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Convergence analysis of Agarwal et al. iterative scheme for Lipschitzian hemicontractive mappings

Abstract

In this paper, we establish strong convergence for the Agarwal et al. iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

MSC:47H10, 47J25.

1 Introduction and preliminaries

Let K be a nonempty subset of a Hilbert space H and T:KK be a mapping.

The mapping T is called Lipschitzian if there exists L>0 such that

TxTyLxy,x,yK.

If L=1, then T is called nonexpansive and if 0L<1, then T is called contractive.

The mapping T:KK is said to be pseudocontractive (see, for example, [1, 2]) if

T x T y 2 x y 2 + ( I T ) x ( I T ) y 2 ,x,yK,

and it is said to be strongly pseudocontractive if there exists k(0,1) such that

T x T y 2 x y 2 +k ( I T ) x ( I T ) y 2 ,x,yK.

Let F(T):={xH:Tx=x}, and the mapping T:KK is called hemicontractive if F(T) and

T x x 2 x x 2 + x T x 2 ,xK, x F(T).

It is easy to see that the class of pseudocontractive mappings with fixed points is a subclass of the class of hemicontractions. For the importance of fixed points of pseudocontractions, the reader may consult [1].

In 1974, Ishikawa [3] proved the following result.

Theorem 1.1 Let K be a compact convex subset of a Hilbert space H, and let T:KK be a Lipschitzian pseudocontractive mapping. For arbitrary x 1 K, let { x n } be a sequence defined iteratively by

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

where { α n } and { β n } are sequences satisfying

  1. (i)

    0 α n β n 1;

  2. (ii)

    lim n β n =0;

  3. (iii)

    n = 1 α n β n =.

Then the sequence { x n } converges strongly to a fixed point of T.

Another iteration scheme has been studied extensively in connection with fixed points of pseudocontractive mappings.

In 2007, Agarwal et al. [4] introduced the new iterative scheme as in the following.

The sequence { x n } defined by, for arbitrary x 1 K,

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

where { α n } and { β n } are sequences in [0,1], is known as the Agarwal et al. iterative scheme.

In this paper, we establish the strong convergence for the Agarwal et al. iterative scheme associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

2 Main results

We need the following lemma.

Lemma 2.1 [5]

For all x,yH and λ[0,1], we have

( 1 λ ) x + λ y 2 =(1λ) x 2 +λ y 2 λ(1λ) x y 2 .

Now we prove our main results.

Theorem 2.2 Let K be a compact convex subset of a real Hilbert space H, and let T:KK be a Lipschitzian hemicontractive mapping satisfying

xTyTxTy,x,yK.
(C)

Let { α n } and { β n } be sequences in [0,1] satisfying

  1. (ii)

    lim n β n =0;

  2. (iii)

    n = 1 α n β n =;

  3. (iv)

    lim n α n =1.

For arbitrary x 1 K, let { x n } be a sequence iteratively defined by

{ x n + 1 = ( 1 α n ) T x n + α n T y n , y n = ( 1 β n ) x n + β n T x n , n 1 .
(2.1)

Then the sequence { x n } converges strongly to the fixed point x of T.

Proof From Schauder’s fixed point theorem, F(T) is nonempty since K is a convex compact set and T is continuous, let x F(T).

By using condition (C), we have

x T x x T y + T x T y 2 T x T y .
(2.2)

Using the fact that T is hemicontractive, we obtain

T x n x 2 x n x 2 + x n T x n 2
(2.3)

and

T y n x 2 y n x 2 + y n T y n 2 .
(2.4)

With the help of (2.1), (2.3) and Lemma 2.1, we obtain

y n x 2 = ( 1 β n ) x n + β n T x n x 2 = ( 1 β n ) ( x n x ) + β n ( T x n x ) 2 = ( 1 β n ) x n x 2 + β n T x n x 2 β n ( 1 β n ) x n T x n 2 ( 1 β n ) x n x 2 + β n ( x n x 2 + x n T x n 2 ) β n ( 1 β n ) x n T x n 2 = x n x 2 + β n 2 x n T x n 2
(2.5)

and

y n T y n 2 = ( 1 β n ) x n + β n T x n T y n 2 = ( 1 β n ) ( x n T y n ) + β n ( T x n T y n ) 2 = ( 1 β n ) x n T y n 2 + β n T x n T y n 2 β n ( 1 β n ) x n T x n 2 .
(2.6)

