The equivalence among new multistep iteration, s-iteration and some ...

arXiv:1211.5701v3 [math.FA] 1 Aug 2013

THE EQUIVALENCE AMONG NEW MULTISTEP ITERATION, S-ITERATION AND SOME OTHER ITERATIVE SCHEMES ¨ FAIK GURSOY, VATAN KARAKAYA, AND B. E. RHOADES Abstract. In this paper, we show that Picard, Krasnoselskij, Mann, Ishikawa, new two step, Noor, multistep, new multistep, SP and S-iterative schemes are equivalent for contractive-like mappings.

1. Introduction and Preliminaries In the last four decades, attention of researchers has been focused on the introduction and the convergences of various iteration procedures for approximate fixed points of certain classes of self- nonlinear mappings, e.g. see [7, 12, 16, 18, 20, 22, 24, 29, 34]. The most celebrated fixed point iterative procedures are the Picard [11], Mann [34], and Ishikawa [24] iterative procedures. Numerous convergence results have been proved through these iterative procedures for approximating fixed points of different type nonlinear mappings, e.g. see [24, 31, 32, 34, 36, 37]. But in some cases, some particular iteration procedure may fail to converge for some class of nonlinear mappings. For instance, (i) the Picard iteration procedure [11] does not convergence to the fixed point of nonexpansive mappings, (for more detail see pp.8, Example 1.8 in [33]),while the Ishikawa iteration [24] and Mann iteration [34] converges. (ii) By providing a counter example, Chidume and Mutangadura [9] showed that the Mann iteration [34] fails to converge for the class of Lipschitzian pseudocontractive mappings while the Ishikawa iteration [24] converges. In the light of the above facts, a conjecture was put forwad in [5, 7] as follows: While the Mann iteration [34] converges to a fixed point of a particular class of mappings, does the Ishikawa iteration [24] converges too? During the past 11 years, this conjecture was proven affirmatively by many researchers and consequently a large literature has developed around the theme of establishing the equivalence among convergences of some well-known iterative schemes deal with various classes of mappings. Some authors who have made contributions to the study of equivalence among various iterative schemes are Rhoades and S ¸ oltuz [1, 2, 3, 4, 5, 6, 7, 8], Berinde [31], S¸oltuz [25, 26], Olaleru and Akewe [14], Chang et al [23] and several of the references therein. The main objective of this paper is attepmt to verify the above conjecture for a new multistep iteration [12] and some other well-known iterative procedures in the literature. As a background for our exposition, we now mention some contractive mappings and iteration schemes. 2000 Mathematics Subject Classification. Primary 47H10. Key words and phrases. New multistep iteration, S-iteration, Equivalence of iterations, Contractive-like mappings. 1

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¨ FAIK GURSOY, VATAN KARAKAYA, AND B. E. RHOADES

In [30] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition: For a mapping T : E → E, there exist real numbers a, b, c satisfying 0 < a < 1, 0 < b, c < 1/2 such that, for each pair x, y ∈ X, at least one of the following is true:  kT x − T yk ≤ a kx − yk ,  (z1 ) (z2 ) kT x − T yk ≤ b (kx − T xk + ky − T yk) , (1.1)  (z3 ) kT x − T yk ≤ c (kx − T yk + ky − T xk) .

A mapping T satisfying the contractive conditions (z1 ), (z2 ) and (z3 ) in (1.1) is called a Zamfirescu mapping. As shown in [32], the contractive condition (1.1) leads to  (b1 ) kT x − T yk ≤ δ kx − yk + 2δ kx − T xk if one use (z2 ),  and (1.2)  (b2 ) kT x − T yk ≤ δ kx − yk + 2δ kx − T yk if one use (z3 ), o n c b , 1−c , δ ∈ [0, 1), and it was shown that this for all x, y ∈ E where δ := max a, 1−b class of mappings is wider than the class of Zamfirescu mappings. Any mapping satisfying condition (b1 ) or (b2 ) is called a quasi-contractive mapping. Extending the above definition, Osilike and Udomene [17] considered mappings T for which there exist real numbers L ≥ 0 and δ ∈ [0, 1) such that for all x, y ∈ E, (1.3)

kT x − T yk ≤ δ kx − yk + L kx − T xk .

