Posts by Mira Anisiu

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Mira-Cristiana Anisiu
Institutul de Matematica, Cluj-Napoca, Romania

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M.-C. Anisiu, Fixed points theorems for retractible mappings, Seminar on Functional Analysis and Numerical Methods, 1-10, Preprint, 89-1, Univ. Babeş-Bolyai Cluj-Napoca, 1989 (pdf file here)

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https://ictp.acad.ro/anisiu/papers/1989-Anisiu-FixedPointTheorems.pdf

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Babes-Bolyai University Cluj-Napoca, Romania

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[1] M.C. Anisiu, Fixed points of retractible mappings with respect to the metric projection, ”Babes-Bolyai” Univ., Fac. of Math. Phys., Preprint nr.7, 1988, 87-96.
[2] M.C. Anisiu, V. Anisiu, On some conditions for the existence of the fixed points in Hilbert spaces, ”Babes-Bolyai” Univ., Fac. of Math. Phys., Preprint nr.6, 1989, 93-100.
[3] R.F. Brown, Retraction methods in Nielses fixed point theory, Pacific J. Math. 115 (1984), 277-298.
[4] W. Cheney, A.H. Goldstein, Proximity maps for convex sets, Proc. Amer. Math. Soc. 10 (1959), 448-450.
[5] G. Darbo, Punti uniti in transformazioni a condominio noncompatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92.
[6] T.C. Lin, C.L.Yen, Applications of the proximity map to fixed point theorems in Hilbert spaces, J. Approx. Theory 52 (1968), 141-148.
[7] I.A. Rus, The fixed point structures and the retraction mappings principle, ”Babes-Bolyai” Univ., Fac. of Math. Preprint nr.3, 1986, 175-184.
[8] S.P. Singh, B. Yatson, On approximating fixed points, Proc. of Symp. in Pure Math., Vol. 45 (1986), Part 2, 393-395.
[9] T.E. Willianson, The Leray-Schauder condition is necessary for existence of fixed points, Vol. 886, Lecture Notes in Math., Springer-Verlag, Berlin

1989-Anisiu-FixedPointTheorems

"BASES-BOLYAT" UNTVERSITT
Faculty of Cathematics and Digisies
Resuarch Seminars
Seminar on Punctional Analysis and Numerical Uetiods
Preprint Nr. 1, 1989, pp. 1 - 10.

FIXED POTNT THEORENS FOR
BÉTHACTITHETE MAPPINGS

Mira-Cristlana Anisiu

The fixed point theorems in this paper axtand some results in

/ 6 /

; if the retract is chosen to be the metric projection one abtains same known theorems.

We state firstly some dofinitions to be used in the following.
Iret

X

be a nonvoid set and

l \notin A \leq X

A

map

r t X \to Λ

is a reEract of

X

A

if the restriction of

r

to the sot

A

is tha identity map id

_{A} : A \to A . A

map

f : A \to X

is retractible on

A

with respect to the retract

x

if Fix rof

= F i x f

, where "Fix" donotes the sot of the fixed points of 2 map /3,?,1/.

It is obvious that always Fix rof

\subseteq

Fix

f

, so in the definition. of the rotractible map one can darand only Fix

f \subseteq F i x

rof. Brown in /3/ has given the following necossary and sufficient condition for the map

f : A \to X

to ba ratractible on

A

with raspect te, the retract r :

\dot{x} \in r (f (A) ∖ A)

impliss

f (x) = x

f (x) \notin r^{- 1} (x)

.
Lat

B

be a bounded nonampty sat in tha patric spaca

X

. The gassure of noncompactness in tha zanse of Kuratowsty, denated

a (B)

, is the infimum of the numbers

α

such that the set

B

can be covered
by 4 finite namber of subsets of

X

of diameter leas than or equal to

α

Let

A

be a monvoid subset of the metric space

\bar{X}

and

I : A \to X

a aap. If there exists

k, c ⩽ k ⩽ 1

such that for each nonempty subset

B

of A, B baing bounded, wo have

a (f (B)) ⩽ k a (B)

,
then

f

is called

x

-set-contractive.
Bach nonexpansive map

f : Λ \to X (

i.e.

d (f (x), f (y)) ⩽ d (x, y)

for each

x, y

in A) is obviously l-eiet-contractive.

