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Algorithms with variant anchors for pseudocontractive mappings

Abstract

The purpose of this paper is to find the fixed points of pseudocontractive mappings by using the iterative technique. Two algorithms with variant anchors have been introduced. Strong convergence results are given. Especially, we can find the minimum-norm fixed point of pseudocontractive mappings.

MSC:47H05, 47H10, 47H17.

1 Introduction

In this paper, we assume that H is a real Hilbert space and CH is a nonempty closed convex subset. Recall that a mapping T:CC is said to be Lipschitzian if

T u T u κ u u ,u, u C,

where κ>0 is a constant, which is in general called the Lipschitz constant. If κ=1, T is called nonexpansive.

A mapping T:CC is said to be pseudocontractive if

T u T u , u u u u 2 ,u, u C.

We use Fix(T) to denote the set of fixed points of T.

In the literature, there are a large number references associated with the fixed point algorithms for the pseudocontractive mappings. See, for instance, [124]. (The interest of pseudocontractions lies in their connection with monotone operators; namely, T is a pseudocontraction if and only if the complement IT is a monotone operator.)

Now there exists an example which shows that Mann iteration does not converge for the pseudocontractive mappings [2]. At present, it is still an interesting topic to construct algorithms for finding the fixed points of the pseudocontractive mappings.

On the other hand, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, in order to find the fixed points of the nonexpansive mappings, Yao and Shahzad [25] introduced the following algorithms with perturbations and obtained the strong convergence results.

Algorithm 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:CC be a nonexpansive mapping. For given x 0 C, define a sequence { x m } in the following manner:

x m = proj C [ α m u m + ( 1 α m ) T x m ] ,m0,
(1.1)

where { α m } is a sequence in [0,1] and the sequence { u m }H is a small perturbation for the m-step iteration satisfying u m 0 as m.

Theorem 1.2 Suppose Fix(T). Then, as α m 0, the sequence { x m } generated by the implicit method (1.1) converges to x ˜ Fix(T), which is the minimum-norm fixed point of T.

Algorithm 1.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:CC be a nonexpansive mapping. For given x 0 C, define a sequence { x n } in the following manner:

x n + 1 =(1 β n ) x n + β n proj C [ α n u n + ( 1 α n ) T x n ] ,n0,
(1.2)

where { α n } and { β n } are two sequences in (0,1) and the sequence { u n }H is a perturbation for the n-step iteration.

Theorem 1.4 Suppose that Fix(T). Assume the following conditions are satisfied:

  1. (i)

    lim n α n =0 and n = 0 α n =;

  2. (ii)

    0< lim inf n β n lim sup n β n <1;

  3. (iii)

    n = 0 α n u n <.

Then the sequence { x n } generated by the explicit iterative method (1.2) converges to x ˜ Fix(T), which is the minimum-norm fixed point of T.

Note that the idea of the iterative algorithms with perturbations has been extended to the other topics, see, for example, [26].

Motivated by the above ideas and the results in the literature, in the present paper, we present two algorithms with variant anchors for finding the fixed points of the pseudocontractive mappings in Hilbert spaces. Strong convergence results are given. As special cases, we can find the minimum-norm fixed point of the pseudocontractive mappings.

2 Preliminaries

Recall that the metric projection proj C :HC is defined by

proj C x:=arg min y C xy,xH.

It is obvious that proj C satisfies

x proj C xxy,yC,

and is characterized by

proj C xC,x proj C x,y proj C x0,yC.

The following two lemmas will be useful for our main results.

Lemma 2.1 ([24])

Let C be a closed convex subset of a Hilbert space H. Let T:CC be a Lipschitzian pseudocontractive mapping. Then Fix(T) is a closed convex subset of C and the mapping IT is demiclosed at 0, i.e., whenever { x n }C is such that x n x and (IT) x n 0, then (IT)x=0.

