Open Access

Convergence theorem for an iterative algorithm of λ-strict pseudocontraction

Fixed Point Theory and Applications20112011:95

https://doi.org/10.1186/1687-1812-2011-95

Received: 14 July 2011

Accepted: 6 December 2011

Published: 6 December 2011

Abstract

In this article, we prove strong convergence of sequence generated by the following iteration sequence for a class of Lipschitzian pseudocontractive mapping T:

x n + 1 = β n u + ( 1 - β n ) [ α n T x n + ( 1 - α n ) x n ]

whenever {α n } and {β n } satisfy the appropriate conditions.

2000 AMS Subject Classification: 47H06; 47J05; 47J25; 47H10; 47H17.

Keywords

λ-strict pseudocontraction 2-uniformly smooth Banach space modified Mann iteration strong convergence

1. Introduction

Let T be a pseudocontractive mapping defined on a real smooth Banach space E. We consider the problem of finding a solution z E of the fixed point equation x = Tx. One classical way to study pseudocontractive mappings is to use a strong pseudocontraction to approximate a pseudocontractive mapping T. More precisely take t (0, 1) and u E define a strong pseudocontraction T t by T t x = tu + (1 - t)Tx. In [1, Corollary 2],Deimling proves that T t has a unique fixed point x t , i.e.,
x t = t u + ( 1 - t ) T x t .
(1.1)
This implicit iteration was introduced by Browder [2] for a nonexpansive mapping T in Hilbert space. Halpern [3] was the first who introduced the following explicit iteration scheme for a nonexpansive mapping T which was referred to as Halpern iteration: for u, x0 K, α n [0, 1],
x n + 1 = α n u + ( 1 - α n ) T x n .
(1.2)

Convergence of this two schemes have been studied by many researchers with various types of additional conditions. For the studies of a nonexpansive mapping T, see Bruck [4, 5], Reich [6, 7], Song-Xu [8], Takahashi-Ueda [9], Suzuki [10], and many others. For the studies of a continuous pseudocontractive mapping T, see Morales-Jung [11], Schu [12], Chidume-Zegeye [13], Chidume-Udomene [14], Udomene [15], Chidume-Ofoedu [16], Chen-Song-Zhou [17, 18], Song [1921], Song-Chen [22, 23] and others. The following results play a key role in proving strong convergence of Halpern iteration.

Theorem 1.1 [11, 22, 23] Let E be a reflexive Banach space which has both the fixed point property for nonexpansive self-mappings and a uniformly Gâteaux differ-entiable norm or be a reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Assume that K is a nonempty, closed and convex subset of E. Suppose that T is a continuous pseudocontractive mapping from K into E with F ( T ) . Then, as t → 0, x t , defined by (1.1) converges strongly to a fixed point of T.

Theorem 1.2. [22] Let K be nonempty, closed and convex subset of a Banach space E with a uniformly Gâteaux differentiable norm and let T : KK be a continuous pseudocontractive mapping with a fixed point. Assume that there exists a bounded sequence {x n } such that limn→∞x n - Tx n = 0 and p = limt→0z t exists, where {z t } is defined by (1.1). Then,
limsup n u - p , J ( x n - p ) 0 .
Mann [24] introduced the following iteration for T in a Hilbert space:
x n + 1 = α n x n + ( 1 - α n ) T x n ,
(1.3)
where {α n } is a sequence in [0, 1]. Latterly, Reich [25] studied this iteration in a uniformly convex Banach space with a Fré chet differentiable norm, and obtained that if T has a fixed point and n = 0 α n ( 1 - α n ) = , then the sequence {x n } converges weakly to a fixed point of T. This Mann's iteration process has extensively been studied over the last 20 years for constructions of fixed points of nonlinear mappings and for solving nonlinear operator equations involving monotone, accretive and pseudocontractive operators (see, e.g., [16, 2634] and others). In an infinite-dimensional Hilbert space, the classical Mann's iteration algorithm (1.3) has, in general, only weak convergence, even for nonexpansive mappings. In order to get strong convergence result, one has to modify the Mann's iteration algorithm. Several attempts have been made and many important results have been reported (see, e.g., [1216, 3537] and others). Recently, Zhou [37] obtained strong convergence theorem for the following iterative sequence in a 2-uniformly smooth Banach space: for u, x0 E and λ-strict pseudocontraction T,
x n + 1 = β n u + γ n x n + ( 1 - β n - γ n ) [ α n T x n + ( 1 - α n ) x n ] ,
(1.4)
where {α n }, {β n } and {γ n } in (0, 1) satisfy:
  1. (i)

    a α n λ K 2 for some a > 0 and for all n ≥ 0;

