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Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space, and let T:CC be an asymptotically k-strictly pseudocontractive mapping with F(T)={xC:Tx=x}. Let { α n } n = 1 and { t n } n = 1 be real sequences in (0,1). Let { x n } n = 1 be the sequence generated from an arbitrary x 1 C by

{ ν n = P C ( ( 1 t n ) x n ) , n 1 , x n + 1 = ( 1 α n ) ν n + α n T n ν n , n 1 ,

where P C :HC is the metric projection. Under some appropriate mild conditions on { α n } n = 1 and { t n } n = 1 , we prove that { x n } n = 1 converges strongly to a fixed point of T. Furthermore, if T:CC is uniformly L-Lipschitzian and asymptotically pseudocontractive with F(T), we first prove that (IT) is demiclosed at 0, and then prove that under some suitable conditions on the real sequences { α n } n = 1 , { β n } n = 1 and { t n } n = 1 in (0,1), the sequence { x n } n = 1 generated from an arbitrary x 1 C by

{ ν n = P C ( ( 1 t n ) x n ) , n 1 , y n = ( 1 β n ) ν n + β n T n ν n , n 1 , x n + 1 = ( 1 α n ) ν n + α n T n y n , n 1 ,

converges strongly to a fixed point of T. No compactness assumption is imposed on T or C and no further requirement is imposed on F(T).

MSC:47H09, 47J25, 65J15.

1 Introduction

Let H be a real Hilbert space with the inner product , and the induced norm . Let C be a nonempty closed convex subset of H. A mapping T:CC is said to be L-Lipschitzian if there exists L0 such that

TxTyLxy,x,yC.
(1.1)

T is said to be a contraction if L[0,1), and T is said to be nonexpansive if L=1. T is said to be asymptotically nonexpansive (see, for example, [1]) if there exists a sequence { k n } n = 1 [1,) with lim n k n =1 such that

T n x T n y k n xy,x,yC.
(1.2)

It is well known (see, for example, [1]) that the class of nonexpansive mappings is a proper subclass of the class of asymptotically nonexpansive mappings. T is said to be asymptotically k-strictly pseudocontractive (see, for example, [2]) if there exist k[0,1) and a sequence { k n } n = 1 [1,), lim n k n =1 such that

T n x T n y 2 k n x y 2 +k ( x T n x ) ( y T n y ) 2 ,x,yC.
(1.3)

T is said to be asymptotically pseudocontractive if there exists a sequence { k n } n = 1 [1,), lim n k n =1 such that

T n x T n y 2 k n x y 2 + ( x T n x ) ( y T n y ) 2 ,x,yC.
(1.4)

It is well known that in real Hilbert spaces, the class of asymptotically nonexpansive maps is a proper subclass of the class of asymptotically k-strictly pseudocontractive maps. Furthermore, the class of asymptotically k-strictly pseudocontractive mappings is a proper subclass of the class of asymptotically pseudocontractive maps. T is said to be uniformly L-Lipschitzian if there exists L0 such that

T n x T n y Lxy,x,yC.

T is said to be demiclosed at p if whenever { x n } n = 1 is a sequence in C which converges weakly to x C and { T x n } n = 1 converges strongly to p, then T x =p. It is well known that if T:CC is asymptotically k-strictly pseudocontractive, then T is uniformly L-Lipschitzian (see, for example, [3, 4]), and (IT) is demiclosed at 0 (see, for example, [5]). The modified Mann iteration scheme { x n } n = 1 generated from an arbitrary x 1 C by

x n + 1 =(1 α n ) x n + α n T n x n ,n1,
(1.5)

where the control sequence { α n } n = 1 is a real sequence in (0,1) satisfying some appropriate conditions, has been used by several authors for the approximation of fixed points of asymptotically k-strictly pseudocontractive maps (see, for example, [210]). The iteration algorithm (1.5) is a modification of the well-known Mann iterative algorithm (see [11]) generated from an arbitrary x 1 C by

x n + 1 =(1 α n ) x n + α n T x n ,n1,
(1.6)

where the control sequence { α n } n = 1 is a real sequence in (0,1) satisfying some appropriate conditions.

In real Hilbert spaces, it is known (see, for example, [35]) that if C is a nonempty closed convex subset of a real Hilbert space H, and T:CC is an asymptotically k-strictly pseudocontractive mapping with a sequence { k n } n = 1 [1,), n = 1 ( k n 1)<, and a nonempty fixed point set F(T), then the modified iteration sequence { x n } generated by (1.5) is an approximate fixed point sequence (i.e., x n T x n 0 as n) if α n [a,b](0,1k). This together with the demiclosedness property of (IT) at 0 yields that { x n } converges weakly to a fixed point of T.

