On some conditions for the existence of the fixed points in Hilbert spaces

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

Valeriu Anisiu
Babes-Bolyai University, Cluj-Napoca, Romania

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M.-C. Anisiu, V. Anisiu, On some conditions for the existence of the fixed points in Hilbert spaces, Itinerant Seminar on Functional Equations, Approximation and Convexity (Cluj-Napoca, 1989), 93-100, Preprint, 89-6, Univ. Babeş-Bolyai, Cluj-Napoca, 1989 (pdf file here)

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[1] M.C. Anisiu, Fixed point of retractible mappings with respect to the metric projection, “Babs-Bolyai” Univ., Fac. of Math., Preprint nr.7, 1988, 87-96.
[2] R.F. Brown, Retraction methods in Nielsen fixed point theory, Pacific J. Math. 115(1984), 277-298.
[3] Ey Fan, Extensions of two fixed point theorem of F.E. Browder, Math. Z. 112(1969), 234-240
[4] Tzu-Chu Lin, Chi-Lin Yen, Applications of the proximity map to fixed point theorem in Hilbert spaces, J. Approx.T heory 52(1988), 141-148.
[5] S. Reich, Some problems and results in fixed point theory, Contemporany Math. 21, 1983, 179-188.
[6] I.A. Rus, The fixed point structures and the retraction mapping principle, “Babs-Bolyai” Univ., Fac. of Math., Preprint nr.3, 1986, 175-184.
[7] S.P.Singh, B. Watson, Proximity maps and fixed points, J.Approx. Theory 39(1983), 72-76.
[8] T.E. Williamson, A geometric approach to fixed points of non-self mappings T:D→X, Contemporary Math. 18, 1983, 247-253.

1989AnisiuAnisiuOnSomeConditionsIttinSeminar
ITIEERAVT SEMINAR ON FUNCTIONAL BQUATIONS, APPROXIMATION ARD CONVEXITY, CLUJ-Napoca, 1989

ON SOWE CONDITIONS FOR THE EYISTENCE OF THE FIXWD POINTS IN HUBERP SPACES by MIRA-CRISTIANA ANISTU "and VALERIU ANISIV (Cluj-Napoca)

In the paper 13 / K y 13 / K y 13//K_(y)13 / K_{y}13/Ky Fan gave the following theorem:
Let K K KKK be a nonempty compact conver set in a normed linear space X X XXX. For any continuous map f f fff of K K KKK into X X XXX, there exists a point u ∈ K u ∈ K u in Ku \in Ku∈K such that
‖ u − f ( u ) ‖ = d ( f ( u ) , K ) ‖ u − f ( u ) ‖ = d ( f ( u ) , K ) ||u-f(u)||=d(f(u),K)\|u-f(u)\|=d(f(u), K)‖u−f(u)‖=d(f(u),K).
In the case of Eilbert space, similar theorems were proved by Singh and Watson /7/ and Lin and Yen /4/ for closed convex sots and nonexpansive maps, respectively continuous 1 -set-contractive maps subject to some supplementary conditions.
It is interesting to observe that in the case of Eilbert space with K K KKK closed convex set, the conclusion in Fan theorer means exactely u = P K ⋅ f ( u ) u = P K ⋅ f ( u ) u=P_(K)*f(u)u=P_{K} \cdot f(u)u=PK⋅f(u), where P K P K P_(K)P_{K}PK is the metric projection. Consequently, the theorems of this type prove the existence of fixed points for the map P K ⋅ f P K ⋅ f P_(K)*fP_{K} \cdot fPK⋅f. If such a fixed point is aiso a firec point for the map f f fff, we obtain a fixed point theorer for the given map.
In the paper /4/ one gives five conditions, the fourth of ther beins just F i x I = F i x F K ∘ I F i x I = F i x F K ∘ I F_(ix)I=FixF_(K^(@))IF_{i x} I=F i x F_{K^{\circ}} IFixI=FixFK∘I. The ail 0 . 0 . 0.0 .0.. tires acte is to study the relations between the claases of maps satisfying
these conditions. 111 of thon ane surficient conditions for Rix f = f = f=f=f= Fix P z f P z f P_(z)f\mathrm{P}_{\mathrm{z}} \mathrm{f}Pzf. Such problems weme studied also by Reich /E/ and ililliamson /8/.
Te finst prove two auxiliary lemas.
