Coupling Ishikawa algorithms with hybrid techniques for pseudocontractive mappings
© Yao et al.; licensee Springer 2013
Received: 4 June 2013
Accepted: 24 July 2013
Published: 8 August 2013
It is well-known that Mann’s algorithm fails to converge for Lipschitzian pseudocontractions and strong convergence of Ishikawa’s algorithm for Lipschitzian pseudocontractions have not been achieved without compactness assumption on pseudocontractive mapping T or underlying space C. A new algorithm, which couples Ishikawa algorithms with hybrid techniques for finding the fixed points of a Lipschitzian pseudocontractive mapping, is constructed in this paper. Strong convergence of the presented algorithm is shown without any compactness assumption.
MSC:47H05, 47H10, 47H17.
In the present article, we are devoted to finding the fixed points of pseudocontractive mappings. Interest in pseudocontractive mappings stems mainly from their firm connection with the class of nonlinear accretive operators. It is a classical result, see Deimling , that if T is an accretive operator, then the solutions of the equations correspond to the equilibrium points of some evolution systems. This explains why a considerable research effort has been devoted to iterative methods for approximating solutions of the equation above, when T is accretive or corresponding to the iterative approximation of fixed points of pseudocontractions. Results of this kind have been obtained firstly in Hilbert spaces, but only for Lipschitz operators, and then they have been extended to more general Banach spaces (thanks to several geometric inequalities for general Banach spaces developed) and to more general classes of operators. There are still no results for the case of arbitrary Lipschitzian and pseudocontractive operators, even when the domain of the operator is a compact and convex subset of a Hilbert space. It is now well known that Mann’s algorithm  fails to converge for Lipschitzian pseudocontractions. This explains the importance, from this point of view, of the improvement brought by the Ishikawa iteration, which was introduced by Ishikawa  in 1974.
The original result of Ishikawa involves a Lipschitzian pseudocontractive self-mapping T on a convex and compact subset C of a Hilbert space. It establishes sufficient conditions such that Ishikawa iteration converges strongly to a fixed point of T.
However, a strong convergence has not been achieved without a compactness assumption on T or C. Consequently, considerable research efforts, especially within the past 40 years or so, have been devoted to iterative methods for approximating fixed points of T, when T is pseudocontractive (see, for example, [4–17] and the references therein). On the other hand, some convergence results are obtained by using the hybrid method in mathematical programming, see, for example, [14, 18–20]. Especially, Zegeye et al.  assumed that the interior of is nonempty () to achieve a strong convergence, when T is a self-mapping of a nonempty closed convex subset of a real Hilbert space. This appears very restrictive, since even in ℝ with the usual norm, Lipschitz pseudocontractive maps with finite number of fixed points do not enjoy this condition that .
The purpose of this article is to construct a new algorithm, which couples Ishikawa algorithms with hybrid techniques for finding the fixed points of a Lipschitzian pseudocontractive mapping. Strong convergence of the presented algorithm is given without any compactness assumption.
for all .
for all .
for all .
The original result of Ishikawa is stated in the following.
converges strongly to a fixed point of T.
for all and .
Lemma 2.1 
is a closed convex subset of C.
is demiclosed at zero.
In the sequel, we shall use the following notations:
denote the weak ω-limit set of ;
stands for the weak convergence of to x;
stands for the strong convergence of to x.
Lemma 2.2 
3 Main results
In this section, we state our main results.
Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be an L-Lipschitzian pseudocontractive mapping with .
Firstly, we present our new algorithm, which couples Ishikawa’s algorithm (2.2) with the hybrid projection algorithm.
for all , where and are two sequences in .
for all .
Remark 3.1 Without loss of generality, we can assume that the Lipschitz constant . If not, then T is nonexpansive. In this case, algorithm (3.1) is trivial. So, in this article, we assume . It is obvious that for all .
We prove the following several lemmas, which will support our main theorem below.
Lemma 3.1 for and is well defined.
- (ii)Suppose that for some . Take . From (3.1), by using (2.4), we have(3.2)
for all .
for all .
Next, we show that is closed and convex for all .
It is obvious that is closed and convex.
Suppose that is closed and convex for some . For , it is obvious that is equivalent to . So, is closed and convex. Then, for any , the set is closed and convex. This implies that is well defined. □
Lemma 3.2 The sequence is bounded.
This implies that the sequence is bounded. □
Lemma 3.3 .