To obtain strong convergence of the modified Mann algorithm (1.5) to a fixed point of an asymptotically k-strictly pseudocontractive mapping, additional conditions are usually required on T and on the subset C (see, for example, [210]). Even for nonexpansive maps, additional conditions are required on T or C to obtain strong convergence using the Mann algorithm (1.6). In [12], Genel and Lindenstraus provided an example of a nonexpansive mapping defined on a bounded closed convex subset of a Hilbert space for which the Mann iteration does not converge to a fixed point of T. Recently Yao et al. [13] (see also [14, 15]) studied a modified Mann iteration algorithm { x n } generated from an arbitrary x 1 H by

{ ν n = ( 1 t n ) x n , x n + 1 = ( 1 α n ) ν n + α n T ν n ,
(1.7)

where { t n } and { α n } are real sequences in (0,1) satisfying some appropriate conditions. They proved strong convergence of the modified algorithm to a fixed point of a nonexpansive mapping T:HH when F(T). Clearly, the modified Mann iteration algorithm reduces to the normal Mann iteration algorithm when t n 0.

It is our purpose in this paper to modify algorithm (1.7) and prove that the modified algorithm converges strongly to a fixed point of an asymptotically k-strictly pseudocontractive mapping T:CC, where C is a nonempty closed convex subset of a real Hilbert space and F(T). Furthermore, we prove that if T:CC is a uniformly L-Lipschitzian asymptotically pseudocontractive mapping, then (IT) is demiclosed at 0. We then introduce an iterative algorithm which converges strongly to a fixed point of a uniformly L-Lipschitzian asymptotically pseudocontractive mapping T:CC with F(T). The technique of proof of our convergence theorems follows the one proposed by Maingé [14].

2 Preliminaries

In what follows, we shall need the following results.

Lemma 2.1 [16]

Let { a n } n = 1 be a sequence of nonnegative real numbers such that

a n + 1 (1 λ n ) a n + λ n γ n + σ n ,n1,

where { λ n }(0,1), { γ n }, { σ n } is a sequence of nonnegative real numbers and

  1. (i)

    n = 0 λ n =, or equivalently, n = 0 (1 λ n )=0,

  2. (ii)

    lim sup n γ n 0, and

  3. (iii)

    n = 0 σ n <.

Then lim n a n =0.

Let C be a closed convex subset of a real Hilbert space H. Let P C :HC denote the metric projection (the proximity map) which assigns to each point xH the unique nearest point in C, denoted by P C (x). It is well known that

z= P C (x)if and only ifxz,zy0,yC,
(2.1)

and that P C is nonexpansive.

It is also well known that in real Hilbert spaces H, we have the following (see, for example, [17]):

(i) x + y 2 y 2 +2x,x+y,x,yH;
(2.2)
(ii) α x + ( 1 α ) y 2 = α x 2 + ( 1 α ) y 2 α ( 1 α ) x y 2 , x , y H  and  α [ 0 , 1 ] ;
(2.3)
(iii) if  { x n } n = 1  is a sequence in  H  which converges weakly to  z ,  then lim sup n x n y 2 = lim sup n x n z 2 + z y 2 , y H .
(2.4)

3 Main results

3.1 Strong convergence of an iterative algorithm for asymptotically k-strictly pseudocontractive maps

We now introduce the following iterative algorithm analogous to one studied in [13].

Modified averaging Mann algorithm Let C be a nonempty closed convex subset of a real Hilbert space H, and let T:CC be a given mapping. For arbitrary x 1 C, our iteration sequence { x n } is given by

{ ν n = P C ( ( 1 t n ) x n ) , x n + 1 = ( 1 α n ) ν n + α n T n ν n ,
(3.1)

where { t n } and { α n } are suitable real sequences in (0,1) satisfying some appropriate conditions that will be made precise in our strong convergence theorem.

We now prove the following convergence theorem.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space, and let T:CC be an asymptotically k-strictly pseudocontractive mapping with a sequence { k n } n = 1 [1,) such that n = 1 ( k n 1)<. Let F(T), and let { t n } n = 1 and { α n } n = 1 be sequences in (0,1) satisfying the conditions:

  1. (c1)

    lim n t n =0;

  2. (c2)

    n = 1 t n =;

  3. (c3)

    lim n 1 t n ( k n 1)=0;

  4. (c4)

    0<ϵ α n < 1 2 (1 t n )(1k), n1 and for some ϵ.

Then the modified averaging iteration sequence { x n } n = 1 generated from x 1 C by (3.1) converges strongly to a fixed point of T.