Imoru and Olantiwo [10] gave a more general definition: The mapping T is called a contractive-like mapping if there exists a constant δ ∈ [0, 1) and a strictly increasing and continuous function ϕ : [0, ∞) → [0, ∞) with ϕ (0) = 0, such that, for each x, y ∈ E, (1.4)

kT x − T yk ≤ δ kx − yk + ϕ (kx − T xk) .

Remark 1. [12] A map satisfying (1.4) need not have a fixed point. However, using (1.4), it is obvious that if T has a fixed point, then it is unique. Throughout the rest of this paper N denotes the set of all nonnegative integers. Let X be a Banach space and E ⊂ X be a nonempty closed, convex subset of X, and T be a self map on E. Define FT := {p ∈ X : p = T p} to be the set of fixed ∞ ∞ ∞ ∞ points of T . Let {αn }n=0 , {β n }n=0 ,{γ n }n=0 and β in n=0 , i = 1, k − 2, k ≥ 2 be real sequences in [0, 1) satisfying certain conditions. Rhoades and S¸oltuz [7], introduced a multistep iterative algorithm by  x0 ∈ E,     xn+1 = (1 − αn ) xn + αn T yn1 , (1.5) yni = 1 − β in xn + β in T yni+1 ,     y k−1 = 1 − β k−1 xn + β k−1 T xn , n ∈ N. n n n

The following multistep iteration was employed in [12]  x0 ∈ E,     xn+1 = (1 − αn ) yn1 + αn T yn1 , i+1 (1.6) + β in T yni+1 , yni = 1 − β in y n     y k−1 = 1 − β k−1 xn + β k−1 T xn , n ∈ N. n n n

THE EQUIVALENCE AMONG VARIOUS ITERATIVE SCHEMES

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By taking k = 3 and k = 2 in (1.5) we obtain the well-known Noor [16] and Ishikawa [24] iterative schemes, respectively. SP iteration [18] and a new two-step iteration [29] processes are obtained by taking k = 3 and k = 2 in (1.6), respectively. Both in (1.5) and in (1.6), if we take k = 2 with β 1n = 0 and k = 2 with β 1n ≡ 0, αn ≡ λ (const.), then we get the iterative procedures introduced in [34] and [15], which are commonly known as the Mann and Krasnoselskij iterations, respectively. The Krasnoselskij iteration reduces to the Picard iteration [11] for λ = 1. ∞ A sequence {xn }n=0 defined by

(1.7)

  x0 ∈ E, xn+1 = (1 − αn ) T xn + αn T yn,  yn = (1 − β n ) xn + β n T xn , n ∈ N

is known as the S-iteration process [19, 20]. The following lemma will be useful to prove the main results of this work and is important by itself. Lemma 1. [35] Let {an }∞ n=0 be a nonnegative sequence which satisfies the following inequality an+1 ≤ (1 − µn ) an + ρn ,

(1.8)

where µn ∈ (0, 1) , for all n ≥ n0 ,

∞ P

µn = ∞, and ρn = o (µn ). Then limn→∞ an =

n=0

0.

2. Main Results Theorem 1. Let T : E → E be a mapping satisfying condition (1.4) with FT 6= ∅. If x0 = u0 ∈ E and αn ≥ A > 0,∀n ∈ N, then the following are equivalent: (1) The Mann iteration [34] converges to p ∈ FT , (2) The new multistep iteration (1.6) converges to p ∈ FT . Proof. We first prove the implication (1) ⇒ (2): Suppose that the Mann iteration [34] converges to p. Using the Mann iteration [34], (1.6), and (1.4) we have the following estimates: kun+1 − xn+1 k

= ≤ ≤

(2.1)

=

(1 − αn ) un − yn1 + αn T un − T yn1

(1 − αn ) un − yn1 + αn T un − T yn1

(1 − αn ) un − yn1 + αn δ un − yn1 + ϕ (kun − T un k)

