An Application of Opial’s Modulus to the Fixed Point Theory of Semigroups of Lipschitzian Mappings

LUBLIN - POLONIA

VOL. LI.2,17 SECTIO A 1997

TADEUSZ KUCZUMOW and SIMEON REICH

An Application of Opial’s Modulus to the Fixed Point Theory

ofSemigroups of Lipschitzian Mappings

Abstract. In this paper we present a new theorem concerning the exis­

tence of common fixed points of asymptotically regular and uniformly Lip­

schitzian semigroups.

1. Introduction. Let (A, ||-||) be a Banach space and let A be a family of sequences in X. The family A is called a family of convergent sequences [12], [19] if A satisfies the following conditions

(i) A is a linear space,

(ii) each {xn} 6 A is bounded,

(iii) if {i„} G A, then each one of its subsequences {zBj} also belongs to A,

(iv) there exists a limit function A-lim : A —► X which is linear, (v) if xn = x for n = 1,2,..., then {z„} € A and A-lima:n = x,

(vi) if {in} € A and A-limi„ = x, then A-limzni = x for every subse­

quence of {x„},

1991 Mathematics Subject Classification. 46 B 20, 47 H 10, 47 H 20.

Key words and phrases. Opial’s modulus, asymptotically regular and uniformly Lip­

schitzian semigroups, fixed points.

(vii) each norm convergent sequence {xn} has a subsequence {xnj} such that {xn,} G A and limx„. = A-lim xnj,

(viii) the norm ||-|| is lower semicontinuous with respect to A, i.e.,

||A- lim xn|| < lim inf ||xn||

n

for each {xn} € A,

(ix) A has sequences which are not norm convergent.

We say that a nonempty bounded subset C of X is sequentially A - compact if C is closed with respect to A-lim and every sequence {xn} in C has a subsequence {xn<} which belongs to A.

We will use the following notation. If {^n}n>i is a bounded sequence and x G X, then

r (x, {xn}) = lim sup ||x - xn||.

n—*oo

A Banach space X is said to satisfy the non-strict A-Opial condition [14], [22] if whenever a sequence {xn} G A and A-lim xn = x, then

lim inf ||x - xn|| < lim inf \\y - xnII

n—^{►oo n—+OO}

for every y G X.

Now we define the Opial modulus rx,\ of X with respect to the family A [21], [23] by rx,A (c) = inf {lim infn —>oo ||xn + a:|| — 1} , where c > 0 and the infimum is taken over all x G X with ||x|| > c and all sequences {xn) £ A such that A-limxn = 0 and lim inf ||xn|| > 1. The function rx,\ is continuous and nondecreasing [21], [24].

If for s > 0 and c > 0 we denote inf {lim infn_oo ||xn + a:|| — s} by rx,A,s(c), where the infimum is taken over all x G X with ||x|| > c and all sequences {xn} G A such that A-lim xn = 0 and liminfn_oo ||xn|| >

then we have

(L1) 5+ rX,A,»(c) ?= s (l + rX)A •

Now let (X, d) be a metric space and T : X —► X. We use the symbol

|T| to denote the exact Lipschitz constant of T, i.e.,

|T| — inf {k G [0,oo] : d(Tx,Ty) < kd(x,y) for all x,y G X}.

If G is an unbounded subset of [0, oo) satisfying t + h G G for all t, h G G, t — h G G for all t,h G G with t > h, and 5 = {Tt : t G G} is a family of

self-mappings on X such that Ts+tx = TaTtx for all s,t € G and x € C, then E is called a semigroup of mappings on X.

5 is said to be uniformly Lipschitzian if there exists k € R+ such that

|TJ < k for each t 6 G [13], [14].

We also use the following notation:

a (E) = lim inf |T(|.

n—*oo

If E satisfies, in addition, lim^oo d(Tt+hX, Ttx} = 0 for each x € E and h G G, then E is said to be asymptotically regular [4].

