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Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings
Pergamon
Nonlinear Analysis, Theory, Methods & Applications, Vol. 26, No. 11, pp. 1835-1842, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00 + 0.00
0362-546X(94)00351-3 CONVERGENCE THEOREMS OF THE SEQUENCE OF ITERATES FOR ASYMPTOTICALLY DEMICONTRACTIVE AND HEMICONTRACTIVE MAPPINGS LIU QIHOU Department of Mathematics, Beijing Universityof Aeronautics and Astronautics, Beijing 100083, People's Republic of China (Received 1 September 1993; received in re~sed form 8 November 1994; received for publication 29 December 1994)
Let C be a nonempty subset of a normal space B. (1) A mapping T : C - ~ C is said to be k-strict asymptotically pseudocontractive with sequence {a n} : ** lim n _~~ a n = 1 for some constant k, 0 < k < 1, and IIT"x - Tnyll e < aZ, llx - y [ I z + kll(x - T n x ) - ( y - Tny)[I e
for all n c N and all x, y c C.
If k = 0, then, T is said to be asymptotically nonexpansive with sequence {an}. (2) A mapping T : C ~ C is said to be asymptotically demicontractive with sequence {a n} : ** lim n ~ ~a n = 1 for some constant k, 0 < k < 1, and for all n ~ N and all x ~ C, Vp ~ F(T)
IlTnx - p l l e < a]llx - p l [ e + kllx - Tnxll 2
( F ( T ) denotes the fixed point set of T). (3) A mapping T : C - ~ C is said to be asymptotically hemicontractive with sequence
{an}: ¢~ lim n _. ~ a n = 1 and IlZnx-pll2<_anllx-plle+llx-Znxll
2
for all n ~ N and all x ~ C ,
Vp~F(T).
(4) A mapping T : C - ~ C is said to be asymptotically pseudocontractive with sequence {an}: ~ l i m n_~ a n = 1 and [ITnx - Tnylle < anllx - ylle + ll( x - T n x ) - ( y - T n y )[[ 2
(5)
for all x , y ~ C.
A mapping T : C ~ C is said to be uniformly L-Lipschitzian:
¢* IITnx - Znyll < t l l x - y l l
for some constant L > 0
for all n ~ N and allx, y ~ C.
Schu [1] has proved the convergence of sequence of Mann iterates for asymptotically nonexpansive mappings. In this manuscript, we will extend Schu's results [1] and will prove convergence theorems of Mann iterative sequences for uniformly L-Lipschitzian asymptotically demicontractive mappings and uniformly L-Lipschitzian k-strict asymptotically pseudocontractive mappings. Schu [1] has also proved the convergence of sequence of Ishikawa iterates for uniformly L-Lipschitzian asymptotically pseudocontractive mappings. In this manuscript, we will also extend this result and will prove convergence theorems of Ishikawa iterative sequences for uniformly L-Lipschitzian asymptotically pseudocontractive mappings and uniformly L-Lipschitzian asymptotically hemidocontractive mappings. THEOREM 1. Let H be a Hilbert space, C c H nonempty closed bounded and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically demicontractive with sequence {an}, a n ~ [1, + ~ ), E ~n = 0 ( a2n - 1 ) < + o % e _ < a n _ < l - k - e , f o r V n ~ N 1835
1836
LIU QIHOU
and some e > 0 .
Vx 0 ~ C Xn+ 1 = a . T " x . + (1 - a . ) x . ,
Vn~N.
Then, {x.}~= 0 converges strongly to s o m e fixed point of T. THEOREM 2. Let H be a Hilbert space, C c H n o n e m p t y closed b o u n d e d and convex; T : C - - , C completely continuous and uniformly L-Lipschitzian and k-strict asymptotically pseudocontractive with sequence {an} , a n ~ [1, + ~), E n=~ 0 ( a ,2 - 1 ) < +0% e~_~_Oln~_<~_ 1 - k - e , for all n ~ N and s o m e e > 0. Vx 0 c C x , + 1 = a , T " x n + (1 - a n ) x n ,
Vn ~ N.
Then, {x,}~= 0 converges strongly to some fixed point of T. Remark. N o t e that Schu's t h e o r e m 1.5 [1] is a special case, when k = 0, (see t h e o r e m 2 above).
THEOREM 3. Let H be a Hilbert space, C c H n o n e m p t y closed b o u n d e d and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a.}, a . ~ [1, +oo) ; Vn ~ N , E : = l ( a . - 1) < +oo; {a.}, {/3.} ~ [1,0]; e < a n < / 3 . < b for Vn ~ N, some e > 0, and some b ~ (0, L - z [ ( 1 + L2) 1/2 - 1]); x 1 E C for Vn ~ N define z. = /3.Tn(x.) + (1-
and
/3n)X.
Xn+ 1 = a . T " ( z . )
+ (1 - a n ) x . .
T h e n {x.} converges strongly to s o m e fixed point of T. R e m a r k Note that the condition "b ~ (0, L - 2 [ ( 1 + L 2 ) 1 / 2 - 1])" is equivalent to "b > 0 and 1 - 2b - LZb 2 > 0".
COROLLARY OF THEOREM 3. Let H be a Hilbert space; C c H n o n e m p t y closed b o u n d e d and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence {an} , a . ~- [1, + ~); Vn ~ N, E~= l(a2. - 1) < + o0; {a.}, {/3.} [1,0]; • _< a . < / 3 . _< b for V n c N, s o m e e > 0, and some b ~ (0, L - 2 [ ( 1 + L2) 1/2 - 1]); x 1 ~ C for Vn ~ N define z. =/3.T"(x.)
+ (1 - f l . ) x ,
and
x.+ 1 = a.T"(z.)
+ (1 - a . ) x . .
T h e n {x n} converges strongly to some fixed point of T. It should be indicated that this corollary is just Schu's t h e o r e m 2.3 [1]. Let us first prove l e m m a 1. LEMMA 1. L e t sequence {a.}~=l, {b.}~= 1 satisfy that a . + 1 < a . + b., a . >_ O, V n o N , Y',n = l bn is convergent and {a.}~= 1 has a subsequences {a.k}~= 1 converging to 0. Then, we must have
lima. n
= 0.