[1 − αn (1 − δ)] un − yn1 + αn ϕ (kun − T un k) ,


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un − y 1 = un − 1 − β 1 y 2 − β 1 T y 2 n n n n n

= un − β 1n un + β 1n un − 1 − β 1n yn2 − β 1n T yn2

= 1 − β 1n un − yn2 + β 1n un − T yn2

≤ 1 − β 1n un − yn2 + β 1n un − T yn2

= 1 − β 1n un − yn2 + β 1n un − T un + T un − T yn2

≤ 1 − β 1n un − yn2 + β 1n T un − T yn2 + β 1n kun − T un k

≤ 1 − β 1n un − yn2 + β 1n δ un − yn2 + β 1n ϕ (kun − T unk)

+β 1n kun − T un k

= 1 − β 1n (1 − δ) un − yn2 + β 1n {kun − T un k + ϕ (kun − T unk)} ,

(2.2)

un − yn2 = 1 − β 2n un − yn3 + β 2n un − T yn3

≤ 1 − β 2n un − yn3 + β 2n un − T yn3




= 1 − β 2n (1 − δ) un − yn3 + β 2n {kun − T un k + ϕ (kun − T unk)} ,

(2.3)

un − yn3 = 1 − β 3n un − yn4 + β 3n un − T yn4

≤ 1 − β 3n un − yn4 + β 3n un − T yn4




= 1 − β 3n (1 − δ) un − yn4 + β 3n {kun − T un k + ϕ (kun − T unk)} .

(2.4)

By combinig (2.1), (2.2), (2.3), and (2.4) we obtain kun+1 − xn+1 k ≤ [1 − αn (1 − δ)] 1 − β 1n (1 − δ) 1 − β 2n (1 − δ)

1 − β 3n (1 − δ) un − yn4 + [1 − αn (1 − δ)] 1 − β 1n (1 − δ) 1 − β 2n (1 − δ) β 3n + 1 − β 1n (1 − δ) β 2n + β 1n {kun − T un k + ϕ (kun − T un k)} +αn ϕ (kun − T un k)

(2.5)

Continuing the above process we have kun+1 − xn+1 k ≤

(2.6)

i

h [1 − αn (1 − δ)] 1 − β 1n (1 − δ) · · · 1 − β nk−2 (1 − δ) un − ynk−1 n i h + [1 − αn (1 − δ)] 1 − β 1n (1 − δ) · · · 1 − β nk−3 (1 − δ) β nk−2 + · · · + 1 − β 1n (1 − δ) β 2n + β 1n {kun − T unk + ϕ (kun − T unk)} +αn ϕ (kun − T un k) .


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Again using (1.6),

un − ynk−1 =

and (1.4) we get

1 − β nk−1 (un − xn ) + β nk−1 (un − T xn ) ≤ 1 − β nk−1 kun − xn k + β nk−1 kun − T xn k ≤ 1 − β nk−1 kun − xn k + β nk−1 kT un − T xn k + β nk−1 kun − T unk h i (2.7) ≤ 1 − β nk−1 (1 − δ) kun − xn k + β nk−1 {kun − T un k + ϕ (kun − T un k)} . ∞ ∞ Since δ ∈ [0, 1) and {αn }n=0 , β in n=0 ⊂ [0, 1) for i = 1, k − 1, we have i h (2.8) [1 − αn (1 − δ)] 1 − β 1n (1 − δ) · · · 1 − β nk−1 (1 − δ) ≤ [1 − αn (1 − δ)] . Using inequality (2.8) and the assumption αn ≥ A > 0,∀n ∈ N in the resultant inequality obtained by substituting (2.7) in (2.6) we get kun+1 − xn+1 k ≤

(2.9)

[1 − A (1 − δ)] kun − xn k n i h + [1 − A (1 − δ)] 1 − β 1n (1 − δ) · · · 1 − β nk−2 (1 − δ) β nk−1 + · · · + 1 − β 1n (1 − δ) β 2n + β 1n {kun − T unk + ϕ (kun − T unk)} +αn ϕ (kun − T un k) .