Theorem 3.2 The sequence defined by (3.1) converges strongly to .
Remark 3.3 Note that is closed and convex. Thus, the projection is well defined.
Now, (3.7) and Lemma 2.1 guarantee that every weak limit point of is a fixed point of T. That is, . This fact, inequality (3.6) and Lemma 2.2 ensure the strong convergence of to . This completes the proof. □
Remark 3.4 It is easily seen that all of the results above hold for nonexpansive mappings.
Remark 3.5 It is nowadays quite clear that, for large classes of contractive type operators, it suffices to consider the simpler Mann iteration, even if the Ishikawa iteration, which is more general but also computationally more complicated than the Mann iteration, could always be used. But if T is only a pseudocontractive mapping, then generally, the Mann iterative process does not converge to the fixed point, and strong convergence of the Ishikawa iteration has not been achieved without the compactness assumption on T or C. However, our algorithm (3.1) has a strong convergence without the compactness assumption.
Yonghong Yao was supported in part by NSFC 71161001-G0105 and LQ13A010007. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
- Deimling K: Zeros of accretive operators. Manuscr. Math. 1974, 13: 365–374. 10.1007/BF01171148MathSciNetView ArticleGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
- Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleGoogle Scholar
- Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudo-contractions. Proc. Am. Math. Soc. 2001, 129: 2359–2363. 10.1090/S0002-9939-01-06009-9MathSciNetView ArticleGoogle Scholar
- Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. Am. Math. Soc. 2003, 132: 831–840.MathSciNetView ArticleGoogle Scholar
- Zhang QB, Cheng CZ: Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space. Math. Comput. Model. 2008, 48: 480–485. 10.1016/j.mcm.2007.09.014View ArticleGoogle Scholar
- Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039–4046. 10.1016/j.na.2008.08.012MathSciNetView ArticleGoogle Scholar
- Zegeye H, Shahzad N, Mekonen T: Viscosity approximation methods for pseudocontractive mappings in Banach spaces. Appl. Math. Comput. 2007, 185: 538–546. 10.1016/j.amc.2006.07.063MathSciNetView ArticleGoogle Scholar
- Zegeye H, Shahzad N, Alghamdi MA: Minimum-norm fixed point of pseudocontractive mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 926017Google Scholar
- Ofoedu EU, Zegeye H: Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces. J. Math. Anal. Appl. 2010, 372: 68–76. 10.1016/j.jmaa.2010.07.020MathSciNetView ArticleGoogle Scholar
- Ofoedu EU, Zegeye H: Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal. 2012, 75: 153–162. 10.1016/j.na.2011.08.015MathSciNetView ArticleGoogle Scholar
- Udomene A: Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudo-contractions in Banach spaces. Nonlinear Anal. 2007, 67: 2403–2414. 10.1016/j.na.2006.09.001MathSciNetView ArticleGoogle Scholar
- Yao Y, Liou YC, Chen R: Strong convergence of an iterative algorithm for pseudo-contractive mapping in Banach spaces. Nonlinear Anal. 2007, 67: 3311–3317. 10.1016/j.na.2006.10.013MathSciNetView ArticleGoogle Scholar
- Yao Y, Liou YC, Marino G: A hybrid algorithm for pseudo-contractive mappings. Nonlinear Anal. 2009, 71: 4997–5002. 10.1016/j.na.2009.03.075MathSciNetView ArticleGoogle Scholar
- Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc. Am. Math. Soc. 2000, 128: 3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView ArticleGoogle Scholar
- Ceng LC, Petrusel A, Yao JC: Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings. Appl. Math. Comput. 2009, 209: 162–176. 10.1016/j.amc.2008.10.062MathSciNetView ArticleGoogle Scholar
- Berinde V Lecture Notes in Mathematics. In Iterative Approximation of Fixed Points. Springer, Berlin; 2007.Google Scholar
- Martinez-Yanes C, Xu HK: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 2006, 64: 2400–2411. 10.1016/j.na.2005.08.018MathSciNetView ArticleGoogle Scholar
- Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 2008, 341: 276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleGoogle Scholar
- Nadezhkina N, Takahashi W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 2006, 16: 1230–1241. 10.1137/050624315MathSciNetView ArticleGoogle Scholar
- Zegeye H, Shahzad N, Alghamdi MA: Convergence of Ishikawas iteration method for pseudocontractive mappings. Nonlinear Anal. 2011, 74: 7304–7311. 10.1016/j.na.2011.07.048MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.