Proof Observe that (1.3) is equivalent to each of the following inequalities:

2 ( I T n ) x ( I T n ) y , x y ( 1 k ) ( I T n ) x ( I T n ) y 2 ( k n 1 ) x y 2 ,
(3.2)
2 T n x T n y , x y ( k n +1) x y 2 (1k) ( I T n ) x ( I T n ) y 2 .
(3.3)

Let pF(T) be arbitrary. Then, using (1.3), (2.3), (3.1) and (3.3), we obtain

x n + 1 p 2 = ( 1 α n ) ( ν n p ) + α n ( T n ν n p ) 2 = ( 1 α n ) ν n p 2 + α n T n ν n p 2 α n ( 1 α n ) ν n T n ν n 2 [ 1 + α n ( k n 1 ) ] ν n p 2 α n ( 1 α n k ) ν n T n ν n 2 .
(3.4)

Hence

x n + 1 p [ 1 + α n ( k n 1 ) ] ν n p = [ 1 + α n ( k n 1 ) ] P C ( ( 1 t n ) x n ) p [ 1 + α n ( k n 1 ) ] ( 1 t n ) x n p = [ 1 + α n ( k n 1 ) ] ( 1 t n ) ( x n p ) t n p [ 1 + α n ( k n 1 ) ] [ ( 1 t n ) x n p + t n p ] [ 1 + α n ( k n 1 ) ] max { x n p , p } j = 1 n [ 1 + α j ( k j 1 ) ] max { x 1 p , p } .

Since n = 1 ( k n 1)<, it follows from (3.4) that { x n } n = 1 is bounded. Hence { ν n } n = 1 is also bounded. Furthermore, it follows from (2.2) that

x n x n + 1 2 = x n ν n + ν n x n + 1 2 ν n x n + 1 2 + 2 x n ν n , x n x n + 1 ν n x n + 1 2 + 2 x n ν n x n x n + 1 ν n x n + 1 2 + 2 t n x n x n x n + 1 .
(3.5)

From (3.1) and (3.5) we obtain

ν n T n ν n 2 = 1 α n 2 ν n x n + 1 2 1 α n 2 [ x n x n + 1 2 2 t n x n x n x n + 1 ] .
(3.6)

Since { ν n } n = 1 is bounded, then

ν n p 2 D,n1 and for some D>0,

and hence using condition (c4) and (3.6) in (3.4), we obtain

x n + 1 p 2 [ 1 + α n ( k n 1 ) ] ν n p 2 α n ( 1 α n k ) ν n T n ν n 2 ν n p 2 ( 1 α n k ) α n [ x n x n + 1 2 2 t n x n x n x n + 1 ] + α n ( k n 1 ) D ( 1 t n ) x n p 2 ( 1 α n k ) α n [ x n x n + 1 2 2 t n x n x n x n + 1 ] + α n ( k n 1 ) D x n p 2 2 t n x n , x n p + t n 2 x n 2 σ 1 x n x n + 1 2 + 2 σ 2 t n x n x n x n + 1 + α n ( k n 1 ) D ( where  σ 1 : = 1 2 ( 1 k ) > 0 ; σ 2 = 1 ϵ ) = x n p 2 σ 1 x n x n + 1 2 + t n [ t n x n 2 + 2 σ 2 x n x n x n + 1 2 x n , x n p ] + α n ( k n 1 ) D .
(3.7)

Since { x n } n = 1 is bounded, we have that there exists M>0 such that

t n x n 2 +2 σ 2 x n x n x n + 1 2 x n , x n pM,n1.
(3.8)

From (3.7) and (3.8) we obtain

x n + 1 p 2 x n p 2 + σ 1 x n x n + 1 2 M t n + α n ( k n 1)D.
(3.9)

To complete the proof, we now consider the following two cases.

Case 1. Suppose that { x n p } n = 1 is a monotone sequence, then we may assume that { x n p} is monotone decreasing. Then lim n x n p exists and it follows from (3.9), conditions (c1) and lim n k n =1 that

lim n x n x n + 1 =0.
(3.10)

Furthermore,

ν n x n t n x n 0 as  n ,  and ν n x n + 1 ν n x n + x n x n + 1 0 as  n .

Hence

ν n T n ν n 1 α n ν n x n + 1 1 ϵ ν n x n + 1 0as n,

and

x n T n x n x n ν n + ν n T n ν n + T n ν n T n x n ( 1 + k n ) x n ν n + ν n T n ν n 0 as  n .

Observe also that since T is uniformly L-Lipschitzian, we obtain

ν n T ν n ν n T n ν n + T n ν n T ν n ν n T n ν n + L T n 1 ν n ν n ν n T n ν n + L T n 1 ν n T n 1 ν n 1 + L T n 1 ν n 1 ν n 1 + L ν n 1 ν n ν n T n ν n + L T n 1 ν n 1 ν n 1 + L ( 1 + L ) ν n ν n 1 ν n T n ν n + L T n 1 ν n 1 ν n 1 + L ( 1 + L ) [ ν n x n + x n x n 1 + x n 1 ν n 1 ] 0 as  n .
(3.11)

Furthermore,

x n T x n x n T n x n + T n x n T x n x n T n x n + L T n 1 x n x n x n T n x n + L T n 1 x n T n 1 x n 1 + L T n 1 x n 1 x n 1 + L x n 1 x n x n T n x n + L T n 1 x n 1 x n 1 + L ( 1 + L ) x n x n 1 0 as  n .
(3.12)

Since lim n x n T x n = lim n ν n T ν n = lim n ν n x n =0, then the demiclosedness property of (IT), (2.4) and the usual standard argument yield that { x n } n = 1 and { ν n } n = 1 converge weakly to some x F(T).

