Skip to main content

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. □

References

  1. Ceng LC, Petruşel A, Yao JC: Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings. Appl. Math. Comput. 2009, 209: 162-176. 10.1016/j.amc.2008.10.062

    Article  MathSciNet  MATH  Google Scholar 

  2. Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudo-contractions. Proc. Am. Math. Soc. 2001, 129: 2359-2363. 10.1090/S0002-9939-01-06009-9

    Article  MathSciNet  MATH  Google Scholar 

  3. Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudo-contractive maps. Proc. Am. Math. Soc. 2004, 132: 831-840. 10.1090/S0002-9939-03-07101-6

    Article  MathSciNet  MATH  Google Scholar 

  4. Cho SY, Qin X, Kang SM: Hybrid projection algorithms for treating common fixed points of a family of demicontinuous pseudocontractions. Appl. Math. Lett. 2012, 25: 854-857. 10.1016/j.aml.2011.10.031

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Kang SM, Cho SY, Qin X: Hybrid projection algorithms for approximating fixed points of asymptotically quasi-pseudocontractive mappings. J. Nonlinear Sci. Appl. 2012, 5: 466-474.

    MathSciNet  MATH  Google Scholar 

  7. Kang SM, Rafiq A: On convergence results for Lipschitz pseudocontractive mappings. J. Appl. Math. 2012., 2012: Article ID 902601 10.1155/2012/902601

    Google Scholar 

  8. Kang SM, Rafiq A, Lee S: Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 132 10.1186/1029-242X-2013-132

    Google Scholar 

  9. Li XS, Kim JK, Huang NJ: Viscosity approximation of common fixed points for L -Lipschitzian semigroup of pseudocontractive mappings in Banach spaces. J. Inequal. Appl. 2009., 2009: Article ID 936121 10.1155/2009/936121

    Google Scholar 

  10. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506-510. 10.1090/S0002-9939-1953-0054846-3

    Article  MathSciNet  MATH  Google Scholar 

  11. Morales CH, Jung JS: Convergence of paths for pseudocontractive mappings in Banach spaces. Proc. Am. Math. Soc. 2000, 128: 3411-3419. 10.1090/S0002-9939-00-05573-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Ofoedu EU, Zegeye H: Further investigation on iteration processes for pseudocontractive mappings with application. Nonlinear Anal. 2012, 75: 153-162. 10.1016/j.na.2011.08.015

    Article  MathSciNet  MATH  Google Scholar 

  13. Qin X, Cho YJ, Kang SM, Zhou H: Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions. Nonlinear Anal. 2009, 71: 685-690. 10.1016/j.na.2008.10.102

    Article  MathSciNet  MATH  Google Scholar 

  14. Song YS, Chen R: An approximation method for continuous pseudocontractive mappings. J. Inequal. Appl. 2006., 2006: Article ID 28950 10.1155/JIA/2006/28950

    Google Scholar 

  15. Udomene A: Path convergence, approximation of fixed points and variational solutions of Lipschitz pseudocontractions in Banach spaces. Nonlinear Anal. 2007, 67: 2403-2414. 10.1016/j.na.2006.09.001

    Article  MathSciNet  MATH  Google Scholar 

  16. Wen DJ, Chen YA: General iterative method for generalized equilibrium problems and fixed point problems of k -strict pseudo-contractions. Fixed Point Theory Appl. 2012., 2012: Article ID 125 10.1186/1687-1812-2012-125

    Google Scholar 

  17. Yao Y, Colao V, Marino G, Xu HK: Implicit and explicit algorithms for minimum-norm fixed points of pseudocontractions in Hilbert spaces. Taiwan. J. Math. 2012, 16: 1489-1506.

    MathSciNet  MATH  Google Scholar 

  18. Yao Y, Liou YC: Strong convergence of an implicit iteration algorithm for a finite family of pseudocontractive mappings. J. Inequal. Appl. 2008., 2008: Article ID 280908 10.1155/2008/280908

    Google Scholar 

  19. Yao Y, Liou YC, Chen R: Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces. Nonlinear Anal. 2007, 67: 3311-3317. 10.1016/j.na.2006.10.013

    Article  MathSciNet  MATH  Google Scholar 

  20. Yao Y, Liou YC, Marino G: A hybrid algorithm for pseudo-contractive mappings. Nonlinear Anal. 2009, 71: 4997-5002. 10.1016/j.na.2009.03.075

    Article  MathSciNet  MATH  Google Scholar 

  21. Yao Y, Marino G, Xu HK, Liou YC: Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces. J. Inequal. Appl. 2014., 2014: Article ID 206 10.1186/1029-242X-2014-206

    Google Scholar 

  22. Zegeye H, Shahzad N, Alghamdi MA: Convergence of Ishikawa’s iteration method for pseudocontractive mappings. Nonlinear Anal. 2011, 74: 7304-7311. 10.1016/j.na.2011.07.048

    Article  MathSciNet  MATH  Google Scholar 

  23. Zegeye H, Shahzad N, Alghamdi MA: Minimum-norm fixed point of pseudocontractive mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 926017 10.1155/2012/926017

    Google Scholar 

  24. Zhou H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear Anal. 2009, 70: 4039-4046. 10.1016/j.na.2008.08.012

    Article  MathSciNet  MATH  Google Scholar 

  25. Yao Y, Shahzad N: New methods with perturbations for non-expansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 79 10.1186/1687-1812-2011-79

    Google Scholar 

  26. Yu ZT, Lin LJ, Chuang CS: Mathematical programming with multiple sets split monotone variational inclusion constraints. Fixed Point Theory Appl. 2014., 2014: Article ID 20 10.1186/1687-1812-2014-20

    Google Scholar 

  27. Xu HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2002, 66: 240-256. 10.1112/S0024610702003332

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Shin Min Kang or Chahn Yong Jung.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-386

Keywords