General Mathematics Vol. 17, No. 3 (2009), 81–90
Convergence of an iterative algorithm to a
fixed point of uniformly L-Lipschitzian
mapping 1
Nikhat Bano, Nazir Ahmad Mir
Abstract
Let K be a non-empty closed convex subset of an arbitrary Banach space E. Let T : K → K be uniformly L Lipschitzian with
F (T ) 6= ∅. Let {kn } ⊆ [1, ∞) be a sequence with lim kn = 1.For
n→∞
any x0 ∈ K and fixed u ∈ K, define the sequence {xn } by
xn+1 = αn u + β n xn + γ n T n xn ,
where {αn } , {β n } , {γ n } are real sequences in (0, 1) satisfying some
conditions.Then the sequence {xn } converges strongly to a fixed
point of T.
Asymptotically nonexpansive, Asymptotically pseudocontractive, uniformly L-Lipschitzian, Banach space.
1
Received 26 August, 2008
Accepted for publication (in revised form) 22 January, 2009
81
82
1
N. Bano, N. A. Mir
Introduction and preliminaries
Let K be the closed convex subset of a real Banach space E with the dual
∗
space E ∗ . The normalized duality mapping J : E → 2E is defined as
©
ª
J (x) = f ∈ E ∗ : hx, f i = kxk2 = kf k2 , kf k = kxk
∀x ∈ E,
where h., .i denotes the duality pairing between E and E ∗ . The single valued
normalized duality mapping is denoted by j. Let F (T ) denotes the set of
fixed points of the mapping T : K → K.
Definition 1 The mapping T : K → K is said to be non-expansive if
kT x − T yk ≤ kx − yk for all x, y ∈ K.
Definition 2 The mapping T : K → K is said to be uniformly L- Lipschitzian if there exists L > 0 such that ∀n ≥ 1
kT n x − T n yk ≤ L kx − yk for all x, y ∈ K.
Definition 3 The mapping T : K → K is said to be asymptotically non
expansive if there exists a sequence {kn } ⊆ [1, ∞) with kn → 1 such that
∀n ≥ 1
kT n x − T n yk ≤ kn kx − yk for all x, y ∈ K.
Definition 4 The mapping T : K → K is said to be asymptotically pseudocontractive if there exists a sequence {kn } ⊆ [1, ∞) with kn → 1 and
j (x − y) ∈ J (x − y) such that ∀n ≥ 1
hT n x − T n y, j (x − y)i ≤ kn kx − yk2 for all x, y ∈ K.
Convergence of an iterative algorithm ....
83
It is easy to see that every asymptotically nonexpansive mapping is
uniformly L- Lipschitzian and every asymptotically non-expansive mapping
is asymptotically pseudocontractive but the converse is not true. The class
of asymptotically pseudocontractive mappings was introduced by Schu[6]
who proved the strong convergence theorem for the iterative approximation
of fixed points of asymptotically pseudocontractive mappings in Hilbert
space. Chang [2] extended the result of Schu to real uniformly smooth
Banach space. In[8], there has been introduced an iteration scheme as
xn+1 = αn u + β n xn + γ n T xn , n ≥ 0,
for given x0 ∈ K and arbitrary but fixed u ∈ K. The result proved in [8] is
as under:
Theorem 1 [8] Let C be a nonempty closed convex subset of a uniformly
smooth Banach space E. Let T : C → C be a non expansive mapping
such that F (T ) 6= ∅. Let {αn } , {β n } , {γ n } be three real sequences in (0, 1)
satisfying the following control conditions
(i) αn + β n + γ n = 1 for all n ≥ 0;
(ii) lim αn = 0 and
n→∞
∞
P
αn = ∞;
n=1
(iii) lim γ n = 0, for given x0 ∈ C arbitrarily and fixed u ∈ C, let the
n→∞
sequence {xn } be generated iteratively by
(1)
xn+1 = αn u + β n xn + γ n T xn , n ≥ 0.
Then the sequence {xn } converges strongly to a fixed point of T.
84
N. Bano, N. A. Mir
Our main motive here is to generalize the result of [8] to arbitrary Banach
space and for uniformly L-Lipschitzian mappings. Our iteration scheme is
a modification of the iteration scheme given in (1) . Our result improves and
generalizes the results proved in [2, 4, 6, 9, 5, 1, 3, 7].
Lemma 1 Let E be real Banach space and J be the normalized duality
mapping. Then for any given x, y ∈ E, we have
kx + yk2 ≤ kxk2 + 2 hy, j (x + y)i ,
∀j (x + y) ∈ J (x + y) .
Lemma 2 Let {an } and {bn } be two nonnegative real sequences satisfying
the following condition
an+1 ≤ (1 + λn ) an + bn , ∀n ≥ n0,
where {λn } is a sequence in (0, 1) with
∞
P
λn < ∞. If
n=0
lim an exists.