Lemma 2.2 ([27])

Assume { a n } is a sequence of nonnegative real numbers such that

a n + 1 (1 γ n ) a n + γ n δ n ,n0,

where { γ n } is a sequence in (0,1) and { δ n } is a sequence in R such that

  1. (i)

    n = 0 γ n =;

  2. (ii)

    lim sup n δ n 0 or n = 0 | δ n γ n |<.

Then lim n a n =0.

3 Main results

In the sequel, we assume that C is a nonempty closed convex subset of a real Hilbert space H and T:CC is a κ-Lipschitzian pseudocontractive mapping with nonempty fixed points set Fix(T).

The first result is on the convergence of the path for the pseudocontractive mappings. Now, we define our path as follows.

For fixed ζ,t(0,1) and u t H, we define a mapping G t :CC by

G t x=(1ζ) proj C [ t u t + ( 1 t ) x ] +ζTx,xC,

where proj C :HC is the metric projection from H on C.

Next, we show that the mapping G t is strongly pseudocontractive. Indeed, for x,yC, we have

G t x G t y , x y = ( 1 ζ ) proj C [ t u t + ( 1 t ) x ] proj C [ t u t + ( 1 t ) y ] , x y + ζ T x T y , x y ( 1 ζ ) proj C [ t u t + ( 1 t ) x ] proj C [ t u t + ( 1 t ) y ] x y + ζ x y 2 ( 1 ζ ) ( 1 t ) x y 2 + ζ x y 2 = [ 1 ( 1 ζ ) t ] x y 2 .

Since ζ,t(0,1), 1(1ζ)t(0,1). Hence, G t is a strongly pseudocontractive mapping. By [2], G t has a unique fixed point x t C. That is, x t satisfies

x t =(1ζ) proj C [ t u t + ( 1 t ) x t ] +ζT x t ,t(0,1).
(3.1)

Remark 3.1 u t H can be seen as a perturbation.

Next, we prove the convergence of the path (3.1).

Theorem 3.2 If lim t 0 u t =uH, then the path { x t } defined by (3.1) converges strongly to proj Fix ( T ) (u).

Proof Let pFix(T). We get from (3.1) that

x t p 2 = ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] p , x t p + ζ T x t p , x t p ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] p x t p + ζ x t p 2 ( 1 ζ ) t ( u t p ) + ( 1 t ) ( x t p ) x t p + ζ x t p 2 ( 1 ζ ) [ ( 1 t ) x t p + t u t p ] x t p + ζ x t p 2 .

It follows that

x t p u t p.

Since lim t 0 u t =uH, there exists a constant M>0 such that sup t ( 0 , 1 ) u t uM. So,

x t p u t p u t u+upM+up.

Thus, { x t } is bounded.

By (3.1), we have

x t T x t = ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] + ζ T x t T x t ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] T x t ( 1 ζ ) [ x t T x t + t u t x t ] .

Therefore,

x t T x t ( 1 ζ ) t ζ u t x t ( 1 ζ ) t ζ ( u t u + x t u ) 0 ( as  t 0 ) .
(3.2)

Let { t n }(0,1) be a sequence satisfying t n 0 + as n. Put x n := x t n . By (3.2), we get

lim n x n T x n =0.
(3.3)

By (3.1), we obtain

x t p 2 = ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] p , x t p + ζ T x t p , x t p ( 1 ζ ) proj C [ t u t + ( 1 t ) x t ] p x t p + ζ x t p 2 1 ζ 2 ( proj C [ t u t + ( 1 t ) x t ] p 2 + x t p 2 ) + ζ x t p 2 .

Hence,

x t p 2 proj C [ t u t + ( 1 t ) x t ] p 2 x t p + t ( u t x t ) 2 = x t p 2 + 2 t u t x t , x t p + t 2 u t x t 2 = x t p 2 2 t x t p , x t p + 2 t u t p , x t p + t 2 u t x t 2 = ( 1 2 t ) x t p 2 + 2 t u t p , x t p + t 2 u t x t 2 .