In. the following we give a generalization of Theorem I in /6/, using in the proof the fixed point theoren of Darbot

THEORS: 1 /5/. Let

X

be a Banach space,

A \subseteq X

a closed convex nonvoid set,

f : A ⟶ A

continuous

k

-set-contractive with

0 < k < j

. such that

Z (A)

is a bounded set. Then Fix

f \neq \emptyset

In the initial form of Darbo'a theorem A is bounded, but it

∵

suffices to roquiro

f (A)

bounded, taking

A_{1} = c l c o f (A) \subseteq A

and

f : A_{1} \to A_{1}

Me nead also the following
Insuld 1. Let

X

be a normed space,

C \subseteq X

and

f : C ⟶ X

, such that

(I - f) (C)

is a closed sot. If there oxiats a sequonce of ㄱapsf

n, C \to X, n ⩾ 1

, each of them having e fixed point

x_{n} (f_{n} (x_{n}) = x_{n}), f_{n}

converging uniformly to

f_{1}

then

f_{has also a fixed}

point.

Proof. Denote

s, g_{n} : C \to I, g = I - f, g_{n} = I - f_{n}, n \geq 1

. Fach

I_{n}

having a fixed point, it follows that each

g_{n}

has a zero, heace

o \in g_{n} (d)

. Because

f_{n} \overset{n}{\to} f

uniforaly;

g_{n} \overset{n}{\to} g

uniformly.

Int

ε > 0

be arbitrarily chosen; Irom the uniform convergenció of

g_{n}

g

one obtains

g_{6} ⩾ 1

such that

‖ g_{x} (x) - g (x) ‖ < ε

for each

n ⩾ n_{ε}

and

x

C

.
It follows that for each

n ⩾ n_{E}

we have

E_{n} (C) \subseteq E (C) + B (0, E)

.
Therafore

0 \in E (C) + B (0, ε)

for each

ε > 0

and

0 \in

g (C)

= g (C)

. This means'that there exists

x \in C

such that

f (x) = \dot{x}

Now we can prove the following
THEOREM 2. Let

X

be a Banach spēce,

A \subseteq X

a nonvoid closéd convex set;

f : A \to X

and

r : X \to A

such that

F =

rof

: A \to A

is a continuous 1-set-contractive map having

P (A)

bounded and

(I - f) (A)

a closed set. Then Fix rof

\notin \emptyset

Proof. Let

t \in (0, 1), x_{0} \in A

and

F_{t} = t F + (1 - t) x_{0}

. We show that

F_{t}

t

-set contractive.

Let

B \subseteq A

a bounded subset ; then

a (F_{t} (B)) = a (t P (B) + (I - t) x_{0}) ⩽ a (t P (B)) = t a (P (B)) ⩽ t a (B)

.
Applying Darbo's theorem, each

F_{t}

has a fixed point

x_{t}

. Considering

t_{n} \to 1, t_{n} < 1

we obtain for each

x

A

‖ F_{t_{n}} (x) - F (x) ‖ = (1 - t_{n}) ‖ F (x) - x_{0} ‖ ⩽

⩽ (1 - t_{n}) d (x_{0}^{'}, F (A)) \overset{t_{n} \to 1}{\to} 0,

.
bence

F_{t_{n}} ⟶ F

uniformly on A .
Now Lema 1 applies and Fix

F \notin \emptyset

.
In the above theorem, instead of

(I - F) (Λ)

to be closod, one could require (

I - F

) (cl co

F (A)

) to be closed to

X

, considering the restriction of

F

on el co

F (A)

, whose range is also in cl co

\vec{r} (A)

In the terms of

F

, Theorem 2 is exactely Lemma 1 in

/ 6 /

, given there without proof.

