New iteration scheme for numerical reckoning fixed points of nonexpansive mappings
© Thakur et al.; licensee Springer. 2014
Received: 27 May 2014
Accepted: 19 August 2014
Published: 2 September 2014
The purpose of this paper is to introduce a new three step iteration scheme for approximation of fixed points of the nonexpansive mappings. We show that our iteration process is faster than all of the Picard, the Mann, the Agarwal et al., and the Abbas et al. iteration processes. We support our analytic proof by a numerical example in which we approximate the fixed point by a computer using Matlab program. We also prove some weak convergence and strong convergence theorems for the nonexpansive mappings.
The iterative method (1.2) is also called a Richardson iteration, a Picard iteration, or the method of successive substitution. The standard result for a fixed point iteration is the contraction mapping theorem. Indeed, the contraction mapping theorem holds on an arbitrary complete metric space; that is, if E is a complete metric space with metric d and such that for some and all , then T has a unique fixed point and the iterates (1.2) converge to the fixed point . The Picard iteration has been successfully employed in approximating the fixed point of contraction mappings and its variants. This success, however, has not extended to nonexpansive mappings T even when the existence of a fixed point of T is known. Consider the simple example of a self mapping in defined by for . Then T is a nonexpansive mapping with a unique fixed point at . If one chooses as a starting value , , then the successive iterations of T yield the sequence . Thus when a fixed point of nonexpansive mappings exists, other approximation techniques are needed to approximate it.
where is a real control sequence in the interval .
for all , where , , and are sequences in .
where and are in . They showed that this process converges at a rate that is the same as that of the Picard iteration and faster than the Mann iteration for contractions.
where , , and are in . They showed that this process converges faster than the Agarwal et al.  iteration process.
where , , and are real sequences in .
The purpose of this paper is to prove that our process (1.8) converges faster than all of the Picard, the Mann, the Ishikawa, the Noor, the Agarwal et al., and the Abbas et al. iteration processes for contractions in the sense of Berinde . We also prove weak and strong convergence theorems for nonexpansive mapping using iteration (1.8). In the last section, using a numerical example, we compare the behavior of iteration (1.8) with respect to the above mentioned iteration processes.
2 Rate of convergence
Berinde  proposed a method to compare the fastness of two sequences.
If , then it can be said that converges faster to a than to b.
If , then it can be said that and have the same rate of convergence.
are available, where and are sequences of positive numbers (converging to zero).
Then, in view of Definition 2.1, Berinde  adopted the following concept.
Definition 2.2 Let and be two fixed point iteration procedures that converge to the same fixed point p and satisfy (2.2) and (2.3), respectively. If converges faster than , then it can be said that converges faster than to p.
In recent years, Definition 2.2 has been used as a standard tool to compare the fastness of two fixed point iterations. Using this technique Sahu  established that the Agarwal et al. iteration (1.6) converges faster than the Mann (1.3) and the Picard (1.2) iterations and supported the claim by the following example.
Example 1 Let and . Let be a mapping defined by for all . For and , , Agarwal et al. iteration is faster than both the Mann and the Picard iteration.
Using a similar technique Abbas and Nazir  established that the Abbas et al. iteration (1.7) converges faster than the Agarwal et al. iteration (1.6) and hence it converges faster than the Mann (1.3) and the Picard (1.2) iterations also. An example is also given in support of the claim.
Example 2 Let and . Let be a mapping defined by for all . For and , , the Abass et al. iteration (1.7) is faster than the Agarwal et al. iteration (1.6). Since Sahu  already has shown that the iteration (1.6) is faster than the Mann iteration (1.3), the iteration (1.7) is faster than the iterations (1.2), (1.3), and (1.6).
We now show that our process (1.8) converges faster than (1.7) in the sense of Berinde .
Theorem 2.3 Let C be a nonempty closed convex subset of a norm space E. Let T be a contraction with a contraction factor and fixed point p. Let be defined by the iteration process (1.7) and by (1.8), where , , and are in for all and for some ε in . Then converges faster than . That is, our process (1.8) converges faster than (1.7).
Consequently converges faster than . □
Now, we present an example which shows that the new iteration process (1.8) converges at a rate faster than the existing iteration schemes mentioned above.
All sequences converge to . Comparison shows that our iteration process (1.8) converges fastest among all the iterations considered in the example.
3 Convergence theorems
Lemma 3.1 Let C be a nonempty closed convex subset of a norm space E. Let T be a nonexpansive self mapping on C, defined by (1.8) and . Then exists for all .
