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 OALibJ  Vol.6 No.3 , March 2019
Fixed Point Results for K-Iteration Using Non-Linear Type Mappings
Abstract: In this paper we establish convergence and stability results using general contractive condition, quasi-nonexpansive mapping and mean non expansive mapping for K-iteration process. We shall also generalize the K-iteration process for a pair of distinct mappings and with the help of example we claim that the generalized iteration process has better convergence rate than the K-iteration process for single mapping and some of the existing iteration processes. Suitable examples are given in the support of main results.

1. Introduction and Preliminary Definitions

Let ( X , d ) be a metric space and T : X X be a self map defined on X. Let F ( T ) = { z X : T z = z } denote the set of fixed point of T. For x 0 X , the sequence { x n } n = 0 defined by

x n + 1 = T x n , n 0 , (1.1)

is called the Picard iteration.

For x 0 X , the sequence { x n } n = 0 defined by

*Corrosponding author.

x n + 1 = ( 1 α n ) x n + α n T x n , n 0 , (1.2)

where { α n } n = 0 is a sequence in [ 0 , 1 ] such that n = 0 α n = is called the Mann iteration process [1] .

In 2013, Khan [2] produced a new type of iteration process by introducing the concept of the following Picard-Mann hybrid iterative process for a single mapping T. For the initial value x 0 X , the sequence { x n } n = 0 defined by

x n + 1 = T y n ,

y n = ( 1 α n ) x n + α n T x n , n 0 , (1.3)

where { α n } n = 0 is a sequence in [ 0 , 1 ] .

Khan [2] showed that the rate of convergence of Picard-Mann hybrid iterative process is more than the Picard iteration scheme, Mann iteration scheme [1] and Ishikawa iterative schemes [3] .

In this direction Gursoy and Karakaya [4] , gave new iteration process as follows:

For the initial value x 0 X , the sequence { x n } n = 0 defined by

{ z n = ( 1 β n ) x n + β n T x n , y n = ( 1 α n ) T x n + α n T z n , x n + 1 = T y n (1.4)

where { α n } n = 0 , { β n } n = 0 is a sequence in [ 0 , 1 ] is known as Picard-S iterative process. By giving appropriate example, Gursoy and Karakaya [4] proved that their iterative process has better convergence rate than Picard, Mann, Ishikawa, Noor and Normal-S iterative processes.

Karakaya et al. in their paper [5] , introduced a new hybrid iterative process as

{ x 0 X , y n = T ( 1 β n ) x n + β n T x n , x n + 1 = T ( 1 α n ) y n + α n T y n (1.5)

where { α n } n = 0 , { β n } n = 0 is a sequence in [ 0 , 1 ] .

With the help of suitable example it was claimed by Karakaya et al. [5] , that their iteration process converges faster than the iteration process of Gursoy and Karakaya [4] .

In 2016, Thakur et al. [6] introduced a new iteration scheme called Thakur New Iteration Scheme as for the initial value x 0 X , the sequence { x n } n = 0 defined by

{ z n = ( 1 β n ) x n + β n T x n , y n = T ( 1 α n ) x n + α n z n , x n + 1 = T y n (1.6)

where { α n } n = 0 , { β n } n = 0 is a sequence in [ 0 , 1 ] .

In [6] it was claimed that the Thakur New Iteration Scheme has higher convergence rate than the iteration process of Karakaya et al. [7] .

In the recent work of Hussain et al. [8] , a new iteration scheme has been developed and it is claimed that it has better convergence rate than the iterative process Thakur et al. [6] . This iteration process is called K-iteration process and is given as:

For the initial value x 0 X , the sequence { x n } n = 0 defined by

{ z n = ( 1 β n ) x n + β n T x n , y n = T ( 1 α n ) T x n + α n T z n , x n + 1 = T y n (1.7)

where { α n } n = 0 , { β n } n = 0 is a sequence in [ 0 , 1 ] .

In the present work we shall generalize some convergence and stability results for K-iteration process. We shall also prove convergence and stability results for more general form of K-iteration process and K-iteration process for a pair of two distinct mappings.

Definition 1.1 [3] : Let X be a real Banach space. The mapping T : X X is said to be asymptotically quasi-nonexpansive if F ( T ) and there exists a sequence { μ n } [ 0 , ) with μ n 0 as n such that

T n x q ( 1 + μ n ) x q (1.8)

for all x X , q F ( T ) and n 0 .