It follows that

x t p 2 u t p , x t p + t 2 u t x t 2 u t p , x t p + t M 1 .
(3.4)

Here M 1 >0 is a constant such that sup t ( 0 , 1 ) u t x t 2 2 M 1 . In particular, we obtain

x n p 2 u n p, x n p+ t n M 1 ,pFix(T).
(3.5)

Since { x n } is bounded, there exists a subsequence { x n i } of { x n } satisfying x n i x C weakly. By (3.3), we get

lim i x n i T x n i =0.
(3.6)

Applying Lemma 2.1 to (3.6) to deduce x Fix(T).

By (3.5), we derive

x n i x 2 u n i x , x n i x + t n i M 1 .
(3.7)

Since u n i x u x and t n i 0, we deduce that x n i x by (3.7). By (3.5), we have

x p 2 u p , x p ,pFix(T).
(3.8)

Assume that there exists another subsequence { x n j } of { x n } satisfying x n j x weakly. Similarly, we can prove that x n j x Fix(T), which satisfies

x p 2 u p , x p ,pFix(T).
(3.9)

In (3.8), we pick up p= x to get

x x 2 u x , x x .
(3.10)

In (3.9), we pick up p= x to get

x x 2 u x , x x .
(3.11)

Adding (3.10) and (3.11), we deduce

x x 2 0.

Thus, x = x . This indicates that the weak limit set of { x n } is singleton and the path { x t } converges strongly to x = proj Fix ( T ) (u) by (3.8). This completes the proof. □

Corollary 3.3 The path { x t } defined by

x t =(1ζ) proj C [ ( 1 t ) x t ] +ζT x t ,t(0,1),

converges strongly to proj Fix ( T ) (0), which is the minimum-norm fixed point of T.

Now, we introduce another algorithm, which is an explicit manner.

Algorithm 3.4 Let { ς n } and { ζ n } be two real number sequences in (0,1). Let { u n }H be a sequence. For x 0 C arbitrarily, let the sequence { x n } be generated by

x n + 1 =(1 ζ n ) proj C [ ς n u n + ( 1 ς n ) x n ] + ζ n T x n ,n0.
(3.12)

Theorem 3.5 Assume the following conditions are satisfied:

  1. (C1)

    lim n ς n = lim n ς n ζ n = lim n ζ n 2 ς n =0;

  2. (C2)

    lim n u n =uH.

Then we have

  1. (1)

    the sequence { x n } is bounded;

  2. (2)

    the sequence { x n } is asymptotically regular, that is, lim n x n + 1 x n =0.

Further, if n = 0 ς n = and lim n x n + 1 x n ζ n =0, then the sequence { x n } converges strongly to proj Fix ( T ) (u).

Proof By the condition (C1), we can find a sufficiently large positive integer m such that

1 1 1 / 2 ζ m (κ+1)(κ+2) ( ς m + 2 ζ m + ζ m 2 ς m ) >0.
(3.13)

Let pFix(T). For fixed m, we pick up a constant M 2 >0 such that

max { x 0 p , x 1 p , , x m 1 p , 4 x m p + 4 u m p } M 2 .
(3.14)

Next, we show that x m + 1 p M 2 . Set y n = proj C [ ς n u n +(1 ς n ) x n ] for all n0. Thus, we have x n + 1 =(1 ζ n ) y n + ζ n T x n for all n0.

Since IT is monotone, we have

( I T ) x m + 1 , x m + 1 p = ( I T ) x m + 1 ( I T ) p , x m + 1 p 0.