     
  2. (ii)

    lim n β n = 0 and n = 1 β n = ;

     
  3. (iii)

    lim n | α n + 1 - α n | = 0 ;

     
  4. (iv)

    0 < liminf n γ n limsup n γ n < 1 .

     

Very recently, Zhang and Su [38] extended Zhou's results to q-uniformly smooth Banach space. However, the above results excluded γ n ≡ 0 and γ n 1 n + 1 .

In this article, we deal with iterative schemes generated by the following iterative sequence (in (1.4), γ n ≡ 0) for λ-strict pseudocontraction T:
x n + 1 = β n u + ( 1 - β n ) [ α n T x n + ( 1 - α n ) x n ] ,
(1.5)
and obtain its strong convergence whenever {α n } and {β n } satisfy the following conditions:
  1. (i)

    α n a , λ K 2 such that n = 1 | α n + 1 - α n | < ;

     
  2. (ii)

    lim n β n = 0 , n = 1 β n = and n = 1 | β n + 1 - β n | < .

     

Our result not only complements and develops corresponding ones of Zhou [37, Theorem 2.3] (see also Zhang and Su [38, Theorem 4.1], where γ n ≡ 0), but also extend main result of Chidume-Chidume [35] and Kim-Xu [36] from nonexpansive mappings to λ-strict pseudocontractions.

2. Preliminaries

Throughout this article, a Banach space E will always be over the real scalar field. We denote its norm by · and its dual space by E*. The value of x* E* at y E is denoted by 〈y, x〉 and the normalized duality mapping from E into 2 E * is denoted by J, that is, J(x) = {f E* : 〈x, f〉 = xf, x = f}. Let F(T) = {x E : Tx = x} be the set of all fixed point of a mapping T.

Recall that a mapping T with domain D(T) and range R(T) in Banach space E is called strongly pseudo-contractive if, for all x, y D(T), there exist k (0, 1) and j(x - y) J(x - y) such that
T x - T y , j ( x - y ) k | | x - y | | 2
(2.1)
or, equivalently,
( x - T x ) - ( y - T y ) , j ( x - y ) ( 1 - k ) | | x - y | | 2
(2.2)
while T is said to be pseudo-contractive if (2.1) or (2.2) holds for k = 1. A mapping T is said to be Lipschitzian if, for all x, y K, there exists L > 0 such that
| | T x - T y | | L | | x - y | | .
A mapping T is called non-expansive if L = 1 and, further, T is said to be contractive if L < 1. An important class of mappings closely related to the class of pseudo-contractive mappings is that of accretive mappings. A mapping A is accretive if and only if (I - A) is pseudo-contractive. The accretive mappings were independently introduced by Browder [39] and Kato [40] in 1967. The importance of these mappings is well known. A mapping T is called λ-strictly pseudocontractive, if for all x, y D(T), there exists λ (0, 1) and j(x - y) J(x - y) such that
T x - T y , j ( x - y ) | | x - y | | 2 - λ | | x - y - ( T x - T y ) | | 2 .
(2.3)

It is obvious that λ-strictly pseudocontractive mapping is Lipschitzian with L = λ + 1 λ . The class of nonexpansive mappings is a subclass of strictly pseudocontractive mappings in Hilbert space, but the converse implication may be false. We remark that the class of strongly pseudo-contractive mappings is independent from the class of λ-strict pseudo-contractions. This can be seen from the existing examples (see, e.g., [30, 37]).

