Common Fixed Point for Contractive Operators by a Faster Iterative Process in Real Banach Spaces

Abbr:J.Comp.&Math.Sci.

2014, Vol.5(2): Pg.251-257

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

Common Fixed Point for Contractive Operators by a Faster Iterative Process in Real Banach Spaces

M. R. Yadav

School of Studies in Mathematics,

Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, INDIA.

(Received on: April 21, 2014) ABSTRACT

In this article, we introduced a new two step iteration scheme for the class of Zamfirescu operators in real Banach spaces and proved strong convergence theorem to approximate common fixed points of Zamfirescu operators. We showed that our iteration process is faster than generalized Mann iteration scheme is defined by Yao and Chen ⁹ and studied by ^{3, 5, 7, 8} . We also give the comparison table for iteration schemes. An example is given to support our main results and satisfied Zamfirescu operators.

2000 Mathematics Subject Classification: 47H05, 47H09, 47H10.

Keywords and phrases: Two-step iteration scheme, Zamfirescu Operator, real Banach spaces, Strong convergence, Common foxed point.

1. INTRODUCTION

Throughout this paper, N denotes the set of all positive integers. Let K be nonempty subset of real Banach space X and T : K → K a given operator. Let

K

x ₀ ∈ be arbitrary and { α _n } ∈ [0, 1] _a sequence or real numbers. The Mann iteration process starting at x ₀ is defined by

 



∈ α

α

∈

+ = (1 - )x + Tx ; n N

x

K;

x

= x

n n n n 1

n 0

1 (1.1)

is called the Mann ⁶ iteration schemes or Mann iteration procedure and the generalized Mann iteration scheme is defined as follows:

 



∈ γ

+ β α

∈

+ = x + Tx Sx ; n N x

K;

x

= x

n n n n n n 1 n

0

1 (1.2)

where { α _n } _, { β _n } and { γ _n } are real sequence in [0, 1] with α _{n +}

β n + γ _n = 1.

Above iteration scheme studied by Yao and Chen ⁹ for common fixed points of two mappings.

The following Ishikawa type iteration scheme has been studied by various authors for common fixed points of two mappings, see for example ^{3, 5, 7, 8} :

 

 



∈ βα

β

α α

∈

+

, N n

; Tx +

)x - (1

= y

Sy + )x - (1

= x

K;

x

= x

n n n n n

n n n n 1

n 0 1

(1.3)

where { α _n } _and { β _n } are sequences in [0; 1]. This scheme also reduces to Mann iteration scheme (1.1) when T = I or S = I.

The purpose of this paper is to define a new type of two-step iteration scheme for Zemfrarescu operators and showed strong convergence theorem to approximate fixed point in the framework of real Banach space. Let K be a nonempty, closed and convex subset of a real Banach space X.

Suppose S; T : K → K _{be two} nonlinear operators and { x _n } ^∞ _{n =0} be the sequence in [0, 1]:

 

 



∈ β

+ β

=

γ + β

α

∈

+

, N n , Tx )x

- (1 y

Sy Tx

+ x

= x

K;

x

= x

n n n n n

n n n n n n 1 n

0 1

(1.4)

where { α _n } _, { β _n } and { γ _n } are real sequence in [0, 1] with α _{n +}

β n + γ _n = 1.

This scheme also reduces to Mann iteration scheme (1.1) when T = I or β = 0.

We recall the following definitions in a metric space (X, d) . A mapping

X X :

T → is called a-contraction if y)

ad(x, Ty)

d(Tx, ≤ ^, (1.5) for all x, ∈ y X, ^where a ∈ [0, 1) .

The map T is called Kannan mapping [4] if there exists b∈ (0, 1/2) ^such that

Ty)]

d(y, + Tx) b[d(x, Ty)

d(Tx, ≤ , (1.6)

for all x, ∈ y X, ^.

The map T is called Chatterjea ² if there exists c∈ (0, 1/2) such that

Tx)]

d(y, + Ty) c[d(x, Ty)

d(Tx, ≤ , (1.7)

for all.

