- Open Access
Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces
© Hussain et al.; licensee Springer. 2013
- Received: 7 May 2013
- Accepted: 5 September 2013
- Published: 7 November 2013
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.
- modified iterative scheme
- nonexpansive mappings
- ϕ-hemicontractive mappings
- Banach spaces
Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday
Let be an operator.
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.
- (a)the Mann iteration scheme  is defined by the following sequence :
where is a sequence in ;
where , are sequences in is known as the Ishikawa  iteration scheme;
where , are sequences in , is known as the Agarwal-O’Regan-Sahu  iteration scheme;
where , are sequences in , is known as the modified Agarwal-O’Regan-Sahu iteration scheme.
Chidume  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.  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].
The following result is now well known.
Lemma 3 
Now, we prove our main result.
converges strongly to the common fixed point q of S and T.
, 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 .
Thus, . Therefore, , and are bounded.
Second, we need to show that (b) implies (a). Suppose that , and are bounded.
It is clear that . Let . Next, we will prove that .
where the second inequality holds by the convexity of .
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.
converges strongly to the unique fixed point q of T.
All the results can also be proved for the same iterative scheme with error terms.
The known results for strongly pseudocontractive mappings are weakened by the ϕ-hemicontractive mappings.
Our results hold in arbitrary Banach spaces, where as other known results are restricted for (or ) spaces and q-uniformly smooth Banach spaces.
Theorem 4 is more general in comparison to the results of Agarwal et al.  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]).
converges strongly to the solution of the system .
, 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. □
Then defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of S and T.
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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