Faster Multistep Iterations for the Approximation of Fixed Points Applied to Zamfirescu Operators

Abstract

By taking a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

1. Introduction

Let 𝐶 be a nonempty convex subset of a normed space 𝐸, and let 𝑇∶𝐶→𝐶 be a mapping.

(a) For arbitrary 𝑥₀∈𝐶, the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑥_𝑛,𝑛≥0, 𝑀_𝑛 where {𝑏_𝑛} is a sequence in [0,1], is known as the Mann iteration process [1].

(b) For arbitrary 𝑥₀∈𝐶, the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑦_𝑛,𝑦_𝑛=1−𝑏^′_𝑛𝑥_𝑛+𝑏^′_𝑛𝑇𝑥_𝑛,𝑛≥0, 𝐼_𝑛 where {𝑏_𝑛} and {𝑏^′_𝑛} are sequences in [0,1], is known as the Ishikawa iteration process [2].

(c) For arbitrary 𝑥₀∈𝐶, the sequence {𝑥_𝑛} defined by 𝑥_𝑛+1=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑦¹_𝑛,𝑦^𝑖_𝑛=1−𝑏^𝑖_𝑛𝑥_𝑛+𝑏^𝑖_𝑛𝑇𝑦_𝑛^𝑖+1,𝑦_𝑛^𝑝−1=1−𝑏_𝑛^𝑝−1𝑥_𝑛+𝑏_𝑛^𝑝−1𝑇^𝑛𝑥_𝑛,𝑛≥0, 𝑅𝑆_𝑛 where {𝑏_𝑛} and {𝑏^𝑖_𝑛},  𝑖=1,2,…,𝑝−2  (𝑝≥2), are sequences in [0,1] and denoted by 𝑅𝑆_𝑛, is known as the multistep iteration process [3].

Definition 1 (see [4]). Suppose that {𝑎_𝑛} and {𝑏_𝑛} are two real convergent sequences with limits 𝑎 and 𝑏, respectively. Then, {𝑎_𝑛} is said to converge faster than {𝑏_𝑛} if lim_𝑛→∞𝑎_𝑛−𝑎𝑏_𝑛−𝑏=0. (1)

Definition 2 (see [4]). Let {𝑢_𝑛} and {𝑣_𝑛} be two fixed-point iteration procedures which, both, converge to the same fixed point 𝑝, say, with error estimates, 𝑢_𝑛−𝑝≤𝑎_𝑛,𝑣_𝑛−𝑝≤𝑏_𝑛,𝑛≥0, (2) where lim𝑎_𝑛=0=lim𝑏_𝑛. If {𝑎_𝑛} converges faster than {𝑏_𝑛}, then {𝑢_𝑛} is said to converge faster than {𝑣_𝑛}.

Theorem 3 (see [5]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a mapping for which there exist real numbers 𝑎, 𝑏, and 𝑐 satisfying 𝑎∈(0,1) and 𝑏,𝑐∈(0,1/2) such that, for each pair 𝑥,𝑦∈𝑋, at least one of the following is true:(𝑧1)𝑑(𝑇𝑥,𝑇𝑦)≤𝑎𝑑(𝑥,𝑦), (𝑧2)𝑑(𝑇𝑥,𝑇𝑦)≤𝑏[𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)], (𝑧3)𝑑(𝑇𝑥,𝑇𝑦)≤𝑐[𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)].
Then, 𝑇 has a unique fixed point 𝑝, and the Picard iteration {𝑥_𝑛} defined by 𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛≥0, (3) converges to 𝑝 for any 𝑥₀∈𝑋.

Remark 4. An operator 𝑇, which satisfies the contraction conditions (𝑧1)–(𝑧3) of Theorem 3, will be called a Zamfirescu operator [4, 6, 7].
In [6, 7], Berinde introduced a new class of operators on a normed space 𝐸 satisfying 𝑇𝑥−𝑇𝑦≤𝛿𝑥−𝑦+𝐿‖𝑇𝑥−𝑥‖, (4) for any 𝑥,𝑦∈𝐸,  0≤𝛿<1, and 𝐿≥0.
He proved that this class is wider than the class of Zamfirescu operators.
The following results are proved in [6, 7].