Substituting (2.5) and (2.6) in (2.4), we obtain

T y n x 2 x n x 2 + ( 1 β n ) x n T y n 2 + β n T x n T y n 2 β n ( 1 2 β n ) x n T x n 2 .
(2.7)

Also, with the help of conditions (2.2) and (2.7), we have

x n + 1 x 2 = ( 1 α n ) T x n + α n T y n x 2 = ( 1 α n ) ( T x n x ) + α n ( T y n x ) 2 = ( 1 α n ) T x n x 2 + α n T y n x 2 α n ( 1 α n ) T x n T y n 2 ( 1 α n ) ( x n x 2 + x n T x n 2 ) + α n ( x n x 2 + ( 1 β n ) x n T y n 2 + β n T x n T y n 2 β n ( 1 2 β n ) x n T x n 2 ) = x n x 2 + ( 1 α n ) x n T x n 2 + α n β n T x n T y n 2 α n β n ( 1 2 β n ) x n T x n 2 + α n ( 1 β n ) x n T y n 2 x n x 2 + ( 4 ( 1 α n ) + α n β n + α n ( 1 β n ) ) T x n T y n 2 α n β n ( 1 2 β n ) x n T x n 2 x n x 2 + θ α n T x n T y n 2 α n β n ( 1 2 β n ) x n T x n 2 ,
(2.8)

because by (iv), there exists n 0 N such that for all n n 0 ,

1 α n θ 1 θ + 3 ,
(2.9)

where θ>1, which implies that

4(1 α n )+ α n β n + α n (1 β n )θ α n .
(2.10)

Hence (2.8) yields

x n + 1 x 2 x n x 2 + θ α n L 2 x n y n 2 α n β n ( 1 2 β n ) x n T x n 2 = x n x 2 + θ α n β n 2 L 2 x n T x n 2 α n β n ( 1 2 β n ) x n T x n 2 = x n x 2 α n β n ( 1 ( 2 + θ L 2 ) β n ) x n T x n 2 .
(2.11)

Now, by (ii), since lim n β n =0, there exists n 0 N such that for all n n 0 ,

β n 1 2 ( 2 + θ L 2 ) .
(2.12)

With the help of (iii) and (2.12), (2.11) yields

x n + 1 x 2 x n x 2 1 2 α n β n x n T x n 2 ,

which implies that

1 2 α n β n x n T x n 2 x n x 2 x n + 1 x 2 ,

so that

1 2 j = N n α j β j x j T x j 2 x N x 2 x n + 1 x 2 .

The rest of the argument follows exactly as in the proof of theorem of [3]. This completes the proof. □

Theorem 2.3 Let K be a compact convex subset of a real Hilbert space H; let T:KK be a Lipschitzian hemicontractive mapping satisfying condition (C). Let { α n } and { β n } be sequence in [0,1] satisfying conditions (ii)-(iv).

Let P K :HK be the projection operator of H onto K. Let { x n } be a sequence defined iteratively by

{ x n + 1 = P K ( ( 1 α n ) T x n + α n T y n ) , y n = P K ( ( 1 β n ) x n + β n T x n ) , n 1 .

Then the sequence{ x n } converges strongly to a fixed point of T.

Proof The operator P K is nonexpansive (see, e.g., [2]). K is a Chebyshev subset of H so that P K is a single-valued mapping. Hence, we have

x n + 1 x 2 = P K ( ( 1 α n ) T x n + α n T y n ) P K x 2 ( 1 α n ) T x n + α n T y n x 2 = ( 1 α n ) ( x n x ) + α n ( T y n x ) 2 x n x 2 α n β n ( 1 ( 2 + θ L 2 ) β n ) x n T x n 2 .

The set K=KT(K) is compact and so the sequence { x n T x n } is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.2. This completes the proof. □

Example 2.4 The choice for the control parameters is α n = n n + 1 and β n = 1 n .

Remark 2.5 (1) We remove the condition α n β n as introduced in [3].

(2) The condition (C) is not new and it is due to [6].

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Acknowledgements

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

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Kang, S.M., Rafiq, A., Ali, F. et al. Convergence analysis of Agarwal et al. iterative scheme for Lipschitzian hemicontractive mappings. J Inequal Appl 2013, 525 (2013). https://doi.org/10.1186/1029-242X-2013-525

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Keywords

  • Agarwal et al. iterative scheme
  • Lipschitzian mappings
  • continuous mappings
  • pseudocontractive mappings
  • Hilbert spaces