Let S(E) := {x E; x = 1} denote the unit sphere of a Banach space E. The space E is said to have (i) a Gâteaux differentiable norm (we also say that E is smooth), if the limit
lim t 0 | | x + t y | | - | | x | | t
(2.4)
exists for each x, y S(E); (ii) a uniformly Gâteaux differentiable norm, if for any y in S(E), the limit (2.4) is uniformly attained for x S(E); (iii) a Fréchet differentiable norm, if for any x S(E), the limit (2.4) is attained uniformly for y S(E); (iv) a uniformly Fréchet differentiable norm (we also say that E is uniformly smooth), if the limit (2.4) is attained uniformly for all (x, y) S(E) × S(E); (v) fixed point property for non-expansive self-mappings, if each non-expansive self-mapping defined on any bounded, closed convex subset K of E has at least one fixed point. Let ρ E : [0, ∞) → [0, ∞) be the modulus of smoothness of E defined by
ρ E ( t ) = sup 1 2 ( | | x + y | | + | | x - y | | ) - 1 : x S ( E ) , | | y | | t .

Let q > 1. A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρ E (t) < ct q . It is well known that E is uniformly smooth if and only if lim t 0 ρ E ( t ) t = 0 . If E is q-uniformly smooth, then E is uniformly smooth, and hence the norm of E is uniformly Fréchet differentiable, in particular, the norm of E is Fréchet differentiable. Typical example of uniformly smooth Banach spaces is L p (p > 1). More precisely, L p is min{p, 2}-uniformly smooth for every p > 1.

Lemma 2.1.(Zhou [37]) Let E be a real 2-uniformly smooth Banach space with the best smooth constant K, C be a nonempty subset of E, and let T : CC be a λ-strict pseudocontraction. For any α (0, 1), we define T α = (1 - α)x + αTx. Then, as α 0 , λ K 2 , T α : C C is nonexpansive such that F(T α ) = F(T).

Lemma 2.2. (Liu [34] and Xu [41]) Let {a n } be a sequence of nonnegative real numbers satisfying the property:
a n + 1 ( 1 - t n ) a n + b n + t n c n ,

where {t n }, {b n } and {c n } satisfy the restrictions:

(i) n = 0 t n = ; (ii) n = 0 b n < + ; (iii) limsup n c n 0 .

Then, {a n } converges to zero as n → ∞.

3. Main result

Theorem 3.1 Let E be a real 2-uniformly smooth Banach space with the best smooth constant K and let C be a nonempty, closed and convex subset of E. Suppose that T : CC is a λ-strict pseudocontraction with F ( T ) . Given u, x0 C, a sequence {x n } is generated by
y n = α n T x n + ( 1 - α n ) x n , x n + 1 = β n u + ( 1 - β n ) y n ,
(3.1)

where {β n } and {α n } in (0, 1) satisfy the following control conditions:

(i) α n a , λ K 2 for some constant a 0 , λ K 2 such that n = 1 | α n + 1 - α n | < ;

(ii) lim n β n = 0 , n = 1 β n = and n = 1 | β n + 1 - β n | < .

Then, {x n } converges strongly to a fixed point of T.

Proof. The proof will be divided into four steps.

Step 1. The sequence {x n } is bounded. Let T α n = α n T + ( 1 - α n ) I . Then, T α n is nonexpansive for every n by Lemma 2.1 and so, for p F(T), we have
| | x n + 1 - p | | = | | β n ( u - p ) + ( 1 - β n ) ( T α n x n - p ) | | β n | | u - p | | + ( 1 - β n ) | | T α n x n - p | | β n | | u - p | | + ( 1 - β n ) | | x n - p | | max { | | x n - p | | , | | u - p | | } max { | | x 0 - p | | , | | u - p | | } .

Consequently, both {x n } and {y n } are bounded. This implies the boundedness of {Tx n } from the inequality | | T x n - p | | 1 + λ λ | | x n - p | | .

Let M > 0 be a constant such that M ≥ supn{u, x n , Tx n }.