--) cc
P r o o f o f l e m m a 1. Ve > 0, since E~_ 1 b. is convergent, there must exist some natural n u m b e r N 1 so that when n > N1 m
k__~0b.+k < e / 2 ,
Vn ~ N .
(1)
Asymptotically demicontractive and hemicontractive mappings
1837
Since an+ 1 <-an + bn, w e have m-1
an+m<_%an+m_l+bn+m l
E bn+k.
(2)
k=0
{an~:= 1 has s u b s e q u e n c e s {an I~= 1 c o n v e r g e n t to 0 so t h a t t h e r e m u s t exist s o m e n a t u r a l number N > N 1 and when k > N a.~ < e / 2 .
(3)
It follows f r o m (1) a n d (2) t h a t V m ~ N
m-1 bnN+, < s / 2 + s / 2 = s.
O
Therefore, we have lim an = 0.
n--, oc
T h e f o l l o w i n g l e m m a s a r e given in Refs. 1 a n d 2. LEMMA 1 A ( S c h u [1]). L e t E b e a n o r m e d space; C c E , C 4 : 0 convex; L > 0, T : C ~ C u n i f o r m l y L - L i p s c h i t z i a n ; % / 3 . ~ [0, 1]; V x 1 ~ C define, y . = / 3 n T n ( x n ) + (1 - /3n)Xn a n d x . + 1 = a n T n ( z n ) + (1 - an)X n a n d set C . = I I T n ( x . ) - x n l l for all n ~ N. T h e n IIx n - T ( x . ) l l ___C . + C . _ 1L(1 + 3 L + 2 L 2) for all n ~ N. LEMMA 1B ( R e i n e r m a n n
[2]). L e t H b e a H i l b e r t space, a ~ [0, 1], Vz, w c H . T h e n
Ilaz + (1 - a ) w l l 2 = allzll 2 + (1 - c~)llwll 2 - a ( 1 - a ) l I z - w l l 2. Proof of theorem 1. T is a c o m p l e t e l y c o n t i n u o u s m a p p i n g in a c l o s e d b o u n d e d c o n v e x s u b s e t C o f H i l b e r t space. F r o m S c h a u d e r ' s t h e o r e m , F ( T ) is n o n e m p t y . It follows f r o m l e m m a 1B that Ilxn+ 1 - p l l 2 = IlanT"xn + (1 - % ) x .
_pl]2 = ila.(Tnx. _p)
+ (1 - a n ) ( x . - p ) l [ 2
= % l l Z " x . - p l l e + (1 - a.)llx n - p l l 2 - a . ( 1 -- a n ) l l x n -- Tnxnll 2,
Vp ~ F(T). (4)
Since
IIT"x. - p l l e _< a~lhx. - e l l 2 + kIlx. - Z"x.ll 2
(5)
s u b s t i t u t i n g (5) i n t o (4) gives [[Xn+ 1 - P l l e ---
ana2,llXn - p I I 2 + a . k l l x . - Tnx.[I 2 + (1 - a . ) l l x n - p l l 2 - a . ( 1 - a . ) l l x n - T " x . II2
= IIx. - e l l 2 + a n ( a 2 . - 1 ) l l X n - e l l 2 - % ( 1 - k - a n ) l l x n - Z n x . ll2.
(6)
O n the o t h e r h a n d , since 0 < s < a n _~ 1 - k - e, 1 - k - a n >__e. T h u s , a n ( 1 -- k - a n) > e 2. F r o m (6), w e have tlxn+ 1 - p l l e -< IIx. - p l l 2 + a.(aZ. - 1)llx. - e l l 2 - e Z l l x . - Znxnll 2.
(7)
1838
LIU QIHOU
C is b o u n d e d and T is a self-mapping in C. Thus, there must exist some M > 0 so that IIxn - p l l = _
(8)
Therefore ~2llxn - T nxnll 2 _
2- 1)-Ilxn+ 1 -pll 2
m
]~ eZllxn - Z " x , II2 -< Ilxl - p l l 2 -IlXm+lPll z + ~ M ( a ] - 1) n=l
n=l ~c
<2M+
EM(a2-1), n=l
for all natural numbers m. Since oo
oo
E ~211xn - Z"xnll 2 < + ~ .
]~ (a2, - 1) < +0% n=l
n=l
Therefore lim Ilxn - Tnxnll 2 = 0
and
n ---~ ~o
lim Ilxn - Tnx, II = 0.
(9)
n --..~ oo
T is uniformly L-Lipschitzian, it follows from lemma A that (10)
lim IIx~ - Zxnll = O. n--,
ac
{xn)~= 0 is a b o u n d e d sequence and T is completely continuous so that {Txn}~= 0 must have a convergent subsequence {TxnkJ~= 0. Therefore, from (10), {xn}~= 0 must have a convergent subsequence {xn~}~=0. Let limk_~xn~ = q. It follows from the continuity of T and (10) that q = Tq, i.e., q is a fixed point of T. Therefore, {xn}~= 0 has a subsequence which converges to the fixed point q of T. Let p = q in the inequality (8). Since E . = l e 2 IlXn- Tnxnll 2 < + ~ and E =n = l M(a2n - 1 ) < + ~, from lemma 1, we have lira 1Ix.
- qll 2 = 0 .
Therefore, lim n _. ~ x n = q. This completes the proof of t h e o r e m 1.
•
Proof o f theorem 2. C is a closed b o u n d e d convex subset of a Hilbert space and T is a completely continuous mapping in C. Thus, it follows from the Schauder's t h e o r e m that T must have a fixed point in C. Since T is k-strict asymptotically pseudocontractive, T must be asymptotically demicontractive. Therefore, t h e o r e m 2 can be proved by using t h e o r e m 1. In order to prove t h e o r e m 3, we will first prove lemma 2 and lemma 3.
LEMMA 2. Let E be a nonempty convex subset of a Hilbert space, T : E ~ E uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a n} ~ [1, ~), Vn c N; and F ( T ) nonempty; 0 < a n, /3n < 1, Vn ~ N define z, = flnTn(xn) + (1 - fin)
and
Xn+ 1 = o t n Z n ( z n ) +
(1
-
Oln)X n
Asymptotically demicontractive and hemicontractive mappings
1839
T h e n , for Vp ~ F ( T ) , the following inequality holds IIx.+ 1 - p l l 2 -< [1 + o~.(a. - 1)(a./3n + 1)]llx. - p l l 2 - or./3.(1 - / 3 . - a n/3. - Z2/32)lix.