Since α n (1 α n k) 1 ϵ (1k)>0, and since ν n x 2 D 2 , n1, and for some D 2 >0, then using (3.4) we obtain

x n + 1 x 2 ν n x 2 + α n ( k n 1 ) D 2 ( 1 t n ) ( x n x ) t n x 2 + α n ( k n 1 ) D 2 = ( 1 t n ) 2 x n x 2 2 t n ( 1 t n ) x n x , x + t n 2 x 2 + α n ( k n 1 ) D 2 ( 1 t n ) x n x 2 2 t n ( 1 t n ) x n x , x + t n 2 x 2 + α n ( k n 1 ) D 2 .
(3.13)

Thus

x n + 1 x 2 (1 t n ) x n x 2 + t n γ n + σ n ,n1,

where γ n :=2(1 t n ) x n x , x + t n x 2 , and σ n = α n ( k n 1) D 2 , with n = 1 σ n <. Since { x n } n = 1 converges weakly to x , then lim n x n x , x =0, and this together with condition (c1) (i.e., lim n t n =0) implies that γ n :=2(1 t n ) x n x , x + t n x 2 0 as n. It now follows from Lemma 2.1 that { x n } n = 1 converges strongly to x . Consequently, { ν n } n = 1 converges strongly to x .

Case 2. Suppose that { x n p } n = 1 is not a monotone decreasing sequence, then set Γ n := x n p 2 , and let τ:NN be a mapping defined for all n N 0 for some sufficiently large N 0 by

τ(n):=max{kN:kn, Γ k Γ k + 1 }.

Then τ is a non-decreasing sequence such that τ(n) as n and Γ τ ( n ) Γ τ ( n ) + 1 for n N 0 . Using (c1) and (c2) in (3.9), we obtain

x τ ( n ) + 1 x τ ( n ) 2 1 σ 1 [ M t τ ( n ) + α τ ( n ) ( k τ ( n ) 1 ) D ] 0as n.
(3.14)

Following the same argument as in Case 1, we obtain

ν τ ( n ) T ν τ ( n ) 0as nand x τ ( n ) T x τ ( n ) 0as n.

As in Case 1, we also obtain that { x τ ( n ) } and { ν τ ( n ) } converge weakly to some x in F(T). Furthermore, for all n N 0 , we obtain from (3.13) that

0 x τ ( n ) + 1 x 2 x τ ( n ) x 2 t τ ( n ) [ 2 ( 1 t τ ( n ) ) x τ ( n ) x , x + t τ ( n ) x 2 + D 2 α τ ( n ) ( k τ ( n ) 1 ) t τ ( n ) x τ ( n ) x 2 ] .
(3.15)

It follows from (3.15) that

x τ ( n ) x 2 2 ( 1 t τ ( n ) ) x x τ ( n ) , x + t τ ( n ) x 2 + D 2 α τ ( n ) ( k τ ( n ) 1 ) t τ ( n ) 0 as  n .

Thus

lim n Γ τ ( n ) = lim n Γ τ ( n ) + 1 .

Furthermore, for n N 0 , we have Γ n Γ τ ( n ) + 1 if nτ(n) (i.e., τ(n)<n), because Γ j > Γ j + 1 for τ(n)+1jn. It then follows that for all n N 0 we have

0 Γ n max{ Γ τ ( n ) , Γ τ ( n ) + 1 }= Γ τ ( n ) + 1 .

This implies lim n Γ n =0, and hence { x n } n = 1 converges strongly to x F(T). □

Corollary 3.1 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H with 0C. Let T:CC be an asymptotically k-strictly pseudocontractive mapping with a sequence { k n } n = 1 [1,) such that n = 1 ( k n 1)<. Let F(T), and let { t n } n = 1 and { α n } n = 1 be sequences in (0,1) satisfying the conditions:

  1. (c1)

    lim n t n =0;

  2. (c2)

    n = 1 t n =;

  3. (c3)

    lim n 1 t n ( k n 1)=0;

  4. (c4)

    0<ϵ α n < 1 2 (1 t n )(1k), n1 and for some ϵ.

Then the modified averaging iteration sequence { x n } n = 1 , generated from x 1 C by

{ ν n : = ( 1 t n ) x n , x n + 1 : = ( 1 α n ) ν n + α n T n ν n ,

converges strongly to a fixed point of T.

