AN N ALES SOCIETATIS M A T H E M A TIC A E P O L O N A E Series I: C O M M ENT ATION ES M A T H E M A TIC A E X X IX (1989) R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A T E M A T Y C Z N E G O
Séria I: PRACE M A T E M A T Y C ZN E X X IX (1989)
L. Dr e w n o w s k i (Poznan), M. Ki s i e l e w i c z and L. Ry b i n s k i (Zielona Gôra)
Minimal selections and fixed point sets
of multivalued contraction mappings in uniformly convex Banach spaces
Abstract. In the present paper we extend the result of Markin [4 ] concerning the stability of fixed point sets of multivalued contraction mappings.
Let Z be a nonempty closed subset of a uniformly convex Banach space (Y, |*|). Denote by Cl(Z) (resp. C(Z)) the space of all nonempty closed (nonempty closed and convex) subsets of Z equipped with the generalized Hausdorff distance D given by
D(B, C) = max {d(B, C), d{C, B)}, where for E, H c Y
d(E, H) = sup{d(w, H): ueE}
and for и e Y
d(u, H) = inffitt — v\: veH}.
We will prove the following parametric version of the contraction principle:
Th e o r e m. Let A be a topological space (uniform space), and let A be a nonempty closed subset of a uniformly convex Banach space (Y, |-|). Suppose a multivalued mapping H: A x A->C(A) is such that:
(i) for every X e A
r(X) = sup{d(x, H(X, x)): x eX } < oo,
(ii) the multivalued mappings Я(-, x): X-+H{X, x), x e A , are equicontinu- ous (equiuniformly continuous on the sets Ar = (Ae A: r(X) ^ rj, r ^ 0),
OO
(iii) there exists a nondecreasing function q> : R+ ->R+ with £ <p"(r) < oo
n = 1
for every r e R + and such that for every XeA, x, y e A, D(H(X, x), H(X, y)) ^ <p(|x-y|).
Then for each XeA the set S(X) = {xeA\ хеН(Л, x)} is nonempty and closed, and the multivalued mapping S: A-+Cl{A) is continuous (uniformly continuous on the sets Ar, r ^ 0). Moreover, there exists a continuous mapping h: A x A-* A (uniformly continuous on the sets Ar x A) such that for every AeA the mapping h(A, •) is a retraction of A onto S{A), i.e., h(A, x) = x iff xeH(A, x), and h{A, x) eH(A, h(A, x)) for x e A .
(In (iii) q>°(r) = r, cpn{r) = <р(<рп_1(г)))-
This type of result concerning stability of fixed point sets was given in Markin’s paper [4] for a uniformly convergent sequence of multivalued mappings Fn: A-*C(A), Fnz$F0 satisfying
D(F„(x), Fn(y)) K\x-y\ for n = 0, 1, . . . , x , y e A,
where K e[0, 1) and A is a nonempty closed bounded subset of a Hilbert space.
In this note we resort to the methods presented in [1], [2], [4]. We would like to stress that the proof of the theorem (as well as Markin’s proof) depends heavily on the properties of the selection which assigns to each set В e C{Y) the unique element s(B)eB with \s(B)\ = d(0, B) (see [8]). In the case of an arbitrary Banach space Y one can prove that the multivalued mapping S is lower semicontinuous assuming that A is paracompact, the mappings H(-, x), x e A , in (ii) are continuous (not necessarily equicontinuous), and assuming additionally in (iii) that lim <p(s) < r (see [6] for details).
s - * r +
1. Minimal selection. We start by stating the basic property of the metric projection in uniformly convex Banach spaces. The property in essence but in a different form has been established by Holmes ([3], Theorem 3). His proof resorts to a lemma due to Ruston [5]. Below we give an elementary proof.
Another elementary proof can be found in [7].
For ZeCl(Y) and r ^ 0, let
Cr(Z) = {BeC(Z): d(0, B) ^ r}. •
1.1. Pr o p o s i t i o n. The mapping s: C(Y)->Y, which assigns to every B eC( Y) the unique point s(B)eB with |s(B)| = d(0, B), is continuous, and its restriction to each of the sets Cr(Y), r ^ 0, is uniformly continuous.
Proof. Fix R = max{r, 1} and choose an arbitrary ee(0> 1). Let k eN be such that
(l-5(e/3R ))-(l+2/fc)< 1,
where £>(•) denotes the modulus of convexity for Y. We show that (0) if B , C e C r{Y) and D(B, C) < q = e/3k,
then \s{B) — s(C)| < e.
Fixed point sets of multivalued contraction mappings 45
Suppose that (0) is false and let B, C e C r(Y) be such that D(B, C) < rj and for b = s(B), c = s(C) we have
(1) \b-c\>8.
Observe that
(2) ||fo| — |c|| < X>(S, C) < 17.
Hence
1Ы > з£, |c| > ie.
Indeed, if e.g. |b| ^ ^e, then by (2) we have |c| < §£• Hence \b — c\ < s, but this contradicts (1).
Now, let b = s(B — c) + c. Clearly, UeB and
(3) |b —c| = d( c,B)^D(C,B).
