Open Access

Strong convergence of an new iterative method for a zero of accretive operator and nonexpansive mapping

Fixed Point Theory and Applications20122012:98

https://doi.org/10.1186/1687-1812-2012-98

Received: 28 November 2011

Accepted: 15 June 2012

Published: 15 June 2012

Abstract

Let E be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence {x n } by the algorithm x n + 1 = α n γ ϕ ( x n ) + ( I - α n F ) J r n x n , where {a n } and {r n } are two sequences satisfying certain conditions, J r n denotes the resolvent (I + r n A)-1 for r n > 0, F be a strongly positive bounded linear operator on E is 0 < γ < γ ̄ , and ϕ be a MKC on E. Strong convergence of the algorithm {x n } is proved assuming E either has a weakly continuous duality map or is uniformly smooth.

MSC: 47H09; 47H10

Keywords

MKC accretive operators the resolvent operator iterative method weakly continuous duality map

1 Introduction

Let E be a real Banach space, C a nonempty closed convex subset of E, and T : CC a mapping. Recall that T is nonexpansive if Tx - Tyx - y for all x, y C. A point x C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T, that is, F(T) = {x C, Tx = x}.

It is assumed throughout the paper that T is a nonexpansive mapping such that F ( t ) . The normalized duality mapping J from a Banach space E into 2 E * is given by J(x) = {f E* : 〈x, f〉 = x2 = f2}, x E, where E* denotes the dual space of E and 〈.,.〉 denotes the generalized duality pairing.

Theorem 1.1. (Banach [1]). Let (X, d) be a complete metric space and let f be a contraction on X, that is, there exists r (0, 1) such that d(f(x), f(y)) ≤ rd(x, y) for all x, y X. Then f has a unique fixed point.

Theorem 1.2. (Meir and Keeler [2]). Let (X, d) be a complete metric space and let ϕ be a Meir-Keeler contraction (MKC, for short) on X, that is, for every ε > 0, there exists δ > 0 such that d(x, y) < ε + δ implies d(ϕ(x), ϕ(y)) < ε for all x, y X. Then ϕ has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

Let F be a strongly positive bounded linear operator on E, that is, there exists a constant γ ̃ > 0 such that
F x , J ( x ) γ ̃ x 2 , a I - b F = sup x 1 ( a I - b F ) x , J ( x ) : a [ 0 , 1 ] , b [ 0 , 1 ] ,

where I is the identity mapping and J is the normalized duality mapping.

Let D be a subset of C. Then Q : CD is called a retraction from C onto D if Q(x) = x for all x D. A retraction Q : CD is said to be sunny if Q(x + t(x - Q(x))) = Q(x) for all x C and t ≥ 0 whenever x + t(x - Q(x)) C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. In a smooth Banach space E, it is known (cf. [[3], p. 48]) that Q : CD is a sunny nonexpansive retraction if and only if the following condition holds:
x - Q ( x ) , J ( z - Q ( x ) ) 0 , x C , z D .
(1.1)
Recall that an operator A with domain D(A) and range R(A) in E is said to be accretive, if for each x i D(A) and y i Ax i , i = 1, 2, there is a j J(x2 - x1) such that
y 2 - y 1 , j 0 .
An accretive operator A is m-accretive if R(I + λA) = E for all λ > 0. Denote by N(A) the zero set of A; i.e.,
N ( A ) : = A - 1 0 = { x D ( A ) : A x = 0 } .
Throughout the rest of this paper it is always assumed that A is m-accretive and N(A) is nonempty. Denote by J r the resolvent of A for r > 0:
J r = ( I + r A ) - 1 .

Note that if A is m-accretive, then J r : EE is nonexpansive and F(J r ) = N(A) for all r > 0. We also denote by A r the Yosida approximation of A, i.e., A r = 1 r ( I - J r ) . It is well known that J r is a nonexpansive mapping from E to C := D(A).