Definition 1.2 [9] : Let X be a real Banach space. The mapping T : X X is said to be mean non-expansive if there exists two non negative real numbers a , b such that a + b 1 and for all x , y X ,

T x T y = a x y + b x T y

Definition 1.3 [10] : Let { z n } n = 0 be any sequence in X. Then the iterative process x n + 1 = f ( T , x n ) which converges to a fixed point q, is said to be stable with respect to the mapping T if for φ n = z n + 1 f ( T , z n ) , n = 0 , 1 , 2 , , we have lim n φ n = 0 if and only if lim n z n = q .

Definition 1.4 [7] : A space X is said to satisfy Opial’s condition if for each sequence { x n } n = 0 in X such that x n converges weakly to x we have for all y X , x y following holds:

1) lim inf n x n x < lim inf n x n y ,

2) lim sup n x n x < lim sup n x n y .

Lemma 1.5 [11] : Let { a n } n = 0 and { b n } n = 0 be non-negative real sequences satisfying the inequality:

a n + 1 ( 1 b n ) a n + b n ,

where b n ( 0 , 1 ) , for all n N , n = 1 b n = and b n a n 0 as n , then lim n a n = 0 .

Lemma 1.6 [12] : Let δ be a real number such that 0 δ < 1 , and { ϵ n } n = 0 be a sequence of positive numbers such that lim n ϵ n = 0 . Then for any sequence of positive numbers { a n } n = 0 satisfying a n + 1 δ a n + ϵ n , n = 0 , 1 , 2 , , we have lim n a n = 0 .

Lemma 1.7 [13] : Let X be a real Banach space and { g n } be any sequence in X such that 0 < g n < 1 for all n N . Let { a n } n = 0 and { b n } n = 0 be non-negative real sequences satisfying lim sup n a n c , lim sup n b n c and lim sup n g n a n + ( 1 g n ) b n = c holds for some c 0 . Then lim sup n a n b n = 0 .

2. Main Results

Theorem 2.1: Let X be a Banach space and T : X X be a mapping satisfying the condition

T x q δ x q (2.1)

where q F , x X and 0 δ < 1 . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (1.7). Then the sequence { x n } n = 0 converges strongly to q F ( T ) .

Proof: From (1.7) and (2.1) we have,

x n + 1 q = T y n q δ T y n q (2.2)

And

y n q = T ( ( 1 α n ) T x n + α n T z n ) q δ ( 1 α n ) T x n + α n T z n q δ ( 1 α n ) ( T x n q ) + α n ( T z n q ) δ [ ( 1 α n ) T x n q + α n T z n q ] δ [ ( 1 α n ) T x n q + α n T z n q ] δ 2 [ ( 1 α n ) x n q + α n z n q ] (2.3)

Again using (1.7) and (2.1) we get,

z n q = ( 1 β n ) x n + β n T x n q ( 1 β n ) x n q + β n T x n q ( 1 β n ) x n q + β n δ x n q (2.4)

Using (2.4) in (2.3) we get,

y n q δ 2 [ ( 1 α n ) x n q + α n ( 1 β n ) x n q + α n β n δ x n q ] δ 2 ( 1 α n + α n ( 1 β n ) + α n β n δ ) x n q δ 2 ( 1 α n β n ( 1 δ ) ) x n q (2.5)

Using (2.5) in (2.2) we get,

x n + 1 q δ 3 ( 1 α n β n ( 1 δ ) ) x n q

Since 0 δ < 1 , α n [ 0 , 1 ) and n = 0 α n = . Hence by using lemma (1.6), we have

lim n x n + 1 q = 0.

Hence the sequence { x n } n = 0 converges strongly to q.

Corollary 2.2: (Akewe and Okeke [14] ) Let X be a Banach space and T : X X be a mapping satisfying the condition

T x q δ x q

where q F , x X and 0 δ < 1 . Let { x n } n = 0 be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the sequence { x n } n = 0 converges strongly to q.

Remark 2.3: Theorem 2.1 gives generalization to many results in the literature by considering a wider class of contractive type operators and more general iterative process, including the results of Chidume [15] , Bosede and Rhoades [16] and Akewe and Okeke [14] .