We have
(4) |b — b\ ^ \b — c| — |b — c| ^ £ — rj ^ fe.
Moreover,
(5) |Ь/|Ь||<1, \В/Ц = 1
and by (4), (5):
b__b_
Ш ~ \ ь \
By the uniform convexity of Y,
(6) $(Ъ + Б)\*:(1-0(г/щу\Б\.
From (5) and (2) it follows that
|b| ^ \c\ + D(B, C) ^ |b| + 2D(J3, C).
Hence, taking into account that |b| ^ ^e, we get
\B\ ^ |b|4-2e/3k ^ \b\(l+2/k).
2 e l e
\B\3> R3'
Applying (6) gives
\i(b + b)\^\b\(l-0(e/3R)){l+2/k).
Thus %(Ь + Б)еВ and \г{Ь + Ь)\ < d(0, B). But this contradicts the definition of d(0, B).
Let AeCl(Y) be fixed and let j: A -*■ Y be the identity mapping on A. We
define (ЦЛ, У) to be the space od all continuous multivalued mappings F : A->C(Y) such that
d(j, F) = sup{d(x, F(x)): x e A ) < oo.
(ЦЛ, У) will be considered with the generalized supremum metric q defined by
q(F, G) = sup{Z)(F(x), G(x)): хеЛ ).
Clearly, Y is embedded into С(У) in a natural way: y->{y}, so every mapping / : A -> У may be identified with the multivalued mapping/: x { / (x)}. Below we do not distinguish between / and / and we use the same symbol / in both meanings. We define <%(.A, У) to be the subspace of (£(Л, У) consisting of all single-valued mappings. Clearly, for f, g e (é{A, У) we have
d{f, 0) = Q{f, g) = su p {| /(x)-0(x)|: xe Л}.
Furthermore, for r e R + we set
\
е г(Л, Y) = {Fe<£^, У): rftf, F) < г}, VA*, Y) = { f e « ( A , У): d ( j , f ) < r } .
We are interested in the properties of the operator I under which to every multivalued mapping F : Л->С(У) there corresponds a mapping / = IF: A —► У defined by
/(x ) = s(F (x)-x) + x,
where s: С(У)->У is the mapping considered in Proposition 1.1.
Let us make the following simple observations. For F, G : Л->С(У) and x, y e A we have:
( + ) \(ZF)(x)~(ZF)(y)\ ^ \s(F(x)-x)-s(F(y)-y)\ + \x-y\
and
D(F( x) -x, F( y) -y) sS |x — >'| + D[Fix), F(y));
( + + ) I (IF)(x) - (£G)(x)| = |s(F(x) - x) - s(G(x) - x)\
and
D(F(x) — x, G(x) —x) = D(F(x), G(x)) < q(F, G).
It is also obvious that d{j, IF) = d{j, F).
1.2. Pr o p o s i t i o n. The operator I is a retraction о/ЩА, У) onto 4!{A, У).
Moreover, for every r e R + the restriction of I to (£Г(Л, У) is a uniformly continuous retraction onto # Г(Л, У), and i / g c (£Г(Л, У) is a set of equiuniformly continuous multivalued mappings, then F(5) <= ^ r(A, Y) is a set of equiuniformly continuous mappings.
Fixed point sets of multivalued contraction mappings 47
Proof. It is obvious that Zf —f forf e ^ ( A , 7). The fact that F еЩА, 7) implies I F e^(A, 7) follows from ( + ) and Proposition 1.1 (and the condition d(j, F) < oo is not needed here). Clearly, Fe&r(A, 7) implies I F e ^ r(A, 7), since d(j, ZF) = d(j, F). The uniform continuity of Z restricted to (£r(A, 7) for every r e R + follows from (+ + ) and Proposition 1.1. Hence Z is continuous on (£(A, 7). Finally, from ( + ) and Proposition 1.1 it follows easily that T(3r) is a set of equiuniformly continuous mappings, whenever ^ is a set of equiuniformly continuous multivalued mappings.
Note that for F: A-*C(Y) and its set of fixed points S(F) = {xeA: xeF(x)}
we have
S(F) = S(ZF) = { x e A : x = (ZF)(x)}.
Hence for Fe&(A, 7) the set S(F) is a closed subset of A.
2. Iteration and fixed point sets. Below we shall deal exclusively with multivalued mappings F: A-*C{A). We define (£(Л), &(A), &r(A), <€r(A) to be the subsets of Ci(A, 7), %>(A, 7), &r(A, 7), r(A, 7), respectively, consisting of multivalued mappings F: A-+C(A). Note that të(A) and &r(A) are complete metric spaces with the metric q.
Let a set W and a mapping/: W -> W be given. For every n e N we denote by / " the n-fold composition of / with itself, i.e.,
Z1 = / , / " = / o/ " -1 forn = 2, 3 ,...
For n e N , let the operator Kn: %>(A)^%>(A) be defined by
* „ / = / " •
2.1 Le m m a. Let SF c i %>(A) he a set of equiuniformly continuous mappings.
Then, for every neN, the restriction of the operator Kn to is uniformly continuous and the set Kn(,!F) consists of equiuniformly continuous mappings.