Recall that a gauge is a continuous strictly increasing function φ : [0, ∞) → [0, ∞) such that φ(0) = 0 and φ(t) → ∞ as t → ∞. Associated to a gauge φ is the duality mapping J φ : EE* defined by
J φ ( x ) = x * E * : x , x * = x φ x , x * = φ x , x E .
Following Browder [4], we say that a Banach space E has a weakly continuous duality map if there exists a gauge φ for which the duality map J φ is single-valued and weak-to-weak* sequentially continuous(i.e., if {x n } is a sequence in E weakly convergent to a point x, then the sequence J φ (x n ) converges weakly* to J φ (x)). It is known that l p has a weakly continuous duality map for all 1 < p < ∞, with gauge φ(t) = tp-1. Set
Φ ( t ) = 0 t φ ( τ ) d τ , t 0 .
(1.2)
Then
J φ ( x ) = Φ x , x E ,

where denotes the subdifferential in the sense of convex analysis.

Recently, Hong-Kun Xu [5] introduced the following iterative scheme: for x1 = x C,
x n + 1 = α n u + ( 1 + α n ) J r n x n , n 1 ,
(1.3)

where {a n } and {r n } are two sequences satisfying certain conditions, and J r n denotes the resolvent (I + r n A)-1 for r n > 0. He proved the strong convergence of the algorithm {x n } assuming E either has a weakly continuous duality map or is uniformly smooth.

Motivated and inspired by the results of Hong-Kun Xu, we introduce the following iterative scheme: for any x0 E,
x n + 1 = α n γ ϕ ( x n ) + ( I - α n F ) J r n x n , n 0 ,
(1.4)

where {a n } and {r n } are two sequences satisfying certain conditions, J r n denotes the resolvent (I + r n A)-1 for r n > 0, F be a strongly positive bounded linear operator on E is 0 < γ < γ ̄ , and ϕ be a MKC on E. Strong convergence of the algorithm {x n } is proved assuming E either has a weakly continuous duality map or is uniformly smooth. Our results extend and improve the corresponding results of Hong-Kun Xu [5] and many others.

2 Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1. [5]. Assume that E has a weakly continuous duality map J φ with gauge φ,
  1. (i)
    For all x, y E, there holds the inequality
    Φ x + y Φ x + y , J φ ( x + y ) .
     
  2. (ii)
    Assume a sequence {x n } in E is weakly convergent to a point x, then there holds the equality
    lim sup n Φ x n - y = lim sup n Φ x n - x + Φ y - x , x , y E .
     
Lemma 2.2. [6, 7]. Let {s n } be a sequence of nonnegative real numbers satisfying
s n + 1 ( 1 - λ n ) s n + λ n δ n + γ n , n 0 ,
where {λ n }, {δ n } and {γ n } satisfy the following conditions:
  1. (i)

    {λ n } [0,1] and n = 0 λ n = ,

     
  2. (ii)

    lim supn→∞δ n ≤ 0 or n = 0 λ n δ n < (iii) γ n 0 ( n 0 ) , n = 0 γ n < . Then limn→∞s n = 0.