-
Tnxnl]
2
- ~ . ( / 3 . - ~ . ) l l x . - T"z.II e. P r o o f o f l e m m a 2. It follows f r o m l e m m a B that
[[Xn+ 1 - p l l e = I1(1 - % ) x . + o t . T " z . - p l l e = I l a . ( T " z . - p )
+ (1 - a . ) ( x .
-p)ll 2 Vp~F(T).
= a . l l T " z . - p l l 2 + (1 - a . ) l l x . - p l l 2 - ~ . ( 1 - a . ) l l T " z . --xnll 2,
(11) However IIT"z. - p l l e _< a.llzn - p l l e + I I z . - T"z.[] 2 IlZn --ell 2 = II/3nT"X. + (1 - / 3 . ) X .
(12)
- e l l 2 = II/3n(Tnxn - - P ) + (1 - - / 3 n ) ( X n - p ) l l e
=/3.1IT"x. - p l l 2 + (1 - / 3 . ) l l x .
-ell 2 -/3.(1 -/3.)l[T"x.
- x . [ [ 2.
(13)
In addition IIT"x. - p l l 2 < a . l l x . - p l l 2 + Ilx. - T"x.[I e.
(14)
Substituting (14) into (13) gives IIz. - p l l e _< a . & l l x . - p l l 2 + / 3 ° l l x . - T"x.II 2 + (1 - / 3 . ) l l x . = [1 + ( a . - 1)/3n]llx" - p l l 2 +/321[x
n -
-pll 2 -/3.(1 -/3.)llx.
- T"x.II 2
T°x.ll 2.
(15)
O n the o t h e r h a n d IIz. - T"z.II 2 = II/3nT"x. + (1 - / 3 . ) x .
- Tnz.[] 2 = I I / 3 . ( T " x . - TnZn) + (1 - / 3 . ) ( x .
=/3nllZ"xn - T"Zn[[ 2 + (1 - / 3 n ) l [ x . -- Z"Zn[[ 2 - / 3 . ( 1 - / 3 . ) l l x .
- T"z.)ll 2
- T"x.[[ 2.
(16)
Substituting (15) and (16) into (12) gives IIT"z. - p l l 2 _ an[1 +
(a n -
1)/3n]llx. - p l l 2 + a . / 3 f l l x .
+ (1 - / 3 . ) l l x .
- T"z.II 2 - / 3 . ( 1
- T"x.II 2 +/3.11T"x. - Z"z.II 2
-/3.)llx.
- T"x.II 2.
(17)
W h e n (17) goes into (11), we have IIx.+a - p l [ e < a . a . [ 1 + ( a . - 1)/3.]11x. - p l l 2 + a . a . / 3 2 1 1 x . - T"x.[I z + c~./3.11T"x. - T"znll 2 + % ( 1 - / 3 . ) l l x ° - T"znl] e - or,,/3.(1 - / 3 n ) l l x . - T"x.II e + (1 - %(1 -
~.)llx.
-
Oln)llx n - p l l e
- T"z.ll e
= {1 + a n [ ( a . - 1) + a . ( a . - 1)/3.]}llx. - p l l e - a n / 3 . ( 1 --/3. -- a . / 3 . ) [ I x n -
-
T"x.ll 2 - a . ( / 3 . - o~.)llx. - T"zn[I e + % / 3 n [ I T n x . - Tnz.]] 2
= [1 + a . ( a n - 1 ) ( a . / 3 . + 1)]llx. - p l l 2 -
o~n / 3 . ( 1
-/3. -
~ . ( / 3 . - a . ) l l x . - T"z.II e + ol./3.11T"x n - T"znll e.
a n/3.)llx.
- T"x.II 2 (18)
1840
LIU QIHOU
Since T is uniformly L-Lipschitzian IITnxn - Z n z n l l <_ t l l x n - Znll = Z f n l l X n - T ° x n l I .
(19)
Substituting (19) into (18) gives [JXn+ 1 - p I I 2 < [1 + a n ( a n - 1)(a n fin + 1)]IlXn --plle -- a n fin( 1 -- fin -- an 3 n ) l l x n - T n x n l l 2 - - a n ( fin -- an)[[Xn -- TnznH 2 + an fl3LZ[lXn - Tnxnll 2
= [1 + a n ( a n - 1)(a n fin + 1)]IlXn --plle -
an fin( 1 - f n - an f n - Z 2 f f f ) I l X n - Znxnll 2
- an( f ° -- O~n)Ilxo - TnZnll 2
This completes the p r o o f of l e m m a 2.
•
LEMMA 3. Let E be a n o n e m p t y b o u n d e d convex subset of a Hilbert space; T : E ~ E uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a n} ~ [1, ~), Vn ~ N; E~= l(a, - 1) < + ~ , and F ( T ) nonempty; e < a n < Bin < b, for Vn ~ N, s o m e s > 0 and s o m e b ~ (0, L-2[(1 + L 2 ) 1/2 - 1]); x I ~ E define z, = fnTn(xn)
+ (1-
fln)x,,
Xn+l=anTn(zn)+(1-an)Xn
.
for Vn ~ N, then, lim . . . . IlXn -- Zx.II = O. Proof of lemma
3. We will first prove the following equality, lim IlXn - TnxnII = O.
It follows f r o m l e m m a 2 that
[[Xn+ 1 - p l l 2 _< [1 + a n ( a n -- 1)(a n Bin + 1)]IlXn - e l l 2 - an fin(1 -- f n - - a n fin - t 2 f 2 ) l l X n -- an( f n -- an)llXn -- Znznll 2
- T n x n 112
Vp ~ F(T).
Since 0 < a n < fin, it can be d e d u c e d that IIx,+ 1 - p l l e -< [1 + a n ( a n - 1)(a n fin + 1)]llxn - p l l e - an 3 . ( 1 -- 3n - - a n f n -- Lefff)llxn - T°Xnll 2.
