Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings
© Zegeye and Shahzad; licensee Springer 2013
Received: 21 September 2012
Accepted: 28 November 2012
Published: 2 January 2013
We introduce an iterative process which converges strongly to the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings. As a consequence, convergence result to a common minimum-norm fixed point of a finite family of nonexpansive mappings is proved.
MSC:47H09, 47H10, 47J05, 47J25.
where A is bounded linear operator from into . A split feasibility problem in finite dimensional Hilbert spaces was first studied by Censor and Elfving  for modeling inverse problems which arise in medical image reconstruction, image restoration and radiation therapy treatment planing (see, e.g., [1–3]).
Recall that a point is said to be a fixed point of T if . We denote the set of fixed points of T by , i.e., . Therefore, finding a solution to the split feasibility problem (1.1) is equivalent to finding the minimum-norm fixed point of the mapping .
The class of asymptotically nonexpansive mappings was introduced as a generalization of the class of nonexpansive mappings by Goebel and Kirk  who proved that if K is a nonempty closed convex bounded subset of a real uniformly convex Banach spaces which includes Hilbert spaces as a special case and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.
In , Browder proved that, as , converges strongly to the nearest point projection of u onto .
converges strongly to the element of which is nearest to u under certain conditions on .
where satisfies certain conditions, converges strongly to the element of which is nearest to u.
It is worth mentioning that the methods studied above are used to approximate the fixed point of T which is closest to the point . These methods can be used to find the minimum-norm fixed point of T if . If, however, , any of the methods above fails to provide the minimum-norm fixed point of T.
They proved that under appropriate conditions on and β, the sequence converges strongly to the minimum-norm fixed point of T in real Hilbert spaces.
More recently, Yao and Xu  have also shown that the explicit scheme , , converges strongly to the minimum-norm fixed point of a nonexpansive self-mapping T provided that satisfies certain conditions.
Let K be a closed convex subset of a real Hilbert space H and let , be a finite family of asymptotically nonexpansive mappings.
It is our purpose in this paper to introduce an explicit iteration process which converges strongly to the common minimum-norm fixed point of . Our theorems improve several results in this direction.
In what follows, we shall make use of the following lemmas.
Lemma 2.2 
Lemma 2.3 
Lemma 2.4 
Let H be a real Hilbert space, K be a closed convex subset of H and be an asymptotically nonexpansive mapping, then is demiclosed at zero, i.e., if is a sequence in K such that and , as , then .
Lemma 2.5 
where , and satisfying the following conditions: , , and , as . Then .
Lemma 2.6 
In fact, .
Proposition 2.7 Let H be a real Hilbert space, let K be a closed convex subset of H, and let T be an asymptotically nonexpansive mapping from K into itself. Then is closed and convex.
and hence, since as , we get that , which implies that . Now, by the continuity of T, we obtain that . Hence, and that is convex. □
3 Main result
We now state and proof our main theorem.
where such that , , for each and , for , satisfying for each . Then converges strongly to the common minimum-norm point of F.
for some , where for all .
Now, we consider the following two cases.
for some . But note that satisfies and . Thus, it follows from (3.15) and Lemma 2.5 that , as . Consequently, .
Thus, from (3.16) and the fact that , we obtain that as . This together with (3.17) gives as . But for all , thus we obtain that . Therefore, from the above two cases, we can conclude that converges strongly to a point of F which is the common minimum-norm fixed point of the family and the proof is complete. □
If in Theorem 3.1 we assume that , then we get the following corollary.
where such that , and , for each . Then converges strongly to the minimum-norm fixed point of T.
If in Theorem 3.1 we assume that each is nonexpansive for , then the method of proof of Theorem 3.1 provides the following corollary.
where such that and , , for , satisfying for each . Then converges strongly to the common minimum-norm point of F.
If in Corollary 3.3 we assume that , then we have the following corollary.
where such that and , for each . Then converges strongly to the minimum-norm point of .
In this section, we study the problem of finding a minimizer of a continuously Fréchet-differentiable convex functional which has the minimum norm in Hilbert spaces.
for all .
Now, we have the following corollary deduced from Corollary 3.2.
where such that , and , for each . Then converges strongly to the minimum-norm solution of the minimization problem (4.1).
Remark 4.2 Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem 3.1 improves Theorem 3.2 of Yang et al.  and of Yao and Xu  to a more general class of a finite family of asymptotically nonexpansive mappings.
The second author gratefully acknowledges the sup- port provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research.
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