     
Lemma 2.3. (The Resolvent Identity [8, 9]). For λ > 0 and ν > 0 and x E,
J λ x = J ν ν λ + 1 - ν λ J λ x .
Lemma 2.4. (see [ [10], Lemma 2.3]). Assume that F is a strongly positive linear bounded operator on a smooth Banach space E with coefficient γ ̄ > 0 and 0 < ρF-1. Then,
I - ρ F 1 - ρ γ ̄ .
Lemma 2.5. (see [ [11], Lemma 2.3]). Let ϕ be a MKC on a convex subset C of a Banach space E. Then for each ε > 0, there exists r (0,1) such that
x - y ε implies ϕ x - ϕ y r x - y x , y C .
Lemma 2.6. Let E be a reflexive Banach space which admits a weakly continuous duality map J φ with gauge φ. Let T : EE be a nonexpansive mapping. Now given ϕ : EE be a MKC, F be a strongly positive linear bounded operator with coefficient γ ̄ > 0 . Assume that 0 < γ < γ ̄ , the sequence {x t } defined by x t = tγϕ(x t ) + (I - tF)Tx t . Then T has a fixed point if and only if {x t } remains bounded as t → 0+, and in this case, {x t } converges as t → 0+ strongly to a fixed point of T. If x ̃ : = lim t 0 x t , then x ̃ uniquely solves the variational inequality
( F - γ ϕ ) x ̃ , J ( x ̃ - p ) 0 , P F ( T ) .
Proof. The definition of {x t } is well defined. Indeed, from the definition of MKC, we can see MKC is also a nonexpansive mapping. Consider a mapping S t on E defined by
S t ( x ) = t γ ϕ ( x ) + ( I - t F ) T x , x E .
It is easy to see that S t is a contraction. Indeed, by Lemma 2.4, we have
S t x - S t y t γ ϕ ( x ) - ϕ ( y ) + ( I - t F ) ( T x - T y ) t γ x - y + ( 1 - t γ ̄ ) T x - T y [ 1 - t ( γ ̄ - γ ) ] x - y ,
for all x, y E. Hence S t has a unique fixed point, denoted as x t , which uniquely solves the fixed point equation
x t = t γ ϕ ( x t ) + ( I - t F ) T x t , x t E .
(2.1)
We next show the sequence {x t } is bounded. Indeed, we may assume F ( t ) and with no loss of generality t < F-1. Take p F(T) to deduce that, for t (0, 1),
x t - p = t γ ϕ ( x t ) + ( I - t F ) T x t - p = t ( γ ϕ ( x t ) - F p ) + ( I - t F ) ( T x t - p ) ( 1 - t γ ̄ ) x t - p + t γ x t - p + t γ ϕ ( p ) - F p [ 1 - t ( γ ̄ - γ ) ] x t - p + t γ ϕ ( p ) - F p .
Hence
x t - p 1 γ ̄ - γ γ ϕ ( p ) - F p

and {x t } is bounded.

Next assume that {x t } is bounded as t → 0+. Assume t n → 0+ and { x t n } is bounded. Since E is reflexive, we may assume that x t n z for some z E. Since J φ is weakly continuous, we have by Lemma 2.1,
lim sup n Φ x t n - x = lim sup n Φ x t n - z + Φ x - z , x E .
Put
f ( x ) = lim sup n Φ x t n - x , x E .
It follows that
f ( x ) = f ( z ) + Φ x - z , x E .
Since
x t n - T x t n = t n γ ϕ ( x t n ) - F T x t n 0 ,
we obtain
f ( T z ) = lim  sup n Φ x t n - T z = lim  sup n Φ T x t n - T z lim  sup n Φ x t n - z = f ( z ) .
(2.2)
On the other hand, however,
f ( T z ) = f ( z ) + Φ T z - z
(2.3)
Combining Equations (2.2) and (2.3) yields
Φ T z - z 0 .

Hence, Tz = z and z F(T).

Finally, we prove that {x t } converges strongly to a fixed point of T provided it remains bounded when t → 0.

Let {t n } be a sequence in (0, 1) such that t n → 0 and x t n z as n → ∞. Then the argument above shows that z F(T). We next show that x t n z . By contradiction, there is a number ε0 > 0 such that x t n - z ε 0 . Then by Lemma 2.8, there is a number r (0, 1) such that
ϕ ( x t n ) - ϕ ( z ) r x t n - z , x t n - z φ x t n - z = x t n - z , J φ ( x t n - z ) = t n ( γ ϕ ( x t n ) - F z ) + ( I - t n F ) ( T x t n - z ) , J φ ( x t n - z ) t n γ ϕ ( x t n ) - F z , J φ ( x n - z ) + ( I - t n F ) ( T x t n - z ) φ x t n - z ( 1 - t n γ ̄ ) x t n - z φ x t n - z + t n γ ϕ ( x t n ) - F z , J φ ( x n - z ) .
It follows that
x t n - z φ x t n - z 1 γ ̄ γ ϕ ( x t n ) - F z , J φ ( x t n - z ) = 1 γ ̄ γ ϕ ( x t n ) - γ ϕ ( z ) , J φ ( x t n - z ) + γ ϕ ( z ) - F z , J φ ( x t n - z ) γ γ ̄ r x t n - z φ x t n - z + 1 γ ̄ γ ϕ ( z ) - F z , J φ ( x t n - z ) .
Therefore,
x t n - z φ x t n - z 1 γ ̄ - γ r γ ϕ ( z ) - F z , J ϕ ( x t n - z ) .
Now observing that x t n z implies J φ ( x t n - z ) 0 , we conclude from the last inequality that
lim n x t n - z φ x t n - z = 0 .

