Abstract
The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.
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Dedication
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
1 Preliminaries
Let K be a nonempty subset of an arbitrary Banach space X, and let be its dual space. Let be an operator. The symbols and stand for the domain and the range of T, respectively. We denote by the set of fixed points of a single-valued mapping . We denote by J the normalized duality mapping from X to defined by
Let be an operator.
Definition 1 T is called L-Lipschitzian if there exists such that
for all . If , then T is called non-expansive, and if , T is called contraction.
-
(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 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 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.
For a nonempty convex subset K of a normed space and ,
-
(a)
the Mann iteration scheme [4] is defined by the following sequence :
where is a sequence in ;
-
(b)
the sequence defined by
where , are sequences in is known as the Ishikawa [2] iteration scheme;
-
(c)
the sequence defined by
where , are sequences in , is known as the Agarwal-O’Regan-Sahu [5] iteration scheme;
-
(d)
the sequence defined by
where , are sequences in , is known as the modified Agarwal-O’Regan-Sahu iteration scheme.
Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of (or ) into itself. Afterwards, several authors generalized this result of Chidume in various directions [3, 6–12].
The purpose of this paper is to characterize conditions for the convergence of the iterative scheme in the sense of Agarwal et al. [5] associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results improve and generalize most results in recent literature [1, 3, 5, 6, 8, 9, 11, 12].
2 Main result
The following result is now well known.
Lemma 3 [13]
For all and ,
Now, we prove our main result.
Theorem 4 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, let be nonexpansive, and let be a uniformly continuous ϕ-hemicontractive mapping such that S and T have the common fixed point. Suppose that and are sequences in satisfying conditions
-
(i)
,
-
(ii)
.
For any , define the sequence inductively as follows:
Then the following conditions are equivalent:
-
(a)
converges strongly to the common fixed point q of S and T.
-
(b)
, and are bounded.
Proof First, we prove that (a) implies (b).
Since T is ϕ-hemicontractive, it follows that is a singleton. Let for some .
Suppose that . Then the continuity of S and T yields that
and
Thus, . Therefore, , and are bounded.
Second, we need to show that (b) implies (a). Suppose that , and are bounded.
Put
It is clear that . Let . Next, we will prove that .
Note that
Thus, we can conclude that the sequence is bounded, and hence, there is a constant satisfying
Let for each . The uniform continuity of T ensures that
because
By virtue of Lemma 3 and (2.1), we infer that
The real function , is increasing and convex. For all and , we have
Hence,
where the second inequality holds by the convexity of .
By substituting (2.5) in (2.4), we get
where
as .
Let . We claim that . Otherwise, . Thus, (2.7) implies that there exists a positive integer such that for each . In view of (2.6), we conclude that
which implies that
which contradicts (ii). Therefore, . Thus, there exists a subsequence of such that
Let be a fixed number. By virtue of (2.7) and (2.9), we can select a positive integer such that
Let . By induction, we show that
Observe that (2.6) means that (2.11) is true for . Suppose that (2.11) is true for some . If , by (2.6) and (2.10), we know that
which is impossible. Hence, . That is, (2.11) holds for all . Thus, (2.11) ensures that . This completes the proof. □
Taking in Theorem 4, we get the following.
Corollary 5 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, and let be a uniformly continuous ϕ-hemicontractive mapping. Suppose that and are sequences in satisfying conditions (i)-(ii) of Theorem 4. For any , define the sequence inductively as follows:
Then the following conditions are equivalent:
-
(a)
converges strongly to the unique fixed point q of T.
-
(b)
is bounded.
Remark 6
-
1.
All the results can also be proved for the same iterative scheme with error terms.
-
2.
The known results for strongly pseudocontractive mappings are weakened by the ϕ-hemicontractive mappings.
-
3.
Our results hold in arbitrary Banach spaces, where as other known results are restricted for (or ) spaces and q-uniformly smooth Banach spaces.
-
4.
Theorem 4 is more general in comparison to the results of Agarwal et al. [5] in the context of the class of ϕ-hemicontractive mappings. Theorem 4 extends convergence results coercing ϕ-hemicontractive mappings in the literature in the framework of Agarwal-O’Regan-Sahu iteration process (see also [14–21]).
3 Applications
Theorem 7 Let X be an arbitrary real Banach space, be nonexpansive, and let be uniformly continuous ϕ-strongly accretive operators, respectively. Suppose that and are sequences in satisfying conditions (i)-(ii) of Theorem 4. For any , define the sequence inductively as follows:
where , and I is the identity operator. Then the following conditions are equivalent:
-
(a)
converges strongly to the solution of the system .
-
(b)
, and are bounded.
Proof Suppose that is the solution of the system . Define by and , respectively. Since S and T are nonexpansive and uniformly continuous ϕ-strongly accretive operators, respectively, so are G and , then is the common fixed point of G and . Thus, Theorem 7 follows from Theorem 4. □
Example 8 Let be the reals with the usual norm and . Define by
and by
By the mean value theorem, we have
Noticing that and . Hence,
where . It is easy to verify that T is ϕ-hemicontractive mapping with defined by for all . Moreover, 0 is the common fixed point of S and T. Let and be sequences in defined by
Then defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of S and T.
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Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.
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Hussain, N., Sahu, D. & Rafiq, A. Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces. Fixed Point Theory Appl 2013, 247 (2013). https://doi.org/10.1186/1687-1812-2013-247
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DOI: https://doi.org/10.1186/1687-1812-2013-247