Define an µn ρn

: = kun − xn k , : = A (1 − δ) ∈ (0, 1) , i n h : = [1 − A (1 − δ)] 1 − β 1n (1 − δ) · · · 1 − β nk−2 (1 − δ) β nk−1 + · · · + 1 − β 1n (1 − δ) β 2n + β 1n {kun − T unk + ϕ (kun − T un k)} +αn ϕ (kun − T un k) .

Since limn→∞ kun − pk = 0 and T p = p ∈ FT , it follows from (1.4) that 0

≤ kun − T un k ≤ kun − pk + kT p − T unk ≤ kun − pk + δ kp − un k + ϕ (kp − T pk)

(2.10)

= (1 + δ) kun − pk → 0 as n → ∞,

which implies limn→∞ kun − T un k = 0; namely ρn = o (µn ). Hence an application of Lemma 1 to (2.10) yields limn→∞ kun − xn k = 0. Since un → p as n → ∞ by assumption, we derive (2.11)

kxn − pk ≤ kxn − un k + kun − pk

and this implies that limn→∞ xn = p. (2) ⇒ (1) : Assume that xn → p as n → ∞. Using the Mann iteration [34], (1.6), and (1.4), we have the following estimates:

kxn+1 − un+1 k = (1 − αn ) yn1 − un + αn T yn1 − T un

≤ (1 − αn ) yn1 − un + αn T yn1 − T un

≤ (1 − αn ) yn1 − un + αn δ yn1 − un + ϕ yn1 − T yn1

= [1 − αn (1 − δ)] yn1 − un + αn ϕ yn1 − T yn1 , (2.12)


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1

y − un = n

= = ≤ ≤ ≤

(2.13)

=

2

yn − un =

= ≤ ≤ =

(2.14)

1 − β 1 y 2 + β 1 T y 2 − un n n n n

1 − β 1n yn2 + β 1n T yn2 − un 1 − β 1n + β 1n

1 − β 1n yn2 − un + β 1n T yn2 − un

1 − β 1n yn2 − un + β 1n T yn2 − un

1 − β 1n yn2 − un + β 1n T yn2 − yn2 + yn2 − un

1 − β 1n yn2 − un + β 1n yn2 − un + β 1n T yn2 − yn2

2

y − un + β 1 T y 2 − y 2 , n n n n

1 − β 2n yn3 + β 2n T yn3 − un

1 − β 2n yn3 − un + β 2n T yn3 − un

1 − β 2n yn3 − un + β 2n T yn3 − un

1 − β 2n yn3 − un + β 2n yn3 − un + β 2n T yn3 − yn3

3

yn − un + β 2n T yn3 − yn3 .

By combining (2.12), (2.13), and (2.14) we obtain

kxn+1 − un+1 k ≤ [1 − αn (1 − δ)] yn3 − un + [1 − αn (1 − δ)] β 2n T yn3 − yn3

+ [1 − αn (1 − δ)] β 1n T yn2 − yn2 + αn ϕ yn1 − T yn1 (2.15) In a similar way, we have kxn+1 − un+1 k (2.16)

≤ [1 − αn (1 − δ)] ynk−1 − un

+ [1 − αn (1 − δ)] β nk−2 T ynk−1 − ynk−1

+ · · · + [1 − αn (1 − δ)] β 1n T yn2 − yn2 + αn ϕ yn1 − T yn1

Using now (1.6) we

k−1

yn − un =

have

1 − β nk−1 xn + β nk−1 T xn − un ≤ 1 − β nk−1 kxn − un k + β nk−1 kT xn − un k ≤ 1 − β nk−1 kxn − un k + β nk−1 kxn − un k + β nk−1 kT xn − xn k ≤ kxn − un k + β nk−1 kT xn − xn k .

(2.17)

Substituting (2.17) in (2.16) and utilizing the assumption αn ≥ A > 0,∀n ∈ N we get kxn+1 − un+1 k ≤

(2.18) Now define an µn ρn

[1 − A (1 − δ)] kxn − un k n

+ [1 − A (1 − δ)] β nk−1 kT xn − xn k + β nk−2 T ynk−1 − ynk−1

+ · · · + β 1n T yn2 − yn2 + αn ϕ yn1 − T yn1 .