Thus Ilxn+ 1 - p l l e - Ilxn - p l l e ~ a n ( a n -- 1)(an fin + 1)llxn - p l l e - - a n f n ( 1 -- fin --an fin - L2fl2)llxn
- Tnxnl[ 2"
(20)
En= l ( a n - 1 ) < Woo implies that lim n_,o~(a n - 1 ) = 0. Hence, {an} n_ 1 is bounded. In addition, f r o m the b o u n d e d n e s s of E and 0 < a n < fin < 1, { a n ( a n f n + 1)IIXn --pIIZ~= 1 must be bounded. T h e r e f o r e , there exists a constant M > 0 such that 0 < an(a n
fn
-~- 1)llxn - p l l 2 < M .
(21)
Asymptoticallydemicontractive and hemicontractive mappings
1841
F r o m (20) and (21), the following inequality holds
IlXn+ 1 - P I I 2 -
I[Xn - p I I 2 < M ( a n
- 1) - a n/3n(1 - [3n - a n [3n - L2/32)llxn - r " x n l l 2.
(22)
O n the other hand, 1 - 2b - L Z b 2 = C > 0 can be d e d u c e d f r o m b ~ (0, L - 2 [ ( 1 + L2) ~/a - 1]). Since lim n _. sa n = 1, there exists s o m e natural n u m b e r N, w h e n n > N 1 - fin - an ¢tn - L Z ~
> 1 - b - anb - LZb 2 > C/2.
(23)
Supposing lim n ~ l l x n - T n x n l l - ~ O, then, there must exist a e0 > 0 and a subsequence {xnk}~= 1 of {xn~=l such that
IIx.k -
rn*x.kll 2 >- So.
(24)
W i t h o u t losing generality, we can suppose n 1 > N. F r o m (22)
Olnfin (1
-- fin - - a n fin - g2/3f)llxn - Tnx,,[I 2 n - 1) + Ilx.
-pll 2 -Ilxn+l -Pll 2
nk
>( ~ m=n
O t m f l , n ( 1 - flrn - a r n f l m
- g2fl2m)llxm - Tmxmll 2
1
nk
<
M ( a m - 1) +
~ m=n
Ilxn,~-pll 2 -Ilxnk+~ -pll e
1
k X E °tn,[Jn,(]-[~n,-an;[~nt-g2[j2)lIXnt l=1
- Tn'xn; 112
nk
<-- E m=n
°tmflm(1-
fl-am[3m-
L2fl2)IIxm - Tmxmll2
I
nk
<
~_. M ( a m - 1) m=n
+lIx,-pll 2-11xn~+a
- p l l 2.
(25)
l
F r o m (23), (24), (25) and 0 _< ¢ < a n < Bin, it can be easily seen that n k
ke2(C/2)eo
<_ ~.. M ( a m - 1) + IIx,,-Pll 2 -Ilxnk+l - p l l 2. m=n
(26)
l
E~ = ~(an - 1) < + ~ and the b o u n d e d n e s s of E imply that the right side of (29) is bounded. However, in fact, the left side of (26) is positively u n b o u n d e d , when k - ~ ~. This is a contradiction. T h e r e f o r e lim IIx. - Z n x . I I = 0. n
--~ o¢
Since T is uniformly L-Lipschitzian, from l e m m a A, it follows that lim IIx~ - Zx,,ll = O. This completes the p r o o f of l e m m a 3.
•
1842
LIU QIHOU
Proof of theorem 3. E is a nonempty closed bounded convex subset of a Hilbert space and T is a completely continuous self-mapping in E. Thus, it follows from the Schauder's theorem that F ( T ) must be nonempty. Using lemma 3 lim IIx. - Tx.II = 0
n--*~
(27)
Since T is completely continuous, there exists a convergent subset {Tx.k}~= 1 of {Tx.}~=l. Let lim k .~Tx.~ =p. From (27) lim x.~ = p .
(28)
k-~
Using the continuity of T and (27), lip - Tell = 0, that is, p is a fixed point of T. Using lemma 2 again IIx.+l - P l l 2 -< [1 + a n ( a n - 1)(a./3. + 1)]llx. - p l l 2 --O/.
/3n(1
-- /3. -a
n/3n
- Z2/3Zn)llx. - Z " x . II2
- ol.(/3. - ~.)llx. - T"z.ll 2.
(29)
(23) shows that there exists some natural number N, when n > N, 1 - 13.- a . / 3 . - L2/32 > C / 2 > 0. In addition, from (21), 0 _< a.(a./3. + 1)[[xn _pl12 _
(30)
E . - ~ ( a . - 1) < + shows that E . _ 1M(a. - 1) < + . On the other hand, (28) Indicates that there exists a subsequency {fix.,-pll2}~_ a of {[[x. _pl[2}~= 1, which converges to 0. Therefore, ,2 from (30), and lemma 1, lira. __,=[[xn - p l [ = 0, that is, lira. _~=x. = p . This completes the proof of theorem 3. •
Proof of corollary of theorem 3, E is a nonempty closed bounded convex subset of a Hilbert space and T is a completely continuous self-mapping in E. Thus, it follows from the Schauder's theorem that F ( T ) must be nonempty. T is an asymptotically psuedocontractive mapping with sequence {a.}, a. a [1, oo). Hence, T must be an asymptotically hemicontractive mapping with {an}, a. ~ [1, oo). Since a. ~ [1, ~), a n2 - 1 > a. - 1 > 0. Obviously, E~ = l ( a n 1 ) < + ~ , f r o m E ~. = l(a.2 - 1) < + ~. Therefore, it follows from theorem 3 that lim x. = p ,
n ---~oo
p ~ F(T).
This completes the proof of corollary of theorem 3. Acknowledgement--The author expresses his sincere thanks for the financial support from the Scientific Fund of Aeronautics and Astronautics Ministry of the People's Republic of China.
REFERENCES 1. SCHU, J. Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. Analysis Applic., 158, 407-413 (1991). 2. REINERMANN, J. LJber Fixpunkte Kontrahieruder Abbidungen und Schwach Konvergente Toeplitz-Verfahren, Arch. Math., 20, 59-64 (1969).