Proof. For every neN, let &>„(•) be the common modulus of uniform continuity of the set Kn(,$F), i.e., for b > 0,
(on(8) = swp{\h(x)-h(y)\: h e K n{3?), x , y e A , \x-y\ < b}.
There is nothing to prove when n = 1. Assume that the lemma is valid for all n < m (m ^ 1).
Let f, g e2F and x e A . Then
\(Km+1f ) { x ) - { K m + 1g)(x)\ = \f(fm(x))-g(gm{x))\
^ \f{fm( x ) ) - g { f m(x))\ + \g(fm(x))-g(gm(x))\
so that
Q{Km + if, Km + l g) < g(f, g) + co1(Q(Kmf, K mg)) and this shows the uniform continuity of Kn for n = m + 1.
L et/eJ * and x , y e A . Then
|(Km+1/ ) W - ( K m+1/)ty)| = \ f ( f " ( x ) ) - f ( f m(y))\
^ <»1(|/"(x)-/"Cv)|) « CO,(<o„(|x->>!)) so that
OJm+1{0) < for all S > 0,
and the equiuniform continuity of the elements of K„{#") for n = m+ 1 is immediate.
Let (p: R+ -+R+ be a nondecreasing function such that
00
£ (pn(r) < oo for every r e R +.
Л=1
It follows easily that q>(r) < r for r > 0 and <p(0) = 0.
Let (^(Л) denote the set of all Fed{A) satisfying the condition (C) D(F(x), F(y)) ^ <p{\x — y\) for all x, ye A,
and let G?(A) = G*(A)n<Zf(A) for r e R +.
Consider the operator I„ = K„o I on d<p(A). Clearly, (£?(Л) consists of equiuniformly continuous multivalued mappings. Therefore, using Proposition
1.2 and Lemma 2.1, we get the following:
2.2. Co r o l l a r y. For every neN the operator I n: а <й{А)-^<ё(А) is con
tinuous, its restriction to 6^(Л) is uniformly continuous, and Zn{(£f(A)) is a set of equiuniformly continuous mappings for every reR + .
2.3. Pr o p o s i t i o n. The sequence of operators (En) converges to a continuous operator Г : d<p(A)-^(ë(A) such that for every Т еС <р(Л) the mapping TF is a retraction of A onto 5(F). Moreover, for every r e R + the restriction of Г to
&f(A) is uniformly continuous and the set Г {(if (A)) consists of equiuniformly continuous mappings.
Proof. Let Тб(£ф(Л), rF = d(j, F) and / = IF. Then for every x e A
l / 2( * ) - /1MI = |s (f(/(4 -/M )l = à(f{x), F(f(x)))
« D(F(x), F{f(x))) « q>(\x-f(x)\) and by induction
|/*+1(x)-/"(x)| < <p"(|x-/(x)|).
Fixed point sets of multivalued contraction mappings 49
Hence
Q{fn+\ f n) ^ <Pn(d(j,f )) = (pn{rF) for all neN.
Clearly,/" = Z„F, so from the above inequalities and the assumption on qyit follows immediately that the sequence (FnF) converges in Я>(А) to some mapping g; we set Г F = g. It is also clear that the convergence is uniform for all F such that rF ^ r, i.e., for F e(£?(A), when r e R + is fixed. This together with Corollary 2.2 imply that F is uniformly continuous on &?(A) and Г {(if (A)) is a set of equiuniformly continuous mappings for every r e R +. From this it follows easily that Г is continuous on (^(Л).
If xeS(F) = S(f), then /"(x) = x for every neN and g(x) = lim/"(x) п-*-о0
= x. On the other hand, if x e A and у — g(x) = lim/"(x), then f(y)
n —* 00
= lim /(/"(x)) = g(x) = y, so g (A) a S(F).
/!“►00
Thus for every F e C4’ {A) its set of fixed points S(F) = S(IF) = S{rF) = (rF)(A)
is a nonempty closed subset of A and S(F) is connected (arcwise connected) if A is connected (arcwise connected).
It is easy to verify that for / , g: A->A
D(f(A), g (A)) < s u p {| /(x )-0(x)|: x eA}.
Hence for any F, Ge&^iA) we have
D{S(F), G(G)) = D((rF)(A), (.ГС){А)) < Q(rF, ГС).
From this and Proposition 2.3 we have the following
2.4. Co r o l l a r y. The mapping S: ^ ( A ) - * Cl(A) defined by S(F)
= { x e A : xeF(x)} is continuous and for every r e R + its restriction to (£>(T) is uniformly continuous.
P r o o f o f the theorem. Consider the mapping F: Л-»Я(А, •)• By assumptions (i) and (iii) we have that F(A)e£<p(A) for every XeA. Hence by (ii) F is a continuous mapping (respectively: uniformly continuous on the sets Ar) from A into £ </>(A).
We define the desired mapping h: A x A - > A by setting Л(Я, X) = (rFU))(x),
and then we clearly have
S(A) = S(F(A)) = (FF{X)){A) = h(X, A).
Now the assertions of the Theorem follow immediately from Proposition 2.3 and Corollary 2.4.
4 — Comment. Math. 29.1
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