Let X X XXX ie a noraed space, A ⊆ X A ⊆ X A sube XA \subseteq XA⊆X a nonvoid conver set and a 0 a 0 a_(0)a_{0}a0 in A. Denote I A ( a 0 ) = { a 0 + t ( a − a 0 ) : t > 0 , a ∈ 2 } I A a 0 = a 0 + t a − a 0 : t > 0 , a ∈ 2 I_(A)(a_(0))={a_(0)+t(a-a_(0)):t > 0,a in2}I_{A}\left(a_{0}\right)=\left\{a_{0}+t\left(a-a_{0}\right): t>0, a \in 2\right\}IA(a0)={a0+t(a−a0):t>0,a∈2} and
d ( x , A ) = i n f { ‖ d ( x , A ) = i n f { ‖ d(x,A)=inf{||d(x, A)=i n f\{\|d(x,A)=inf{‖ m-a ‖ : a ∈ Λ } ‖ : a ∈ Λ } ||:a in Lambda}\|: a \in \Lambda\}‖:a∈Λ} for any x x xxx in X X XXX.
Leama 1. For encit x x xxx in x , lim t → 0 r 1 t d ( a 0 + 1 ; ( x − a 0 ) , A ) = x , lim t → 0 r   1 t d a 0 + 1 ; x − a 0 , A = x,lim_(t rarr0r)(1)/(t)d(a_(0)+1;(x-a_(0)),A)=x, \lim _{t \rightarrow 0 r} \frac{1}{t} d\left(a_{0}+1 ;\left(x-a_{0}\right), A\right)=x,limt→0r1td(a0+1;(x−a0),A)=
= inf t → 0 l t d ( a 0 + t ( x − a 0 ) , A ) = d ( x , I A ( a 0 ) ) = inf t → 0   l t d a 0 + t x − a 0 , A = d x , I A a 0 =i n f_(t rarr0)(l)/(t)d(a_(0)+t(x-a_(0)),A)=d(x_(,)I_(A)(a_(0)))=\inf _{t \rightarrow 0} \frac{l}{t} d\left(a_{0}+t\left(x-a_{0}\right), A\right)=d\left(x_{,} I_{A}\left(a_{0}\right)\right)=inft→0ltd(a0+t(x−a0),A)=d(x,IA(a0)).
Broof, Eecause of the convexity of A A AAA, the map d ( 0 , A ) : X ⟶→ R + d ( 0 , A ) : X ⟶→ R + d(0,A):X longrightarrow rarrR_(+)d(0, A): X \longrightarrow \rightarrow R_{+}d(0,A):X⟶→R+is convex. It follows that φ : R + → R + , φ ( t ) == d ( α 0 + t ( x − a 0 ) , d ) φ : R + → R + , φ ( t ) == d α 0 + t x − a 0 , d varphi:R_(+)rarrR_(+),varphi(t)==d(alpha_(0)+t(x-a_(0)),d)\varphi: R_{+} \rightarrow R_{+}, \varphi(t)= =d\left(\alpha_{0}+t\left(x-a_{0}\right), d\right)φ:R+→R+,φ(t)==d(α0+t(x−a0),d) is a convex map, kence ψ : R + ∖ { 0 } ⟶ R + ψ : R + ∖ { 0 } ⟶ R + psi:R_(+)\\{0}longrightarrowR_(+)\psi: R_{+} \backslash\{0\} \longrightarrow R_{+}ψ:R+∖{0}⟶R+, ψ ( t ) = 1 t d ( a 0 + i ( x − a 0 ) , λ ) ψ ( t ) = 1 t d a 0 + i x − a 0 , λ psi(t)=(1)/(t)d(a_(0)+i(x-a_(0)),lambda)\psi(t)=\frac{1}{t} d\left(a_{0}+i\left(x-a_{0}\right), \lambda\right)ψ(t)=1td(a0+i(x−a0),λ) is an increasing map (we have ψ ( t ) = 1 t ( ψ ( t ) − φ ( 0 ) ) ψ ( t ) = 1 t ( ψ ( t ) − φ ( 0 ) ) psi(t)=(1)/(t)(psi(t)-varphi(0))\psi(t)=\frac{1}{t}(\psi(t)-\varphi(0))ψ(t)=1t(ψ(t)−φ(0)) ) , That is why the first equality in the Ingma holds.
