Abstract
In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.
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1 Introduction
Let K be a nonempty subset of an arbitrary Banach space X and be its dual space. The symbols , and stand for the domain, the range and the set of fixed points of T respectively (for a single-valued map , is called a fixed point of T iff ). We denote by J the normalized duality mapping from E to defined by
Let be an operator. The following definitions can be found in [1–15] for example.
Definition 1 T is called Lipshitzian if there exists such that
for all . If , then T is called nonexpansive, and if , T is called contraction.
Definition 2
-
(i)
T is said to be strongly pseudocontractive if there exists a such that for each , there exists satisfying
-
(ii)
T is said to be strictly hemicontractive if is nonempty and if there exists a such that for each and , there exists satisfying
-
(iii)
T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function with such that for each , there exists satisfying
-
(iv)
T is said to be ϕ-hemicontractive if is nonempty and if there exists a strictly increasing function with such that for each and , there exists satisfying
Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.
Definition 3
-
(i)
T is called accretive if the inequality
holds for every and for all .
-
(ii)
T is called strongly accretive if, for all , there exists a constant and such that
-
(iii)
T is called ϕ-strongly accretive if there exists and a strictly increasing function with such that for each ,
Remark 4 It has been shown in [11, 12] that the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators. If I denotes the identity operator, then T is called strongly pseudocontractive (respectively, ϕ-strongly pseudocontractive) if and only if is strongly accretive (respectively, ϕ-strongly accretive).
Chidume [1] showed that the Mann iterative method can be used to approximate fixed points of Lipschitz strongly pseudocontractive operators in (or ) spaces for . Chidume and Osilike [4] proved that each strongly pseudocontractive operator with a fixed point is strictly hemicontractive, but the converse does not hold in general. They also proved that the class of strongly pseudocontractive operators is a proper subclass of the class of ϕ-strongly pseudocontractive operators and pointed out that the class of ϕ-strongly pseudocontractive operators with a fixed point is a proper subclass of the class of ϕ-hemicontractive operators. These classes of nonlinear operators have been studied by various researchers (see, for example, [7–25]). Liu et al. [26] proved that, under certain conditions, a three-step iteration scheme with error terms converges strongly to the unique fixed point of ϕ-hemicontractive mappings.
In this paper, we study a three-step iterative scheme with error terms for nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.
2 Preliminaries
We need the following results.
Lemma 5 [27]
Let , and be three sequences of nonnegative real numbers with and . If
then the limit exists.
Lemma 6 [28]
Let . Then for every if and only if there is such that .
Lemma 7 [9]
Suppose that X is an arbitrary Banach space and is a continuous ϕ-strongly accretive operator. Then the equation has a unique solution for any .
3 Strong convergence of a three-step iterative scheme to a solution of the system of nonlinear operator equations
For the rest of this section, L denotes the Lipschitz constant of , and , and denote the ranges of , and respectively. We now prove our main results.
Theorem 8 Let X be an arbitrary real Banach space and Lipschitz ϕ-strongly accretive operators. Let and generate from an arbitrary by
where , and are bounded sequences in X and , , , , , , , are sequences in and in satisfying the following conditions: (i) , (ii) , (iii) , , (iv) , and . Then the sequence converges strongly to the solution of the system ; .
Proof By Lemma 7, the system ; has the unique solution . Following the techniques of [5, 8–12, 26, 29], define by ; ; then each is demicontinuous and is the unique fixed point of ; , and for all , we have
where for all ; . Let be the fixed point set of , and let . Thus
It follows from Lemma 6 and inequality (3.2) that
for all and for all ; .
Set , and , then (3.1) becomes
We have
Furthermore,
so that
Hence,
Hence,
Furthermore, we have the following estimates:
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Using (3.4) and (3.6),
Using (3.7),
Again, using (3.7),
Substituting (3.10)-(3.12) in (3.9), we obtain
Substituting (3.8), (3.12) and (3.13) in (3.5), we obtain
Since , and are bounded, we set
Then it follows from (3.14) that
where
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Since , the conditions (iii) and (iv) imply that and . It then follows from Lemma 5 that exists. Let . We now prove that . Assume that . Then there exists a positive integer such that for all . Since
for all , it follows from (3.15) that
Hence,
This implies that
Since ,
yields , contradicting the fact that . Hence, . □
Corollary 9 Let X be an arbitrary real Banach space and be three Lipschitz ϕ-strongly accretive operators, where ϕ is in addition continuous. Suppose or as ; . Let , , , , , , , , , , , , and be as in Theorem 8. Then, for any given , the sequence converges strongly to the solution of the system ; .
Proof The existence of a unique solution to the system ; follows from [9] and the result follows from Theorem 8. □
Theorem 10 Let X be a real Banach space and K be a nonempty closed convex subset of X. Let be three Lipschitz ϕ-strong pseudocontractions with a nonempty fixed point set. Let , , , , , , , , , , and be as in Theorem 8. Let be the sequence generated iteratively from an arbitrary by
Then converges strongly to the common fixed point of , , .
Proof As in the proof of Theorem 8, set , , to obtain
Since each ; is a ϕ-strong pseudocontraction, is ϕ-strongly accretive so that for all , there exist and a strictly increasing function with such that
The rest of the argument now follows as in the proof of Theorem 8. □
Remark 11 The example in [4] shows that the class of ϕ-strongly pseudocontractive operators with nonempty fixed point sets is a proper subclass of the class of ϕ-hemicontractive operators. It is easy to see that Theorem 8 easily extends to the class of ϕ-hemicontractive operators.
Remark 12
-
(i)
If we set for all in our results, we obtain the corresponding results for the Ishikawa iteration scheme with error terms in the sense of Xu [15].
-
(ii)
If we set for all in our results, we obtain the corresponding results for the Mann iteration scheme with error terms in the sense of Xu [15].
Remark 13 Let and be real sequences satisfying the following conditions:
-
(i)
, ,
-
(ii)
,
-
(iii)
,
-
(iv)
, and
-
(v)
.
If we set , , , , , , for all in Theorems 8 and 10 respectively, we obtain the corresponding convergence theorems for the original Ishikawa [18] and Mann [30] iteration schemes.
Remark 14
-
(i)
Gurudwan and Sharma [29] studied a strong convergence of multi-step iterative scheme to a common solution for a finite family of ϕ-strongly accretive operator equations in a reflexive Banach space with weakly continuous duality mapping. Some remarks on their work can be seen in [31].
-
(ii)
All the above results can be extended to a finite family of ϕ-strongly accretive operators.
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Acknowledgements
The last author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.
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Khan, S.H., Rafiq, A. & Hussain, N. A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces. Fixed Point Theory Appl 2012, 149 (2012). https://doi.org/10.1186/1687-1812-2012-149
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DOI: https://doi.org/10.1186/1687-1812-2012-149