Step 2. Since y n = T α n x n = α n T x n + ( 1 - α n ) x n , then
| | y n | | = | | α n T x n + ( 1 - α n ) x n | | α n | | T x n | | + ( 1 - α n ) | | x n | | M .
Furthermore, we have
| | y n + 1 - y n | | = | | T α n + 1 x n + 1 - T α n x n | | | | T α n + 1 x n + 1 - T α n + 1 x n | | + | | T α n + 1 x n - T α n x n | | | | x n + 1 - x n | | + | | α n + 1 T x n + ( 1 - α n + 1 ) x n - α n T x n - ( 1 - α n ) x n | | | | x n + 1 - x n | | + | α n + 1 - α n | | | x n - T x n | | | | x n + 1 - x n | | + 2 M | α n + 1 - α n | .
(3.2)
From (3.1), it follows
| | x n + 2 - x n + 1 | | = | | β n + 1 u + ( 1 - β n + 1 ) y n + 1 - β n u - ( 1 - β n ) y n | | | β n + 1 - β n | ( | | u | | + | | y n + 1 | | ) + ( 1 - β n ) | | y n + 1 - y n | | 2 M | β n + 1 - β n | + ( 1 - β n ) | | y n + 1 - y n | | .
(3.3)
Substituting (3.2) into (3.3) yields
| | x n + 2 - x n + 1 | | ( 1 - β n ) | | x n + 1 - x n | | + 2 M | α n + 1 - α n | + 2 M | β n + 1 - β n | .
From the assumptions on {α n } and {β n } and using Lemma 2.3, we conclude that
lim n | | x n + 1 - x n | | = 0 .
(3.4)
From the definition of x n and since limn→∞β n = 0, it follows
lim n | | x n + 1 - y n | | = lim n β n | | u - T x n | | = 0 .
Combining (3.4), we have
lim n | | x n - y n | | = 0 .
(3.5)
Thus, we obtain
lim n | | x n - T x n | | = lim n | | x n - y n | | α n = 0 .
Step 3. There exists z F(T) such that
limsup n u - z , J ( x n + 1 - z ) 0 .

Since E is 2-uniformly smooth, then E is a reflexive Banach space which has both the fixed point property for non-expansive self-mappings and a uniformly Gâteaux differentiable norm. Then, from Theorem 1.1, as t → 0, x t , defined by (1.1) converges strongly to a fixed point z of T. The desired conclusion follows from Theorem 1.2.

Step 4. lim n x n = z . In fact,
| | x n + 1 - z | | 2 = β n ( u - z ) + ( 1 - β n ) ( y n - z ) , J ( x n + 1 - z ) β n u - z , J ( x n + 1 - z ) + ( 1 - β n ) | | T α n x n - z | | | | J ( x n + 1 - z ) | | β n u - z , J ( x n + 1 - z ) + ( 1 - β n ) | | x n - z | | 2 + | | x n + 1 - z | | 2 2 ( 1 - β n ) | | x n - z | | 2 2 + | | x n + 1 - z | | 2 2 + β n u - z , J ( x n + 1 - z ) ,
which implies that
| | x n + 1 - z | | 2 ( 1 - β n ) | | x n - z | | 2 + 2 β n u - z , J ( x n + 1 - z ) ,
(3.6)

and hence limn→∞x n - z = 0 because of Lemma 2.2. This completes the proof.

Remark 1. Theorem 3.1 is applicable to l p and L p for all p ≥ 2, however, we do not know whether it works for L p for 1 < p < 2.

Remark 2. In Theorem 3.1, if the condition n = 1 | β n + 1 - β n | < is replaced by lim n β n + 1 β n = 1 , the conclusion still holds.

Remark 3. Theorem 3.1 not only complements and develops corresponding result of Zhou [37, Theorem 3.2] (see also Zhang and Su [38, Theorem 4.1] where γ n ≡ 0), but also extend main result of Chidume-Chidume [35] and Kim-Xu [36] from nonexpansive mappings to λ-strict pseudocontractions.