It is obvious that
1 t d ( a 0 + t ( x − a 0 ) , Δ ) = J t d ( t ( x − a 0 ) , A − a 0 ) = d ( x − a 0 , I t ( A − a 0 ) ) 1 t d a 0 + t x − a 0 , Δ = J t d t x − a 0 , A − a 0 = d x − a 0 , I t A − a 0 (1)/(t)d(a_(0)+t(x-a_(0)),Delta)=(J)/(t)d(t(x-a_(0)),A-a_(0))=d(x-a_(0),(I)/(t)(A-a_(0)))\frac{1}{t} d\left(a_{0}+t\left(x-a_{0}\right), \Delta\right)=\frac{J}{t} d\left(t\left(x-a_{0}\right), A-a_{0}\right)=d\left(x-a_{0}, \frac{I}{t}\left(A-a_{0}\right)\right)1td(a0+t(x−a0),Δ)=Jtd(t(x−a0),A−a0)=d(x−a0,It(A−a0)),
bence
imf t → 0 l t d ( a 0 + t ( π − a 0 ) , A ) = inf t → 0 d ( z − a 0 , t ( A − a 0 ) ) = imf t → 0 l t d a 0 + t π − a 0 , A = inf t → 0   d z − a 0 , t A − a 0 = imf_(t rarr0)((l)/(t)d)(a_(0)+t(pi-a_(0)),A)=i n f_(t rarr0)d(z-a_(0),t(A-a_(0)))=\operatorname{imf}_{t \rightarrow 0} \frac{l}{t} d\left(a_{0}+t\left(\pi-a_{0}\right), A\right)=\inf _{t \rightarrow 0} d\left(z-a_{0}, t\left(A-a_{0}\right)\right)=imft→0ltd(a0+t(π−a0),A)=inft→0d(z−a0,t(A−a0))=
= inf t → 0 d ( x , a 0 + t ( a − a 0 ) ) = inf i → 0 ini a ∈ A ‖ π − a 0 − t ( a − a 0 ) ‖ = = inf t → 0   d x , a 0 + t a − a 0 = inf i → 0   ini a ∈ A π − a 0 − t a − a 0 = =i n f_(t rarr0)d(x,a_(0)+t(a-a_(0)))=i n f_(i rarr0)ini_(a in A)||pi-a_(0)-t(a-a_(0))||==\inf _{t \rightarrow 0} d\left(x, a_{0}+t\left(a-a_{0}\right)\right)=\inf _{i \rightarrow 0} \operatorname{ini}_{a \in A}\left\|\pi-a_{0}-t\left(a-a_{0}\right)\right\|==inft→0d(x,a0+t(a−a0))=infi→0inia∈A‖π−a0−t(a−a0)‖=
= d ( x 1 I Λ ( a 0 ) ) = d x 1 I Λ a 0 =d(x_(1)I_(Lambda)(a_(0)))=d\left(x_{1} I_{\Lambda}\left(a_{0}\right)\right)=d(x1IΛ(a0)) 。
Eemax. The nonotony of ψ ψ psi\psiψ implies d ( x 1 x 1 ( a 0 ) ) = d x 1 x 1 a 0 = d(x_(1)x_(1)(a_(0)))=d\left(x_{1} x_{1}\left(a_{0}\right)\right)=d(x1x1(a0))=
= lim 0 < t ≤ 1 1 τ d ( a 0 + t ( Σ − a 0 ) , A ) = lim 0 < t ≤ 1   1 τ d a 0 + t Σ − a 0 , A =lim_(0 < t <= 1)(1)/(tau)d(a_(0)+t(Sigma-a_(0)),A)=\lim _{0<t \leq 1} \frac{1}{\tau} d\left(a_{0}+t\left(\Sigma-a_{0}\right), A\right)=lim0<t≤11τd(a0+t(Σ−a0),A).
Let X X XXX be a prehilbertian apace, A ⊆ X A ⊆ X A sube XA \subseteq XA⊆X a monvoid conplete cunver sot, a 0 a 0 a_(0)a_{0}a0 in Λ , x Λ , x Lambda,x\Lambda, xΛ,x in X X XXX and P ; X → A P ; X → A P;X rarr AP ; X \rightarrow AP;X→A the proximity wap.
