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 .

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Panwar, A. and Bhokal, R.P. (2019) Fixed Point Results for K-Iteration Using Non-Linear Type Mappings. Open Access Library Journal, 6: e5245. https://doi.org/10.4236/oalib.1105245