An operator T which satisfies at least one of the contractive condition (1.5), (1.6) and (1.7) is called Zamfirescu Operator or a Z-operator ¹⁰ .

In 1972 Zamfirescu ¹⁰ obtained very interesting fixed point Theorem, by combining (1.5), (1.6) and (1.7) given below:

Theorem 1.1. Let (X, d) be complete metric space and T : X → X a map for which there exist the real number a, b, and c satisfying 0 < a < 1, 0 < b, c < 0.5 such that for each pair x, ∈ y X, , at least one the following is true:

(i) d(Tx, Ty) ≤ ad(x, y) ^,

(ii) d(Tx, Ty) ≤ b[d(x, Tx) + d(y, Ty)] ^, (iii) d(Tx, Ty) ≤ c[d(x, Ty) + d(y, Tx)]

Then T is a Picard operator x _{n + 1} = Tx _n ^. 2. MAIN RESULTS

In this section, we have proved

strong convergence theorem and find

approximate common fixed points of two

self mapping S and T.

Theorem 2.1. Let K be a nonempty closed convex subset of real Banach space E and

X X :

T → be a Zemfrarescu operator.

Suppose { x _n } be the sequence defined by iteration (1.4) for arbitrary x ₀ ∈ K ^with

∑ ^∞ β = ∞

n =0 n , where { α _n } _, { β _n } and { γ _n } are sequences in [0, 1] with

n 1

n

n + β + γ =

α . Then { x _n } converges strongly to a common fixed point of T and S.

Proof. Suppose F(T) ≠ φ and p∈ F(T) . Since T is Zamfirescu operator, then T satisfies at least one of the conditions (1.5), (1.6) and (1.7). If (1.6) holds, then for any

K,

x, ∈ y , we obtain

||]

Ty Tx

||

||

Tx x

||

||

x y

||

||

Tx x [||

b

||]

Ty y

||

||

Tx x [||

b

||

Ty - Tx

||

− +

− +

− +

−

≤

− +

−

≤ which yield that

||

Tx - x

||

2b +

||

y - x

||

b[

||

Ty - Tx

||

b) -

(1 ≤ ^,

since 0 ≤ b < 1/2 _{, we have} || x Tx ||

b 1

b

|| 2 y x b ||

1

|| b Ty Tx

|| −

+ −

− −

≤

− ^. (2.1)

Similarly, if (1.7) holds, then we obtain for any x, ∈ y K,

||]

Tx x

||

||

x y

||

||

Ty Tx

||

||

Tx x [||

b

||]

Tx y

||

||

Ty x [||

c

||

Ty - Tx

||

− +

− +

− +

−

≤

− +

−

≤ which yield that

||

Tx - x

||

2c +

||

y - x

||

c[

||

Ty - Tx

||

c) -

(1 ≤ ^,

since 0 ≤ c < 1/2 _{, we have}

||

Tx x c ||

1 c

|| 2 y x c ||

1

|| c Ty Tx

|| −

+ −

− −

≤

− ^. (2.2)

Suppose,

 



 



−

= −

δ 1 c

, c b 1 , b a

max (2.3)

Then we have 0 ≤ δ < 1 and from (2.1), (2.2) and (2.3), we get

||

Tx x

||

2

||

y x

||

||

Ty Tx

|| − ≤ δ − + δ − (2.4)

Similarly, since S is a Zamfirescu operator, we obtain

||

Sx x

||

2

||

y x

||

||

Sy Sx

|| − ≤ δ − + δ − (2.5)

holds for all x, ∈ y K, . Let p is a fixed point of T, then if x = p and y = x _n in (2.4), we obtain

||

p x

||

||

p Tx

|| _n − ≤ δ _n − (2.6)

and put x = p and y = y _n in (2.5), we get

||

p y

||

||

p Sy

|| _n − ≤ δ _n − (2.7)

Suppose { x _n } ^∞ _{n =0} be the sequences defined by (1.4). Then we have

||

p Sy

||

||

p Tx

||

||

p x

||

||

) p Sy ( ) p Tx ( ) p x (

||

||

p Sy Tx

x

||

p x

||

n n n

n n

n

n n n

n n

n

n n n n n n 1

n

− γ

+

− β

+

− α

≤

− γ

+

− β

+

− α

≤

− γ + β

+ α

=

+ −

(2.8)