Theorem 5 (see [7]). Let 𝐶 be a nonempty closed convex subset of a normed space 𝐸. Let 𝑇∶𝐶→𝐶 be an operator satisfying (4). Let {𝑥_𝑛} be defined through the iterative process 𝑀_𝑛 and 𝑥₀∈𝐶. If 𝐹(𝑇)≠∅ and ∑^∞_𝑛=0𝑏_𝑛=∞, then {𝑥_𝑛} converges strongly to the unique fixed point of 𝑇.

Theorem 6 (see [6]). Let 𝐶 be a nonempty closed convex subset of an arbitrary Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be an operator satisfying (4). Let {𝑥_𝑛} be defined through the iterative process 𝐼_𝑛 and 𝑥₀∈𝐶, where {𝑏_𝑛} and {𝑏^′_𝑛} are sequences of positive numbers in [0,1] with {𝑏_𝑛} satisfying ∑^∞_𝑛=0𝑏_𝑛=∞. Then, {𝑥_𝑛} converges strongly to the fixed point of 𝑇.

The following result can be found in [8].

Theorem 7. Let 𝐶 be a closed convex subset of an arbitrary Banach space 𝐸. Let the Mann and Ishikawa iteration processes with real sequences {𝑏_𝑛} and {𝑏^′_𝑛} satisfy 0≤𝑏_𝑛, 𝑏^′_𝑛≤1, and ∑^∞_𝑛=0𝑏_𝑛=∞. Then, 𝑀_𝑛 and 𝐼_𝑛 converge strongly to the unique fixed point of 𝑇. Let   𝑇∶𝐶→𝐶 be a Zamfirescu operator, and, moreover, the Mann iteration process converges faster than the Ishikawa iteration process to the fixed point of 𝑇.

In [4], Berinde proved the following result.

Theorem 8. Let 𝐶 be a closed convex subset of an arbitrary Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be a Zamfirescu operator. Let {𝑦_𝑛} be defined by 𝑀_𝑛 and 𝑦₀∈𝐶 with a sequence {𝑏_𝑛} in [0,1] satisfying ∑^∞_𝑛=0𝑏_𝑛=∞. Then, {𝑦_𝑛} converges strongly to the fixed point of 𝑇, and, moreover, the Picard iteration {𝑥_𝑛} converges faster than the Mann iteration.

Remark 9. In [9], Qing and Rhoades by taking a counterexample showed that the Mann iteration process converges more slowly than the Ishikawa iteration process for Zamfirescu operators.

In this paper, we establish a general theorem to approximate fixed points of quasi-contractive operators in a Banach space through the multistep iteration process. Our result generalizes and improves upon, among others, the corresponding results of Babu and Vara Prasad [8] and Berinde [4, 6, 7].

We also prove that the Mann iteration process and the Ishikawa iteration process converge more slowly than the multistep iteration process for Zamfirescu operators.

2. Main Results

We now prove our main results.

Theorem 10. Let 𝐶 be a nonempty closed convex subset of an arbitrary Banach space 𝐸, and let 𝑇∶𝐶→𝐶 be an operator satisfying (4). Let {𝑥_𝑛} be defined through the iterative process 𝑅𝑆_𝑛 and 𝑥₀∈𝐶, where {𝑏_𝑛} and {𝑏^𝑖_𝑛},  𝑖=1,2,…,𝑝−2  (𝑝≥2), are sequences in [0,1] with ∑^∞_𝑛=0𝑏_𝑛=∞. If 𝐹(𝑇)≠∅, then 𝐹(𝑇) is a singleton, and the sequence {𝑥_𝑛} converges strongly to the fixed point of 𝑇.