Theorem 2.4: Let X be a Banach space and T : X X be a mapping satisfying the condition

T x q δ x q

where q F , x X and 0 δ < 1 . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (1.7). Then the iteration process (1.7) is T-stable.

Proof: By theorem 2.1, the sequence { x n } n = 0 converges strongly to q. Let { u n } n = 0 , { v n } n = 0 and { w n } n = 0 be real sequences in X.

Let φ n = u n + 1 T v n , n = 0 , 1 , 2 , , where

w n = ( 1 β n ) u n + β n T u n ,

v n = T ( ( 1 α n ) T u n + α n T w n ) ,

u n + 1 = T v n ,

and let lim n φ n = 0 .

We shall prove that lim n u n = q .

Now,

u n + 1 q = u n + 1 T v n + T v n q φ n + δ v n q (2.6)

v n q = T ( ( 1 α n ) T u n + α n T w n ) q δ ( 1 α n ) T u n + α n T w n q δ ( 1 α n ) ( T u n q ) + α n ( T w n q ) δ [ ( 1 α n ) T u n q + α n T w n q ] δ [ ( 1 α n ) T u n q + α n T w n q ] δ 2 [ ( 1 α n ) u n q + α n w n q ] (2.7)

Again using (1.7) and (2.1) we get,

w n q = ( 1 β n ) u n + β n T u n q ( 1 β n ) u n q + β n T u n q ( 1 β n ) u n q + β n δ u n q ( 1 β n ( 1 δ ) ) u n q (2.8)

Using (2.8) in (2.7) we get,

v n q δ 2 [ ( 1 α n ) u n q + α n ( 1 β n ( 1 δ ) ) u n q ] δ 2 ( 1 α n β n ( 1 δ ) ) u n q (2.9)

Using (2.9) in (2.6) we get,

u n + 1 q φ n + δ 3 ( 1 α n β n ( 1 δ ) ) u n q (2.10)

Since 0 δ < 1 and since 0 α n , β n 1 we have by lemma (1.6)

lim n u n = q .

Conversely let lim n u n = q . We shall show that lim n φ n = 0 .

Now

φ n = u n + 1 T v n u n + 1 q + T q T v n u n + 1 q + δ v n q (2.11)

Substituting (2.9) in (2.11),

φ n u n + 1 q + δ 3 ( 1 α n β n ( 1 δ ) ) u n q (2.12)

Since lim n u n = q , we have from (2.12) lim n φ n = 0 . Hence the K-iteration scheme is T-stable.

From theorem 2.4, we have the following corollary.

Corollary 2.5: Let X be a Banach space and T : X X be a mapping satisfying the condition

T x q δ x q ,

where q F , x X and 0 δ < 1 . Let { x n } n = 0 be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the iteration process (1.3) is T-stable.

Example 2.6: Let X = [ 0 , 1 ] and consider the mapping T x = x 2 . The clearly the mapping T satisfies the inequality (2.1). Now F ( T ) = 0 . Now we claim that the K-iteration scheme (1.7) is T-stable. Let us take α n = β n = 1 2 and consider the sequences x n = y n = z n = 1 n . Then clearly lim n x n = 0 .

Now

φ n = x n + 1 T y n = x n + 1 y n 2 = x n + 1 T ( ( 1 α n ) T x n + α n T z n ) 2 = x n + 1 ( 1 α n ) T x n + α n T z n 4 = x n + 1 ( ( 1 α n ) x n 8 + α n z n 8 ) = x n + 1 ( ( 1 α n ) x n 8 + α n ( 1 β n ) x n 8 + α n β n T x n 8 ) = x n + 1 ( ( 1 α n ) x n 8 + α n ( 1 β n ) x n 8 + α n β n x n 16 ) = 1 n + 1 ( 1 16 n + 1 32 n + 1 64 n ) = 1 n + 1 1 8 n

= 7 n 1 8 n ( n + 1 ) = 7 1 n 8 ( n + 1 ) (2.13)

Taking limit n in (2.13), we have lim n φ n = 0 . Hence the K-iteration process is T-stable.