Remark 3.1 Prototypes for our real sequences { t n } n = 1 and { α n } n = 1 are:

t n := k n 1 + 1 n + 1 ,n1; α n := n 2 ( n + 1 ) (1k)(1 t n ),n1.

Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space, and let T:CC be an asymptotically nonexpansive mapping with a sequence { k n } n = 1 [1,). Let F(T), and let { t n } n = 1 and { α n } n = 1 be sequences in (0,1) satisfying the conditions:

  1. (c1)

    lim n t n =0;

  2. (c2)

    n = 1 t n =;

  3. (c3)

    0<ϵ α n < 1 2 (1 t n ), n1 and for some ϵ.

Then the modified averaging iteration sequence { x n } n = 1 , generated from x 1 C by (3.1), converges strongly to a fixed point of T.

3.2 Demiclosedness principle and strong convergence results for uniformly Lipschitzian asymptotically pseudocontractive maps

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. For uniformly L-Lipschitzian asymptotically pseudocontractive maps T:CC, we first prove that (IT) is demiclosed at 0 and then introduce a modified averaging Ishikawa iteration algorithm and prove that it converges strongly to a fixed point of T:CC without any compactness assumption on T or C and without further requirement on F(T). Our demiclosedness principle does not require the boundedness of C imposed in the result of [18].

Theorem 3.2 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let T:CC be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping. Then (IT) is demiclosed at 0.

Proof Let { x n } n = 1 be a sequence in C which converges weakly to p and { x n T x n } n = 1 converges strongly to 0. We prove that pF(T). Since { x n } n = 1 converges weakly, it is bounded. For each xH, define f:H[0,) by

f(x):= lim sup n x n x 2 .
(3.16)

Observe that for arbitrary but fixed integer m1, we have

x n T m x n x n T x n + T x n T 2 x n + + T m 1 x n T m x n m L x n T x n 0 as  n .

Set

G m x:= T m ( ( 1 β ) x + β T m x ) ,

where β(0, 2 ( 1 + λ ) + ( 1 + λ ) 2 + 4 L 2 ), and λ:= sup n 1 k n . Then

( 1 β ) x n + β T m x n T m x n =(1β) x n T m x n 0as n,

and

T m x n G m x n = T m x n T m ( ( 1 β ) x n + β T m x n ) L β x n T m x n 0 as  n .

Hence

( 1 β ) x n + β T m x n G m x n ( 1 β ) x n + β T m x n T m x n + T m x n G m x n 0 as  n .

Also

x n G m x n x n T m x n + T m x n G m x n 0as n.

From (2.4) we obtain

f(x)= lim sup n x n p 2 + p x 2 ,xH.

Thus

f(x)=f(p)+ p x 2 ,xH,

and hence

f( G m p)=f(p)+ p G m p 2 .
(3.17)

Observe that

f ( G m p ) = lim sup n x n G m p 2 = lim sup n x n G m x n + G m x n G m p 2 = lim sup n G m x n G m p 2 = lim sup n T m ( ( 1 β ) x n + β T m x n ) T m ( ( 1 β ) p + β T m p ) 2 lim sup n [ k m ( 1 β ) x n + β T m x n ( ( 1 β ) p + β T m p ) 2 + ( 1 β ) x n + β T m x n G m x n ( ( 1 β ) p + β T m p G m p ) 2 ] = lim sup n [ k m ( 1 β ) ( x n p ) + β ( T m x n T m p ) 2 + ( 1 β ) ( p G m p ) + β ( T m p G m p ) 2 ] = lim sup n [ k m ( ( 1 β ) x n p 2 + β T m x n T m p 2 β ( 1 β ) x n T m x n ( p T m p ) 2 ) + ( 1 β ) p G m p 2 + β T m p G m p 2 β ( 1 β ) p T m p 2 ] lim sup n [ k m ( 1 β + k m β ) x n p 2 + k m β x n T m x n ( p T m p ) 2 k m β ( 1 β ) x n T m x n ( p T m p ) 2 + ( 1 β ) p G m p 2 + β 3 L 2 p T m p 2 β ( 1 β ) p T m p 2 ] = lim sup n [ k m ( 1 + β ( k m 1 ) ) x n p 2 + ( 1 β ) p G m p 2 β [ 1 β ( 1 + k m ) β 2 L 2 ] p T m p 2 ] k m ( 1 + β ( k m 1 ) ) f ( p ) + ( 1 β ) p G m p 2 .
(3.18)

Equations (3.17) and (3.18) imply that

f(p)+ p G m p 2 k m ( 1 + β ( k m 1 ) ) f(p)+(1β) p G m p 2 ,

from which it follows that

β p G m p 2 [ k m ( 1 + β ( k m 1 ) ) 1 ] f(p)0as m.

Thus

p G m p 0 as  m , p T m p p G m p + G m p T m p p G m p + L ( 1 β ) p + β T m p p = p G m p + L β p T m p .