Proof. Assume that 𝐹(𝑇)≠∅ and 𝑤∈𝐹(𝑇). Then, using 𝑅𝑆_𝑛, we have 𝑥_𝑛+1−𝑤=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑦¹_𝑛=−𝑤1−𝑏_𝑛𝑥_𝑛−𝑤+𝑏_𝑛𝑇𝑦¹_𝑛≤−𝑤1−𝑏_𝑛𝑥_𝑛−𝑤+𝑏_𝑛𝑇𝑦¹_𝑛−𝑤. (5)
Now, for 𝑥=𝑤 and 𝑦=𝑦¹_𝑛, (4) gives 𝑇𝑦¹_𝑛−𝑤≤𝛿𝑦¹_𝑛−𝑤. (6) By substituting (6) in (5), we obtain 𝑥_𝑛+1−𝑤≤1−𝑏_𝑛𝑥_𝑛−𝑤+𝛿𝑏_𝑛𝑦¹_𝑛−𝑤. (7)
In a similar fashion, again by using 𝑅𝑆_𝑛, we can get 𝑦^𝑖_𝑛−𝑤≤1−𝑏^𝑖_𝑛𝑥_𝑛−𝑤+𝛿𝑏^𝑖_𝑛𝑦_𝑛^𝑖+1−𝑤, (8) where 𝑖=1,2,…,𝑝−2  (𝑝≥2) and 𝑦_𝑛^𝑝−1−𝑤≤1−(1−𝛿)𝑏_𝑛^𝑝−1𝑥_𝑛−𝑤. (9) It can be easily seen that, for 𝑖=1,2,…,𝑝−2  (𝑝≥2), we have 𝑦¹_𝑛−𝑤≤1−𝑏¹_𝑛𝑥_𝑛−𝑤+𝛿𝑏¹_𝑛𝑦²_𝑛,⋮𝑦−𝑤_𝑛𝑝−3−𝑤≤1−𝑏_𝑛^𝑝−3𝑥_𝑛−𝑤+𝛿𝑏_𝑛^𝑝−3𝑦_𝑛^𝑝−2,𝑦−𝑤_𝑛𝑝−2−𝑤≤1−𝑏_𝑛^𝑝−2𝑥_𝑛−𝑤+𝛿𝑏_𝑛^𝑝−2𝑦_𝑛^𝑝−1−𝑤. (10)
Substituting (9) in (10) gives us 𝑦_𝑛^𝑝−2−𝑤≤1−(1−𝛿)𝑏_𝑛^𝑝−21+𝛿𝑏_𝑛^𝑝−1𝑥_𝑛−𝑤. (11) It may be noted that, for 𝛿∈[0,1) and {𝜂_𝑛}∈[0,1], the following inequality is always true: 1≤1+𝛿𝜂_𝑛≤1+𝛿. (12) From (11) and (12), we get 𝑦_𝑛^𝑝−2−𝑤≤1−(1−𝛿)𝑏_𝑛^𝑝−2𝑥_𝑛−𝑤. (13) By repeating the same procedure, finally from (7) and (10), we yield 𝑥_𝑛+1−𝑤≤1−(1−𝛿)𝑏_𝑛𝑥_𝑛−𝑤. (14) By (14), we inductively obtain 𝑥_𝑛+1−𝑤≤^𝑛_𝑘=01−(1−𝛿)𝑏_𝑘𝑥₀−𝑤,𝑛≥0. (15) Using the fact that 0≤𝛿<1,  0≤𝑏_𝑛≤1, and ∑^∞_𝑛=0𝑏_𝑛=∞, it results that lim^𝑛_{𝑛→∞𝑘=0}1−(1−𝛿)𝑏_𝑘=0, (16) which, by (15), implies that lim_𝑛→∞𝑥_𝑛+1−𝑤=0. (17) Consequently, 𝑥_𝑛→𝑤∈𝐹, and this completes the proof.

Now, by a counterexample, we prove that the multistep iteration process is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

