Journal
and Stochastic Analysis, 1{}:2
of Applied Mathematics
(1997),
187-189.
AN ITERATIVE ALGORITHM ON FIXED POINTS
OF RELAXED LIPSCHITZ OPERATORS
RAM U. VERMA
International Publications, 1205 Coed Drive
Orlando, Florida 32825 USA
and
Istituto per la Ricerca di Base
1-85075 Monteroduni (IS), Molise, Italy
(Received February, 1996; Revised November, 1996)
Fixed points of Lipschitzian relaxed Lipschitz operators based on a generalized iterative algorithm are approximated.
Key words: Iterative Algorithm, Fixed Point, Relaxed Lipschitz
Operator.
AMS subject classifications: 47H10.
1. Introduction
Recently, Wittman [6, Theorem
xn
2], using an iterative procedure
(1 an)X 0 q- anTx n
1
for n
_> 1,
(1)
approximated fixed points of nonexpansive mappings T:K--,K from a nonempty
closed convex subset K of a real Hilbert space H into itself, where x 0 is an element of
g and {an} is an increasing sequence in [0, 1) such that
nliman
1 and
E (1
an)
oo.
(2)
n=l
This result refines a number of results including [1].
Here our aim is to approximate the fixed points of Lipschitzian relaxed Lipschitz
operators in a Hilbert space setting. As such, the iterative algorithm (1) is not
suitable for our purpose, so we apply a modified iterative algorithm which reduces to
(1).
I
Let H be a Hilbert space and (u,v) and u [I denote, respectively, the inner
product and norm on H for u, v in H.
An operator T:H---.H is said to be relaxed Lipschitz if, for all u,v in H, there
exists a constant r > 0 such that
Printed in the U.S.A.
()1997 by North
Atlantic Science Publishing Company
187
RAM U. VERMA
188
The operator T is called Lipschitz continuous
constant s > 0 such that
IITu-Tvll <_s[lu-vII
(or Lipschitzian)
if there exists a
(4)
for allu, vinH.
Next, we consider the main result on the approximation of the fixed points of Lipschitzian relaxed Lipschitz operators using a modified iterative algorithm which contains a number of iterative schemes including those considered by the author [4, 5] as
special cases.
2. The Main Result
_
Theorem 1: Let H be a real Hilbert space and K be a nonempty closed convex subset
of H. Let T:K-,K be a relaxed Lipschitz and Lipschitz continuous operator on K.
Let r >_ 0 and s >_ 1 be constants for relaxed Lipschitzity and Lipschitz continuity of
T, respectively. Let F- {x in K:Tx- x} be nonempty, and let {an} be a sequence
in [0, 1] such that
E an
Then
for
-
any x o in K the sequence
Xn
oc
for
all n
n’-O
1
{Xn} defined
by
(1 an)x n + an[(1 t)x n + tTxn] for n >_ 0,
((1 t) 2- 2t(1 t)r + t2s2) 1/2 < 1 for all t
(1 + 2r + s 2) and r <_ s, converges to an element ofF.
0
<k
(5)
O.
such that 0
(6)
< t < 2(1 +
For {an}- 1, Theorem 1 reduces to:
Corollary 1: Let T:K--K be relaxed Lipschitz and Lipschitz continuous.
F--{x in K:Tx-x} be a nonempty set. Then, for x o in K, the sequence
generated by an iterative algorithm
Xn
for 0 < t < 2(1 + r)/(1 + 2r + s 2)
-
1
(1 t)x n + tTx n
-
converges to a unique fixed point
Proof of Theorem 1" For an element z in F, we have
I Xn + 1
z
I[
Let
{xn}
(7)
of T.
II( 1 an)Xn an[(1 t)Xn + tTxn]- z I
_< (1 an)II (xn z)II + an ]l( 1 t)(xn z) + t(Tx n- Tz) I
Using the relaxed Lipschitzity and Lipschitz continuity of T, we find that
I t(Tx,- Tz) + (1 t)(x n- z)II 2
z I 2 + 2t(1 t)(Tx n z, x n z} + t 2 I Txn- z I 2
(1 t) 2 I
_< (1 t) 2 ]] x n- z I 2- 2t(1 t)r I xn- z I] + t2s 2 I xn- z I 2
An Iterative Algorithm
on Fixed Points
of Relaxed Lipschitz Operators
((1 t) 2 2t(1 t)r + t2s 2) I] Xn
It follows that
I Xn + 1
Z
_
I]
(1
an
(x)
j=0
]] 2.
+ a,((1 t) 2 2t(1 t)r + t2s2) 1/2) I xn
z
I
(1 -(1- k)an)II xn- z I
n
H(1-(1-k)aj)llx0-zll’
-< j=O
where 0 < k ((1- t) 2- 2t(1
(1 + 2r + s 2) and r s.
Since
z
189
.
t)r + t2s2) 1/2 <
aj diverges and k
1 for all t such that 0
< t < 2(1
n
< 1, nli__,m
(1-(1- k)aj)- 0 and,
as a
result,
converges strongly to z. This completes the proof.
References
[1]
[4]
Halpern, B., Fixed points of nonexpanding maps, Bull. A mer. Math. Soc. 73
(9), 9V-91.
Reich, S., Approximating fixed points of nonexpansive maps, PanAmerican
Math J. 4:2 (1994), 23-28.
Verma, R.U., Iterative algorithms for approximating fixed points of strongly
monotone operators, Boletin A cad. Cien. Fis. Mat. Natur. (to appear).
Verma, R.U., A fixed point theorem involving Lipschitzian generalized pseudocontractions, Proc. Royal Irish A cad. (to appear).
Verma, R.U., An iterative procedure for approximating fixed points of relaxed
monotone operators, Numer. Functional Anal. Optimiz. 17 (1996), 1045-1051.
Wittmann, R., Approximation of fixed points of nonexpansive mappings Arch.
Math. 58 (1992), 486-491.