Hence

(1Lβ) p T m p p G m p0as m.

It now follows that T m pp as m. Since T is continuous, we have that T m + 1 pTp as m, and hence Tp=p. □

We now introduce the following iterative algorithm for uniformly L-Lipschitzian asymptotically pseudocontractive maps.

Modified averaging Ishikawa algorithm For arbitrary x 1 C, the sequence { x n } n = 1 is given by

{ ν n = P C ( ( 1 t n ) x n ) , n 1 , y n = ( 1 β n ) ν n + β n T n ν n , n 1 , x n + 1 = ( 1 α n ) ν n + α n T n y n , n 1 .
(3.19)

Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let T:CC be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence { k n } n = 1 [1,), n = 1 ( k n 1)< and F(T). Let { t n } n = 1 , { α n } n = 1 and { β n } n = 1 be real sequences in (0,1) satisfying the conditions:

  1. (c1)

    lim n t n =0;

  2. (c2)

    n = 1 t n =;

  3. (c3)

    0<ϵ α n (1 t n ) β n β n b< 2 ( 1 + λ ) + ( 1 + λ ) 2 + 4 L 2 , where λ= sup n k n ;

  4. (c4)

    lim n ( k n 1 ) t n =0.

Then the sequence { x n } n = 1 generated from an arbitrary x 1 C by (3.19) converges strongly to a fixed point of T.

Proof Since T is asymptotically pseudocontractive, it follows that

2 ( I T n ) x ( I T n ) y , x y ( k n 1) x y 2
(3.20)

and

2 T n x T n y , x y ( k n +1) x y 2 .
(3.21)

Set

G n ν n = T n ( ( 1 β n ) ν n + β n T n ν n ) ,n1.

Then, for arbitrary pF(T), we obtain

G n ν n p 2 = T n ( ( 1 β n ) ν n + β n T n ν n ) T n p 2 k n ( 1 β n ) ( ν n p ) + β n ( T n ν n p ) 2 + ( 1 β n ) ν n + β n T n ν n G n ν n 2 = k n ( 1 β n ) ν n p 2 + k n β n T n ν n p 2 k n β n ( 1 β n ) ν n T n ν n 2 + ( 1 β n ) ( ν n G n ν n ) + β n ( T n ν n G n ν n ) 2 = [ k n ( 1 β n ) + k n 2 β n ] ν n p 2 + k n β n ν n T n ν n 2 k n ( 1 β n ) β n ν n T n ν n 2 + ( 1 β n ) ν n G n ν n 2 + β n T n ν n G n ν n 2 β n ( 1 β n ) ν n T n ν n 2 [ 1 + ( k n 2 1 ) ] ν n p 2 + k n β n ν n T n ν n 2 k n ( 1 β n ) β n ν n T n ν n 2 + ( 1 β n ) ν n G n ν n 2 + β n 3 L 2 ν n T n ν n 2 β n ( 1 β n ) ν n T n ν n 2 = [ 1 + ( k n 2 1 ) ] ν n p 2 + ( 1 β n ) ν n G n ν n 2 β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T n ν n 2 .

Thus

G n ν n p 2 [ 1 + ( k n 2 1 ) ] ν n p 2 + ( 1 β n ) ν n G n ν n 2 β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T n ν n 2 .
(3.22)

From (3.22) we obtain

2 ν n G n ν n , ν n p β n ν n G n ν n 2 + β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T ν n 2 ( k n 2 1 ) ν n p 2
(3.23)

and

2 G n ν n p , ν n p ( 1 + k n 2 ) ν n p 2 β n ν n G n ν n 2 β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T n ν n 2 .
(3.24)

Observe that

x n + 1 p 2 = ( 1 α n ) ν n + α n G n ν n p 2 = ( 1 α n ) ν n p 2 + α n G n ν n p 2 α n ( 1 α n ) ν n G n ν n 2 ( 1 α n ) ν n p 2 + α n [ [ 1 + ( k n 2 1 ) ] ν n p 2 + ( 1 β n ) ν n G n ν n 2 β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T n ν n 2 ] α n ( 1 α n ) ν n G n ν n 2 [ 1 + ( k n 2 1 ) ] ν n p 2 α n ( β n α n ) ν n G n ν n 2 α n β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] ν n T n ν n 2 .
(3.25)

Hence

x n + 1 p 2 [ 1 + ( k n 2 1 ) ] ν n p 2

and it follows, as in the proof of Theorem 3.1, that { x n } n = 1 is bounded. Observe that

x n x n + 1 2 = x n ν n + ν n x n + 1 2 ν n x n + 1 2 + 2 x n ν n , x n x n + 1 ν n x n + 1 2 + 2 x n ν n x n x n + 1 ν n x n + 1 2 + 2 P C ( ( 1 t n ) x n ) x n x n x n + 1 ν n x n + 1 2 + 2 t n x n x n x n + 1 .