Nonlinear Analysis, Theory, Methods & Applications, Vol. 26, No. 11, pp. 1835-1842, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00 + 0.00
0362-546X(94)00351-3 CONVERGENCE THEOREMS OF THE SEQUENCE OF ITERATES FOR ASYMPTOTICALLY DEMICONTRACTIVE AND HEMICONTRACTIVE MAPPINGS LIU QIHOU Department of Mathematics, Beijing Universityof Aeronautics and Astronautics, Beijing 100083, People's Republic of China (Received 1 September 1993; received in re~sed form 8 November 1994; received for publication 29 December 1994)
Let C be a nonempty subset of a normal space B. (1) A mapping T : C - ~ C is said to be k-strict asymptotically pseudocontractive with sequence {a n} : ** lim n _~~ a n = 1 for some constant k, 0 < k < 1, and IIT"x - Tnyll e < aZ, llx - y [ I z + kll(x - T n x ) - ( y - Tny)[I e
for all n c N and all x, y c C.
If k = 0, then, T is said to be asymptotically nonexpansive with sequence {an}. (2) A mapping T : C ~ C is said to be asymptotically demicontractive with sequence {a n} : ** lim n ~ ~a n = 1 for some constant k, 0 < k < 1, and for all n ~ N and all x ~ C, Vp ~ F(T)
IlTnx - p l l e < a]llx - p l [ e + kllx - Tnxll 2
( F ( T ) denotes the fixed point set of T). (3) A mapping T : C - ~ C is said to be asymptotically hemicontractive with sequence
{an}: ¢~ lim n _. ~ a n = 1 and IlZnx-pll2<_anllx-plle+llx-Znxll
2
for all n ~ N and all x ~ C ,
Vp~F(T).
(4) A mapping T : C - ~ C is said to be asymptotically pseudocontractive with sequence {an}: ~ l i m n_~ a n = 1 and [ITnx - Tnylle < anllx - ylle + ll( x - T n x ) - ( y - T n y )[[ 2
(5)
for all x , y ~ C.
A mapping T : C ~ C is said to be uniformly L-Lipschitzian:
¢* IITnx - Znyll < t l l x - y l l
for some constant L > 0
for all n ~ N and allx, y ~ C.
Schu [1] has proved the convergence of sequence of Mann iterates for asymptotically nonexpansive mappings. In this manuscript, we will extend Schu's results [1] and will prove convergence theorems of Mann iterative sequences for uniformly L-Lipschitzian asymptotically demicontractive mappings and uniformly L-Lipschitzian k-strict asymptotically pseudocontractive mappings. Schu [1] has also proved the convergence of sequence of Ishikawa iterates for uniformly L-Lipschitzian asymptotically pseudocontractive mappings. In this manuscript, we will also extend this result and will prove convergence theorems of Ishikawa iterative sequences for uniformly L-Lipschitzian asymptotically pseudocontractive mappings and uniformly L-Lipschitzian asymptotically hemidocontractive mappings. THEOREM 1. Let H be a Hilbert space, C c H nonempty closed bounded and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically demicontractive with sequence {an}, a n ~ [1, + ~ ), E ~n = 0 ( a2n - 1 ) < + o % e _ < a n _ < l - k - e , f o r V n ~ N 1835
1836
LIU QIHOU
and some e > 0 .
Vx 0 ~ C Xn+ 1 = a . T " x . + (1 - a . ) x . ,
Vn~N.
Then, {x.}~= 0 converges strongly to s o m e fixed point of T. THEOREM 2. Let H be a Hilbert space, C c H n o n e m p t y closed b o u n d e d and convex; T : C - - , C completely continuous and uniformly L-Lipschitzian and k-strict asymptotically pseudocontractive with sequence {an} , a n ~ [1, + ~), E n=~ 0 ( a ,2 - 1 ) < +0% e~_~_Oln~_<~_ 1 - k - e , for all n ~ N and s o m e e > 0. Vx 0 c C x , + 1 = a , T " x n + (1 - a n ) x n ,
Vn ~ N.
Then, {x,}~= 0 converges strongly to some fixed point of T. Remark. N o t e that Schu's t h e o r e m 1.5 [1] is a special case, when k = 0, (see t h e o r e m 2 above).
THEOREM 3. Let H be a Hilbert space, C c H n o n e m p t y closed b o u n d e d and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a.}, a . ~ [1, +oo) ; Vn ~ N , E : = l ( a . - 1) < +oo; {a.}, {/3.} ~ [1,0]; e < a n < / 3 . < b for Vn ~ N, some e > 0, and some b ~ (0, L - z [ ( 1 + L2) 1/2 - 1]); x 1 E C for Vn ~ N define z. = /3.Tn(x.) + (1-
and
/3n)X.
Xn+ 1 = a . T " ( z . )
+ (1 - a n ) x . .
T h e n {x.} converges strongly to s o m e fixed point of T. R e m a r k Note that the condition "b ~ (0, L - 2 [ ( 1 + L 2 ) 1 / 2 - 1])" is equivalent to "b > 0 and 1 - 2b - LZb 2 > 0".
COROLLARY OF THEOREM 3. Let H be a Hilbert space; C c H n o n e m p t y closed b o u n d e d and convex; T : C ~ C completely continuous and uniformly L-Lipschitzian and asymptotically pseudocontractive with sequence {an} , a . ~- [1, + ~); Vn ~ N, E~= l(a2. - 1) < + o0; {a.}, {/3.} [1,0]; • _< a . < / 3 . _< b for V n c N, s o m e e > 0, and some b ~ (0, L - 2 [ ( 1 + L2) 1/2 - 1]); x 1 ~ C for Vn ~ N define z. =/3.T"(x.)
+ (1 - f l . ) x ,
and
x.+ 1 = a.T"(z.)
+ (1 - a . ) x . .
T h e n {x n} converges strongly to some fixed point of T. It should be indicated that this corollary is just Schu's t h e o r e m 2.3 [1]. Let us first prove l e m m a 1. LEMMA 1. L e t sequence {a.}~=l, {b.}~= 1 satisfy that a . + 1 < a . + b., a . >_ O, V n o N , Y',n = l bn is convergent and {a.}~= 1 has a subsequences {a.k}~= 1 converging to 0. Then, we must have
lima. n
= 0.