It contradicts x t n - z φ x t n - z ε 0 φ ( ε 0 ) > 0 . Hence x t n z .

We finally prove that the entire net {x t } converges strongly. Towards this end, we assume that two null sequences {t n } and {s n } in (0, 1) are such that
t n 0 , x t n z and s n 0 , x s n ź .
We have to show z = ź . Indeed, for p F(T). Since
x t = t γ ϕ ( x t ) + ( I - t f ) T x t ,
we derive that
( F - γ ϕ ) x t = - 1 t ( I - t F ) ( I - T ) x t .
(2.4)
Notice
( I - T ) x t - ( I - T ) p , J φ ( x t - p ) = x t - p φ x t - p + T p - T x t , J φ ( x t - p ) x t - p φ x t - p - T p - T x t J φ x t - p x t - p φ x t - p - φ x t - p = 0 .
It follows that,
( F γ ϕ ) x t , J φ ( x t p ) = 1 t ( I t F ) ( I T ) x t , J φ ( x t p ) ( F ( I T ) x t , J φ ( x t p ) .
(2.5)

Now replacing t in (2.5) with t n and letting n → ∞, noticing ( I - T ) x t n ( I - T ) z = 0 for z F(T), we obtain 〈(F - γϕ)z, J φ (z - p)〉 ≤ 0. In the same way, we have ( F - γ ϕ ) ź , J φ ( ź - p ) 0 .

Thus, we have
( F - γ ϕ ) z , J φ ( z - ź ) 0 and ( F - γ ϕ ) ź , J φ ( ź - z ) 0 .
(2.6)
Adding up (2.6) gets
( F - γ ϕ ) z - ( F - γ ϕ ) ź , J φ ( z - ź ) 0 .
On the other hand, without loss of generality, we may assume there is a number ε such that z - ź ε , then by Lemma 2.5 there is a number r1 such that ϕ ( z ) - ϕ ( ź ) r 1 z - ź . Noticing that
J φ = φ x / x J ( x ) x 0 , ( F - γ ϕ ) z - ( F - γ ϕ ) ź , J φ ( z - ź ) = F ( z - ź ) , J φ ( z - ź ) - γ ϕ z - γ ϕ ź , J φ ( z - ź ) γ ̄ z - ź J φ ( z - ź ) - γ r 1 z - ź J φ ( z - ź ) ( γ ̄ - γ r 1 ) z - ź φ z - ź > 0 .
Hence z = ź and {x t } converges strongly. Thus we may assume x t x ̃ . Since we have proved that, for all t (0, 1) and p F(T),
( F γ ϕ ) x t , J φ ( x t p ) ( F ( I T ) x t , J φ ( x t p ) ,
letting t → 0, we obtain that
( F - γ ϕ ) x ̃ , J φ ( x ̃ - p ) 0 .
This implies that
( F - γ ϕ ) x ̃ , J ( x ̃ - p ) 0 .

Lemma 2.7. (see [12]). Assume that C2C1 > 0. Then J C 1 x - x 2 J C 2 x - x for all x E.

Lemma 2.8. [13]. Let C be a nonempty closed convex subset of a reflexive Banach space E which satisfies Opial's condition, and suppose T : CE is a nonexpansive mapping. Then the mapping I - T is demiclosed at zero, that is x n x and x n - Tx n → 0, then x = Tx.