4 natural example of a map

x : X \to A

is the meteric projection， which is well－defined if for mample

x

is aniformly convex．In thls case we obtain obviously

COROLTARY 1．Let

X

be a prifornaly convex Bagach Boach

i \subseteq X

ㅡ nonvoid closed convex fat，

f : A \to X

ang

P = F, t x \to A

the metric projection．If

P = p \circ f : A \to A

as a continuigs 1－getcog－ tructive men with

F (Λ)

bounded and（I．F ）（

A

）cloged get，then Rix pof

\neq 0

．

The conclusion means axactely that there oxists

x

in a ach that par

(x) = x, \frac{1}{4}, 0.1 - f (x) = d (f (x), 1)

this raspht appears in the well－icnomn theorem of

\bar{X} y

Fan（1969）gtyan for a namveld com－ pact convex subset

K

of a normed space

X

and continuous map

I

：

: K ⟶ K

a nonvoid slosed convex set，

f : A \to X

ig continuous

1 -

setchontrac－ tive map．We suppose that althgr

(I - p o f) (A)

is closed gr（

I -

－pof）（cl co pef（A））is closed in

x

, whers

p = P_{A} : x \to A

is the metric profection．If

f (A)

is bounded，then there exists u in such that

‖ u - f (u) ‖ = d (f (u), Λ)

.
Proof．Because

f, t \to \dot{X}

is continneus 1－set－contractive and i

p : X \to A

is nonexpansive

/ 4 /

，it follows that

F = p a r

is a conti－ nuous 1－sht－contractive map．The fact that

f (A)

is bourded implios

F (A)

boundad and Cerellary 1 applies．

It is obvieus that in Caroliary 2 ，ingtend of

1 (1)

to be bounded it is enough the require pof（A）to be boandad．

In the papar／4／there mew given many results which fallow from Corsllary 2，mong mich theorems of Isn，Singh and Hatison．

If in Theorein 1 we can choose

r : X \to Δ

to be a retract such that i is a retractible map with respect to r，we obtain a rixed polyt theoren for

^{1}

．

FHSOBJM 3 ：If in the conditions in Theorem

1, x : x \to 118

a Fetract and

I : A ⟶ I

ls retractible with respect to

r

，then致和

f = 6

．

Froof．From Theoren 1 it follows Fir rof

\notin g

and

f

being retrac－ tible with respect to

r

we have

F i x f = F i x

rof

\notin \emptyset

．
comothary 3 （Theorem 5 in／6／）．Let

\underset{―}{x}

be a Hilbert space，

↑

a nonvoid closed convex set，

f : A \to X

a continuous 1－set－contrac－ tive map．量e suppose that sither

(I \sim p \circ f) (A)

is closed in X or （I－pof）（cl co p－f（A））is closed in

X

，where

P = P_{A}

is the netric projection．If

f (A)

is bounded and

f

satisfies one of the following conditions ：
（1）For each

x

A

，there is a number

λ

（real or complex，de－ pending on whether the vector space

x

is real or complex）such that

| λ | < 1

and

λ x + (1 - λ) f (x) \in Λ

．
（2）For each

x

in 1 with

x \neq r (x)

，there exists

y

I_{A} (x) == {x + c (z - x) : z \in A, c > 0}

such that．

H y - f (x) ‖ < ‖ x - f (x) ‖

.
（3）

f

is weakly inward（i．e．

f (x) \in

I_{A} (x)

for each

x

in A）．
（4）For each u in the boundary of 1 with

u = p \cdot f (u)

，u is a fixed point of 1 ．
（5）For oach

x

in the boundary of

A, ‖ f (x) - y ‖ \leq ‖ x - y ‖

for some

y

A

．
Then

I

has a fixed point in

A

．
Proof．Corollary 2 applies，so Pix pof

& \infty

：Each of the five conditions implies（4）

/ 2 /

，which is in fact exactly Fix fof

=

This means that

f : A \to X

is retractible with respect to

p

and applying Theoxam 3 one has Fix

£ \neq \emptyset

We mention that the condition of the retractibility . Fix pot

==

Pix

f

takes in Eilbert spaces the rollowing equivalent forms

t

(i) For each in in the boundary of

i

which is not a fixed point for

f

, there exists

y

A

such that

(f (u) - u, u - y) < o

.
(ii) Ror each 4 in the boundary of 4 which is not a fixed point for

f

, there exists

y

A

such that

‖ y - f (u) ‖ < ‖ u - f (u) ‖

.
(iii) For aach

u

in the boundary of

A

which is not a fixed potnt for

t

lim_{t \to 0_{t}} \frac{1}{t} a ((1 - t) u + t f (u), \dot{A}) < ‖ f (u) - u!