Hence

ν n x n + 1 2 x n x n + 1 2 2 t n x n x n x n + 1 .
(3.26)

Furthermore,

ν n G n ν n ν n T n ν n + T n ν n G n ν n ν n T n ν n + L β n ν n T n ν n = ( 1 + L β n ) ν n T n ν n .

Thus

ν n T n ν n 2 1 ( 1 + L β n ) 2 ν n G n ν n 2 .
(3.27)

Observe also that

x n + 1 ν n = ( 1 α n ) ν n + α n G n ν n ν n = α n ν n G n ν n .
(3.28)

Using (3.26) and (3.28), we obtain

ν n G n ν n 2 = 1 α n 2 x n + 1 ν n 2 1 α n 2 [ x n x n + 1 2 2 t n x n x n x n + 1 ] .

It now follows from (3.27) that

ν n T n ν n 2 1 ( 1 + L β n ) 2 ν n G n ν n 2 1 α n 2 ( 1 + L β n ) 2 [ x n x n + 1 2 2 t n x n x n x n + 1 ] .
(3.29)

Using (3.29) in (3.25), we obtain

x n + 1 p 2 [ 1 + ( k n 2 1 ) ] ν n p 2 α n β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] { 1 α n 2 ( 1 + L β n ) 2 [ x n x n + 1 2 2 t n x n x n x n + 1 ] } ν n p 2 + ( k n 2 1 ) D β n [ 1 ( 1 + k n ) β n β n 2 L 2 ] α n ( 1 + L β n ) 2 [ x n x n + 1 2 2 t n x n x n x n + 1 ] x n p 2 2 t n x n , x n p + t n 2 x n 2 σ 3 x n x n + 1 2 + 2 σ 4 t n x n x n x n + 1 ( where  σ 3 = ϵ [ 1 ( 1 + λ ) b b 2 L 2 ] b ( 1 + L ϵ ) 2 , σ 4 = 1 ϵ ( 1 + L ϵ ) 2 ) = x n p 2 σ 3 x n x n + 1 + t n [ x n , x n p + t n x n 2 + 2 σ 4 x n x n x n + 1 ] + ( k n 2 1 ) D .
(3.30)

Since { x n } is bounded, we have that there exists M>0 such that

x n , x n p+ t n x n 2 +2 σ 4 x n x n x n + 1 M,n1.
(3.31)

From (3.30) and (3.31) we obtain

x n + 1 p 2 x n p 2 + σ 3 x n x n + 1 M t n + ( k n 2 1 ) D.
(3.32)

To complete the proof, we now consider the following two cases.

Case 1. Suppose that { x n p } n = 1 is a monotone sequence, then we may assume that { x n p} is monotone decreasing. Then lim n x n p exists and it follows from (3.32), conditions (c1) and lim n k n =1 that

lim n x n x n + 1 =0.
(3.33)

Furthermore,

ν n x n t n x n 0 as  n ,  and  ν n x n + 1 ν n x n + x n x n + 1 0 as  n .

Hence

ν n G n ν n = 1 α n ν n G n ν n 1 ϵ ν n x n + 1 0as n.

Furthermore,

ν n T n ν n ν n G n ν n + G n ν n T n ν n ν n G n ν n + L β n ν n T n ν n .

Thus

ν n T n ν n 2 1 1 L β n ν n G n ν n 1 1 L b ν n G n ν n 0as n,

and

x n T n x n x n ν n + ν n T n ν n + T n ν n T n x n ( 1 + k n ) x n ν n + ν n T n ν n 0 as  n .

Observe also that since T is uniformly L-Lipschitzian, we obtain

ν n T ν n ν n T n ν n + T n ν n T ν n ν n T n ν n + L T n 1 ν n ν n ν n T n ν n + L T n 1 ν n T n 1 ν n 1 + L T n 1 ν n 1 ν n 1 + L ν n 1 ν n ν n T n ν n + L T n 1 ν n 1 ν n 1 + L ( 1 + L ) ν n ν n 1 ν n T n ν n + L T n 1 ν n 1 ν n 1 + L ( 1 + L ) [ ν n x n + x n x n 1 + x n 1 ν n 1 ] 0 as  n .
(3.34)

Furthermore,

x n T x n x n T n x n + T n x n T x n x n T n x n + L T n 1 x n x n x n T n x n + L T n 1 x n T n 1 x n 1 + L T n 1 x n 1 x n 1 + L x n 1 x n x n T n x n + L T n 1 x n 1 x n 1 + L ( 1 + L ) x n x n 1 0 as  n .
(3.35)

