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Research | Open | Published:

Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces

Abstract

In this paper, we establish strong convergence for the iterative scheme introduced by Sahu and Petruşel associated with Lipschitzian hemicontractive mappings in Hilbert spaces.

MSC:47H10, 47J25.

1 Introduction

Let H be a Hilbert space, and let T:HH be a mapping. The mapping T is called Lipshitzian if there exists L>0 such that

TxTyLxy,x,yH.

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

The mapping T:HH 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,yH
(1.1)

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,yH.
(1.2)

Let F(T):={xH:Tx=x}, and let K be a nonempty subset of H. A mapping T:KK is called hemicontractive if F(T) and

T x x 2 x x 2 + x T x 2 ,xH, 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 the Ishikawa iterative scheme

{ 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 ,
(1.3)

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

  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 iterative scheme which has been studied extensively in connection with fixed points of pseudocontractive mappings is the S-iterative scheme introduced by Sahu and Petruşel [4] in 2011.

In this paper, we establish strong convergence for the S-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], the following well-known identity holds:

( 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

Let { β n } be a sequence in [0,1] satisfying

  1. (iv)
    n = 1 β n =

    ;

  2. (v)
    lim n β n =0

    .

For arbitrary x 1 K, let { x n } be a sequence defined iteratively by the S-iterative scheme

{ x n + 1 = 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 of T.

Proof From Schauder’s fixed point theorem, F(T) is nonempty since K is a compact convex set and T is continuous. Let x F(T). Using the fact that T is hemicontractive, we obtain

T x n x 2 x n x 2 + x n T x n 2
(2.2)

and

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

With the help of (2.1), (2.2) and Lemma 2.1, we obtain the following estimates:

(2.4)
(2.5)

Substituting (2.4) and (2.5) in (2.3) 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 .
(2.6)

Also, with the help of condition (C) and (2.6), we have

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

Now, by lim n β n =0, there exists n 0 N such that for all n n 0 ,

β n 1 2 ( 2 + L 2 ) ,
(2.8)

and with the help of (2.8), (2.7) yields

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

which implies

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

so that

1 2 j = N n 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, and let T:KK be a Lipschitzian hemicontractive mapping satisfying condition (C). Let { β n } be a sequence in [0,1] satisfying conditions (iv) and (v).

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

{ x n + 1 = P K ( 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 the following estimate:

x n + 1 x 2 = P K ( T y n ) P K x 2 T y n x 2 x n x 2 β n ( 1 ( 2 + 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. □

Remark 2.4 In Theorem 1.1, putting α n =1, 0 α n β n 1 implies β n =1, which contradicts lim n β n =0. Hence the S-iterative scheme is not the special case of Ishikawa iterative scheme.

Remark 2.5 In Theorems 2.2 and 2.3, condition (C) is not new; it is due to Liu et al. [6].

References

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    Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis. Am. Math. Soc., Providence; 1976.

  2. 2.

    Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6

  3. 3.

    Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

  4. 4.

    Sahu DR, Petruşel A: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. 2011, 74: 6012–6023. 10.1016/j.na.2011.05.078

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    Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

  6. 6.

    Liu Z, Feng C, Ume JS, Kang SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwan. J. Math. 2007, 11: 27–42.

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Acknowledgements

The authors would like to thank the referees for useful comments and suggestions.

Author information

Correspondence to Sunhong Lee.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

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Keywords

  • iterative scheme
  • Lipschitzian mappings
  • hemicontractive mappings
  • Hilbert spaces