Lemma 2.9. In a smooth Banach space E there holds the inequality
x + y 2 x 2 + 2 y , J ( x + y ) , x , y E .

3 Main result

Theorem 3.1. Suppose that E is reflexive which admits a weakly continuous duality map J φ with gauge φ and A is an m-accretive operator in E such that F * = N ( A ) . Now given ϕ : EE be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient γ ̄ > 0 , 0 < γ < γ ̄ . Assume
  1. (i)

    lim n α n = 0 , n = 0 α n = ;

     
  2. (ii)

    r n → ∞.

     

Then {x n } defined by (1.4) converges strongly to a point in F*.

Proof. First notice that {x n } is bounded. Indeed, take p F* to get
x n + 1 - p = α n ϕ ( x n ) + ( I - α n F ) J r n x n - p = α n γ ϕ ( x n ) - α n F p + α n F p + ( I - α n F ) J r n x n - p = α n ( γ ϕ ( x n ) - F p ) + ( I - α n F ) ( J r n x n - p ) ( 1 - α n γ ̄ ) x n - p + α n γ x n - p + α n γ ϕ ( p ) - F p [ 1 - α n ( γ ̄ - γ ) ] x n - p + α n γ ϕ ( p ) - F p [ 1 - α n ( γ ̄ - γ ) ] x n - p + α n ( γ ̄ - γ ) γ ϕ ( p ) - F p γ ̄ - γ max x n - p , γ ϕ ( p ) - F p γ ̄ - γ , n 0 .
By induction, we have
x n - p max x 0 - p , γ ϕ ( p ) - F p γ ̄ - γ , n 0 .
This implies that {x n } is bounded and hence
x n + 1 - J r n x n = α n γ ϕ ( x n ) + ( I - α n F ) J r n x n - J r n x n = α n γ ϕ ( x n ) - F J r n x n 0 .

We next prove that

lim supn→∞γϕ(p) - Fp, J φ (x n - p)〉 ≤ 0, where p = limt→0x t with x t = t γ ϕ ( x t ) + ( I - t F ) J r n x t .

Since {x n } is bounded, take a subsequence { x n k } of {x n } such that
lim sup n γ ϕ ( p ) - F p , J ϕ ( x n - p ) = lim k γ ϕ ( p ) - F p , J φ ( x n k - p ) .
(3.1)
Since E is reflexive, we may further assume that x n k x ̃ . Moreover, since
x n + 1 - J r n x n 0 ,
we obtain
J r n k - 1 x n k - 1 x ̃ .
Taking the limit as k → ∞ in the relation
J r n k - 1 x n k - 1 , A r n k - 1 x n k - 1 A ,
we get [ x ̃ , 0 ] A . That is, x ̃ F * . Hence by (3.1) and Lemma 2.6 we have
lim sup n γ ϕ ( p ) - F p , J φ ( x n - p ) = lim k γ ϕ ( p ) - F p , J φ ( x n k - p ) = γ ϕ ( p ) - F p , J φ ( x ̃ - p ) 0 .
Finally to prove that x n p, we apply Lemma 2.1 to get
Φ x n + 1 - p = Φ α n γ ϕ ( x n ) + ( I - α n F ) J r n x n - p = Φ ( I - α n F ) ( J r n x n - p ) + α n ( γ ϕ ( x n ) - F p ) = Φ ( I - α n F ) ( J r n x n - p ) + α n ( γ ϕ ( x n ) - γ ϕ ( p ) ) + α n ( γ ϕ ( p ) - F p ) Φ ( I - α n F ) ( J r n x n - p ) + α n ( γ ϕ ( x n ) - γ ϕ ( p ) ) + α n γ ϕ ( p ) - F p , J φ ( x n + 1 - p ) [ 1 - α n ( γ ̄ - γ ) ] Φ x n - p + α n γ ϕ ( p ) - F p , J φ ( x n + 1 - p ) .