Now we shall prove the convergence and stability results for asymptotically quasi-nonexpansive mapping by considering the more general form of K-iteration process as:

z n = ( 1 β n ) x n + β n T n x n ,

y n = T n ( ( 1 α n ) T n x n + α n T n z n ) ,

x n + 1 = T n y n , where n = 0 , 1 , 2 , , (2.14)

Theorem 2.7: Let H be a non-empty closed convex subset of a Banach space X and T : H H be asymptotically quasi-nonexpansive mapping with real sequence μ n [ 0 , ) . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that n = 0 α n β n μ n = . Then the sequence { x n } n = 0 converges strongly to some fixed point q of the mapping T.

Proof: From the iterative process (2.14) we have,

z n q = ( 1 β n ) x n + β n T n x n q ( 1 β n ) x n q + β n T n x n q ( 1 β n ) x n q + β n ( 1 + μ n ) x n q ( 1 + β n μ n ) x n q (2.15)

and

y n q = T n ( ( 1 α n ) T n x n + α n T n z n ) q ( 1 + μ n ) ( 1 α n ) T n x n + α n T n z n q ( 1 + μ n ) ( 1 α n ) ( T n x n q ) + α n ( T n z n q ) ( 1 + μ n ) [ ( 1 α n ) T n x n q + α n T n z n q ]

( 1 + μ n ) [ ( 1 α n ) ( 1 + μ n ) x n q + α n ( 1 + μ n ) z n q ] ( 1 + μ n ) 2 [ ( 1 α n ) x n q + α n z n q ] ( 1 + μ n ) 2 [ ( 1 α n ) x n q + α n ( 1 + β n μ n ) x n q ] ( 1 + μ n ) 2 ( 1 α n β n μ n ) x n q (2.16)

Again using (2.14) we have,

x n + 1 q T n y n q ( 1 + μ n ) y n q ( 1 + μ n ) 3 ( 1 α n β n μ n ) x n q (2.17)

By repeating the above process, we have the following inequalities

x n + 1 q ( 1 + μ n ) 3 ( 1 α n β n μ n ) x n q

x n q ( 1 + μ n 1 ) 3 ( 1 α n 1 β n 1 μ n 1 ) x n 1 q

x n 1 q ( 1 + μ n 2 ) 3 ( 1 α n 2 β n 2 μ n 2 ) x n 2 q

x 1 q ( 1 + μ 0 ) 3 ( 1 α 0 β 0 μ 0 ) x 0 q

So we can write,

x n + 1 q ( 1 + μ 0 ) 3 ( n + 1 ) x 0 q j = 0 n ( 1 α j β j μ j )

Since 1 x e x for all x [ 0 , 1 ] . Now 1 α j β j μ j < 1 , so we can write,

x n + 1 q ( 1 + μ 0 ) 3 ( n + 1 ) x 0 q e ( 1 α j β j μ j ) ( 1 + μ 0 ) 3 ( n + 1 ) x 0 q e j = 0 n α j β j μ j (2.18)

Taking limit n in (2.18), we have lim n x n q = 0 , that is the sequence { x n } n = 0 converges strongly to fixed point q of the mapping T.

Theorem 2.8: Let H be a non-empty closed convex subset of a Banach space X and T : H H be asymptotically quasi-nonexpansive mapping with real sequence μ n [ 0 , ) . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that n = 0 α n β n μ n = . Then the iterative process (2.14) is T-stable.

Proof: Let { u n } n = 0 X be any arbitrary sequence. Let the sequence generated by the iterative process (2.14) is x n + 1 = f ( T , x n ) converging to the fixed point q.

Let φ n = u n + 1 f ( T , x n ) .

We shall prove that lim n φ n = 0 if and only if lim n u n = q .

First suppose lim n φ n = 0 . Now we have

u n + 1 q = u n + 1 f ( T , u n ) + f ( T , u n ) q = φ n + T n ( T n ( 1 β n ) T n u n + β n T n ( ( 1 α n ) u n + α n T n u n ) ) q φ n + ( 1 + μ n ) 3 ( 1 α n β n μ n ) x n q (2.19)

where α n , β n [ 0 , 1 ] , lim n φ n = 0 and lim n μ n = 0 .

Now using (2.19) together with lemma (1.5), we have lim n u n q = 0 that is lim n u n = q .