By (3.12), we obtain

x m + 1 p 2 = ( 1 ζ m ) y m p , x m + 1 p + ζ m T x m p , x m + 1 p = ( 1 ζ m ) y m ς m u m ( 1 ς m ) x m , x m + 1 p + ( 1 ζ m ) ς m u m + ( 1 ς m ) x m p , x m + 1 p + ζ m T x m p , x m + 1 p = ( 1 ζ m ) y m ς m u m ( 1 ς m ) x m , x m + 1 p + ( 1 ζ m ) x m p , x m + 1 p + ( 1 ζ m ) ς m u m x m , x m + 1 p + ζ m T x m p , x m + 1 p = ( 1 ζ m ) y m ς m u m ( 1 ς m ) x m , x m + 1 p + x m p , x m + 1 p ( 1 ζ m ) ς m x m + 1 p , x m + 1 p ( 1 ζ m ) ς m x m x m + 1 , x m + 1 p ( 1 ζ m ) ς m p u m , x m + 1 p + ζ m T x m T x m + 1 , x m + 1 p + ζ m x m + 1 x m , x m + 1 p ζ m x m + 1 T x m + 1 , x m + 1 p .

Note that

y m ς m u m ( 1 ς m ) x m y m x m + ς m x m u m = proj C [ ς m u m + ( 1 ς m ) x m ] x m + ς m x m u m 2 ς m x m u m .

Then we have

x m + 1 p 2 ( 1 ζ m ) y m ς m u m ( 1 ς m ) x m x m + 1 p + x m p x m + 1 p ( 1 ζ m ) ς m x m + 1 p 2 + ( 1 ζ m ) ς m ( x m + 1 x m + u m p ) x m + 1 p + ζ m ( T x m T x m + 1 + x m + 1 x m ) x m + 1 p 2 ( 1 ζ m ) ς m x m u m x m + 1 p + x m p x m + 1 p + ( 1 ζ m ) ς m ( x m + 1 x m + u m p ) x m + 1 p ( 1 ζ m ) ς m x m + 1 p 2 + ζ m ( κ + 1 ) x m + 1 x m x m + 1 p x m p x m + 1 p + 2 ( 1 ζ m ) ς m ( x m p + u m p ) x m + 1 p ( 1 ζ m ) ς m x m + 1 p 2 + ( ς m + ζ m ) ( κ + 1 ) x m + 1 x m x m + 1 p .

Hence,

[ 1 + ( 1 ζ m ) ς m ] x m + 1 p x m p + 2 ς m ( x m p + u m p ) + ( κ + 1 ) ( ς m + ζ m ) x m + 1 x m .
(3.15)

By (3.12), we have

x m + 1 x m ( 1 ζ m ) proj C [ ς m u m + ( 1 ς m ) x m ] x m + ζ m T x m x m ( 1 ζ m ) ς m ( x m p + u m p ) + ζ m ( T x m p + p x m ) ς m ( x m p + u m p ) + ζ m ( κ + 1 ) x m p ( κ + 1 ) ( ς m + ζ m ) x m p + ς m u m p ( κ + 2 ) ( ς m + ζ m ) M 2 .
(3.16)

From condition (C1), we deduce ς m 0 and ζ m 0 as m. Therefore, we get

lim m x m + 1 x m =0.

That is, the sequence { x m } is asymptotically regular.

By (3.15) and (3.16), we have

[ 1 + ( 1 ζ m ) ς m ] x m + 1 p x m p + ς m ( 2 x m p + 2 u m p ) + ( κ + 1 ) ( κ + 2 ) ( ς m + ζ m ) 2 M 2 ( 1 + 1 2 ς m ) M 2 + ( κ + 1 ) ( κ + 2 ) ( ς m + ζ m ) 2 M 2 .

This together with (3.13) and (3.14) imply that

x m + 1 p [ 1 ( 1 / 2 ζ m ) ς m ( κ + 1 ) ( κ + 2 ) ( ς m + ζ m ) 2 1 + ( 1 ζ m ) ς m ] M 2 = { 1 ( 1 / 2 ζ m ) ς m [ 1 1 1 / 2 ζ m ( κ + 1 ) ( κ + 2 ) ( ς m + 2 ζ m + ( ζ m 2 / ς m ) ) ] 1 + ( 1 ζ m ) ς m } M 2 M 2 .

By induction, we get

x n p M 2 ,n0.

So { x n } is bounded.