The next lema gives a characterization of the fact that a 0 a 0 a_(0)a_{0}a0 is the point of best approximation in Δ Δ Delta\DeltaΔ for x x xxx, using the vellknown one:
P X = a 0 P X = a 0 PX=a_(0)P X=a_{0}PX=a0 i. X X XXX. R e ⟨ X − a 0 , a − a 0 ⟩ ⩽ 0 R e X − a 0 , a − a 0 ⩽ 0 Re(:X-a_(0),a-a_(0):) <= 0R e\left\langle X-a_{0}, a-a_{0}\right\rangle \leqslant 0Re⟨X−a0,a−a0⟩⩽0 for each a in A.
Lenra 2. In a prebilbertian space X X XXX, for a nonvoid complets conver set A ⊆ X A ⊆ X A sube XA \subseteq XA⊆X and a 0 a 0 a_(0)a_{0}a0 in A , x A , x A,xA, xA,x in X X XXX the following assertions are equivalent:
1 0 P x = a 0 2 0 ‖ x − a 0 ‖ = d ( x , I A ( a 0 ) ) 3 0 ‖ x − a 0 ‖ ≤ d ( x , I A ( a 0 ) ) . 1 0 P x = a 0 2 0 x − a 0 = d x , I A a 0 3 0 x − a 0 ≤ d x , I A a 0 . {:[1^(0)quad Px=a_(0)],[2^(0)||x-a_(0)||=d(x,I_(A)(a_(0)))],[3^(0)||x-a_(0)|| <= d(x,I_(A)(a_(0))).]:}\begin{aligned} & 1^{0} \quad P x=a_{0} \\ & 2^{0}\left\|x-a_{0}\right\|=d\left(x, I_{A}\left(a_{0}\right)\right) \\ & 3^{0}\left\|x-a_{0}\right\| \leq d\left(x, I_{A}\left(a_{0}\right)\right) . \end{aligned}10Px=a020‖x−a0‖=d(x,IA(a0))30‖x−a0‖≤d(x,IA(a0)).

Proof.

I 0 ⇒ 2 0 I 0 ⇒ 2 0 I^(0)=>2^(0)I^{0} \Rightarrow 2^{0}I0⇒20. Let a 0 = P z a 0 = P z a_(0)=Pza_{0}=P za0=Pz. Denote A 0 = A − a 0 A 0 = A − a 0 A_(0)=A-a_(0)A_{0}=A-a_{0}A0=A−a0 and x 0 = x − a 0 x 0 = x − a 0 x_(0)=x-a_(0)x_{0}=x-a_{0}x0=x−a0. Then using Iemma 1 and the remark after, one has
a ( x , T A ( a 0 ) ) 2 = inf 0 < t < 1 j t 2 d ( a 0 + t ( x → a 0 ) , A ) 2 = = inf 0 < t ≤ 1 l t 2 d ( t x 0 , A 0 ) 2 = inf 0 < t d ( x 0 , l t A 0 ) 2 = = inf t ≥ 1 d ( x 0 , t A 0 ) 2 = inf t ≥ 1 inf λ ∈ A ‖ x 0 − t a ‖ 2 = = inf t ≥ 1 inf a ∈ A 0 ( t 2 ‖ a ‖ 2 − 2 t R e ⟨ x 0 , a ⟩ + ‖ x 0 ‖ 2 ) = = inf 2 ∈ A 0 inf 2 ( t 2 ‖ a ‖ 2 − 2 t R e ⟨ x 0 , 2 ⟩ + ‖ x 0 ‖ 2 ) a x , T A a 0 2 = inf 0 < t < 1   j t 2 d a 0 + t x → a 0 , A 2 = = inf 0 < t ≤ 1   l t 2 d t x 0 , A 0 2 = inf 0 < t   d x 0 , l t A 0 2 = = inf t ≥ 1   d x 0 , t A 0 2 = inf t ≥ 1   inf λ ∈ A   x 0 − t a 2 = = inf t ≥ 1   inf a ∈ A 0   t 2 ‖ a ‖ 2 − 2 t R e x 0 , a + x 0 2 = = inf 2 ∈ A 0   inf 2   t 2 ‖ a ‖ 2 − 2 t R e x 0 , 2 + x 0 2 {:[a(x,T_(A)(a_(0)))^(2)=i n f_(0 < t < 1)(j)/(t^(2))d(a_(0)+t(x rarra_(0)),A)^(2)=],[=i n f_(0 < t <= 1)(l)/(t^(2))d(tx_(0),A_(0))^(2)=i n f_(0 < t)d(x_(0),(l)/(tA_(0)))^(2)=],[=i n f_(t >= 1)d(x_(0),tA_(0))^(2)=i n f_(t >= 1)i n f_(lambda in A)||x_(0)-ta||^(2)=],[=i n f_(t >= 