: = kun − xn k , : = A (1 − δ) ∈ (0, 1) , n

: = [1 − A (1 − δ)] β nk−1 kT xn − xn k + β nk−2 T ynk−1 − ynk−1

+ · · · + β 1n T yn2 − yn2 + αn ϕ yn1 − T yn1 .


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Since limn→∞ kxn − pk = 0 and T p = p ∈ FT , it follows from (1.4) that 0

≤ kxn − T xn k ≤ kxn − pk + kT p − T xn k ≤ kxn − pk + δ kp − xn k + ϕ (kp − T pk)

(2.19) Utilizing 0

≤ ≤ ≤ = = ≤ ≤ = = ≤ ≤

= (1 + δ) kxn − pk → 0 as n → ∞. ∞ (1.4), (1.6), and the condition β in n=0 ⊂ [0, 1), i = 1, k − 1, we have

1

yn − T yn1 = yn1 − p + p − T yn1

1

yn − p + T p − T yn1

1

yn − p + δ p − yn1 + ϕ (kp − T pk)

(1 + δ) yn1 − p

(1 + δ) 1 − β 1n yn2 + β 1n T yn2 − p 1 − β 1n + β 1n

(1 + δ) 1 − β 1n yn2 − p + β 1n T yn2 − T p

(1 + δ) 1 − β 1n yn2 − p + β 1n δ yn2 − p

(1 + δ) 1 − β 1n (1 − δ) yn2 − p (1 + δ) 1 − β 1n (1 − δ) 1 − β 2n yn3 + β 2n T yn3 − p 1 − β 2n + β 2n

(1 + δ) 1 − β 1n (1 − δ) 1 − β 2n yn3 − p + β 2n T yn3 − T p

(1 + δ) 1 − β 1n (1 − δ) 1 − β 2n (1 − δ) yn3 − p

··· i

h ≤ (1 + δ) 1 − β 1n (1 − δ) · · · 1 − β nk−2 (1 − δ) ynk−1 − p i h ≤ (1 + δ) 1 − β 1n (1 − δ) · · · 1 − β nk−1 (1 − δ) kxn − pk (2.20) ≤ (1 + δ) kxn − pk → 0 as n → ∞.

It is easy to see from (2.20) that this result is also valid for T yn2 − yn2 , . . . , T ynk−1 − ynk−1 . Since ϕ is continuous, we have

lim kxn − T xn k = lim ϕ yn1 − T yn1 n→∞ n→∞

= lim yn2 − T yn2 = · · · = lim ynk−1 − T ynk−1 = 0, (2.21) n→∞

n→∞

that is ρn = o (µn ). Hence an application of Lemma 1 to (2.18) lead to limn→∞ kxn − un k = 0. Since xn → p as n → ∞ by assumption, we derive (2.22)

kun − pk ≤ kun − xn k + kxn − pk

and this implies that limn→∞ un = p.

Theorem 2. Let T : E → E be a mapping satisfying condition (1.4) with FT 6= ∅. If x0 = u0 ∈ E and αn ≥ A > 0,∀n ∈ N, then the following are equivalent: (1) The Mann iteration [34] converges to p ∈ FT , (2) The S-iteration (1.7) converges to p ∈ FT . Proof. To prove the implication (1) ⇒ (2), suppose that the Mann iteration [34] converges to p. Using (1.4), the Mann iteration [34], and (1.7) we have the following


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estimates: kun+1 − xn+1 k = ≤ ≤

(2.23)

k(1 − αn ) (un − T xn ) + αn (T un − T yn )k (1 − αn ) kun − T xn k + αn kT un − T yn k (1 − αn ) kun − T xn k + αn δ kun − yn k + αn ϕ (kun − T un k) ,

kun − yn k =

kun − (1 − β n ) xn − β n T xn k kun − β n un + β n un − (1 − β n ) xn − β n T xn k (1 − β n ) kun − xn k + β n kun − T xn k ,

= ≤

(2.24)

kun − T xn k = ≤ ≤

(2.25)

kun − T un + T un − T xn k kun − T un k + kT un − T xn k kun − T un k + δ kun − xn k + ϕ (kun − T un k) .