Since lim n x n T x n = lim n ν n T ν n = lim n ν n x n =0, then the demiclosedness property of (IT), (2.4) and the usual standard argument yield that { x n } n = 1 and { ν n } n = 1 converge weakly to some x F(T). Since ν n x 2 D 2 , n1, and for some D 2 >0, then using (3.25) we obtain

x n + 1 x 2 ν n x 2 + α n ( k n 2 1 ) D 2 ( 1 t n ) ( x n x ) t n x 2 + α n ( k n 2 1 ) D 2 = ( 1 t n ) 2 x n x 2 2 t n ( 1 t n ) x n x , x + t n 2 x 2 + α n ( k n 2 1 ) D 2 ( 1 t n ) x n x 2 2 t n ( 1 t n ) x n x , x + t n 2 x 2 + α n ( k n 2 1 ) D 2 .
(3.36)

Thus

x n + 1 x 2 (1 t n ) x n x 2 + t n γ n + σ n ,n1,

where γ n :=2(1 t n ) x n x , x + t n x 2 0 as n, and σ n = α n ( k n 2 1) D 2 with n = 1 σ n <. It now follows from Lemma 2.1 that { x n } n = 1 converges strongly to x . Consequently, { ν n } n = 1 converges strongly to x .

Case 2. Suppose that { x n p } n = 1 is not a monotone decreasing sequence, then set Γ n := x n p 2 , and let τ:NN be a mapping defined for all n N 0 for some sufficiently large N 0 by

τ(n):=max{kN:kn, Γ k Γ k + 1 }.

Then τ is a non-decreasing sequence such that τ(n) as n and Γ τ ( n ) Γ τ ( n ) + 1 for n N 0 . Using (c1) and (c2) in (3.32), we obtain

x τ ( n ) + 1 x τ ( n ) 2 1 σ 4 [ M t τ ( n ) + ( k τ ( n ) 2 1 ) D ] 0as n.
(3.37)

Following the same argument as in Case 1, we obtain

ν τ ( n ) T ν τ ( n ) 0as nand x τ ( n ) T x τ ( n ) 0as n.

As in Case 1 we also obtain that { x τ ( n ) } and { ν τ ( n ) } converge weakly to some x in F(T).

Furthermore, for all n N 0 , we obtain from (3.36) that

0 x τ ( n ) + 1 x 2 x τ ( n ) x 2 t τ ( n ) [ 2 ( 1 t τ ( n ) ) x τ ( n ) x , x + t τ ( n ) x 2 + D 2 α τ ( n ) ( k τ ( n ) 2 1 ) t τ ( n ) x τ ( n ) x 2 ] .
(3.38)

It follows from (3.38) that

x τ ( n ) x 2 2 ( 1 t τ ( n ) ) x x τ ( n ) , x + t τ ( n ) x 2 + D 2 α τ ( n ) ( k τ ( n ) 2 1 ) t τ ( n ) 0 as  n .

Thus

lim n Γ τ ( n ) = lim n Γ τ ( n ) + 1 .

Furthermore, for n N 0 , we have Γ n Γ τ ( n ) + 1 if nτ(n) (i.e., τ(n)<n), because Γ j > Γ j + 1 for τ(n)+1jn. It then follows that for all n N 0 we have

0 Γ n max{ Γ τ ( n ) , Γ τ ( n ) + 1 }= Γ τ ( n ) + 1 .

This implies lim n Γ n =0, and hence { x n } n = 1 converges strongly to x F(T). □

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H with 0C, and let T:CC be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence { k n } n = 1 [1,), n = 1 ( k n 1)< and F(T). Let { t n } n = 1 , { α n } n = 1 and { β n } n = 1 be real sequences in (0,1) satisfying the conditions:

  1. (c1)

    lim n t n =0;

  2. (c2)

    n = 1 t n =;

  3. (c3)

    0<ϵ α n (1 t n ) β n β n b< 2 ( 1 + λ ) + ( 1 + λ ) 2 + 4 L 2 , where λ= sup n k n ;

  4. (c4)

    lim n ( k n 1 ) t n =0.

Then the sequence { x n } n = 1 generated from an arbitrary x 1 C by

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

converges strongly to a fixed point of T.

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Acknowledgements

The work was completed when the first author was visiting the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy as an Associate. He is grateful to the Centre for the invaluable facilities at the Centre and for hospitality. The authors are grateful to the referees for their valuable comments.

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Correspondence to Micah Okwuchukwu Osilike.

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MO conceived of the study. All authors carried out the research, read and approved the final manuscript.

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Osilike, M.O., Nwokoro, P.U. & Chima, E.E. Strong convergence of new iterative algorithms for certain classes of asymptotically pseudocontractions. Fixed Point Theory Appl 2013, 334 (2013). https://doi.org/10.1186/1687-1812-2013-334

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Keywords

  • asymptotically pseudocontractive maps
  • fixed points
  • strong convergence
  • Hilbert spaces
  • iterative algorithm