.
The equivalence follows easily from the next lemme, which relies on the fact that in a llibert spiace for a closed conver set 1 and

x

x

one has

p (x) = a

iff Re

(x - a, y - a) \leq 0

for each

\exists

A (p = P_{A}

being the metric projection).

Lisen 2. Let

X

be a Eilbert Brace,

A \subseteq X

a nomvoid closed convex set, a in

A, x

X

. The following assertions are equivalent :

i^{c} p (x) = 2

, i.e.

‖ x - a ‖ ⩽ ‖ x - y ‖

for aach

y

in.

2^{0} lim_{t \to 3} \frac{1}{\frac{1}{t}} d ((1 - t) a + t x, A) =

= lim_{t \to 0,} \frac{1}{τ} d ((1 - t) a + t x, A) =

= inf_{lat 0} \frac{1}{t} d ((1 - t) a + t x, d) = ‖ x - a ‖

3^{2} min_{ini} \frac{1}{t} d ((1 - t) a + σ x, A) ⩾ ‖ x - i ‖ :

\Rightarrow 2^{\circ}

. Tree first equalities in

2^{\circ}

.ase true because
the map

ψ : 3_{+} ∖ {0} \to R_{+}, ψ (t) = \frac{1}{t} d (a + t (x - a)

, i) is increaaine and

\begin{aligned} lim_{i \geq t > 0} Ψ (t) = lim_{t > 0} Ψ (t) . \\ Now inf_{\begin{array}{c} i \geq t > 0 \\ i^{2} (t) \end{array}} inf_{\begin{array}{c} i \geq 0 \\ t \end{array}} \frac{I}{t} d^{2} (t (x - a), A - a) = \\ = \ln_{1 \geq t > 0} d^{2} (x - a, \frac{1}{t} (A - a)) \\ = inf_{t \geq 1} d^{2} (x - a, t (d - a)) = \\ = {ligf}_{t \times 4} inf_{y \in A - 2} ‖ x - a - t y ‖^{2} = \\ = inf_{y \in A - 2} {inr}_{t \geq 1} (t^{2} ‖ y ‖^{2} - 2 t Re (x - a, y) + ‖ x - a ‖^{2}) . \end{aligned}

But

A_{θ} (x - a, y) = R_{θ} (x - a, y + a - a) \leq 0

, because

y + a \in a

y d (x) == a

It follows that the map to be unimized is increasing on

[Re (x - a, y), + \infty)

, hence also on

[1, + \infty)

. The influua will be attained on

t = 1

and

\begin{aligned} inf_{1 \geq t > 0} ψ^{2} (t) & = {int}_{y \in A - a} ‖ x - a - y ‖^{2} = \\ = inf_{y \in A} ‖ x - y ‖^{2} = ‖ x - a ‖^{2} \end{aligned}

and the implication is proved.

2^{\circ} ⟹ 3^{\circ}

in obvious,

3^{0} \Rightarrow 1^{0}

. Te have

‖ x - a ‖ \leq inf_{t \to 0} \frac{1}{t} ((1 - t) a + t x, 1) \leq

\leq ψ (1) = d (x, 1) \leq ‖ x - y ‖

for each

J

Λ

.
Remark. The equivalence in Inama 2 is also true if

λ

is a prenifibertian space and 1 a complete convex set.

The assertions (i) - (iii) are all equivalent to the condition (4) in Cogollary 3.

For (1) one uses the fact that pof(u)

\notin u

is equiralent to the axistance of

y

A

such that

B o (u - f (u), j - u) > 0, u s i n g

the characteriation of the metric projection memtloned before Lawia 2 was given.

