A Picard-Mann hybrid iterative process
© Khan; licensee Springer. 2013
Received: 24 May 2012
Accepted: 4 March 2013
Published: 25 March 2013
We introduce a new iterative process which can be seen as a hybrid of Picard and Mann iterative processes. We show that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde (Iterative Approximation of Fixed Points, 2002) for contractions. We support our analytical proof by a numerical example. We prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. Our weak convergence results are proved when the underlying space satisfies Opial’s condition or has Fréchet differentiable norm or its dual satisfies the Kadec-Klee property.
1 Introduction and preliminaries
Let C be a nonempty convex subset of a normed space E, and let be a mapping. Throughout this paper, ℕ denotes the set of all positive integers, I the identity mapping on C and the set of all fixed points of T.
where is in .
where and are in , is known as the Ishikawa iterative process .
where is in . This process is independent of all Picard, Mann and Ishikawa iterative processes since and are in . Even if it is allowed to take to make it a special case of the Ishikawa iterative process, our process is faster than Ishikawa and ‘faster is better’ rule should prevail.
The purpose of this paper is to prove that our process (1.4) converges faster than all of Picard, Mann and Ishikawa iterative processes for contractions in the sense of Berinde . We support our analytical proof by a numerical example. We also prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. We also prove some weak convergence results when the underlying space satisfies Opial’s condition or has the Fréchet differentiable norm or its dual satisfies the Kadec-Klee property.
for all x, h in E, where J is the Fréchet derivative of the functional at , is the dual pairing between E and , and b is an increasing function defined on such that ; (iii) Opial’s condition  if for any sequence in E, implies that for all with ; and (iv) the Kadec-Klee property if for every sequence in E, and together imply as . Examples of Banach spaces satisfying Opial’s condition are Hilbert spaces and all spaces (). On the other hand, with fail to satisfy Opial’s condition. Uniformly convex Banach spaces, Banach spaces of finite dimension and reflexive locally uniform convex Banach spaces are some of the examples of reflexive Banach spaces which satisfy the Kadec-Klee property.
A mapping is demiclosed at if for each sequence in C and each , and imply that and .
Lemma 1 
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 .
Lemma 2 
for all and .
Lemma 3 
Let E be a uniformly convex Banach space satisfying Opial’s condition, and let C be a nonempty closed convex subset of E. Let be a nonexpansive mapping. Then is demiclosed with respect to zero.
Lemma 4 
Let E be a reflexive Banach space such that has the Kadec-Klee property. Let be a bounded sequence in E and (weak limit set of ). Suppose that exists for all . Then .
then converges faster than .
are available where and are two sequences of positive numbers converging to zero. If converges faster than , then converges faster than to p.
In the sequel, whenever we talk of rate of convergence, we mean the one given by the above definitions.
2 Rate of convergence
We now show that (1.4) converges at a rate faster than all of Picard (1.1), Mann (1.2) and Ishikawa (1.3) iterative processes for contractions.
Proposition 1 Let C be a nonempty closed convex subset of a normed space E, and let T be a contraction of C into itself. Suppose that each of the iterative processes (1.1), (1.2), (1.3) and (1.4) converges to the same fixed point p of T where and are such that for all and for some λ. Then the iterative process given by (1.4) converges faster than all the other three processes.
Now as . Thus converges faster than to p. Similarly, as . It is not difficult to prove that for the Ishikawa iterative process, we have , and hence converges faster than to p.
Summing up, we conclude that our process (1.4) converges faster than all of Picard (1.1), Mann (1.2) and Ishikawa (1.3) iterative processes. □
We support our above analytical proof by a numerical example.
A comparison of our process with other processes
Remark The above calculations have been repeated by taking different values of parameters and . It has been verified every time that our iterative process (1.4) converges faster than all Picard, Mann and Ishikawa iterative processes. Moreover, it has been observed that as the values of and go far below 0.5 and near 0 (above 0.5 and near 1), the convergence gets slower (faster), and it happens with every scheme except Picard as it has nothing to do with these parameters. For example, when , for all n, the values for the above four processes at the tenth iteration become 5.0000582, 5.0011640, 12.3847322, 10.88505847, respectively. The accuracy of seven decimal places is obtained by our process at the 14th iteration. The values at this iteration for the above processes are 5.0000000, 5.0000018, 9.8671409, 8.2715668. By the way, for , for all n, at the 12th iteration, we get 5.0000000, 5.0000465, 5.8689348 and 5.1255429, respectively.
3 Convergence results in uniformly convex Banach spaces
In this section, we use the iterative process (1.4) for nonexpansive mappings to prove some convergence theorems.
exists for all .
To prove part (ii), we first prove that .
Lemma 6 For any , exists for all under the conditions of Lemma 5.
for all . Then and exist. Let .
Since exists for all , as . Hence as .
we have .
so that exists for all . □
Lemma 7 Assume that all the conditions of Lemma 5 are satisfied. Then, for any , exists; in particular, for all , the set of all weak limits of .
Proof The proof is the same as that of Lemma 2.3 of  and is therefore omitted here. □
We now give our weak convergence theorem.
Theorem 1 Let E be a uniformly convex Banach space, and let C, T and be taken as in Lemma 5. Assume that (a) E satisfies Opial’s condition or (b) E has a Fréchet differentiable norm, or (c) the dual of E satisfies the Kadec-Klee property. If , then converges weakly to a point of .
This is a contradiction, so . Next assume (b). By Lemma 7, for all . Therefore implies . Finally, say (c) is true. Since exists for all by Lemma 6, therefore by Lemma 4. Consequently, converges weakly to a point of F and this completes the proof. □
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 .
Theorem 2 Let E be a real Banach space, and let C, T, be taken as in Lemma 5. Then converges to a point of if and only if , where .
Proof Necessity is obvious. Suppose that . As proved in Lemma 5, exists for all ; therefore, exists. But by the hypothesis, , we have . On lines similar to , is a Cauchy sequence in a closed subset C of a Banach space E; therefore, it must converge to a point in C. Let . Now gives that . Thus . □
Note that this condition is weaker than the requirement that T is demicompact or C is compact, see . Applying Theorem 2, we obtain a strong convergence of the process (1.4) under Condition (I) as follows.
Theorem 3 Let E be a real uniformly convex Banach space, and let C, T, be taken as in Lemma 5. Let T satisfy Condition (I), then converges strongly to a fixed point of T.
Now all the conditions of Theorem 2 are satisfied; therefore, by its conclusion, converges strongly to a point of . □
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