SOOCHOW JOURNAL OF MATHEMATICS
Volume 27, No. 4, pp. 401-404, October 2001
ITERATION TO OBTAIN RANDOM SOLUTIONS
AND FIXED POINTS OF OPERATORS
IN UNIFORMLY CONVEX BANACH SPACES
BY
B. E. RHOADES
Abstract. We extend the theorems of 1] from Hilbert space to p-uniformly smooth
separable Banach spaces.
1. Introduction
Let ( ) denote a measurable space, X a p-uniformly convex separable
Banach space. Let C be a subset of X . A function f : ! C is said to
be measurable if f ;1(B \ C ) 2 for each Borel subset B of X . A function
F : C ! C is said to be X -continuous if F (t ) : C ! C is continuous
for all t 2 . A function F : C ! C is said to be a random operator if
F ( x) : ! C is measurable for each x 2 C . A measurable function g : ! C
is said to be a random xed point of the random operator F : C ! C if
F (t g(t)) = g(t) for each t 2 . A measurable function g : ! C is said to be a
solution of the random operator equation
S (t x(t)) = T (t x(t))
where S T : C ! C are random operators, if S (t g(t)) = T (t g(t)) for each
t 2 .
Denition 1.1. Two functions S T : C ! C are said to satisfy the gener-
alized Tricomi condition if
Sp = Tp implies kSx ; pk kx ; pk and kTx ; pk kx ; pk:
Received May 11, 2000
revised April 10, 2001.
AMS Subject Classication. 47H10.
401
(1.1)
402
B. E. RHOADES
Denition 1.2. A random operator S : C ! C on a subset of X is said
to satisfy condition ;A if S (t ) : C ! C is onto for all t 2 , and further, if
S (t x(t)) = f (t) is measurable, then the function x : ! C is measurable.
The random iteration scheme to be used is a variant of the Ishikawa process
and is dened as follows:
S (t gn+1 (t)) = (1 ; n)S (t gn (t)) + nhn (t)
hn(t) = (1 ; n)S (t gn (t)) + n T (t gn(t))
(1.2)
Pn=1 nn = 1.
where 0 < n n < 1 for all n 0 lim sup n = M < 1 and 1
2. Main Results
We shall need the following inequality from 3].
kx + (1 ; )ykp kxkp + (1 ; )kykp ; Wp()ckx ; ykp
(2.1)
Wp() = (1 ; )p + p(1 ; )
(2.2)
where 2 0 1] c = (1) > 0
and where (t) is dened by
n p
yjjp ; jjx + (1 ; )yjjp :
(t) = inf jjxjj + (1 ; )jjW
()
p
o
0 < < 1 x y 2 X and jjx ; yjj = t > 0:
Theorem 2.1. Let S T : C ! C , where C is a closed convex subset of
a p-uniformly convex separable Banach space X , be two random operators such
that
(a) S and T are X -continuous,
(b) S satises condition ;A, and
(c) there exists a function f : ! X (not necessarily measurable) such that, for
each t 2 ,
RANDOM SOLUTIONS AND FIXED POINTS
403
jjT (t x) ; f (t)k kS(t x) ; f (t)jj:
Then the random iteration scheme (1:2), if convergent, converges to a random
solution of the random operator equation S (t x(t)) = T (t x(t)):
Proof. From (2.1), using (1.2) and (c),
jjS(t gn+1 (t)) ; f (t)jjp
(1 ; n)jjS(t gn (t)) ; f (t)jjp + njjhn(t) ; f (t)jjp
;Wp(1 ; n)cjjS(t gn (t)) ; hn(t)jjp
(1 ; n)jjS(t gn (t)) ; f (t)jjp
+n (1 ; n )jjS (t gn (t)) ; f (t)jjp + n jjT (t gn (t)) ; f (t)jjp
;Wp(1 ; n)cjjS(t gn (t)) ; T (t gn(t))jjp ]
;Wp(1 ; n)cjjS(t gn (t)) ; (1 ; n)S(t gn(t)) ; nT (t gn(t))jjp (2.3)
(1 ; n) + n((1 ; n) + n)]jjS(t gn (t)) ; f (t)jjp
;nWp(1 ; n)cjjS(t gn (t)) ; T (t gn(t))jjp
;Wp(1 ; n)cnp jjS(t gn (t)) ; T (t gn(t))jjp
= jjS (t gn (t)) ; f (t)jjp ; cn Wp (1 ; n )
+Wp(1 ; n )np ]jjS (t gn (t)) ; T (t gn (t))jjp :
(2.4)
From (1.1) there exists an integer n0 such that, for each n n0 n M 0 < 1.
Then, for all n n0 , using (2.2),
nWp(1 ; n ) + Wp(1 ; n)np = n (1 ; n)np + (1 ; n)pn ]
+np (1 ; n )pn + (1 ; n )p n ]
= n n (1 ; n )np;1 + (1 ; n )p
+np;1 (1 ; n )pn;1 + (1 ; n )p ]]
nn(1 ; n)p nn(1 ; M 0)p:
Substituting into (2.4) gives
c
1
X
(1 ; M 0)pjjS (t g (t)) ; T (t g (t))jjp
n=n0
n n
n
n
404
B. E. RHOADES
c X nWp(1 ; n) + Wp(1 ; n)np ]jjS(t gn (t)) ; T (t gn(t))jjp
n=n
1
X
jjS(t g (t)) ; f (t)jjp ; jjS(t g (t)) ; f (t)jjp]
1
0
n+1
n
n=n0
jjS(t gn (t)) ; f (t)jjp < 1
0
which implies that
lim
inf jjS (t gn (t)) ; T (t gn (t))jjp = 0:
(2.5)
n!1
The remainder of the proof is the same as that in 2], and will therefore be
omitted.
Theorem 2.1 of 2] is an immediate corollary of Theorem 2.1 of this paper.
Theorem 2.2. If, in Theorem 2:1, S (t ) T (t ) : ! C satisfy Denition
1:1, then the sequence dened by (1:1), if convergent, converges to a common xed
point of S and T .
Proof. Let gn
! g as n ! 1.
Then, from Theorem 2.1, g : ! C is
measurable and S (t g(t)) = T (t g(t)):
For any t 2 , using (2.3) and (1.1), with f replaced by g, we have,
jjS(t gn+1 (t)) ; g(t)jjp (1 ; n)jjgn(t) ; g(t)jjp
+n (1 ; n )kgn (t) ; g(t)kp n kgn (t) ; g(t)kp ]
= jjgn (t) ; g(t)jjp :
Since gn (t) ! g(t) limn S (t gn (t)) = g(t):
Theorem 2.2 of 2] is a corollary of Theorem 2.2 of this paper.
References
1] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung and S. M. Kang, Iterative approximations
of strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math.
Anal. and Appl. 224(1998), 149-165.
2] B. S. Choudhury and A. Upadhyay, An iteration leading to random solutions and xed
points of operators, Soochow J. Math., 25(1999), 395-400.
3] Hong-Kun Xu, Inequalities in Banach spaces and applications, Nonlinear Analsis, 16(1991),
1127-1138.
Department of Mathematics, Indiana University, Bloomington, IN 47405{7106, U.S.A.