By combining (2.23),(2.24), and (2.25) we obtain kun+1 − xn+1 k

≤ {(1 − αn ) δ + αn δ [1 − β n (1 − δ)]} kun − xn k + [1 − αn + αn β n δ] kun − T unk + [1 + αn β n δ] ϕ (kun − T un k) .

(2.26)

Since δ, αn , β n ∈ [0, 1) for all n ∈ N, (1 − αn ) δ < 1 − αn , 1 − β n (1 − δ) < 1.

(2.27)

Using (2.27) and the assumption αn ≥ A > 0,∀n ∈ N in (2.26) we derive kun+1 − xn+1 k

≤ [1 − A (1 − δ)] kun − xn k + [1 − A (1 − δ)] kun − T un k + [1 + αn β n δ] ϕ (kun − T un k) .

(2.28) Define an

: = kun − xn k ,

µn ρn

: = A (1 − δ) ∈ (0, 1) , : = [1 − A (1 − δ)] kun − T un k + [1 + αn β n δ] ϕ (kun − T un k) .

Since limn→∞ kun − pk = 0, limn→∞ kun − T un k = 0 as in the proof of Theorem1. It therefore follows, using the same argument as that employed in the proof of Theorem 1 that limn→∞ xn = p. We will prove now that, if the S-iteration converges, then the Mann iteration does too. Using (1.4), the Mann iteration [34], and (1.7) we have kxn+1 − un+1 k = (2.29)

≤ ≤

k(1 − αn ) (T xn − un ) + αn (T yn − T un )k (1 − αn ) kT xn − un k + αn kT yn − T un k (1 − αn ) kT xn − un k + αn δ kyn − un k + αn ϕ (kyn − T yn k) .

We now have the following estimates kyn − un k = = (2.30)

≤

k(1 − β n ) xn + β n T xn − un k k(1 − β n ) xn + β n T xn − un − β n un + β n un k (1 − β n ) kxn − un k + β n kT xn − un k ,


kT xn − un k

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= kT xn − xn + xn − un k ≤ kT xn − xn k + kxn − un k .

(2.31)

Relations (2.29),(2.30), and (2.31) lead to kxn+1 − un+1 k ≤

[1 − αn (1 − δ)] kxn − un k + [1 − αn + αn β n δ] kT xn − xn k + αn ϕ (kyn − T yn k) .

(2.32)

Since β n ∈ [0, 1) for all n ∈ N, (2.33)

αn β n δ < αn δ.

Utilizing inequality (2.33) and the assumption αn ≥ A > 0,∀n ∈ N in (2.32) we get kun+1 − xn+1 k ≤

[1 − A (1 − δ)] kxn − un k + [1 − A (1 − δ)] kT xn − xn k + αn ϕ (kyn − T yn k) .

(2.34) Now define an µn

: = kxn − un k , : = A (1 − δ) ∈ (0, 1) ,

ρn

: = [1 − A (1 − δ)] kT xn − xn k + αn ϕ (kyn − T yn k) .

Since limn→∞ kxn − pk = 0, limn→∞ kT xn − xn k = 0 as in the proof of Theorem1. Now we have 0

≤ kyn − T yn k ≤ kyn − pk + kT p − T ynk ≤ kyn − pk + δ kp − yn k + ϕ (kp − T pk) = (1 + δ) kyn − pk ≤ (1 + δ) (1 − β n ) kxn − pk + (1 + δ) β n kT xn − T pk

(2.35)