1)i n f_(a inA_(0))(t^(2)||a||^(2)-2tRe(:x_(0),a:)+||x_(0)||^(2))=],[=i n f_(2inA_(0))i n f2(t^(2)||a||^(2)-2tRe(:x_(0),2:)+||x_(0)||^(2))]:}\begin{aligned} a\left(x, T_{A}\left(a_{0}\right)\right)^{2} & =\inf _{0<t<1} \frac{j}{t^{2}} d\left(a_{0}+t\left(x \rightarrow a_{0}\right), A\right)^{2}= \\ & =\inf _{0<t \leq 1} \frac{l}{t^{2}} d\left(t x_{0}, A_{0}\right)^{2}=\inf _{0<t} d\left(x_{0}, \frac{l}{t A_{0}}\right)^{2}= \\ & =\inf _{t \geq 1} d\left(x_{0}, t A_{0}\right)^{2}=\inf _{t \geq 1} \inf _{\lambda \in A}\left\|x_{0}-t a\right\|^{2}= \\ & =\inf _{t \geq 1} \inf _{a \in A_{0}}\left(t^{2}\|a\|^{2}-2 t R e\left\langle x_{0}, a\right\rangle+\left\|x_{0}\right\|^{2}\right)= \\ & =\inf _{2 \in A_{0}} \inf ^{2}\left(t^{2}\|a\|^{2}-2 t R e\left\langle x_{0}, 2\right\rangle+\left\|x_{0}\right\|^{2}\right) \end{aligned}a(x,TA(a0))2=inf0<t<1jt2d(a0+t(x→a0),A)2==inf0<t≤1lt2d(tx0,A0)2=inf0<td(x0,ltA0)2==inft≥1d(x0,tA0)2=inft≥1infλ∈A‖x0−ta‖2==inft≥1infa∈A0(t2‖a‖2−2tRe⟨x0,a⟩+‖x0‖2)==inf2∈A0inf2(t2‖a‖2−2tRe⟨x0,2⟩+‖x0‖2)
But Re ⟨ I 0 , 2 ⟩ = Re ⟨ x ¯ − a 0 , a + a 0 − a 0 ⟩ ≤ 0 Re I 0 , 2 = Re x ¯ − a 0 , a + a 0 − a 0 ≤ 0 Re(:I_(0),2:)=Re(:( bar(x))-a_(0),a+a_(0)-a_(0):) <= 0\operatorname{Re}\left\langle I_{0}, 2\right\rangle=\operatorname{Re}\left\langle\bar{x}-a_{0}, a+a_{0}-a_{0}\right\rangle \leq 0Re⟨I0,2⟩=Re⟨x¯−a0,a+a0−a0⟩≤0 and the function to be minimized ior t ⩾ l t ⩾ l t >= lt \geqslant lt⩾l is increasing on the intervel ( Be ⟨ x 0 , a ⟩ p 0 Be x 0 , a p 0 Be(:x_(0),a:)p_(0)\mathrm{Be}\left\langle x_{0}, a\right\rangle p_{0}Be⟨x0,a⟩p0 ) which includes [ 0 , ∞ ) [ 0 , ∞ ) [0,oo)[0, \infty)[0,∞), hence the infimum is atteined in t = l 0 t = l 0 t=l_(0)t=l_{0}t=l0.
It follows
d ( x , I A ( a 0 ) ) 2 = inf a ∈ A 0 ‖ x 0 − a ‖ 2 = d ( x 0 , A 0 ) 2 = d ( z , A ) 2 = = ‖ x − a 0 ‖ 2 d x , I A a 0 2 = inf a ∈ A 0   x 0 − a 2 = d x 0 , A 0 2 = d ( z , A ) 2 = = x − a 0 2 {:[d(x,I_(A)(a_(0)))^(2)=i n f_(a inA_(0))||x_(0)-a||^(2)=d(x_(0),A_(0))^(2)=d(z","A)^(2)=],[=||x-a_(0)||^(2)]:}\begin{aligned} d\left(x, I_{A}\left(a_{0}\right)\right)^{2} & =\inf _{a \in A_{0}}\left\|x_{0}-a\right\|^{2}=d\left(x_{0}, A_{0}\right)^{2}=d(z, A)^{2}= \\ & =\left\|x-a_{0}\right\|^{2} \end{aligned}d(x,IA(a0))2=infa∈A0‖x0−a‖2=d(x0,A0)2=d(z,A)2==‖x−a0‖2
and d ( x , I 2 ( a 0 ) ) = ‖ x − a 0 ‖ d x , I 2 a 0 = x − a 0 d(x,I_(2)(a_(0)))=||x-a_(0)||d\left(x, I_{2}\left(a_{0}\right)\right)=\left\|x-a_{0}\right\|d(x,I2(a0))=‖x−a0‖.