For (ii) one applias fust the definition of

p (f (a))

and obtains inde inequality in (ii).

For (iii) wa use the equivalence

1^{\circ} ⟺ 3^{\circ}

which was proved in Leana 2.

So the ascertiona (i) - (iii) ure fust reformulations of the Iact that she map

f : Λ \to X

is retractible on

A

with respect to tiau retraction p.

The equivalence of (i) - (iii) was in fact proved in /9/, where (i) is called the Leray-Schauder condition, (ii) the BrowderPetryshy condition and (iii) the Cramer-Ray condition. Hare Te emphazised the rale of the properties of the metric projection in this equivalence.

We flnish the paper giving a mathod of approximation of fixed Folnts for maps with pol nonexpansive in liilbert spaces by a procedure sinilar to that in the proof of Theorem 1 .

THSOEN 4. Let

X

be a Hilbert space, A a nonvoid cloged convex subset of

x, 1; A \to X

a nap retraction an

A

with resp. at to the petric projection such that

F =

pol

: Λ \to Δ

is nonempansive and

P (1)

bounded.

Consider

F_{k} : Λ \to Λ, F_{k} (x) = k F (x) + (1 - k) x_{0}, 0 < k < 1

k \to 1, x_{0} \in C

and

x_{k}

, the fixed point of the contraction

F_{k}

Then

x_{k} \overset{k \to 1}{\to} J_{0}

, whore

J_{0}

is the fixed point of

f

which closest to

x_{0}

Proof.

The flxad point act Fix

F

is nonvoid; let

J_{0}

be the fixad point
of

F

mileh is closest to

x_{0}

. For

F

one applies the approxiation rasult in

/ 8 /

; for

k_{n} \overset{n}{\to}, k_{n} \in (0, 1), n \in M

, one obtains firstly that

{x_{k_{n}}}

N

is a bounded scquence. Phan a subsequence of

{x_{k_{n}}} n \in N

will converge veakij to a paint

x

. Using the dealiclosedness of I-I one obtatns

x = y_{0}

and then the strong convorgenco or

{x_{k_{n}}}_{n \in M}

y_{0}

But Pix

P = P

I

and the theorer is proved.

REFZENCES

M. C. Anisiu, Fixed points of retraotible nappings with respect to the netric projection, "Babes-Bolyal" Univ., Fac. of Nath. Phys., Preprint nr. ?, 1988, 67-96
M.C.Antstu, V.Anisia, On some conditions for the existence of the fixad points in Hilbert spaces, "Babes-Bolyai" Univ., Fac. of With. Phys., Preprint nr. 6, 1989, 93-100
R.F.Brown, Retraction methods in Nielsen fixed point theory, Pacific J. Math. 115 (1984), 277-298
W. Gheney, A.II.Goldstein, Proximity maps for convex sets, Proc. Amer. Math. Soc. lo (1959), 448-450
G.Darbo, Punti uniti in transformazioni a condominio noncompatto, Rend. Sen. Mat. Univ. Padova 24 (1955), 54-92
笙C.Lin, C.L. Yen, Applications of the proximity nap to fixed point thzoreas in Hilbert spaces, J. Approx. Theory 22 (1968), 141-148
I.A.Bus, The fired point structures and the retraction mappings
principla, "Baves-Bolyili" Univ., Fac. of Bath., Freprint ar. 3, 1986, 175-184
S.P.Singh, B. Watson, On approximating fixed points, Proc. of Syap. in Pure Math., Vol. 45 (1986), Part 2, 393-395
F. T. T. Filliaason, The Leray-Schauder condition is necessary for existence of fixed points, Vol. 886, Lecture Notes in Math., Springer-Verlag, Berin.

\begin{aligned} Institutul de Matenaticá \\ C.P. 68 \\ I400 Cluj-Hapoca \\ ROMANIM \end{aligned}

This paper is in final form and no version of it is or will be Edbaitted for publication elsewhers.

Posts by Mira Anisiu

Abstract

Authors

Keywords

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

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References

Proof.

REFZENCES

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