Advertisement

Strong and weak convergence of an implicit iterative process for pseudocontractive semigroups in Banach space

  • Jing Quan
  • Shih-sen Chang
  • Min Liu
Open Access
Research

Abstract

The purpose of this article is to study the strong and weak convergence of implicit iterative sequence to a common fixed point for pseudocontractive semigroups in Banach spaces. The results presented in this article extend and improve the corresponding results of many authors.

Keywords

Banach Space Nonexpansive Mapping Real Banach Space Common Fixed Point Nonempty Closed Convex Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction and preliminaries

Throughout this article we assume that E is a real Banach space with norm ||·||, E* is the dual space of E; 〈·, ·〉 is the duality pairing between E and E*; C is a nonempty closed convex subset of E; ℕ denotes the natural number set; ℜ+ is the set of nonnegative real numbers; The mapping J : E 2 E * Open image in new window defined by
J ( x ) = f * E * : x , f * = x 2 ; f * = x , x E Open image in new window
(1)

is called the normalized duality mapping. We denote a single valued normalized duality mapping by j.

Let T: CC be a nonlinear mapping; F(T) denotes the set of fixed points of mapping T, i.e., F(T) := {xC, x = Tx}. We use "→" to stand for strong convergence and "⇀" for weak convergence. For a given sequence {x n } ⊂ C, let ω w (x n ) denote the weak ω-limit set.

Recall that T is said to be pseudocontractive if for all x, yC, there exists j(x - y) ∈ J(x - y) such that
T x - T y , j ( x - y ) x - y 2 ; Open image in new window
(2)
T is said to be strongly pseudocontr active if there exists a constant α ∈ (0,1), such that for any x, yC, there exists j(x - y) ∈ J(x - y)
T x - T y , j ( x - y ) α x - y 2 . Open image in new window
(3)

In recent years, many authors have focused on the studies about the existence and convergence of fixed points for the class of pseudocontractions. Especially in 1974, Deimling [1] proved the following existence theorem of fixed point for a continuous and strong pseudocontraction in a nonempty closed convex subset of Banach spaces.

Theorem D. Let E be a Banach space, C be a nonempty closed convex subset of E and T: CC be a continuous and strong pseudocontraction. Then T has a unique fixed point in C.

Recently, the problems of convergence of an implicit iterative algorithm to a common fixed point for a family of nonexpansive mappings or pseudocontractive mappings have been considered by several authors, see [2, 3, 4, 5]. In 2001, Xu and Ori [2] firstly introduced an implicit iterative x n = α n xn-1+ (1 - α n )T n x n , n ∈ ℕ, x0C for a finite family of nonexpansive mappings T i i = 1 N Open image in new window and proved some weak convergence theorems to a common fixed point for a finite family of nonexpansive mappings in a Hilbert space. In 2004, Osilike [3] improved the results of Xu and Ori [2] from nonexpansive mappings to strict pseudocontractions in the framework of Hilbert spaces. In 2006, Chen et al. [4] extended the results of Osilike [3] to more general Banach spaces.

On the other hand, the convergence problems of semi-groups have been considered by many authors recently. Suzuki [6] considered the strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Xu [7] gave strong convergence theorem for contraction semigroups in Banach spaces. Chang et al. [8] proved the strong convergence theorem for nonexpansive semi-groups in Banach space. He also studied the weak convergence problems of the implicit iteration process for Lipschitzian pseudocontractive semi-groups in the general Banach spaces [9]. The pseudocontractive semi-groups is defined as follows.

Definition 1.1 (1) One-parameter family T: = {T(t): t ≥ 0} of mappings from C into itself is said to be a pseudo-contraction semigroup on C, if the following conditions are satisfied:

(a). T(0)x = x for each xC;

(b). T(t + s)x = T(s)T(t) for any t, s ∈ ℜ+ and xC;

(c). For any xC, the mapping tT(t)x is continuous;

(d). For all x, yC, there exists j(x - y) ∈ J(x - y) such that
T ( t ) x - T ( t ) y , j ( x - y ) x - y 2 , f o r a n y t > 0 . Open image in new window
(4)

(2) A pseudo-contraction semigroup of mappings from C into itself is said to be a Lipschitzian if the condition (a)-(d) and following condition (f) are satisfied.

(f) there exists a bounded measurable function L: [0, ∞) → [0, ∞) such that for any x, yC,
T ( t ) x - T ( t ) y L ( t ) x - y Open image in new window
for any t > 0. In the sequel, we denote it by
L = sup t 0 L ( t ) < Open image in new window
(5)

Cho et al. [10] considered viscosity approximations with continuous strong pseudocontractions for a pseudocontraction semigroup and prove the following theorem.