By (3.12), we have

x n T x n x n x n + 1 + x n + 1 T x n x n x n + 1 + ( 1 ζ n ) proj C [ ς n u n + ( 1 ς n ) x n ] T x n x n x n + 1 + ( 1 ζ n ) x n T x n + ς n x n u n .

It follows that

x n T x n 1 ζ n x n x n + 1 + ς n ζ n x n u n .

By the condition lim n ς n ζ n =0 and the assumption lim n x n + 1 x n ζ n =0, we deduce

lim n x n T x n =0.
(3.17)

Let the net { z t } be defined by z t =(1ζ) proj C [t u t +(1t) z t ]+ζT z t . By Theorem 3.2, we know that z t converges strongly to proj Fix ( T ) (u). Next, we prove

lim sup n proj Fix ( T ) ( u ) u n , proj Fix ( T ) ( u ) y n 0.

By the definition of { z t }, we have

z t x n =(1ζ) ( proj C [ t u t + ( 1 t ) z t ] x n ) +ζ(T z t T x n )+ζ(T x n x n ).

It follows that

z t x n 2 = ( 1 ζ ) proj C [ t u t + ( 1 t ) z t ] x n , z t x n + ζ T z t T x n , z t x n + ζ T x n x n , z t x n = ( 1 ζ ) proj C [ t u t + ( 1 t ) z t ] t u t ( 1 t ) z t , z t x n + ( 1 ζ ) t u t + ( 1 t ) z t x n , z t x n + ζ T z t T x n , z t x n + ζ T x n x n , z t x n .

Since x n C, by the characteristic inequality of metric projection, we have

proj C [ t u t + ( 1 t ) z t ] t u t ( 1 t ) z t , z t x n 0.

Then

z t x n 2 ( 1 ζ ) t u t + ( 1 t ) z t x n , z t x n + ζ z t x n 2 + ζ T x n x n z t x n = ( 1 ζ ) z t x n 2 ( 1 ζ ) t z t u t , z t x n + ζ z t x n 2 + ζ T x n x n z t x n ,

which implies that

z t u t , z t x n ζ ( 1 ζ ) t T x n x n z t x n .

By (3.17), we deduce

lim sup t 0 lim sup n z t u t , z t x n 0.
(3.18)

Note the fact that the two limits lim sup t 0 and lim sup n are interchangeable. This together with z t proj Fix ( T ) (u), u t u and (3.18) implies that

lim sup n proj Fix ( T ) ( u ) u , proj Fix ( T ) ( u ) x n 0.

Note that y n x n 0 and u n u0. We derive

lim sup n proj Fix ( T ) ( u ) u n , proj Fix ( T ) ( u ) y n 0.

Finally, we prove that x n proj Fix ( T ) (u). Note that

T x n proj Fix ( T ) ( u ) , x n + 1 proj Fix ( T ) ( u ) = T x n proj Fix ( T ) ( u ) , x n proj Fix ( T ) ( u ) + T x n proj Fix ( T ) ( u ) , x n + 1 x n x n proj Fix ( T ) ( u ) 2 + T x n proj Fix ( T ) ( u ) x n + 1 x n
(3.19)

and

y n proj Fix ( T ) ( u ) 2 = y n ς n u n ( 1 ς n ) x n , y n proj Fix ( T ) ( u ) + ς n u n + ( 1 ς n ) x n proj Fix ( T ) ( u ) , y n proj Fix ( T ) ( u ) ς n u n + ( 1 ς n ) x n proj Fix ( T ) ( u ) , y n proj Fix ( T ) ( u ) = ( 1 ς n ) x n proj Fix ( T ) ( u ) , y n proj Fix ( T ) ( u ) ς n proj Fix ( T ) ( u ) u n , y n proj Fix ( T ) ( u ) ( 1 ς n ) 2 x n proj Fix ( T ) ( u ) 2 + 1 2 y n proj Fix ( T ) ( u ) 2 ς n proj Fix ( T ) ( u ) u n , y n proj Fix ( T ) ( u ) .