2 0 ⇒ 3 0 2 0 ⇒ 3 0 2^(0)=>3^(0)2^{0} \Rightarrow 3^{0}20⇒30 being obvious, we have to prove
3 0 ⇒ 1 0 3 0 ⇒ 1 0 3^(0)=>1^(0)3^{0} \Rightarrow 1^{0}30⇒10. But ‖ x − a 0 ‖ ⩽ d ( x , I A ( a 0 ) ) ⩽ a ( x , A ) ⩽ ‖ x − a 0 ‖ x − a 0 ⩽ d x , I A a 0 ⩽ a ( x , A ) ⩽ x − a 0 ||x-a_(0)|| <= d(x,I_(A)(a_(0))) <= a(x,A) <= ||x-a_(0)||\left\|x-a_{0}\right\| \leqslant d\left(x, I_{A}\left(a_{0}\right)\right) \leqslant a(x, A) \leqslant\left\|x-a_{0}\right\|‖x−a0‖⩽d(x,IA(a0))⩽a(x,A)⩽‖x−a0‖, because A ⊆ I A ( a 0 ) A ⊆ I A a 0 A subeI_(A)(a_(0))A \subseteq I_{A}\left(a_{0}\right)A⊆IA(a0) and the lemma is proved.
Let X X XXX be a prehilbertian space, A ⊆ X A ⊆ X A sube XA \subseteq XA⊆X a nonvoid complete convex set. and f : A → X f : A → X f:A rarr Xf: A \rightarrow Xf:A→X.
The following conditions appear: in /4/, where are named also the authors to which they belong:
(1) For each a in A A AAA, there is a number λ λ lambda\lambdaλ (real or complex, depending on whether the vector space X X XXX is real or complex) such that | λ | < 1 | λ | < 1 |lambda| < 1|\lambda|<1|λ|<1 and a + ( 1 − λ ) f ( a ) ∈ A a + ( 1 − λ ) f ( a ) ∈ A a+(1-lambda)f(a)in Aa+(1-\lambda) f(a) \in Aa+(1−λ)f(a)∈A.
(2) For each a ∈ A a ∈ A a in Aa \in Aa∈A with a ≠ f ( a ) a ≠ f ( a ) a!=f(a)a \neq f(a)a≠f(a), there exists b b bbb in I A ( a ) I A ( a ) I_(A)(a)I_{A}(a)IA(a) auch that ‖ b − f ( a ) ‖ < ‖ a − f ( a ) ‖ ‖ b − f ( a ) ‖ < ‖ a − f ( a ) ‖ ||b-f(a)|| < ||a-f(a)||\|\mathrm{b}-\mathrm{f}(\mathrm{a})\|<\|\mathrm{a}-\mathrm{f}(\mathrm{a})\|‖b−f(a)‖<‖a−f(a)‖.
(3) For each a ∈ A , f ( a ) ∈ c I I A ( a ) a ∈ A , f ( a ) ∈ c I I A ( a ) a in A,f(a)in cII_(A)(a)a \in A, f(a) \in c I I_{A}(a)a∈A,f(a)∈cIIA(a), i.e. I is weakly inward.
(4) For each a in the boundary ∂ A ∂ A del A\partial A∂A of A A AAA with a = P ∘ f ( a ) a = P ∘ f ( a ) a=P@f(a)a=P \circ f(a)a=P∘f(a), a is a fixed point of I I III.
(5) For each a a aaa in ∂ A , ‖ f ( a ) − a ′ ‖ ⩽ ‖ a − a ′ ‖ ∂ A , f ( a ) − a ′ ⩽ a − a ′ del A,||f(a)-a^(')|| <= ||a-a^(')||\partial A,\left\|f(a)-a^{\prime}\right\| \leqslant\left\|a-a^{\prime}\right\|∂A,‖f(a)−a′‖⩽‖a−a′‖ for some a ′ a ′ a^(')a^{\prime}a′ in A A AAA.