An application of Lemma 2.2 yields that Φ(x n - p) → 0. That is, x n - p → 0, i.e., x n p. The proof is complete.

Theorem 3.2. Suppose that E is reflexive which admits a weakly continuous duality map J φ with gauge φ and A is an m-accretive operator in E such that F * = N ( A ) . Now given ϕ : EE be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient γ ̄ > 0 , 0 < γ < γ ̄ . Assume
  1. (i)

    lim n α n = 0 , n = 0 α n = , and n = 1 α n + 1 - α n < ( e . g . , α n = 1 n ) ;

     
  2. (ii)

    r n ε for all n and n = 1 r n + 1 - r n < ( e . g . , r n = 1 + 1 n ) .

     

Then {x n } defined by (1.4) converges strongly to a point in F*.

Proof. We only include the differences. We have
x n + 1 = α n γ ϕ ( x n ) + ( I - α n F ) J r n x n , x n = α n - 1 γ ϕ ( x n - 1 ) + ( I - α n - 1 F ) J r n - 1 x n - 1 .
Thus,
x n + 1 - x n = ( I - α n F ) ( J r n x n - J r n - 1 x n - 1 ) + α n γ ϕ ( x n ) + α n - 1 γ ϕ ( x n - 1 ) + ( α n - α n - 1 ) F J r n - 1 x n - 1 .
(3.2)
If rn-1r n , using the resolvent identity
J r n x n = J r n - 1 r n - 1 r n x n + 1 - r n - 1 r n J r n x n ,
we obtain
J r n x n - J r n - 1 x n - 1 r n - 1 r n x n - x n - 1 + 1 - r n - 1 r n J r n x n - x n - 1 x n - x n - 1 + r n - r n - 1 r n J r n x n - x n - 1 x n - x n - 1 + 1 ε r n - 1 - r n J r n x n - x n - 1 .
(3.2a)
It follows from (3.2) that
x n + 1 - x n ( 1 - α n ( γ ̄ - γ ) ) x n - x n - 1 + M α n - α n - 1 + r n - 1 - r n ,
(3.3)
where M > 0 is some appropriate constant. Similarly we can prove (3.3) if rn-1r n . By assumptions (i) and (ii) and Lemma 2.2, we conclude that
x n + 1 - x n 0 .
This implies that
x n - J r n x n x n + 1 - x n + x n + 1 - J r n x n 0 ,
(3.4)
since x n + 1 - J r n x n = α n γ ϕ ( x n ) - F J r n x n 0 . It follows that
A r n x n = 1 r n x n - J r n x n 1 ε x n - J r n x n 0 .
Now if { x n k } is a subsequence of {x n } converging weakly to a point x ̃ , then taking the limit as k → ∞ in the relation
[ J r n k x n k , A r n k x n k ] A ,

we get [ x ̃ , 0 ] A ; i.e., x ̃ F * . We therefore conclude that all weak limit points of {x n } are zeros of A.

The rest of the proof follows that of Theorem 3.1.

Finally, we consider the framework of uniformly smooth Banach spaces. Assume r n ε for some ε > 0 (not necessarily r n → ∞), A is an m-accretive operator in E. Moreover let ϕ : EE be a MKC and F be a strongly positive linear bounded operator on E. Since J r n is nonexpansive, the map S : x E t γ ϕ ( x ) + ( I - t F ) J r n x is a contraction and for each integer n ≥ 1 it has a unique fixed z t,n E. Hence the scheme
z t , n = t γ ϕ ( z t , n ) + ( I - t F ) J r n z t , n
(3.5)

is well defined.

Note that {z t,n } is uniformly bounded; indeed, z t , n - p 1 γ ̄ - γ γ ϕ ( p ) - F p for all t (0, 1), n ≥ 1 and p F*. A key component of the proof of the next theorem is the following lemma.

Lemma 3.1. The limit = lim t 0 z t , n is uniform for all n ≥ 1.