Theorem Cho. Let E be a real uniformly convex Banach space with a uniformly Gâ teaux differentiable norm, and C be a nonempty closed convex subset of E. Let T(t): t ≥ 0 be a strongly continuous L-Lipschitz semigroup of pseudocontractions on C such that Ω Open image in new window, where Ω is the set of common fixed points of semi-group T(t). Let f: CC be a fixed bounded, continuous and strong pseudocontraction with the coefficient α in (0,1), let α n and t n be sequences of real numbers satisfying α n ∈ (0, 1), t n > 0, and lim n t n = lim n α n t n = 0 Open image in new window; Let {x n } be a sequence generated in the following manner:
x n = ( 1 - α n ) f ( x n ) + α n T ( t n ) x n , n 1 . Open image in new window
(6)

Assume that LIM||T(t)x n - T(t)x*|| ≤ ||x n - x*||, ∀x*K, t ≥ 0, where K := {x*C: Φ(x*) = minxCΦ(x)} with Φ(x) = LIM||x n - x||2, ∀xC. Then x n converges strongly to x* ∈ Ω which solves the following variational inequality: 〈(I - f)x*, j(x* - x)〉 ≤ 0, ∀x ∈ Ω.

Qin and Cho [11] established the theorems of weak convergence of an implicit iterative algorithm with errors for strongly continuous semigroups of Lipschitz pseudocontractions in the framework of real Banach spaces.

Theorem Q. Let E be a reflexive Banach space which satisfies Opial's condition and K a nonempty closed convex subset of E. Let T : = { T ( t ) : t 0 } Open image in new window be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with F : = t 0 F ( T ( t ) ) Open image in new window; Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x n } be a sequence generated by the following iterative process:
x 0 K ; x n = α n x n - 1 + β n T ( t n ) x n + γ n u n ; n 1 ; Open image in new window
(7)

where {α n }, {β n }, {γ n } are sequences in (0,1), {t n } is a sequence in (0, ∞) and {u n } is a bounded sequence in K. Assume that the following conditions are satisfied:

(a) α n + β n + γ n = 1;

(b) lim n t n = lim n α n + γ n t n = 0 Open image in new window.

Then the sequence {x n } generated in (7) converges weakly to a common fixed point of the semigroup T : = { T ( t ) : t 0 } Open image in new window;

Agarwal et al. [12] studied strongly continuous semigroups of Lipschitz pseudocontractions and proved the strong convergence theorems of fixed points in an arbitrary Banach space based on an implicit iterative algorithm.

Theorem A. Let E be an arbitrary Banach space and K a nonempty closed convex subset of E. Let T : = { T ( t ) : t 0 } Open image in new window be a strongly continuous semigroup of Lipschitz pseudocontractions from K into itself with F : = t 0 F ( T ( t ) ) Open image in new window. Assume that supt≥0{L(t)} < ∞, where L(t) is the Lipschitz constant of the mapping T(t). Let {x n } be a sequence in
x 0 K ; x n = α n x n - 1 + β n T ( t n ) x n + γ n u n ; n 1 , Open image in new window
(8)

where {α n }, {β n }, {γ n } are sequences in (0,1) such that α n + β n + γ n = 1, {t n } is a sequence in (0, ∞) and {u n } is a bounded sequence in K. Assume that lim n t n = lim n α n + γ n t n = 0 Open image in new window, lim n γ n α n + γ n < Open image in new window and there is a nondecreasing function f: (0, ∞) → (0, ∞) with f(0) = 0 and f(t) > 0 for all t ∈ (0, ∞) such that, for all xC, sup { x - T ( t ) x : t 0 } f ( dist ( x , F ) ) Open image in new window. Then the sequence {x n } converges strongly to a common fixed point of the semigroup T : = { T ( t ) : t 0 } Open image in new window.

The purpose of this article is to prove the strong and weak convergence of implicit iterative process
x n = ( 1 - α n ) x n - 1 + α n T ( t n ) x n , n , x 0 C Open image in new window
(9)

for a pseudocontraction semigroup T: = {T(t): t ≥ 0} in the framework of Banach spaces, which improves and extends the corresponding results of many author's. We need the following Lemma.

Lemma 1.1 [9] Let E be a real reflexive Banach space with Opial condition. Let C be a nonempty closed convex subset of E and T: CC be a continuous pseudocontractive mapping. Then I - T is demiclosed at zero, i.e., for any sequence {x n } ⊂ E, if x n y and ||(I - T)x n || → 0, then (I - T)y = 0.