2. Existence of common fixed points of semigroups of mappings.

Our main result is the following theorem.

Theorem 2.1. Let X be a Banach space with rx,A(l) > 0 and with the non-strict A-Opial property. Let C be a sequentially A- compact subset of X and E = {Tt : t G G} an asymptotically regular semigroup with

(2.1) <r(E) = fc < 1 + tx

,

(1).

Then there exists z in C such that Ttz = z for all t G G.

Proof. Let us select a sequence {tn} and 0 < c < 1 such that cr(E) = k = Kmn-oo |Tfn|, tn -> oo, and

(2-2) sup |Tt„ | < 1 + rx,A (c) < 1 + rX)A (1).

This is possible by (2.1) and the continuity of rx,A- First we claim that if for x G C a subsequence {Ttn x} of the sequence {Ttnx} is A-convergent to V, {Ttn y} is A-convergent to z and all the limits

(2.3) r(y,{Ttny}) = .firn ||li - Tu.3/||, r(MT‘». 3/}) = llz "

exist, then

r (Z> {Ttn,y}) = - Cr(yATiniX})

= c.lirn||j/-r<n.a:||.

(2.4)

Let us suppose this were not the case. Then, after deleting a finite number of indices if necessary - see the limit which appears in (2.3) - we have (2-5) inf ||z - Ttn. j/|| > cr (y, {Tt„. x}) .

Let us observe that r (y, {Tt„ z}) = 0 leads to

||w - < >•(!(, {T,..l}) + r (r,.,!/,

and by the asymptotic regularity of E we obtain ||y - 2/|| = 0 for j = 1,2,..., and therefore y= z. But this contradicts (2.5). Hence

(2.6) r(y,{Ttn.x}) >0.

The asymptotic regularity of E, the non-strict Opial property, the mono­

tonicity and the continuity of rx,\, and the application of (1.1), (2.2), (2.3), (2.5) and (2.6) now yield the following contradiction:

[1 + rX)A (c)] • r (y, {Ttn.i}) > <r(E) • r (y, {7\.z})

> lim sup r (rt y,{Ttn.x}'\

j—>oo x '

+ (»,{T,..4) J]-'(Mr..,4)

^lim^oollz-Tt

> [1 + r%,A (c)] • r (j/, {TtK.x})

Therefore the inequality (2.4) is valid. Now using the standard diagonaliza- tion procedure we can construct a sequence {i/} C C in the following way:

xq £ C arbitrary, z, = A-limT^Zf.j for I = 1,2,..., where all the limits r (z/+1, {Ttn.®/}) = bm ||z,+i - . z/||

and

r («<+!, {T«.,«»+i}) = Bm ||x<+i - Ttn.xl+11|

for I = 0,1,... exist. By (2.4) we have

(2-7) r (xl+1,{Ttn.x,}) < c'r(x1,{Ttn.x0})

for I = 0,1,... . Next by the A-lower semi continuity of the norm, the asymptotic regularity of E and (2.7) we obtain

11*1+1 -*l|| < r (xt+i,{Ttn.x,}}+ r(xh {Ttn.x,}) < r (x,+i, {Ttn>x,}) + limsup lim sup Ttn.x(_i - ^„.x/ < r (x/+i, {Ttn. x;})

»—*OO j—>00 11 11 + lim sup lim sup T( x,_, -Tt„.+tn. ®j-i

i—+OO j—>00

+ U^tn.+ln^. *1-1 - ^„j*!!,] - T (*1+1 » {7tBi*l})

+k-r(xh{Ttn. xi-i < c'_1 • (c + fc) • r (xl5 {Tt„. x0 })

for I = 1,2..., which shows that {x(} is strongly convergent to x. By (2.7) for this x we get

r (x, {Tj x}) < lim lim [||x - x;+i|| + ||x/+i - Ttn x;||

' 1 " /—,oo»-*ooL

+ |r,.,|-||x1-x||]=o.