Proof. It suffices to show that for any positive integer n t (which may depend on t (0, 1)), if z t , n t E is the unique point in E that satisfies the property
z t , n t = t γ ϕ ( z t , n t ) + ( I - t F ) J r n t z t , n t ,
(3.6)
then { z t , n t } converges as t → 0 to a point in F*. For simplicity put
w t = z t , n t and V t = J r n t .
It follows that
w t = t γ ϕ ( w t ) + ( I - t F ) V t w t .
(3.7)
Note that Fix(V t ) = F* for all t. Note also that {w t } is bounded; indeed, we have w t - p 1 γ ̄ - γ γ ϕ ( p ) - F p for all t (0, 1) and p F*. Since {V t w t } is bounded, it is easy to see that
w t - V t w t = t γ ϕ ( w t ) - F V t w t 0 , as t 0 .
Since r n ε for all n, by Lemma 2.7, we have
w t - J ε w t 2 w t - J r n t w t = 2 w t - V t w t 0 .
(3.8)
Let {t k } be a sequence in (0,1) such that t k → 0 as k → ∞. Define a function f on E by
f ( w ) = LI M k 1 2 w t k - w 2 , w E ,
where LIM denotes a Banach limit on l. Let
K : = w E : f ( w ) = min { f ( y ) : y E } .
Then K is a nonempty closed convex bounded subset of E. We claim that K is also invariant under the nonexpansive mapping J ε . Indeed, noting (3.8), we have for w K,
f ( J ε w ) = LI M k 1 2 w t k - J ε w 2 = LI M k 1 2 J ε w t k - J ε w 2 LI M k 1 2 w t k - w 2 = f ( w ) .
Since a uniformly smooth Banach space has the fixed point property for nonexpansive mappings and since J ε is a nonexpansive self-mapping of E, J ε has a fixed point in K, say w'. Now since w' is also a minimizer of f over E, it follows that, for w E,
0 f ( w + λ ( w - w ) ) - f ( w ) λ = LI M k 1 2 w t k - w + λ ( w - w ) 2 - 1 2 w t k - w 2 λ .
Since E is uniformly smooth, the duality map J is uniformly continuous on bounded sets, letting λ → 0+ in the last equation yields
0 LI M k w - w , J ( w t k - w ) , w E .
(3.9)
Since
w t k - w = ( I - t k F ) ( V t k w t k - w ) + t k ( γ ϕ ( w t k ) - F w ) ,
we obtain
w t n - w 2 = t k γ ϕ ( w t k ) - F w , J ( w t k - w ) + ( I - t k F ) ( V t k w t k - w ) , J ( w t k - w ) t k γ ϕ ( w t k ) - F w , J ( w t k - w ) + ( 1 - t k γ ̄ ) w t k - w 2 t k γ ϕ ( w t k ) - γ ϕ ( w ) , J ( w t k - w ) + t k γ ϕ ( w ) - F w , J ( w t k - w ) + ( 1 + t k γ ̄ ) w t k - w 2 [ 1 - t k ( γ ̄ - γ ) ] w t k - w 2 + t k γ ϕ ( w ) - F w , J ( w t k - w ) .
It follows that
w t k - w 2 1 γ ̄ - γ γ ϕ ( w ) - F w , J ( w t k - w ) .
(3.10)
Upon letting w = γϕ(w') - Fw' + w' in (3.9), we see that the last equation implies
LI M k w t k - w 2 0 .
(3.11)

Therefore, { w t k } contains a subsequence, still denoted { w t k } , converging strongly to w1 (say). By virtue of (3.8), w1 is a fixed point of J ε ; i.e., a point in F*.

To prove that the entire net {w t } converges strongly, assume {s k } is another null subsequence in (0, 1) such that w s k w 2 strongly. Then w2 F*.

Repeating the argument of (3.10) we obtain
w t - w 2 1 γ ̄ - γ γ ϕ ( w ) - F w , J ( w t - w ) , w F * .
In particular,
w 1 - w 2 2 1 γ ̄ - γ γ ϕ ( w 1 ) - F w 1 , J ( w 2 - w 1 )
(3.12)
and
w 2 - w 1 2 1 γ ̄ - γ γ ϕ ( w 2 ) - F w 2 , J ( w 1 - w 2 ) .
(3.13)
Adding up the last two equations gives
w 1 - w 2 2 0 .