--) cc
P r o o f o f l e m m a 1. Ve > 0, since E~_ 1 b. is convergent, there must exist some natural n u m b e r N 1 so that when n > N1 m
k__~0b.+k < e / 2 ,
Vn ~ N .
(1)
Asymptotically demicontractive and hemicontractive mappings
1837
Since an+ 1 <-an + bn, w e have m-1
an+m<_%an+m_l+bn+m l
E bn+k.
(2)
k=0
{an~:= 1 has s u b s e q u e n c e s {an I~= 1 c o n v e r g e n t to 0 so t h a t t h e r e m u s t exist s o m e n a t u r a l number N > N 1 and when k > N a.~ < e / 2 .
(3)
It follows f r o m (1) a n d (2) t h a t V m ~ N
m-1 bnN+, < s / 2 + s / 2 = s.
O
Therefore, we have lim an = 0.
n--, oc
T h e f o l l o w i n g l e m m a s a r e given in Refs. 1 a n d 2. LEMMA 1 A ( S c h u [1]). L e t E b e a n o r m e d space; C c E , C 4 : 0 convex; L > 0, T : C ~ C u n i f o r m l y L - L i p s c h i t z i a n ; % / 3 . ~ [0, 1]; V x 1 ~ C define, y . = / 3 n T n ( x n ) + (1 - /3n)Xn a n d x . + 1 = a n T n ( z n ) + (1 - an)X n a n d set C . = I I T n ( x . ) - x n l l for all n ~ N. T h e n IIx n - T ( x . ) l l ___C . + C . _ 1L(1 + 3 L + 2 L 2) for all n ~ N. LEMMA 1B ( R e i n e r m a n n
[2]). L e t H b e a H i l b e r t space, a ~ [0, 1], Vz, w c H . T h e n
Ilaz + (1 - a ) w l l 2 = allzll 2 + (1 - c~)llwll 2 - a ( 1 - a ) l I z - w l l 2. Proof of theorem 1. T is a c o m p l e t e l y c o n t i n u o u s m a p p i n g in a c l o s e d b o u n d e d c o n v e x s u b s e t C o f H i l b e r t space. F r o m S c h a u d e r ' s t h e o r e m , F ( T ) is n o n e m p t y . It follows f r o m l e m m a 1B that Ilxn+ 1 - p l l 2 = IlanT"xn + (1 - % ) x .
_pl]2 = ila.(Tnx. _p)
+ (1 - a n ) ( x . - p ) l [ 2
= % l l Z " x . - p l l e + (1 - a.)llx n - p l l 2 - a . ( 1 -- a n ) l l x n -- Tnxnll 2,
Vp ~ F(T). (4)
Since
IIT"x. - p l l e _< a~lhx. - e l l 2 + kIlx. - Z"x.ll 2
(5)
s u b s t i t u t i n g (5) i n t o (4) gives [[Xn+ 1 - P l l e ---
ana2,llXn - p I I 2 + a . k l l x . - Tnx.[I 2 + (1 - a . ) l l x n - p l l 2 - a . ( 1 - a . ) l l x n - T " x . II2
= IIx. - e l l 2 + a n ( a 2 . - 1 ) l l X n - e l l 2 - % ( 1 - k - a n ) l l x n - Z n x . ll2.
(6)
O n the o t h e r h a n d , since 0 < s < a n _~ 1 - k - e, 1 - k - a n >__e. T h u s , a n ( 1 -- k - a n) > e 2. F r o m (6), w e have tlxn+ 1 - p l l e -< IIx. - p l l 2 + a.(aZ. - 1)llx. - e l l 2 - e Z l l x . - Znxnll 2.
(7)
1838
LIU QIHOU
C is b o u n d e d and T is a self-mapping in C. Thus, there must exist some M > 0 so that IIxn - p l l = _
(8)
Therefore ~2llxn - T nxnll 2 _
2- 1)-Ilxn+ 1 -pll 2
m
]~ eZllxn - Z " x , II2 -< Ilxl - p l l 2 -IlXm+lPll z + ~ M ( a ] - 1) n=l
n=l ~c
<2M+
EM(a2-1), n=l
for all natural numbers m. Since oo
oo
E ~211xn - Z"xnll 2 < + ~ .
]~ (a2, - 1) < +0% n=l
n=l
Therefore lim Ilxn - Tnxnll 2 = 0
and
n ---~ ~o
lim Ilxn - Tnx, II = 0.
(9)
n --..~ oo
T is uniformly L-Lipschitzian, it follows from lemma A that (10)
lim IIx~ - Zxnll = O. n--,
ac
{xn)~= 0 is a b o u n d e d sequence and T is completely continuous so that {Txn}~= 0 must have a convergent subsequence {TxnkJ~= 0. Therefore, from (10), {xn}~= 0 must have a convergent subsequence {xn~}~=0. Let limk_~xn~ = q. It follows from the continuity of T and (10) that q = Tq, i.e., q is a fixed point of T. Therefore, {xn}~= 0 has a subsequence which converges to the fixed point q of T. Let p = q in the inequality (8). Since E . = l e 2 IlXn- Tnxnll 2 < + ~ and E =n = l M(a2n - 1 ) < + ~, from lemma 1, we have lira 1Ix.
- qll 2 = 0 .
Therefore, lim n _. ~ x n = q. This completes the proof of t h e o r e m 1.
•
Proof o f theorem 2. C is a closed b o u n d e d convex subset of a Hilbert space and T is a completely continuous mapping in C. Thus, it follows from the Schauder's t h e o r e m that T must have a fixed point in C. Since T is k-strict asymptotically pseudocontractive, T must be asymptotically demicontractive. Therefore, t h e o r e m 2 can be proved by using t h e o r e m 1. In order to prove t h e o r e m 3, we will first prove lemma 2 and lemma 3.
LEMMA 2. Let E be a nonempty convex subset of a Hilbert space, T : E ~ E uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a n} ~ [1, ~), Vn c N; and F ( T ) nonempty; 0 < a n, /3n < 1, Vn ~ N define z, = flnTn(xn) + (1 - fin)
and
Xn+ 1 = o t n Z n ( z n ) +
(1
-
Oln)X n
Asymptotically demicontractive and hemicontractive mappings
1839
T h e n , for Vp ~ F ( T ) , the following inequality holds IIx.+ 1 - p l l 2 -< [1 + o~.(a. - 1)(a./3n + 1)]llx. - p l l 2 - or./3.(1 - / 3 . - a n/3. - Z2/32)lix.