Remark. Fix I = I = I=I=I= Fix Pof iff (4) holds.
If Pix f = f = f=f=f= Fix Polf, (4) is obvious.
We have always Fix f ⊆ f ⊆ f subef \subseteqf⊆ Fix Pof. Let a ∈ a ∈ a ina \ina∈ Fix Pof, hence a == P ( f ( a ) ) ; a ∈ ∂ A ; a == P ( f ( a ) ) ; a ∈ ∂ A ; a==P(f(a));a in del A;a= =P(f(a)) ; a \in \partial A ;a==P(f(a));a∈∂A; (4) implics a ∈ a ∈ a ina \ina∈ Fix f f fff and Fix Pof = = === Fix f f fff.
We mention that Fix f = F i x f = F i x f=Fixf=F i xf=Fix Pof is exactely the condition given by Brown in /2/ for ± ± +-\pm± to be retractible on A with respect to P P PPP (see also /6/); fixed point theorems for such maps are obtained in 1 / 1 1 / 1 1//11 / 11/1.
THEDREM. The conditions above are related by the following implications :
( 3 ) ⇒ ( 4 ) ( 2 ) ⇐ ( 5 ) ( 3 ) ⇒ ( 4 ) ( 2 ) ⇐ ( 5 ) (3)=>(4)_((2))lArr(5)(3) \Rightarrow \underset{(2)}{(4)} \Leftarrow(5)(3)⇒(4)(2)⇐(5), where in (1') λ ∈ C λ ∈ C lambda in C\lambda \in Cλ∈C. (1')

Proof.

(1) ⇒ ⇒ =>\Rightarrow⇒ (3). Let a ∈ A a ∈ A a in Aa \in Aa∈A and a ′ = λ a + ( 1 − λ ) f ( a ) ∈ Λ , λ ∈ ( − 1 , 1 ) a ′ = λ a + ( 1 − λ ) f ( a ) ∈ Λ , λ ∈ ( − 1 , 1 ) a^(')=lambda a+(1-lambda)f(a)in Lambda,lambda in(-1,1)a^{\prime}=\lambda a+(1-\lambda) f(a) \in \Lambda, \lambda \in(-1,1)a′=λa+(1−λ)f(a)∈Λ,λ∈(−1,1). It follows f ( a ) = 1 I − λ ( a ′ − λ a ) = a + 1 I − λ ( a ′ − a ) ∈ I A ( a ) ⊆ c I I A ( a ) f ( a ) = 1 I − λ a ′ − λ a = a + 1 I − λ a ′ − a ∈ I A ( a ) ⊆ c I I A ( a ) f(a)=(1)/(I-lambda)(a^(')-lambda a)=a+(1)/(I-lambda)(a^(')-a)inI_(A)(a)sube cII_(A)(a)f(a)=\frac{1}{I-\lambda}\left(a^{\prime}-\lambda a\right)=a+\frac{1}{I-\lambda}\left(a^{\prime}-a\right) \in I_{A}(a) \subseteq c I I_{A}(a)f(a)=1I−λ(a′−λa)=a+1I−λ(a′−a)∈IA(a)⊆cIIA(a).
( 3 ) ⇒ ( 4 ) ( 3 ) ⇒ ( 4 ) (3)=>(4)(3) \Rightarrow(4)(3)⇒(4). Let a ∈ ∂ A , a = P = I ( a ) a ∈ ∂ A , a = P = I ( a ) a in del A,a=P=I(a)a \in \partial A, a=P=I(a)a∈∂A,a=P=I(a). Voing the implication 1 0 ⇒ 2 0 1 0 ⇒ 2 0 1^(0)=>2^(0)1^{0} \Rightarrow 2^{0}10⇒20 in Lemaa 2, one has ‖ f ( a ) − a ‖ = d ( f ( a ) , I 1 ( a ) ) ‖ f ( a ) − a ‖ = d f ( a ) , I 1 ( a ) ||f(a)-a||=d(f(a),I_(1)(a))\|f(a)-a\|=d\left(f(a), I_{1}(a)\right)‖f(a)−a‖=d(f(a),I1(a)). But
f ( a ) ∈ C I I A ( a ) f ( a ) ∈ C I I A ( a ) f(a)in CII_(A)(a)f(a) \in C I I_{A}(a)f(a)∈CIIA(a), hence ‖ f ( a ) − a ‖ = 0 ‖ f ( a ) − a ‖ = 0 ||f(a)-a||=0\|f(a)-a\|=0‖f(a)−a‖=0 and f ( a ) = a f ( a ) = a f(a)=af(a)=af(a)=a.