Thus exists for all . □
We need following lemma to establish our next result.
Lemma 3.2 Suppose that E is a uniformly convex Banach space and for all . Let and be two sequences of E such that , and hold for some . Then .
We now establish a result which will be of key importance for the main result.
Lemma 3.3 Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Let T be a nonexpansive self mapping on C, defined by (1.8), where , , and are in for all and for some ε in and . Then .
Proof By Lemma 3.1, exists. Assume that .
This completes the proof. □
Lemma 3.4 
for all and .
Lemma 3.5 For any , exists, for all under the conditions of Lemma 3.3.
for all . Then and, from Lemma 3.1, exist.
Now it remains to show that exists for .
for all .
Since exists for all , we get and by the property of g, we get .
This implies that exists for all , i.e., exists for all . □
exists for each x and y in . In this case, the norm of E is called Gâteaux differentiable.
for all , where J is the Fréchet derivative of the function at , is the dual pairing between E and , and b is an increasing function defined on such that .
Lemma 3.6 Assume that the conditions of Lemma 3.3 are satisfied. Then, for any , exists; in particular, for all , the set of all weak limits of .
The proof of Lemma 3.6 is similar to the proof of Lemma 2.3 of Khan and Kim .
for all with .
A Banach space E is said to have the Kadec-Klee property if for every sequence in E, and together imply as .
We need the following to prove our next result.
Definition 3.7 A mapping is demiclosed at if for each sequence in C and each , , and imply that and .
Lemma 3.8 
Let C be a nonempty closed convex subset of a uniformly convex Banach space E, and T a nonexpansive mapping on C. Then is demiclosed at zero.
Lemma 3.9 
Let E be a reflexive Banach space satisfying the Opial condition, C a nonempty convex subset of E, and an operator such that demiclosed at zero and . Let be a sequence in C such that and exists for all . Then converges weakly to a fixed point of T.
Lemma 3.10 
Let E be a real reflexive Banach space such that its dual has the Kadec-Klee property. Let be a bounded sequence in E and , here denotes the w-limit set of . Suppose exists for all . Then .
We now establish a weak convergence result.
E satisfies the Opial condition,
E has a Fréchet differentiable norm,
the dual of E satisfies the Kadec-Klee property.
Then converges weakly to a point of .
Proof Let , by Lemma 3.1, exists.
We prove that has a unique weak subsequential limit in .
Let u and v be weak limits of the subsequences and of , respectively. By Lemma 3.3, , and also is demiclosed with respect to zero, hence by Lemma 3.8, we obtain . In a similar manner, we have .
Next, we prove the uniqueness.
This is a contradiction, so .
Next, assume (b) holds.
By Lemma 3.6, , for all . Therefore, implies .
Finally, assume that (c) is true.
Since exists for all by Lemma 3.5, by Lemma 3.10, and converges weakly to a fixed point of and this completes the proof. □
A mapping is said to be semicompact if any sequence in C, such that , has a subsequence converging strongly to some .
Next we establish the following strong convergence results.
Theorem 3.12 Let E be a uniformly convex Banach space and let C, T, and be as in Lemma 3.3. If T is semicompact and , then converges strongly to a fixed point of T.
This yields . By Lemma 3.1, exists for all , and therefore must itself converge to and this completes the proof. □
Theorem 3.13 Let E be a uniformly convex Banach space and let C, T, , and be as in Lemma 3.3. Then converges to a point of if and only if , where .
Proof Necessity is obvious. Suppose that . As proved in Lemma 3.3, exists for all , therefore exists. But by hypothesis, , therefore .
Hence is a Cauchy sequence in C. Since C is a closed subset of a complete space, . Since is closed, gives , i.e., . □
Definition 3.14 A mapping , where C is a subset of a normed space E, is said to satisfy Condition (I)  if there exists a nondecreasing function with , for all such that for all where .
Applying Theorem 3.13, we obtain strong convergence of the process (1.8) under Condition (I) as follows.
Theorem 3.15 Let e be a uniformly convex Banach space and let C, T, and be as in Lemma 3.3. Let T satisfy Condition (I), then converges strongly to a fixed point of T.
Now all the conditions of Theorem 3.13 are satisfied, therefore, by its conclusion, converges strongly to a point of . □
The second author would like to thank the Rajiv Gandhi National Fellowship of India for the grant (F1-17.1/2011-12/RGNF-ST-CHH-6632).
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