That is, w1 = w2. This concludes the proof.

Theorem 3.3. Suppose that E is a uniformly smooth Banach space and A is an m-accretive operator in E such that F * = N ( A ) . Now given ϕ : EE be a MKC, and let F be a strongly positive linear bounded operator on E with coefficient γ ̄ > 0 , 0 < γ < γ ̄ . Assume
  1. (i)

    lim n α n = 0 , n = 0 α n = , and n = 1 α n + 1 - α n < ( e . g . , α n = 1 n ) ;

     
  2. (ii)

    limn→∞= r n = r,r R+, r n ε for all n and n = 1 r n + 1 - r n < ( e . g . , r n = 1 + 1 n ) .

     

Then {x n } defined by (1.4) converges strongly to a point in F*.

Proof. Since
J r n x n - J r x n J r r r n x n + 1 - r r n J r n x n - J r x n r r n x n + 1 - r r n J r n x n - x n 1 - r r n J r n x n - x n 0 as ( n ) .
(3.14)
Thus
x n - J r x n x n - J r n x n + J r n x n - J r x n 0 .
(3.15)

We next claim that lim  sup n γ ϕ ( ) - F , J ( x n - ) 0 , where = lim t 0 z t , n with z t,n = tγϕ(z t,n ) + (I - tF)J r z t,n .

For this purpose, let { x n k } be a subsequence chosen in such a way that lim sup n γ ϕ ( ) - F , J ( x n - ) = lim k γ ϕ ( ) - F , J ( x n k - ) and x n k x ̃ . Moreover, since x n - J r x n → 0, using Lemma 2.8, we know x ̃ F ( J r ) . Hence by Lemma 2.6, we have
lim sup n γ ϕ ( z ^ ) F z ^ , J ( x n z ^ ) = lim k γ ϕ ( z ^ ) F z ^ , J ( x n k z ^ ) = γ ϕ ( z ^ ) F z ^ , x ˜ z ^ ) 0.
(3.16)
Finally to prove that x n strongly, we write
x n + 1 - = ( I - α n F ) ( J r n x n - ) + α n ( γ ϕ ( x n ) - F ) .
Apply Lemma 2.9 to get
x n + 1 - 2 ( 1 - α n γ ̄ ) 2 x n - 2 + 2 α n γ ϕ ( x n ) - F , J ( x n + 1 - ) ( 1 - α n γ ̄ ) 2 x n - 2 + 2 α n γ ϕ ( x n ) - γ ϕ ( ) , J ( x n + 1 - ) + 2 α n γ ϕ ( ) - F , J ( x n + 1 - ) ( 1 - α n γ ̄ ) 2 x n - 2 + α n γ x n - 2 + x n + 1 - 2 + 2 α n γ ϕ ( ) - F , J ( x n + 1 - )
It follows that
x n + 1 - 2 ( 1 - α n γ ̄ ) 2 + α n γ 1 - α n γ x n - 2 + 2 α n 1 - α n γ γ ϕ ( ) - F , J ( x n + 1 - ) 1 - 2 α n ( γ ̄ - γ ) 1 - α n γ x n - 2 + 2 α n ( γ ̄ - γ ) 1 - α n γ 1 γ ̄ - γ γ ϕ ( ) - F , J ( x n + 1 - ) + α n γ ̄ 2 2 ( γ ̄ - γ ) M 1 ,

where M 1 = sup n 1 x n - 2 . By Lemma 2.2 and (3.16), we see that x n .

Remark 3.4. If γ = 1, F is the identity operator and ϕ(x n ) = u in our results, we can obtain Theorems 3.1, 4.1, 4.2, 4.4 and Lemma 4.3 of Hong-Kun Xu [5].

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Hubei Normal University

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© Wen and Hu; licensee Springer. 2012

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