Then

y n proj Fix ( T ) ( u ) 2 ( 1 ς n ) x n proj Fix ( T ) ( u ) 2 2 ς n proj Fix ( T ) ( u ) u n , y n proj Fix ( T ) ( u ) .
(3.20)

By (3.12), (3.16), and (3.20), we get

x n + 1 proj Fix ( T ) ( u ) 2 = ( 1 ζ n ) ( y n proj Fix ( T ) ( u ) ) + ζ n ( T x n proj Fix ( T ) ( u ) ) 2 ( 1 ζ n ) ( y n proj Fix ( T ) ( u ) ) 2 + 2 ζ n T x n proj Fix ( T ) ( u ) , x n + 1 proj Fix ( T ) ( u ) ( 1 ζ n ) 2 ( 1 ς n ) x n proj Fix ( T ) ( u ) 2 + 2 ζ n x n proj Fix ( T ) ( u ) 2 2 ς n ( 1 ζ n ) 2 proj Fix ( T ) ( u ) u n , y n proj Fix ( T ) ( u ) + 2 ζ n T x n proj Fix ( T ) ( u ) x n + 1 x n [ 1 ( 1 2 ζ n ) ς n ] x n proj Fix ( T ) ( u ) 2 + ζ n 2 x n proj Fix ( T ) ( u ) 2 + 2 ς n ( 1 ζ n ) 2 proj Fix ( T ) ( u ) u n , proj Fix ( T ) ( u ) y n + 2 ζ n T x n proj Fix ( T ) ( u ) ( κ + 2 ) ( ς n + ζ n ) M 2 = ( 1 γ n ) x n proj Fix ( T ) ( u ) 2 + γ n δ n ,
(3.21)

where γ n =(12 ζ n ) ς n and

δ n = 2 ( 1 ζ n ) 2 1 2 ζ n proj Fix ( T ) ( u ) u n , proj Fix ( T ) ( u ) y n + ζ n 2 ( 1 2 ζ n ) ς n x n proj Fix ( T ) ( u ) 2 + 2 ζ n 1 2 ζ n T x n proj Fix ( T ) ( u ) ( κ + 2 ) M 2 + 2 ζ n 2 ( 1 2 ζ n ) ς n T x n proj Fix ( T ) ( u ) ( κ + 2 ) M 2 .

It is clear that n = 0 γ n = and lim sup n δ n 0. We can therefore apply Lemma 2.2 to (3.21) and conclude that x n proj Fix ( T ) (u) as n. This completes the proof. □

Corollary 3.6 Let { ς n } and { ζ n } be two real number sequences in (0,1). For x 0 C arbitrarily, let the sequence { x n } be generated by

x n + 1 =(1 ζ n ) proj C [ ( 1 ς n ) x n ] + ζ n T x n ,n0.
(3.22)

Assume lim n ς n = lim n ς n ζ n = lim n ζ n 2 ς n =0. Then we have

  1. (1)

    the sequence { x n } is bounded;

  2. (2)

    the sequence { x n } is asymptotically regular, that is, lim n x n + 1 x n =0.

Further, if n = 0 ς n = and lim n x n + 1 x n ζ n =0, then the sequence { x n } converges strongly to proj Fix ( T ) (0), which is the minimum-norm fixed point of T.

Proof Letting u n =u=0 in (3.12), we obtain (3.22). Consequently, by Theorem 3.5, we find that the sequence { x n } generated by (3.22) converges strongly to proj Fix ( T ) (0), which is the minimum-norm fixed point of T. □

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Acknowledgements

The authors are very grateful to the reviewers for their valuable comments and suggestions.

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Correspondence to Shin Min Kang or Chahn Yong Jung.

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Guo, L., Kang, S.M. & Jung, C.Y. Algorithms with variant anchors for pseudocontractive mappings. J Inequal Appl 2014, 386 (2014). https://doi.org/10.1186/1029-242X-2014-386

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Keywords

  • algorithms
  • pseudocontractive mapping
  • fixed point
  • strong convergence