The asymptotic regularity of E and |TtB. | < oo imply that Ttn.x = x for i = 1,2,... . Now we apply the asymptotic regularity of c once more to obtain

||Ttx - x|| = .lim ||Tt+t>. x - Ttn. x|| = 0 for each t £ G. The proof is complete.

Remark 2.1. Theorem 2.1 is a generalization of Theorem 3.2 in [20].

3. The case of LJ(0,1) with the topology ofpointwise convergence.

H. Brezis and E. Lieb [3] (see also [2]) proved that in Ll (0,1) if /n ~* / a-e- and the sequence {/„} is bounded in Ll (0,1), then

lim (||/n||-||/n- /II) = 11/11- n

This implies that for the family A of pointwise convergent and bounded in L1 (0,1) sequences we have rjr,i(0,i),A (c) = c and

(3.1) 1 + rL»(o,i),A (1) = 2.

We obtain the same result if we consider the family A of convergent in measure and bounded in Ll (0,1) sequences [16]. It is obvious that A A, but the family of all sequentially A-compact sets and the family of all sequentially A -compact sets coincide in Lx (0,1) [11]. In view of the equality (3.1) , Theorem 2.1 is especially interesting if we recall Alspach’s example [1]

of a fixed point free nonexpansive selfmapping of a convex weakly compact subset of L1 (0,1).

4. Common fixed points of commuting asymptotically regular and uniformly Lipschitzian mappings. In this section we establish the existence of common fixed points of commuting asymptotically regular and uniformly Lipschitzian mappings.

Theorem 4.1. Let (X,d) be a metric spa.ce and let fc > 0 be a constant such that every asymptotically regular and uniformly Lipschitzian selfmap­

ping T :X -> X with supn |Tn| < fc has a fixed point. If Ty, T2 : X —► X are two commuting asymptotically regular and uniformly Lipschitzian map­

pings such that supn |T"| < fcy, supn |T2n| < fc2 and fcy • fcj < fc, then Ty and T2 have a common fixed point.

Proof. First we observe that T = T2 o Ty is an asymptotically regular and uniformly Lipschitzian mapping. Indeed, for each x,y € X and n = 1,2,...

we have

d(Tnx,Tny) = d(T?T?x,T?T?y) < (*2 • fcy) • d(x,y), and

d (Tn+1x,Tnx) = d (T?+1T?+lx,T?T?x)

< d (T?+1T?+lx,T?+iT?x) + d (T?+1T?x,T?T?x)

< k2d (T?+1x,T?x) + d (T?T?+lx,T?T?x)

< fc2d (T?+lx,T?x) + fcyd (T2n+1z,T2nz).

Hence |Tn| < fcyfc2 < fc for all n and limn d (Tn+1z,Tnx) = 0.

By assumption there exists a fixed point of T. Now we show that every such point x0 is a common fixed point of 1\ and 72. To this end, we observe that

d(Ti®o,«o) = = d x0,T?T?x0)

< k2d (T"+1 xo,T"xo) —>0.

n

Hence T'iZo = xo- Similarly we prove that T^zo = xo- ■ Remark 4.1. For common fixed point results for nonexpansive mappings see [5], [6], [17] and [18].

Remark 4.2. For up-to-date references about fixed points of asymptoti­

cally regular and uniformly Lipschitzian mappings see [7], [8], [9], [10], [15]

and [20].

5. Acknowledgments. The second author was partially supported by the Fund for the Promotion of Research at the Technion. Part of the work on this paper was done when the second author visited the Institute of Mathematics at UMCS. He thanks the Institute and its members for their hospitality.

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Instytut Matematyki UMCS received September 7, 1997 20-031 Lublin, Poland

e-mail: tadek@golem.umcs.lublin.pl Department of Mathematics

The Technion-Israel Institute of Technology 32000 Haifa, Israel

e-mail: sreich@techunix.technion.ac.il