≤ (1 + δ) [1 − β n + β n ] kxn − pk + (1 + δ) β n ϕ (kp − T pk) = (1 + δ) kxn − pk → 0 as n → ∞,

that is, limn→∞ kyn − T ynk = 0, threfore using the same argument as in the proof of Theorem 1, it can be shown that limn→∞ un = p. As shown by S¸oltuz and Grosan ([27], Theorem 3.1), in a real Banach space X, the Ishikawa iteration [24] converges to the fixed point of T , where T : E → E is a mapping satisfying condition (1.4). In 2007, S ¸ oltuz ([28], Corollary 2) proved that the Krasnoselskij [15], Mann [34], Ishikawa [24], Noor [16] and multistep (1.5) iterations are equivalent for quasicontractive mappings in a normed space setting. In 2011, Chugh and Kumar ([21], Corollary 3.2) proved that the Picard [11], Mann [34], Ishikawa [24], new two step [29], Noor [16] and SP [18] iterations are equivalent for quasi-contractive mappings in a Banach space setting. From the argument used in the proofs of ([27], Theorem 3.1), ([28], Corollary 2) and ([21], Corollary 3.2) we easily obtain the following corollary: Corollary 1. T : E → E be a mapping satisfying condition (1.4) with FT 6= ∅. If the initial point is the same for all iterations, αn ≥ A > 0, ∀n ∈ N, then the following are equivalent: (1) The Picard iteration [11] converges to p ∈ FT ;


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(2) The (3) The (4) The (5) The (6) The (7) The (8) The Together corollary:

Krasnoselskij iteration [15] converges to p ∈ FT ; Mann iteration [34] converges to p ∈ FT ; Ishikawa iteration [24] converges to p ∈ FT ; new two step iteration [29] converges to p ∈ FT ; Noor iteration [16] converges to p ∈ FT ; SP iteration [18] converges to p ∈ FT ; Multistep iteration (1.5) converges to p ∈ FT ; with Theorem 1 and Theorem 2,Corollary 1 leads to the following

Corollary 2. T : E → E be a mapping satisfying condition (1.4) with FT 6= ∅. If the initial point is the same for all iterations, αn ≥ A > 0, ∀n ∈ N, then the following are equivalent: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

The The The The The The The The The The

Picard iteration [11] converges to p ∈ FT ; Krasnoselskij iteration [15] converges to p ∈ FT ; Mann iteration [34] converges to p ∈ FT ; Ishikawa iteration [24] converges to p ∈ FT ; new two step iteration [29] converges to p ∈ FT ; Noor iteration [16] converges to p ∈ FT ; SP iteration [18] converges to p ∈ FT ; Multistep iteration (1.5) converges to p ∈ FT ; new multistep iteration (1.6) converges to p ∈ FT ; S-iteration (1.7) converges to p ∈ FT .

Acknowledgement 1. The first two authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Department under project number BAPK 2012-07-03-DOP02 for financial support during the preparation of this manuscript. References [1] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci. 42(2003) 2645–2651 [2] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map, J. Math. Anal. Appl. 283(2003) 681– 688. [3] B.E. Rhoades, S.M. S ¸ oltuz, On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci. 7(2001) 451–459 . [4] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian Ψ-uniformly pseudocontractive and Ψ-uniformly accretive maps, Tamkang J. Math. 35(2004) 235–245. [5] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J.Math. Anal. Appl. 289(2004) 266–278. [6] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence of Mann and Ishikawa iteration for Ψ-uniformly pseudocontractive or Ψ-uniformly accretive maps, Int. J. Math. Math. Sci. 46(2004) 2443– 2452. [7] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Analysis 58(2004) 219-228. [8] B.E. Rhoades, S.M. S ¸ oltuz, The equivalence of Mann and Ishikawa iteration dealing with strongly pseudocontractive or strongly accretive maps, Panamer. Math. J. 14(2004) 51–59. [9] C.E. Chidume, S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Am. Math. Soc. 129 (2001) 2359–2363.


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[37] Z. Huang, Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings, Computers & Mathematics with Applications 37(1999) 1-7. Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey E-mail address: [email protected];[email protected] URL: http://www.yarbis.yildiz.edu.tr/fgursoy Current address: Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul E-mail address: [email protected];[email protected] URL: http://www.yarbis.yildiz.edu.tr/vkkaya Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA E-mail address: [email protected] URL: http://www.math.indiana.edu/people/profile.phtml?id=rhoades