∞
P
bn < ∞, then
n=0
n→∞
Lemma 3 Let {θn } be a sequence of nonnegative real numbers and {λn } ⊆
[0, 1] be the real sequence satisfying the following condition
∞
X
λn = ∞.
n=0
If there exists a strictly increasing function φ : [0, ∞) → [0, ∞) such that
θ2n+1 ≤ θ2n − λn φ (θn+1 ) + σ n , ∀n ≥ n0,
where n0 is some non negative integer and {σ n } is a sequence of nonnegative
numbers such that σ n = o (λn ) , then θn → 0 as n → ∞.
Convergence of an iterative algorithm ....
2
85
Main results
Theorem 2 Let K be a non empty closed convex subset of an arbitrary
Banach space E. Let T : K → K be a uniformly L- Lipschitzian mapping
with F (T ) 6= ∅. Let {kn } ⊆ [1, ∞) be a sequence with lim kn = 1.For any
n→∞
x0 ∈ K, and fixed u ∈ K define the sequence {xn } by
xn+1 = αn u + β n xn + γ n T n xn ,
(2)
where {αn } , {β n } , {γ n } are real sequences in (0, 1) satisfying the following
conditions
∞
∞
∞
∞
P
P
P
P
γ n kn < ∞;
γ 2n < ∞; (iii)
αn < ∞; (iv)
γ n = ∞; (ii)
(i)
n=0
n=0
(v) αn = o (γ n ) ; (vi) lim γ n = 0.
n=0
n=0
n→∞
If there exists a strictly increasing function φ : [0, ∞) → [0, ∞) with
φ (0) = 0 such that
hT n x − p, j (x − p)i ≤ kn kx − pk2 − φ (kx − pk) ,
∀x ∈ K,
then the iterative scheme {xn } defined by (2) converges strongly to a fixed
point of T.
Proof. Let p ∈ F (T ) . First we prove that {xn } is bounded. By using
Lemma 6 we have:
kxn+1 −pk2 = kαn (u − p) + β n (xn − p) + γ n (T n xn − p)k2
≤ β 2n kxn − pk2 + 2 hαn (u − p) + γ n (T n xn − p) , j (xn+1 − p)i
86
N. Bano, N. A. Mir
≤ β 2n kxn − pk2 + 2 hαn (u − p) , j (xn+1 − p)i
+2 hγ n (T n xn − p) , j (xn+1 − p)i
≤ β 2n kxn −pk2 +2αn ku−pkkxn+1 −pk+2γ n kT n xn+1 −T n xnkkxn+1 −pk
+2γ n hT n xn+1 − p, j (xn+1 − p)i
≤ β 2n kxn −pk2 +2αn ku−pkkxn+1 −pk+2γ n kT n xn+1 −T n xnkkxn+1 −pk
¡
¢
+2γ n kn kxn+1 − pk2 − φ (kxn+1 − pk)
≤ β 2n kxn −pk2 +2αn ku−pkkxn+1 −pk+2γ n L kxn+1 −xn kkxn+1 −pk
¡
¢
(3)
+2γ n kn kxn+1 − pk2 − φ (kxn+1 − pk)
Now consider
kxn+1 − xn k = kαn (u − xn ) + γ n (T n xn − xn )k
≤ αn ku − xn k + γ n kT n xn − xn k
≤ αn (ku − pk + kxn − pk) + γ n (kT n xn − pk + kxn − pk)
(4)
≤ [αn + γ n (L + 1)] kxn − pk + αn ku − pk .
Substituting (4) into (3)
kxn+1 −pk2 ≤ β 2n kxn −pk2 +2αn ku − pk kxn+1 − pk
+2γ n L ((αn +γ n (L+1)) kxn −pk + αn ku − pk) kxn+1 −pk
¡
¢
+2γ n kn kxn+1 −pk2 − φ (kxn+1 − pk)
Convergence of an iterative algorithm ....
87
≤ β 2n kxn − pk2 + αn ku − pk2 + αn kxn+1 − pk2
+γ n L (αn + γ n (L + 1)) kxn − pk2
+γ n L (αn + γ n (L + 1)) kxn+1 − pk2
+γ n Lαn ku − pk2 + γ n Lαn kxn+1 − pk2
¡
¢
+2γ n kn kxn+1 − pk2 − φ (kxn+1 − pk)
¢
¡
= β 2n +γ nαn L+γ 2n L (L+1) kxn −pk2 +(αn +γ nαn L) ku−pk2
¡
¢
+ αn +2γ nαnL+γ 2nL (L+1)+2γ nkn kxn+1 −pk2 −2γ n φ (kxn+1 −pk)
¡
¢
≤ β 2n +γ n αn L+γ 2n L (L+1) kxn −pk2 +αn (1+L) ku−pk2
¡
¢
(5)
+ αn +2γ nαnL+γ 2nL (L+1)+2γ nkn kxn+1 −pk2 −2γ nφ(kxn+1 −pk) .