Corollary 3.2 Let E be a reflexive Banach space which has both the fixed point property for non-expansive self-mappings and a uniformly Gâteaux differentiable norm and let C be a nonempty, closed and convex subset of E. Suppose that T : CC is a nonexpansive mapping with F ( T ) . Given u, x0 C, a sequence {x n } is generated by (3.1), where {α n } and {β n } in (0,1) satisfy the following control conditions:

(i) α n (0, 1) such that n = 1 | α n + 1 - α n | < ;

(ii) lim n β n = 0 , n = 1 β n = ;

(iii) either n = 1 | β n + 1 - β n | < or lim n β n + 1 β n = 1 .

Then, {x n } converges strongly to a fixed point of T.

Proof. Let T α n = α n T + ( 1 - α n ) I . Clearly, T α n is nonexpansive and F ( T ) = F ( T α n ) for each n. Therefore, following the same proof technique of Theorem 3.1, the desired result is obtained.

Remark 4. Theorem 3.1 of Chidume-Chidume [35] and Theorem 1 of Kim-Xu [36] can be regarded as a special case of Corollary 3.2, respectively. In fact, if α n δ (0, 1) in Corollary 3.2, then Theorem 3.1 of Chidume-Chidume [35] is reached; if in Corollary 3.2, E is a uniformly smooth Banach space and the conditions limn→∞a n = 1 and n = 1 ( 1 - α n ) = are added, then Theorem 1 of Kim-Xu [36] is obtained.

Declarations

Acknowledgements

The authors were thank the editor and the anonymous referee for useful comments and valuable suggestions on the language and structure of our manuscript.