Conversely let lim n u n = q . we have

φ n = u n + 1 f ( T , u n ) u n + 1 q + f ( T , u n ) q u n + 1 q + ( 1 + μ n ) 3 ( 1 α n β n μ n ) u n q

Taking limit n both sides of (6) we have lim n φ n = 0 . Hence (2.14) is T-stable.

Now we shall prove the convergence results for mean non-expansive mapping by modifying the K-iteration process for two mappings as:

z n = ( 1 β n ) x n + β n S x n ,

y n = T ( ( 1 α n ) S x n + α n T z n ) ,

x n + 1 = T y n , where n = 0 , 1 , 2 , , (2.20)

Lemma 2.9: Let H be a non-empty closed convex subset of a Banach space X and S , T : H H be two mean non-expansive mapping such that F = F ( T ) F ( S ) ϕ . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (2.20). Then lim n x n q exists for some q F .

Proof: We have

z n q = ( 1 β n ) x n + β n S x n q ( 1 β n ) x n q + β n S x n q ( 1 β n ) x n q + β n ( a 1 x n q + b 1 x n q ) ( 1 β n ) x n q + β n ( a 1 + b 1 ) x n q x n q (2.21)

Again using (2.20) and (2.21)

y n q = T ( ( 1 α n ) S x n + α n T z n ) q a 2 ( ( 1 α n ) S x n + α n T z n ) q + b 2 ( ( 1 α n ) S x n + α n T z n ) q ( a 2 + b 2 ) ( ( 1 α n ) S x n + α n T z n ) q ( 1 α n ) S x n q + α n T z n q

( 1 α n ) ( a 1 x n q + b 1 x n q ) + α n ( a 2 z n q + b 2 z n q ) ( 1 α n ) ( a 1 + b 1 ) x n q + α n ( a 2 + b 2 ) z n q ( 1 α n ) x n q + α n z n q x n q (2.22)

Again using (2.20) and (2.22)

x n + 1 q T y n q a 2 y n q + b 2 y n q ( a 2 + b 2 ) y n q y n q x n q (2.23)

This shows that { x n q } is non-increasing and bounded sequence for q F . Hence lim n x n q exists.

Lemma 2.10: Let be a non-empty closed convex subset of a Banach space and S , T : H H be two mean non-expansive mapping such that F = F ( T ) F ( S ) ϕ . Let { x n } n = 0 be the sequence defined by the K-iterative process given by (2.20). Also consider that lim n S x n q = lim n T x n q = 0 for some q F . Then lim n T x n x n = 0 .

Proof: Let q F . In lemma (2.9) we have proved the existence of

lim n x n q . Let lim n x n q = c . (2.24)

W.L.O.G. let c > 0 .

Now from (2.20) and (2.24) we have,

lim sup n z n q lim sup n x n q = c (2.25)

Now

S x n q a 1 x n q + b 1 x n q ( a 1 + b 1 ) x n q x n q

Implies that lim sup n S x n q lim sup n x n q = c (2.26)

Now

x n + 1 q T y n q a 2 y n q + b 2 y n q ( a 2 + b 2 ) y n q y n q T ( ( 1 α n ) S x n + α n T z n ) q a 2 ( ( 1 α n ) S x n + α n T z n ) q + b 2 ( ( 1 α n ) S x n + α n T z n ) q ( a 2 + b 2 ) ( ( 1 α n ) S x n + α n T z n ) q

( 1 α n ) S x n q + α n T z n q ( 1 α n ) ( a 1 x n q + b 1 x n q ) + α n ( a 2 z n q + b 2 z n q ) ( 1 α n ) ( a 1 + b 1 ) x n q + α n ( a 2 + b 2 ) z n q ( 1 α n ) x n q + α n z n q x n q α n x n q + α n z n q

x n + 1 q x n q α n = z n q x n q

and hence

x n + 1 q x n q x n + 1 q x n q α n = z n q x n q

which implies that x n + 1 q z n q (2.27)

Taking limit inferior in (2.27) we obtain

c lim inf n z n q (2.28)

From (2.20) and (2.28) we have

c = lim n z n q = lim n ( 1 β n ) x n + β n S x n q = lim n β n ( S x n q ) + ( 1 β n ) ( x n q ) (2.29)

Now from (2.24), (2.26), (2.29) and lemma (1.7), we have lim n S x n x n = 0 .