From (2.6), (2.7) and (2.8), we obtain

||

p y

||

||

p x

||

) (

||

p y

||

||

p x

||

||

p x

||

||

p x

||

n n n

n n

n n n

n n

n 1

n

− δ γ +

− δ

β + α

≤

− δ γ +

− δ β +

− α

≤

+ −

(2.9) and,

||

p x

||

||

p x

||

) 1 (

||

p Tx

||

||

p x

||

) 1 (

||

) p Tx ( ) p x )(

1 (

||

||

p Tx x

) 1 (

||

p y

||

n n n

n

n n n

n

n n n

n

n n n n n

− δ β +

− β

−

≤

− β

+

− β

−

≤

− β

+

− β

−

≤

− β

+ β

−

=

−

(2.10)

Combining (2.9) and (2.10), we get

.

||

p x

||

] )(

1 ( 1 [

||

p x

||

)]

1 ( )

1 ( ) 1 ( 1 [

||

p x

||

)]

1 ( )

1 ( 1

1 [

||

p x

||

)]

1 ( )

1 ( 1

1 [

||

p x

||

)]

1 ( [

||

p x

||

)]

1 ( )

1 ( [

||

p x

||

) (

||

p x

||

||

p x

||

) 1 (

||

p x

||

) (

||}

p x

||

||

p x

||

) 1 {(

||

p x

||

) (

||

p x

||

n n n n n

n n

n n

n n

n n

n n

n n

n n

n n n

n n

n n n

2 n n n n n n n n

2 n n n n

n n

n n n

n n n

n n

n n n 1

n

− δ

γ β + + γ + β δ

−

−

≤

− δ

− δ γ β

− δ

− α + δ

−

−

≤

− δ

− δ γ β

− δ

− α + δ +

−

≤

− δ

− δ γ β

− δ

− α + δ +

−

≤

− δ

− δ γ β + δ α

− α + δ

≤

− δ

− δ γ β

− δ α

− + α

≤

− δ

β γ + δ β γ

− δ γ + δ β + α

≤

− δ

β γ +

− β

− δ γ +

− δ

β + α

≤

− δ β +

− β

− δ γ +

− δ

β + α

=

+ −

Now,

] ) 1 ( 1 [

)]

1 ( ) 1 ( ) 1 ( 1 [ )}]

1 ( ){

1 ( 1 [

n

n n n

n n n

β δ

−

−

≤

δ β + γ δ

−

− β δ

−

−

= δ β + γ + β δ

−

−

By the induction method, we have

∏ − − δ β − =

−

= +

n 0

m n 0

1

n p || [ 1 ( 1 ) ]. || x p ||, n 0 , 1 , 2 ,...

x

|| (2.11)

Now, since 0 ≤ δ < 1 _, β _m ∈ [ 0 , 1 ] and

∑ ^∞ β = ∞

m =0 m , then it follows that

∏ − − δ β =

∞ =

→ n

0

m n

m lim [ 1 ( 1 ) ] 0 ,

which implies by (2.11), we obtain 0

||

p x

||

lim _{n 1}

m _{→ ∞} ⁺ − = ^.

Therefore, { x _n } ^∞ _{n =0} converges strongly to p, which is the common fixed point of T and S.

This complete the proof.

When T = I then Berinde result ([1], Theorem 2.1) is a corollary to our result.

Corollary 2.2. Let K be a nonempty closed convex subset of real Banach space E and

K K :

S → _{be a} Zemfrarescu operator.

Suppose { x _n } be the sequence defined by iteration (1.1) for arbitrary x ⁰ ∈ K ^with

∑ ^∞ γ = ∞

n =0

n , where { γ _n } is sequence in 1]

[0, . Then { x _n } converges strongly to a common fixed point of S.

Following examples illustrating our main results.

Example : Consider the mappings define by S , T : R → R as follows

3 ) x 2 Tx ( −

= ^and

4 ) 1 y 2

Sy ( +

= ^{for all} x , y ∈ R

It is clear that, both S and T are satisfied the condition of Zamfirescu operators with the common fixed point

2 1 _.