By (5) we get
¡
¢
(1−σ n) kxn+1 −pk2≤ β 2n +γ nαn L+γ 2n L(L+1) kxn −pk2 +αn (1+L) ku−pk2
− 2γ n φ (kxn+1 − pk)
¡
¢
≤ β n +γ nαn L+γ 2nL (L+1) kxn −pk2 +αn (1+L) ku−pk2
(6)
where σ n = αn +2γ n αn L+γ 2n L (L + 1)+2γ n kn . By conditions (i) and (ii) it
is obvious that lim σ n = 0, therefore there exists a positive integer N such
that 0 < σ n <
n→∞
1
for
2
γn
all n ≥ N, which implies 1 − σ n > 0, hence 1−σ
> 0.
n
By (6) we have
µ
kxn+1 − pk
2
3Lγ n αn + 2γ n kn + 2γ 2n L (L + 1)
γn
≤
1+
−
1 − σn
1 − σn
2γ n
αn (1 + L)
ku − pk2 −
φ (kxn+1 − pk)
+
1 − σn
1 − σn
¶
kxn − pk2
88
N. Bano, N. A. Mir
µ
¶
3Lγ n αn + 2γ n kn + 2γ 2n L (L + 1)
≤
1+
kxn − pk2
1 − σn
αn (1 + L)
2γ n
+
ku − pk2 −
φ (kxn+1 − pk)
1 − σn
1 − σn
¢
¡
≤ 1 + 6Lγ n αn + 4γ n kn + 4γ 2n L (L + 1) kxn − pk2
2γ n
+2αn (1 + L) ku − pk2 −
φ (kxn+1 − pk)
1 − σn
¡
¢
≤ 1 + 6Lαn + 4γ n kn + 4γ 2n L (L + 1) kxn − pk2
(7)
+2αn (1 + L) ku − pk2 .
By conditions (ii) , (iii) , (iv) we deduce that
∞
X
¡
¢
6Lαn + 4γ n kn + 4γ 2n L (L + 1) < ∞,
n=0
and
∞
X
2αn (1 + L) ku − pk2 < ∞.
n=0
Hence by Lemma 7 and inequality ( 7), we deduce that lim kxn − pk exists.
½n→∞
¾
Therefore the sequence {xn }is bounded. Let M = min sup kxn − pk , ku − pk ,
n
for a positive constant M. Secondly we prove that xn → p as n → ∞.Considering
inequality (7) as n → ∞, we have:
kxn+1 − pk2 ≤ kxn − pk2 − 2γ n φ (kxn+1 − pk)
¶
µ
αn
αn
+ 2kn + 2γ n L (L + 1) +
(L + 1) M 2 .
+2γ n 3L
γn
γn
³
´
Let θn := kxn − pk , λn := γ n , δ n := 3L γαn +2kn +2γ n L (L+1)+ αγ n (L+1) M 2 ,
n
n
then by conditions (i) to (vi) δ n = o (λn ) . Therefore using Lemma 3 it follows that kxn − pk → 0. Hence xn → p as n → ∞. This completes the
proof.
Convergence of an iterative algorithm ....
89
References
[1] S.C.Bose, Weak convergence to a fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc. 68(1978) 305-308.
[2] S.S.Chang, Some results for asymptotically pseudocontractive mappings,
Proc. Amer. Math. Soc. 129(2000) 845-853.
[3] S.S.Chang, Y.J.Cho, H.Zhou, Demi-closed principle and weak convergence problems for asymptotically non expansive mappings, J.Korean
Math. Soc. 38(2001)1245-1260.
[4] Y.J.Cho, J.I.Kang, H.Y.Zhou, Approximating common fixed points
of asymptotically nonexpansive mappings,
Bull. Korean Math.
Soc.42(2005)661-670
[5] B.E.Rhoades, Fixed point iterations for certain non linear mappings,
J.Math. Anal. Appl. 183(1994)118-120.
[6] J.Schu, Iterative construction of fixed points of asymptotically non expansive mapping, J.Math.Anal.Appl.158(1991)407-413.
[7] K.K.Tan, H.K.Xu, Fixed point iteration process for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc.122(1994)733-739.
[8] Y. Yao, Y. C. Liou, H. Zhou,Strong convergence of an iterative method
for non expansive mapppingswith new control conditions, Nonlinear Nalysis(2008),doi:10.1016/j.na.2008.03.014.
90
N. Bano, N. A. Mir
[9] L.C.Zeng, On the approximation of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Math.Sci, 23(2003)31-37.
Nikhat Bano
Centre for Advanced Studies in Pure and Applied Mathematics
Bahauddin Zakariya University
Multan 60800, Pakistan
e-mail: nikhat.bano@gmail.com
Nazir Ahmad Mir
Department of Mathematics
COMSATS Institute of Information Technology
Plot No. 30, Sector H-8/1, Islamabad 44000, Pakistan
e-mail: nazirahmad.mir@gmail.com