Now,

T x n q a 2 x n q + b 2 x n q x n q

lim sup n T x n q lim sup n x n q c (2.30)

Using the conditions of the lemma in (2.30), we can write

C = lim n β n ( T x n q ) + ( 1 β n ) ( x n q ) (2.31)

Using (2.24), (2.30), (2.31) along with the lemma (1.7), we have

lim n T x n x n = 0.

Theorem 2.11: Let H be a non-empty closed convex subset of a Banach space X satisfying Opial’s condition and S, T and { x n } n = 0 be same as defined in the lemma (2.10) .Then the sequence { x n } n = 0 converges weakly to some q F .

Proof: From lemma (2.10) we have, lim n T x n x n = 0 .

Since X is uniformly convex and hence it is reflexive so there exists a subsequence { x n m } of { x n } such that { x n m } converges weakly to some q 1 F . Since H is closed so q 1 H . Now we claim the weak convergence of { x n } to q 1 . Let it is not true, then there exists a subsequence of { x n i } of { x n } which converges weakly to q 2 and let q 1 q 2 . Also q 2 F . Now from lemma (2.9) lim n x n q 1 and lim n x n q 2 both exist. Using Opial’s condition we have,

lim n x n q 1 lim n x n m q 1 < lim n x n m q 2 = lim n x n q 2 = lim n x n i q 2 < lim n x n i q 1 lim n x n q 1

This is a contradiction, so we must have q 1 = q 2 . Thus the sequence { x n } n = 0 converges weakly to some q F .

Theorem 2.12: Let H be a non-empty closed compact subset of a Banach space X and S, T and { x n } n = 0 be same as defined in the lemma (2.10). Then the sequence { x n } n = 0 converges strongly to some q F .

Proof: Since H is compact and hence it is sequentially compact. So there exists a subsequence { x n i } of { x n } which converges to q H .

Now

x n i T q = x n i T x n i + T x n i T q x n i T x n i + a 2 x n i q + b 2 x n i q x n i T x n i + x n i q (2.32)

Taking limit n in (2.32) we have, T q = q that is q F . We have earlier proved that lim n x n q exists for q F . Hence the sequence { x n } n = 0 converges strongly to some q F .

In [8] it is proves that the K-iteration process converges faster than Picard-S, Thakur-New and Vatan two-step iterative process. Now we shall compare the rate of convergence the K-iteration process defined in [8] and our new modified K-iteration process for two mappings.

Table 1. Iterative values of K-iteration process and Modified K-iteration process.

Example 2.13: Let S , T : [ 0 , 3 ] [ 0 , 3 ] be two mappings defined by T ( x ) = x + 2 2 and s ( x ) = ( x + 2 ) 1 2 . Let α n , β n be the sequences defined by α n = β n = 1 4 . Let the initial approximation be x 0 = 2.25 . Clearly S, T has

unique common fixed point 2. The convergence pattern of K-iteration process and modified K-iteration process is shown in Table 1.

Clearly we can conclude from Table 1, that the modified K-iteration process has better rate of convergence than the k-iteration process.

Cite this paper: Panwar, A. and Bhokal, R. (2019) Fixed Point Results for K-Iteration Using Non-Linear Type Mappings. Open Access Library Journal, 6, 1-14. doi: 10.4236/oalib.1105245.
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[13]   Sahu, J. (1991) Weak and Strong Convergence to Fixed Points of Asymptotically Non-Expansive Mappings. Bulletin of the Australian Mathematical Society, 43, 153-159. https://doi.org/10.1017/S0004972700028884

[14]   Akewe, H. and Okeke, G.A. (2015) Convergence and Stability Theorems for the Picard-Mann Hybrid Iterative Scheme for a General Class of Contractive-Like Operators. Fixed Point Theory and Applications, 2015, 66.

[15]   Chidume, C.E. (2014) Strong Convergence and Stability of Picard Iteration Sequence for General Class of Contractive-Type Mappings. Fixed Point Theory and Applications, 2014, 233. https://doi.org/10.1186/1687-1812-2014-233

[16]   Bosede, A.O. and Rhoades, B.E. (2010) Stability of Picard and Mann Iteration for a General Class of Functions. Journal of Advanced Mathematical Studies, 3, 23.

 
 
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