2 Main results

Theorem 2.1 Let E be a real Banach space and C be a nonempty compact convex subset of E. Let T: = {T(t): t ≥ 0}: CC be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose F ( T ) : = t 0 F ( T ( t ) ) Open image in new window. Let α n and t n be sequences of real numbers satisfying t n > 0, α n ∈ [a, 1) ⊂ (0, 1) and limn→∞α n = 1. Then the sequence {x n } defined by (9) converges strongly to a common fixed point x*F(T) in C.

Proof. We divide the proof into five steps.

(I). The sequence {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x n , n ∈ ℕ, x0C is well defined.

In fact for all n ∈ ℕ, we define a mapping S n as follows:
S n x = ( 1 - α n ) x n - 1 + α n T ( t n ) x , n , x C . Open image in new window
(10)
Then we have
S n x - S n y , j ( x - y ) = α n T ( t n ) x - T ( t n ) y , j ( x - y ) α n x - y 2 . Open image in new window
(11)

So S n is strongly pseudo-contraction, thus from Theorem D, there exists a point x n such that x n = (1 - α n )xn-1+ α n T(t n )x n , that is the sequence {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x n , n ∈ ℕ, x0C is well defined.

(II). Since the common fixed-point set F(T) is nonempty let pF(T). For each pF(T), we prove that limn→∞||x n - p|| exists.

In fact
x n - p 2 = x n - p , j ( x n - p ) = ( 1 - α n ) ( x n - 1 - p ) + α n ( T ( t n ) x n - p ) , j ( x - p ) ( 1 - α n ) x n - 1 - p x n - p + α n x n - p 2 . Open image in new window
(12)
So we get ||x n - p|| ≤ (1 - α n )||xn-1- p|| + α n ||x n - p||, that is
x n - p x n - 1 - p . Open image in new window

This implies that the limit limn→∞||x n - p|| exists.

(III). We prove limn→∞||T(t n )x n - x n || = 0.

The sequence {||x n - p||n∈ℕ} is bounded since limn→∞||x n - p|| exists, so the sequence {x n } is bounded. Since
T ( t n ) x n = x n - ( 1 - α n ) x n - 1 α n x n α n + ( 1 - α n ) x n - 1 α n x n a + ( 1 - α n ) x n - 1 a , Open image in new window
(13)
This shows that {T(t n )x n } is bounded. In view of
x n - T ( t n ) x n = ( 1 - α n ) ( x n - 1 - T ( t n ) x n ) = 1 - α n x n - 1 - T ( t n ) x n Open image in new window
and condition limn→∞α n = 1, we have
lim n T ( t n ) x n - x n = 0 . Open image in new window
(14)

(IV). Now we prove that for all t > 0, limn→∞||T(t)x n - x n || = 0.

Since pseudocontraction semigroup T: = {T(t) : t ≥ 0} is Lipschitian, for any k ∈ ℕ,
T ( ( k + 1 ) t n ) x n - T ( k t n ) x n = T ( k t n ) T ( t n ) x n - T ( k t n ) x n L ( k t n ) T ( t n ) x n - x n L T ( t n ) x n - x n . Open image in new window
(15)
Because limn→∞||T(t n )x n - x n || = 0, so for any k ∈ ℕ,
lim n T ( ( k + 1 ) t n ) x n - T ( k t n ) x n = 0 . Open image in new window
(16)
Since
T ( t ) x n - T t t n t n x n = T t t n t n T t - t t n t n x n - T t t n t n x n L T t - t t n t n x n - x n Open image in new window
(17)
and T(·) is continuous, we have
lim n T t t n t n x n - T ( t ) x n = 0 . Open image in new window
(18)
So from
x n - T ( t ) x n k = 0 t t n - 1 T ( ( k + 1 ) t n ) x n - T ( k t n ) x n + T t t n t n x n - T ( t ) x n , Open image in new window
(19)
and limn→∞||T((k+1)t n )x n - T(kt n )x n || = 0 as well as lim n T t t n t n x n - T ( t ) x n = 0 Open image in new window, we can get
lim n T ( t ) x n - x n = 0 . Open image in new window
(20)

(V). We prove {x n } converges strongly to an element of F(T).

