable. In 1997, Xu [32] introduced the following more satisfactory definitions.
(I-c) The sequence defined iteratively by
for each , where , are the bounded sequences in K and , , , , and are the sequences in such that for each , is called the Ishikawa iteration sequence with errors in the sense of Xu.
(I-d) If, with the same notations and definitions as in (I-c), for each , then the sequence now defined by
for each is called the Mann iteration sequence with errors in the sense of Xu.
We remark that if K is bounded (as is generally the case), then the error terms , are arbitrary in K.
In [11], Chidume and Chika Moore proved the following theorem.
Theorem 1.2 Let K be a compact convex subset of a real Hilbert space H and be a continuous hemicontractive mapping. Let , , , , and be the real sequences in satisfying the following conditions:
-
(g)
;
-
(h)
;
-
(i)
; ;
-
(j)
and , where ;
-
(k)
for each , where and .
For arbitrary , define the sequence iteratively by
for each , where and are the arbitrary sequences in K. Then converges strongly to a fixed point of T.
They also gave the following remark in [11].
Remark 1.1 (1) In connection with the iterative approximation of fixed points of pseudocontractive mappings, the following question is still open.
Does the Mann iteration process always converge for continuous pseudocontractive mappings or for even Lipschitz pseudocontractive mappings?
-
(2)
Let E be a Banach space and K be a nonempty compact convex subset of E. Let be a Lipschitz pseudocontractive mapping. Under this setting, even for , a Hilbert space, the answer to the above question is not known. There is, however, an example [34] of a discontinuous pseudocontractive mapping T with a unique fixed point for which the Mann iteration process does not always converge to the fixed point of T.
Let H be the complex plane and . Define a mapping by
Then zero is the only fixed point of T. It is shown in [20] that T is pseudocontractive and, with , the sequence defined by
for each does not converge to zero. Since the T in this example is not continuous, the above question remains open.
In [14], Chidume and Mutangadura provide an example of a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iteration sequence failed to converge and they stated that ‘This resolves a long standing open problem’. However, in [38, 39], Rafiq provided affirmative answers to the above questions (see also [40]) and proved the following result.
Theorem 1.3 Let K be a compact convex subset of a real Hilbert space H and be a continuous hemicontractive mapping. Let be a real sequence in satisfying for some . For arbitrary , define the sequence by
for each . Then converges strongly to a fixed point of T.
The purpose of this paper is to introduce the following Mann-type implicit iteration process associated with a family of continuous hemicontractive mappings to have a strong convergence in the setting of Hilbert spaces.
Let K be a closed convex subset of a real normed space H and , be a family of mappings. Then we define the sequence in the following way:
for each , where , , are such that and some appropriate conditions hold.
2 Main results
In the sequel, we will use following results.
Lemma 2.1 [29]
Suppose that , are two sequences of nonnegative numbers such that, for some real number ,
for all . Then we have the following:
-
(1)
If , then exists.
-
(2)
If and has a subsequence converging to zero, then .
Lemma 2.2 [31]
For all and , the following well-known identity holds:
Now, we prove our main results.
Lemma 2.3 Let H be a Hilbert space. Then, for all , ,
where , , and .
Proof For any , , it can be easily seen that
Consider the following:
For all , we have
and
Substituting (2.4) and (2.5) in (2.3), we get
This completes the proof. □
Remark 2.1 Lemma 2.2 is now the special case of our result.
Theorem 2.1 Let K be a compact convex subset of a real Hilbert space H and , , be a family of continuous hemicontractive mappings. Let be such that and satisfying for some , .
Then, for arbitrary , the sequence defined by (1.9) converges strongly to a common fixed point in .
Proof Let . Using the fact that , are hemicontractive, we obtain
With the help of (1.9), Lemma 2.3 and (2.6), we obtain the following estimates:
Substituting (2.6) in (2.7), we get
Also, we have
Substituting (2.9) in (2.8), we get
which implies that
Thus, from the condition for some , , we obtain
for all fixed points . Moreover, we have
and thus, for all ,
Hence, for all , we obtain
for each , which implies that
for each . From (2.9), it further implies that
By the compactness of K, this immediately implies that there is a subsequence of which converges to a common fixed point of , say . Since (2.10) holds for all fixed points of , we have
and, in view of (2.11) and Lemma 2.1, we conclude that as , that is, as . This completes the proof. □
Theorem 2.2 Let H, K, , , be as in Theorem 2.1 and be such that and satisfying for some , .
If is the projection operator of H onto K, then the sequence defined iteratively by
for each converges strongly to a common fixed point in .
Proof The mapping is nonexpansive (see [2]) and K is a Chebyshev subset of H and so is a single-valued mapping. Hence, we have the following estimate:
which implies that
The set is compact and so the sequence is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.1. This completes the proof. □
Theorem 2.3 Let K be a compact convex subset of a real Hilbert space H and , , be a family of Lipschitz hemicontractive mappings. Let be such that and satisfying for some , .
Then, for arbitrary , the sequence defined by (1.9) converges strongly to a common fixed point in .
Theorem 2.4 Let H, K, , , be as in Theorem 2.3 and be such that and satisfying for some , .
If is the projection operator of H onto K, then the sequence defined iteratively by
for each converges strongly to a common fixed point in .
Example For , we can choose the following control parameters: , and .
References
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. of Symposia in Pure Math., Vol. XVIII, Part 2 1976.