Furthermore, since 0 ≤ δ < 1 and from (2.4) and (2.5), we have

||

Tx - x

||

2 +

||

y - x

||

| Ty - Tx

|| ≤ δ δ _{, and}

||

Sx - x

||

2 +

||

y - x

||

| Sy - Sx

|| ≤ δ δ _,

for all x , y ∈ R . In fact,

3 y x 3

) y 2 ( 3

) x 2

|| ( Ty Tx

|| −

− =

− −

=

− , and

3 . 4 y 3 x 11 3

4 x 8 y 3 x 3

3 1 x 4 2 y 3 x

2 x 2 4 y x

3 x 2 x 2 3 y 3 x

x x 2 2 y x

||

Tx x

||

2

||

y x

||

− δ −

− = + δ −

=

δ − +

− δ

− = δ +

− δ

=

+ δ −

+

− δ

− =

− δ +

− δ

=

− δ +

−

δ

Similarly, we conclude that

2 , y

|| x Sy Sx

|| −

=

− and

[ ^{4 x 2 y 1} ] ^.

2

|| 1 Sx x

||

2

||

y x

|| − + δ − = δ − −

δ

Put x = 0.25, y = 0.14 and δ = 0.5, we have

036 . 3 0

y

|| x Ty Tx

|| − =

=

− ^,

and

388 . 3 0

4 y 3 x

|| 11 Tx x

||

2

||

y x

|| − − =

δ

=

− δ +

−

δ _.

Table 1. A comparison table of our process with other processes

Steps Iter.1 Steps Iter.1 Steps Iter.2 Steps Iter.2 Steps Iter.3 Steps Iter.4 1 0.2500 16 0.5742 1 0.2500 16 0.5834 1 0.2500 16 0.2136 2 9.5100 17 0.5742 2 10.5825 17 0.5829 2 11.4900 17 0.2118 3 4.6846 18 0.5741 3 5.7325 18 0.5827 3 6.6393 18 0.2107 4 2.4649 19 0.5741 4 3. 2347 19 0.5826 4 3.8744 19 0.2101 5 1.4439 20 0.5741 5 1.9484 20 0.5825 5 2.2984 20 0.2098 6 0.9742 21 0.5741 6 1.2859 21 0.5825 6 1.4001 21 0.2096 7 0.7581 22 0.5741 7 0.9447 22 0.5825 7 0.8881 22 0.2095 8 0.6587 23 0.5741 8 0.7690 23 0.5825 8 0.5962 23 0.2094 9 0.6130 24 0.5741 9 0.6786 24 0.5825 9 0.4298 24 0.2094 10 0.5920 25 0.5741 10 0.6320 25 0.5825 10 0.3350 25 0.2093 11 0.5823 26 0.5741 11 0.6080 26 0.5825 11 0.2810 26 0.2093 12 0.5779 17 0.5741 12 0.5956 17 0.5825 12 0.2501 17 0.2093 13 0.5758 28 0.5741 13 0.5892 28 0.5825 13 0.232 28 0.2093 14 0.5749 29 0.5741 14 0.5860 29 0.5825 14 0.2226 29 0.2093 15 0.5744 30 0.5741 15 0.5843 30 0.5825 15 0.2169 30 0.2093

Hence, for x = 0.25, y = 0.14 and δ

= 0.5 the condition (2.4) is satisfied.

Similarly we show that the condition (2.5) is also satisfied. It is not difficult to show that T and S are contraction.

Now, choose α = 0 . 5 , β = 0 . 33 and γ = 0 . 25 for all n with initial value

20

x 1 = . The comparison table given in the following table shows that our iterative process (1.4) converges faster than all Yao et al. ⁹ and Das et al. ³ iterative processes up to the accuracy of forth decimal places.

Here we observe that the values the

above calculations have been repeated by

taking different values of parameters α _, β and γ . It has been verified every time that our iterative process Iter1 converges faster than all studied by Yao and Chen ⁹ and ^{3, 5, 7, 8} iterative processes.

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