Since C is a compact convex subset of E, we know there exists a subsequence x n j x n Open image in new window, such that x n j x C Open image in new window. So we have lim j T ( t ) x n j - x n j = 0 Open image in new window from limn→∞||T(t)x n - x n || = 0, and
x - T ( t ) x = lim j T ( t ) x n j - x n j = 0 . Open image in new window
(21)

This manifests that xF(T). Because for any pF(T), limn→∞||x n - p|| exists, and lim n x n - x = lim j x n j - x = 0 Open image in new window, we have that {x n } converges strongly to an element of F(T). This completes the proof of Theorem 2.1.

Theorem 2.2 Let E be a reflexive Banach space satisfying the Opial condition and C be a nonempty closed convex subset of E. Let T: = {T(t): t ≥ 0}: CC be a Lipschitian and pseudocontraction semigroup defined by Definition 1.1 with a bounded measurable function L: [0, ∞) → [0, ∞). Suppose F ( T ) : = t 0 F ( T ( t ) ) Open image in new window. Let α n and t n be sequences of real numbers satisfying t n > 0, α n ∈ [a, 1) ⊂ (0,1) and limn→∞α n = 1. Then the sequence {x n } defined by x n = (1 - α n )xn-1+ α n T(t n )x n , x0C, n ∈ ℕ, converges weakly to a common fixed point x*F(T) in C.

Proof. It can be proved as in Theorem 2.1, that for each pF(T), the limit limn→∞||x n - p|| exists and {T(t n )x n } is bounded, for all t > 0, limn→∞||T(t)x n - x n || = 0. Since E is reflexive, C is closed and convex, {x n } is bounded, there exist a subsequence x n j x n Open image in new window such that x n j x Open image in new window. For any t > 0, we have lim n j T ( t ) x n j - x n j = 0 Open image in new window. By Lemma 1.1, xF(T(t)), ∀t > 0. Since the space E satisfies Opial condition, we see that ω w (x n ) is a singleton. This completes the proof.

Remark 2.1 There is no other condition imposed on t n in the Theorems 2.1 and 2.2 except that in the definition of pseudo-contraction semigroups. So our results improve corresponding results of many authors such as [10, 11, 12], of cause extend many results in [4, 5, 6, 7, 8].

Notes

Acknowledgements

This work was supported by National Research Foundation of Yibin University (No.2011B07).

References

  1. 1.
    Deimling K: Zeros of accretive operators. Manuscripta Math 1974, 13: 365–374. 10.1007/BF01171148MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Numer Funct Anal Optim 2001, 22: 767–773. 10.1081/NFA-100105317MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Osilike MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J Math Anal Appl 2004, 294: 73–81. 10.1016/j.jmaa.2004.01.038MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen RD, Song YS, Zhai HY: Convergence theorems for implicit iteration press for a finite family of continuous pseudocontractive mappings. J Math Anal Appl 2006, 314: 701–709. 10.1016/j.jmaa.2005.04.018MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhou HY: Convergence theorems of common fixed points for a finite family of Lipschitzian pseudocontractions in Banach spaces. Nonlinear Anal 2008, 68: 2977–2983. 10.1016/j.na.2007.02.041MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc Am Math Soc 2003, 131: 2133–2136. 10.1090/S0002-9939-02-06844-2MATHCrossRefGoogle Scholar
  7. 7.
    Xu HK: A strong convergence theorem for contraction semigruops in Banach spaces. Bull Aust Math Soc 2005, 72: 371–379. 10.1017/S000497270003519XMATHCrossRefGoogle Scholar
  8. 8.
    Chang SS, Yang L, Liu JA: Strong convergence theorem for nonexpansive semigroups in Banach spaces. Appl Math Mech 2007, 28: 1287–1297. 10.1007/s10483-007-1002-xMathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhang SS: Convergence theorem of common fixed points for Lipschitzian pseudo-contraction semigroups in Banach spaces. Appl Math Mech (English Edition) 2009, 30(2):145–152. 10.1007/s10483-009-0202-yMATHCrossRefGoogle Scholar
  10. 10.
    Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Qin X, Cho SY: Implicit iterative algorithms for treating strongly continuous semigroups of Lipschitz pseudocontractions. Appl Math Lett 2010, 23: 1252–1255. 10.1016/j.aml.2010.06.008MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Agarwal RP, Qin X, Kang SM: Strong convergence theorems for strongly continuous semigroups of pseudocontractions. Appl Math Lett 2011, 24: 1845–1848. 10.1016/j.aml.2011.05.003MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Quan et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of MathematicsYibin UniversityYibinChina
  2. 2.College of Statistics and MathematicsYunnan University of Finance and EconomicsKunmingChina

Personalised recommendations