Example 11. Suppose that 𝑇∶[0,1]→[0,1] is defined by 𝑇𝑥=(1/2)𝑥;  𝑏_𝑛=0=𝑏_𝑛^𝑝−1=𝑏^𝑖_𝑛, 𝑖=1,2,…,𝑝−1  (𝑝≥2), and 𝑛=1,2,…,15;  𝑏_𝑛=4/√𝑛=𝑏_𝑛^𝑝−1=𝑏^𝑖_𝑛,  𝑖=1,2,…,𝑝−1  (𝑝≥2), and 𝑛≥16. It is clear that 𝑇 is a Zamfirescu operator with a unique fixed point 0 and that all of the conditions of Theorem 10 are satisfied. Also, 𝑀_𝑛=𝑥₀=𝐼_𝑛=𝑅𝑆_𝑛,  𝑛=1,2,…,15. Suppose that 𝑥₀≠0. For the Mann and Ishikawa iteration processes, we have 𝑀_𝑛=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑥_𝑛=41−√𝑛𝑥_𝑛+4√𝑛12𝑥_𝑛=21−√𝑛𝑥_𝑛=⋯=^𝑛_𝑖=1621−√𝑖𝑥₀,𝐼_𝑛=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇1−𝑏^′_𝑛𝑥_𝑛+𝑏^′_𝑛𝑇𝑥_𝑛=41−√𝑛𝑥_𝑛+4√𝑛1221−√𝑛𝑥_𝑛=21−√𝑛−4𝑛𝑥_𝑛=⋯=^𝑛_𝑖=1621−√𝑖−4𝑖𝑥₀,𝑅𝑆_𝑛=1−𝑏_𝑛𝑥_𝑛+𝑏_𝑛𝑇𝑦¹_𝑛,𝑦^𝑖_𝑛=1−𝑏^𝑖_𝑛𝑥_𝑛+𝑏^𝑖_𝑛𝑇𝑦_𝑛^𝑖+1,𝑦_𝑛^𝑝−1=1−𝑏_𝑛^𝑝−1𝑥_𝑛+𝑏_𝑛^𝑝−1𝑇𝑥_𝑛,𝑛≥0, (18) where 𝑖=1,2,…,𝑝−2  (𝑝≥2) implies that 𝑅𝑆_𝑛=1−^𝑝_𝑗=12√𝑛^𝑗𝑥_𝑛=⋯=^𝑛_𝑖=161−^𝑝_𝑗=12√𝑖^𝑗𝑥₀. (19) Now, consider 𝑅𝑆_𝑛−0𝑀_𝑛=∏−0^𝑛𝑖=161−∑^𝑝_𝑗=12/√𝑖^𝑗𝑥₀∏^𝑛_𝑖=161−2/√𝑖𝑥₀=∏^𝑛_𝑖=161−∑^𝑝_𝑗=12/√𝑖^𝑗∏^𝑛_𝑖=161−2/√𝑖=^𝑛_𝑖=16∑1−^𝑝𝑘=22/√𝑖^𝑘1−2/√𝑖=^𝑛_𝑖=16∑1−^𝑝𝑘=22/√𝑖^𝑘√𝑖√.𝑖−2 (20) It is easy to see that 0≤lim^𝑛_{𝑛→∞𝑖=16}∑1−^𝑝𝑘=22/√𝑖^𝑘√𝑖√𝑖−2≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0. (21) Hence, lim_𝑛→∞𝑅𝑆_𝑛−0𝑀_𝑛−0=0. (22) Thus, the Mann iteration process converges more slowly than the multistep iteration process to the fixed point 0 of 𝑇.
Similarly, 𝑅𝑆_𝑛−0𝐼_𝑛=∏−0^𝑛𝑖=161−∑^𝑝_𝑗=12/√𝑖^𝑗𝑥₀∏^𝑛_𝑖=161−2/√𝑖−(4/𝑖)𝑥₀=∏^𝑛_𝑖=161−∑^𝑝_𝑗=12/√𝑖^𝑗∏^𝑛_𝑖=161−2/√𝑖−()=4/𝑖_𝑛𝑖=16∑1−^𝑝𝑘=32/√𝑖^𝑘1−2/√𝑖−()=4/𝑖_𝑛𝑖=16∑1−^𝑝𝑘=32/√𝑖^𝑘𝑖√𝑖𝑖√𝑖−2𝑖−4√𝑖, (23) with 0≤lim^𝑛_{𝑛→∞𝑖=16}∑1−^𝑝𝑘=32/√𝑖^𝑘𝑖√𝑖𝑖√𝑖−2𝑖−4√𝑖≤lim^𝑛_{𝑛→∞𝑖=16}11−𝑖=lim_𝑛→∞15𝑛=0, (24) implies that lim_𝑛→∞𝑅𝑆_𝑛−0𝐼_𝑛−0=0. (25) Thus, the Ishikawa iteration process converges more slowly than the multistep iteration process to the fixed point 0 of 𝑇.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the referees for their useful comments and suggestions. This study was supported by research funds from Dong-A University.