-
Tnxnl]
2
- ~ . ( / 3 . - ~ . ) l l x . - T"z.II e. P r o o f o f l e m m a 2. It follows f r o m l e m m a B that
[[Xn+ 1 - p l l e = I1(1 - % ) x . + o t . T " z . - p l l e = I l a . ( T " z . - p )
+ (1 - a . ) ( x .
-p)ll 2 Vp~F(T).
= a . l l T " z . - p l l 2 + (1 - a . ) l l x . - p l l 2 - ~ . ( 1 - a . ) l l T " z . --xnll 2,
(11) However IIT"z. - p l l e _< a.llzn - p l l e + I I z . - T"z.[] 2 IlZn --ell 2 = II/3nT"X. + (1 - / 3 . ) X .
(12)
- e l l 2 = II/3n(Tnxn - - P ) + (1 - - / 3 n ) ( X n - p ) l l e
=/3.1IT"x. - p l l 2 + (1 - / 3 . ) l l x .
-ell 2 -/3.(1 -/3.)l[T"x.
- x . [ [ 2.
(13)
In addition IIT"x. - p l l 2 < a . l l x . - p l l 2 + Ilx. - T"x.[I e.
(14)
Substituting (14) into (13) gives IIz. - p l l e _< a . & l l x . - p l l 2 + / 3 ° l l x . - T"x.II 2 + (1 - / 3 . ) l l x . = [1 + ( a . - 1)/3n]llx" - p l l 2 +/321[x
n -
-pll 2 -/3.(1 -/3.)llx.
- T"x.II 2
T°x.ll 2.
(15)
O n the o t h e r h a n d IIz. - T"z.II 2 = II/3nT"x. + (1 - / 3 . ) x .
- Tnz.[] 2 = I I / 3 . ( T " x . - TnZn) + (1 - / 3 . ) ( x .
=/3nllZ"xn - T"Zn[[ 2 + (1 - / 3 n ) l [ x . -- Z"Zn[[ 2 - / 3 . ( 1 - / 3 . ) l l x .
- T"z.)ll 2
- T"x.[[ 2.
(16)
Substituting (15) and (16) into (12) gives IIT"z. - p l l 2 _ an[1 +
(a n -
1)/3n]llx. - p l l 2 + a . / 3 f l l x .
+ (1 - / 3 . ) l l x .
- T"z.II 2 - / 3 . ( 1
- T"x.II 2 +/3.11T"x. - Z"z.II 2
-/3.)llx.
- T"x.II 2.
(17)
W h e n (17) goes into (11), we have IIx.+a - p l [ e < a . a . [ 1 + ( a . - 1)/3.]11x. - p l l 2 + a . a . / 3 2 1 1 x . - T"x.[I z + c~./3.11T"x. - T"znll 2 + % ( 1 - / 3 . ) l l x ° - T"znl] e - or,,/3.(1 - / 3 n ) l l x . - T"x.II e + (1 - %(1 -
~.)llx.
-
Oln)llx n - p l l e
- T"z.ll e
= {1 + a n [ ( a . - 1) + a . ( a . - 1)/3.]}llx. - p l l e - a n / 3 . ( 1 --/3. -- a . / 3 . ) [ I x n -
-
T"x.ll 2 - a . ( / 3 . - o~.)llx. - T"zn[I e + % / 3 n [ I T n x . - Tnz.]] 2
= [1 + a . ( a n - 1 ) ( a . / 3 . + 1)]llx. - p l l 2 -
o~n / 3 . ( 1
-/3. -
~ . ( / 3 . - a . ) l l x . - T"z.II e + ol./3.11T"x n - T"znll e.
a n/3.)llx.
- T"x.II 2 (18)
1840
LIU QIHOU
Since T is uniformly L-Lipschitzian IITnxn - Z n z n l l <_ t l l x n - Znll = Z f n l l X n - T ° x n l I .
(19)
Substituting (19) into (18) gives [JXn+ 1 - p I I 2 < [1 + a n ( a n - 1)(a n fin + 1)]IlXn --plle -- a n fin( 1 -- fin -- an 3 n ) l l x n - T n x n l l 2 - - a n ( fin -- an)[[Xn -- TnznH 2 + an fl3LZ[lXn - Tnxnll 2
= [1 + a n ( a n - 1)(a n fin + 1)]IlXn --plle -
an fin( 1 - f n - an f n - Z 2 f f f ) I l X n - Znxnll 2
- an( f ° -- O~n)Ilxo - TnZnll 2
This completes the p r o o f of l e m m a 2.
•
LEMMA 3. Let E be a n o n e m p t y b o u n d e d convex subset of a Hilbert space; T : E ~ E uniformly L-Lipschitzian and asymptotically hemicontractive with sequence {a n} ~ [1, ~), Vn ~ N; E~= l(a, - 1) < + ~ , and F ( T ) nonempty; e < a n < Bin < b, for Vn ~ N, s o m e s > 0 and s o m e b ~ (0, L-2[(1 + L 2 ) 1/2 - 1]); x I ~ E define z, = fnTn(xn)
+ (1-
fln)x,,
Xn+l=anTn(zn)+(1-an)Xn
.
for Vn ~ N, then, lim . . . . IlXn -- Zx.II = O. Proof of lemma
3. We will first prove the following equality, lim IlXn - TnxnII = O.
It follows f r o m l e m m a 2 that
[[Xn+ 1 - p l l 2 _< [1 + a n ( a n -- 1)(a n Bin + 1)]IlXn - e l l 2 - an fin(1 -- f n - - a n fin - t 2 f 2 ) l l X n -- an( f n -- an)llXn -- Znznll 2
- T n x n 112
Vp ~ F(T).
Since 0 < a n < fin, it can be d e d u c e d that IIx,+ 1 - p l l e -< [1 + a n ( a n - 1)(a n fin + 1)]llxn - p l l e - an 3 . ( 1 -- 3n - - a n f n -- Lefff)llxn - T°Xnll 2.