Authors’ Affiliations

(1)
College of Mathematics and Information Science, Henan Normal University

References

  1. Deimling K: Zero of accretive operators. Manuscripta Math 1974, 13: 365–374.MathSciNetView ArticleGoogle Scholar
  2. Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proc Natl Acad Sci USA 1965, 53: 1272–1276.MathSciNetView ArticleGoogle Scholar
  3. Halpern B: Fixed points of nonexpansive maps. Bull Amer Math Soc 1967, 73: 957–961.MathSciNetView ArticleGoogle Scholar
  4. Bruck RE: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J Math 1979, 32: 107–116.MathSciNetView ArticleGoogle Scholar
  5. Bruck RE: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces. Israel J Math 1981, 38: 304–314.MathSciNetView ArticleGoogle Scholar
  6. Reich S: Approximating zeros of accretive operators. Proc Amer Math Soc 1975, 51: 381–384.MathSciNetView ArticleGoogle Scholar
  7. Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math Anal Appl 1980, 75: 287–292.MathSciNetView ArticleGoogle Scholar
  8. Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. J Math Anal Appl 2008, 338: 152–161.MathSciNetView ArticleGoogle Scholar
  9. Takahashi W, Ueda Y: On Reich's strong convergence for resolvents of accretive operators. J Math Anal Appl 1984, 104: 546–553.MathSciNetView ArticleGoogle Scholar
  10. Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl 2005,2005(1):103–123.View ArticleGoogle Scholar
  11. Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc Amer Math Soc 2000, 128: 3411–3419.MathSciNetView ArticleGoogle Scholar
  12. Schu J: Approximating fixed points of Lipschitzian pseudocontractive mappings. Houston J Math 1993, 19: 107–115.MathSciNetGoogle Scholar
  13. Chidume CE, Zegeye H: Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc Amer Math Soc 2004, 132: 831–840.MathSciNetView ArticleGoogle Scholar
  14. Chidume CE, Udomene A: Strong convergence theorems for uniformly continuous pseudo-contractive maps. J Math Anal Appl 2006, 323: 88–99.MathSciNetView ArticleGoogle 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.MathSciNetView ArticleGoogle Scholar
  16. Chidume CE, Ofoedu EU: A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings. Nonlinear Anal 2007, 67: 307–315.MathSciNetView ArticleGoogle Scholar
  17. Chen R, Song Y, Zhou H: Viscosity approximation methods for continuous pseudocontrac-tive mappings. Acta Math Sin Chin Ser 2006, 49: 1275–1278.MathSciNetView ArticleGoogle Scholar
  18. Chen R, Song Y, Zhou H: Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings. J Math Anal Appl 2006, 314: 701–709.MathSciNetView ArticleGoogle Scholar
  19. Song Y: A note on the paper "A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings". Nonlinear Anal 2008, 68: 3047–3049.MathSciNetView ArticleGoogle Scholar
  20. Song Y: On a Mann type implicit iteration process for continuous pseudo-contractive mappings. Nonlinear Anal 2007, 67: 3058–3063.MathSciNetView ArticleGoogle Scholar
  21. Song Y: Strong convergence of viscosity approximation methods with strong pseudocontrac-tion for Lipschitz pseudocontractive mappings. Positivity 2009, 13: 643–655.MathSciNetView ArticleGoogle Scholar
  22. Song Y, Chen R: Convergence theorems of iterative algorithms for continuous pseudocon-tractive mappings. Nonlinear Anal 2007, 67: 486–497.MathSciNetView ArticleGoogle Scholar
  23. Song Y, Chen R: An approximation method for continuous pseudocontractive mappings. J Inequal Appl 2006: 1–9. Article ID 28950,Google Scholar
  24. Mann WR: Mean value methods in iteration. Proc Amer Math Soc 1953, 4: 506–510.MathSciNetView ArticleGoogle Scholar
  25. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl 1979, 67: 274–276.MathSciNetView ArticleGoogle Scholar
  26. Chidume CE, Moore C: Fixed point iteration for pseudo-contractive maps. Proc Amer Math Soc 1999,127(4):1163–1170.MathSciNetView ArticleGoogle Scholar
  27. Chidume CE: Iterative approximation of Lipschitz strictly pseudo-contractive mappings. Proc Amer Math Soc 1987,99(2):283–288.MathSciNetGoogle Scholar
  28. Chidume CE: Approximation of fixed points of strongly pseudo-contractive mappings. Proc Amer Math Soc 1994,120(2):545–551.MathSciNetView ArticleGoogle Scholar
  29. Chidume CE: Global iteration schemes for strongly pseudo-contractive maps. Proc Amer Math Soc 1998,126(9):2641–2649.MathSciNetView ArticleGoogle Scholar
  30. Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudocontractions. Proc Amer Math Soc 2001,129(8):2359–2363.MathSciNetView ArticleGoogle Scholar
  31. Deng L: On Chidume's open problems. J Math Anal Appl 1993,174(2):441–449.MathSciNetView ArticleGoogle Scholar
  32. Deng L, Ding XP: Iterative approximation of Lipschitz strictly pseudo-contractive mappings in uniformly smooth Banach spaces. Nonlinear Anal 1995,24(7):981–987.MathSciNetView ArticleGoogle Scholar
  33. Hicks TL, Kubicek JR: On the Mann iteration process in Hilbert space. J Math Anal Appl 1977, 59: 498–504.MathSciNetView ArticleGoogle Scholar
  34. Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 1995, 194: 114–125.MathSciNetView ArticleGoogle Scholar
  35. Chidume CE, Chidume CO: Iterative approximation of fixed points of nonexpansive mappings. J Math Anal Appl 2006, 318: 288–295.MathSciNetView ArticleGoogle Scholar
  36. Kim TH, Xu HK: Strong convergence of modified Mann iterations. Nonlinear Anal 2005, 61: 51–60.MathSciNetView ArticleGoogle Scholar
  37. Zhou H: Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces. Nonlinear Anal 2008,69(9):3160–3173.MathSciNetView ArticleGoogle Scholar
  38. Zhang H, Su Y: Convergence theorems for strict pseudo-contractions in q-uniformly smooth Banach spaces. Nonlinear Anal 2009, 71: 4572–4580.MathSciNetView ArticleGoogle Scholar
  39. Browder FE: Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces. Bull Amer Math Soc 1967, 73: 470–475. Part 2 (1976)MathSciNetView ArticleGoogle Scholar
  40. Kato T: Nonlinear semigroups and evolution equations. J Math Soc Japan 1967, 19: 508–520.MathSciNetView ArticleGoogle Scholar
  41. Xu HK: Iterative algorithms for nonlinear operators. J London Math Soc 2002, 66: 240–256.MathSciNetView ArticleGoogle Scholar

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© Chai and Song; licensee Springer. 2011

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