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Schu J: On a theorem of C.E. Chidume concerning the iterative approximation of fixed points. Math. Nachr. 1991, 153: 313–319. 10.1002/mana.19911530127
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158: 407–413. 10.1016/0022-247X(91)90245-U
Xu ZB, Roach GF: A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations. J. Math. Anal. Appl. 1992, 167: 340–354. 10.1016/0022-247X(92)90211-U
Qihou L: The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. J. Math. Anal. Appl. 1990, 148: 55–62. 10.1016/0022-247X(90)90027-D
Chidume CE: Iterative approximation of Lipschitz strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1987, 99: 283–288.
Chidume CE: Approximation of fixed points of strongly pseudocontractive mappings. Proc. Am. Math. Soc. 1994, 120: 545–551. 10.1090/S0002-9939-1994-1165050-6
Chidume CE: Global iteration schemes for strongly pseudocontractive maps. Proc. Am. Math. Soc. 1998, 126: 2641–2649. 10.1090/S0002-9939-98-04322-6
Chidume CE: Iterative solution of nonlinear equations of strongly accretive type. Math. Nachr. 1998, 189: 49–60. 10.1002/mana.19981890105
Chidume CE, Moore C: Fixed point iteration for pseudocontractive maps. Proc. Am. Math. Soc. 1999, 127: 1163–1170. 10.1090/S0002-9939-99-05050-9
Chidume CE, Osilike MO: Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings. J. Math. Anal. Appl. 1995, 192: 727–741. 10.1006/jmaa.1995.1200
Chidume CE, Osilike MO: Nonlinear accretive and pseudocontractive operator equations in Banach spaces. Nonlinear Anal. 1998, 31: 779–789. 10.1016/S0362-546X(97)00439-2
Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc. Am. Math. Soc. 2001, 129: 2359–2363. 10.1090/S0002-9939-01-06009-9
Cho YJ, Kang SM, Qin X: Strong convergence of an implicit iterative process for an infinite family of strict pseudocontractions. Bull. Korean Math. Soc. 2010, 47: 1259–1268. 10.4134/BKMS.2010.47.6.1259
Cho YJ, Kim JK, Lan HY: Three step procedure with errors for generalized asymptotically quasi-nonexpansive mappings in Banach spaces. Taiwan. J. Math. 2008, 12: 2155–2178.
Crandall MG, Pazy A: On the range of accretive operators. Isr. J. Math. 1977, 27: 235–246. 10.1007/BF02756485
Deng L: On Chidume’s open problems. J. Math. Anal. Appl. 1993, 174(2):441–449. 10.1006/jmaa.1993.1129
Deng L: Iteration process for nonlinear Lipschitzian strongly accretive mappings in L p spaces. J. Math. Anal. Appl. 1994, 188: 128–140. 10.1006/jmaa.1994.1416
Deng L, Ding XP: Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces. Nonlinear Anal. 1995, 24: 981–987. 10.1016/0362-546X(94)00115-X
Hicks TL, Kubicek JR: On the Mann iteration process in Hilbert space. J. Math. Anal. Appl. 1977, 59: 498–504. 10.1016/0022-247X(77)90076-2
Reich S: Constructing zeros of accretive operators, II. Appl. Anal. 1979, 9: 159–163. 10.1080/00036817908839264
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6
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)90323-6
Rhoades BE: Comments on two fixed point iteration procedures. J. Math. Anal. Appl. 1976, 56: 741–750. 10.1016/0022-247X(76)90038-X
Schu J: Iterative construction of fixed points of strictly pseudocontractive mappings. Appl. Anal. 1991, 40: 67–72. 10.1080/00036819108839994
Song Y, Cho YJ: Some notes on ishikawa iteration for multi-valued mappings. Bull. Korean Math. Soc. 2011, 48: 575–584. 10.4134/BKMS.2011.48.3.575
Suantai S, Cho YJ, Cholamjiak P: Composite iterative schemes for maximal monotone operators in reflexive Banach spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 7
Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309
Weng XL: Fixed point iteration for local strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1991, 113: 727–731. 10.1090/S0002-9939-1991-1086345-8
Xu HK: Inequality in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987
Yao Y, Cho YJ: A strong convergence of a modified Krasnoselskii-Mann method for non-expansive mappings in Hilbert spaces. Math. Model. Anal. 2010, 15: 265–274. 10.3846/1392-6292.2010.15.265-274
Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 4(1):147–150.
Qihou L: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 1996, 26: 1835–1842. 10.1016/0362-546X(94)00351-H
Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–610. 10.1090/S0002-9939-1953-0054846-3
Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289
Rafiq A: On Mann iteration in Hilbert spaces. Nonlinear Anal. 2007, 66: 2230–2236. 10.1016/j.na.2006.03.012
Rafiq A: Implicit fixed point iterations for pseudocontractive mappings. Kodai Math. J. 2009, 32: 146–158. 10.2996/kmj/1238594552
Ćirić LB, Rafiq A, Cakić N, Ume JS: Implicit Mann fixed point iterations for pseudo-contractive mappings. Appl. Math. Lett. 2009, 22: 581–584. 10.1016/j.aml.2008.06.034
Acknowledgements
We are grateful to the editor and the referees for their valuable suggestions for the improvement of this manuscript. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.
The second author is supported by the Ministry of Science, Technology and Development, Republic of Serbia. The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Hussain, N., Ćirić, L.B., Cho, Y.J. et al. On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces. J Inequal Appl 2013, 41 (2013). https://doi.org/10.1186/1029-242X-2013-41
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-41