Thus Ilxn+ 1 - p l l e - Ilxn - p l l e ~ a n ( a n -- 1)(an fin + 1)llxn - p l l e - - a n f n ( 1 -- fin --an fin - L2fl2)llxn
- Tnxnl[ 2"
(20)
En= l ( a n - 1 ) < Woo implies that lim n_,o~(a n - 1 ) = 0. Hence, {an} n_ 1 is bounded. In addition, f r o m the b o u n d e d n e s s of E and 0 < a n < fin < 1, { a n ( a n f n + 1)IIXn --pIIZ~= 1 must be bounded. T h e r e f o r e , there exists a constant M > 0 such that 0 < an(a n
fn
-~- 1)llxn - p l l 2 < M .
(21)
Asymptoticallydemicontractive and hemicontractive mappings
1841
F r o m (20) and (21), the following inequality holds
IlXn+ 1 - P I I 2 -
I[Xn - p I I 2 < M ( a n
- 1) - a n/3n(1 - [3n - a n [3n - L2/32)llxn - r " x n l l 2.
(22)
O n the other hand, 1 - 2b - L Z b 2 = C > 0 can be d e d u c e d f r o m b ~ (0, L - 2 [ ( 1 + L2) ~/a - 1]). Since lim n _. sa n = 1, there exists s o m e natural n u m b e r N, w h e n n > N 1 - fin - an ¢tn - L Z ~
> 1 - b - anb - LZb 2 > C/2.
(23)
Supposing lim n ~ l l x n - T n x n l l - ~ O, then, there must exist a e0 > 0 and a subsequence {xnk}~= 1 of {xn~=l such that
IIx.k -
rn*x.kll 2 >- So.
(24)
W i t h o u t losing generality, we can suppose n 1 > N. F r o m (22)
Olnfin (1
-- fin - - a n fin - g2/3f)llxn - Tnx,,[I 2 n - 1) + Ilx.
-pll 2 -Ilxn+l -Pll 2
nk
>( ~ m=n
O t m f l , n ( 1 - flrn - a r n f l m
- g2fl2m)llxm - Tmxmll 2
1
nk
<
M ( a m - 1) +
~ m=n
Ilxn,~-pll 2 -Ilxnk+~ -pll e
1
k X E °tn,[Jn,(]-[~n,-an;[~nt-g2[j2)lIXnt l=1
- Tn'xn; 112
nk
<-- E m=n
°tmflm(1-
fl-am[3m-
L2fl2)IIxm - Tmxmll2
I
nk
<
~_. M ( a m - 1) m=n
+lIx,-pll 2-11xn~+a
- p l l 2.
(25)
l
F r o m (23), (24), (25) and 0 _< ¢ < a n < Bin, it can be easily seen that n k
ke2(C/2)eo
<_ ~.. M ( a m - 1) + IIx,,-Pll 2 -Ilxnk+l - p l l 2. m=n
(26)
l
E~ = ~(an - 1) < + ~ and the b o u n d e d n e s s of E imply that the right side of (29) is bounded. However, in fact, the left side of (26) is positively u n b o u n d e d , when k - ~ ~. This is a contradiction. T h e r e f o r e lim IIx. - Z n x . I I = 0. n
--~ o¢
Since T is uniformly L-Lipschitzian, from l e m m a A, it follows that lim IIx~ - Zx,,ll = O. This completes the p r o o f of l e m m a 3.
•
1842
LIU QIHOU
Proof of theorem 3. E is a nonempty closed bounded convex subset of a Hilbert space and T is a completely continuous self-mapping in E. Thus, it follows from the Schauder's theorem that F ( T ) must be nonempty. Using lemma 3 lim IIx. - Tx.II = 0
n--*~
(27)
Since T is completely continuous, there exists a convergent subset {Tx.k}~= 1 of {Tx.}~=l. Let lim k .~Tx.~ =p. From (27) lim x.~ = p .
(28)
k-~
Using the continuity of T and (27), lip - Tell = 0, that is, p is a fixed point of T. Using lemma 2 again IIx.+l - P l l 2 -< [1 + a n ( a n - 1)(a./3. + 1)]llx. - p l l 2 --O/.
/3n(1
-- /3. -a
n/3n
- Z2/3Zn)llx. - Z " x . II2
- ol.(/3. - ~.)llx. - T"z.ll 2.
(29)
(23) shows that there exists some natural number N, when n > N, 1 - 13.- a . / 3 . - L2/32 > C / 2 > 0. In addition, from (21), 0 _< a.(a./3. + 1)[[xn _pl12 _
(30)
E . - ~ ( a . - 1) < + shows that E . _ 1M(a. - 1) < + . On the other hand, (28) Indicates that there exists a subsequency {fix.,-pll2}~_ a of {[[x. _pl[2}~= 1, which converges to 0. Therefore, ,2 from (30), and lemma 1, lira. __,=[[xn - p l [ = 0, that is, lira. _~=x. = p . This completes the proof of theorem 3. •
Proof of corollary of theorem 3, E is a nonempty closed bounded convex subset of a Hilbert space and T is a completely continuous self-mapping in E. Thus, it follows from the Schauder's theorem that F ( T ) must be nonempty. T is an asymptotically psuedocontractive mapping with sequence {a.}, a. a [1, oo). Hence, T must be an asymptotically hemicontractive mapping with {an}, a. ~ [1, oo). Since a. ~ [1, ~), a n2 - 1 > a. - 1 > 0. Obviously, E~ = l ( a n 1 ) < + ~ , f r o m E ~. = l(a.2 - 1) < + ~. Therefore, it follows from theorem 3 that lim x. = p ,
n ---~oo
p ~ F(T).
This completes the proof of corollary of theorem 3. Acknowledgement--The author expresses his sincere thanks for the financial support from the Scientific Fund of Aeronautics and Astronautics Ministry of the People's Republic of China.
REFERENCES 1. SCHU, J. Iterative construction of fixed points of asymptotically nonexpansive mappings, J. math. Analysis Applic., 158, 407-413 (1991). 2. REINERMANN, J. LJber Fixpunkte Kontrahieruder Abbidungen und Schwach Konvergente Toeplitz-Verfahren, Arch. Math., 20, 59-64 (1969).