(5) ⇒ ⇒ =>\Rightarrow⇒ (4). Let a ∈ ∂ A , a ∉ f ( a ) a ∈ ∂ A , a ∉ f ( a ) a in del A,a!in f(a)a \in \partial A, a \notin f(a)a∈∂A,a∉f(a). Applying (5), one obtains a point a' in & such that ‖ r ˙ ( a ) − a ′ ‖ ≤ ‖ a − a ′ ‖ r ˙ ( a ) − a ′ ≤ a − a ′ ||(r^(˙))(a)-a^(')|| <= ||a-a^(')||\left\|\dot{r}(a)-a^{\prime}\right\| \leq\left\|a-a^{\prime}\right\|‖r˙(a)−a′‖≤‖a−a′‖. Then o y ≥ ‖ a − a ′ ‖ 2 − o y ≥ a − a ′ 2 − o_(y) >= ||a-a^(')||^(2)-o_{y} \geq\left\|a-a^{\prime}\right\|^{2}-oy≥‖a−a′‖2−
− ‖ f ( a ) − a ‖ 2 = ‖ a − f ( a ) + f ( a ) − a ‖ 2 − ‖ f ( a ) − a ‖ 2 = ‖ a − f ( a ‖ 2 + − ‖ f ( a ) − a ‖ 2 = ‖ a − f ( a ) + f ( a ) − a ‖ 2 − ‖ f ( a ) − a ‖ 2 = ‖ a − f a ‖ 2 + -||f(a)-a||^(2)=||a-f(a)+f(a)-a||^(2)-||f(a)-a||^(2)=||a-f(a||^(2)+:}-\|f(a)-a\|^{2}=\|a-f(a)+f(a)-a\|^{2}-\|f(a)-a\|^{2}=\| a-f\left(a \|^{2}+\right.−‖f(a)−a‖2=‖a−f(a)+f(a)−a‖2−‖f(a)−a‖2=‖a−f(a‖2+
  • 2Re<a-f(a), f ( a ) − a ′ ⟩ f ( a ) − a ′ {:f(a)-a^('):)\left.f(a)-a^{\prime}\right\ranglef(a)−a′⟩. Ifor this at one has
2 Re ⟨ a − f ( a ) , a ⊤ − f ( a ) ⟩ ⩽ − ‖ a − f ( a ) ‖ 2 < 0 2 Re a − f ( a ) , a ⊤ − f ( a ) ⩽ − ‖ a − f ( a ) ‖ 2 < 0 2Re(:a-f(a),a^(TT)-f(a):) <= -||a-f(a)||^(2) < 02 \operatorname{Re}\left\langle a-f(a), a^{\top}-f(a)\right\rangle \leqslant-\|a-f(a)\|^{2}<02Re⟨a−f(a),a⊤−f(a)⟩⩽−‖a−f(a)‖2<0
and using the characterization of the metric projection it follows
a ≠ P ∘ f ( a ) ≠ P ∘ f ( a ) !=P@f(a)\neq P \circ f(a)≠P∘f(a).
(4) ⇒ ⇒ =>\Rightarrow⇒ (2). Liet a ∈ A , a ≠ f ( a ) a ∈ A , a ≠ f ( a ) a in A,a!=f(a)a \in A, a \neq f(a)a∈A,a≠f(a). For a ∈ a ∈ a ina \ina∈ int A A AAA and t ∈ ( 0 , 1 ) t ∈ ( 0 , 1 ) t in(0,1)t \in(0,1)t∈(0,1) : sufficiently small, b r = a + t ( f ( a ) − a ) ∈ A b r = a + t ( f ( a ) − a ) ∈ A b_(r)=a+t(f(a)-a)in Ab_{r}=a+t(f(a)-a) \in Abr=a+t(f(a)−a)∈A and ∣ b − f ( a ) ‖ = ∣ b − f ( a ) ‖ = ∣b-f(a)||quad=\mid b-f(a) \| \quad=∣b−f(a)‖=
= ( 1 − t ) | a − f ( a ) | < | a − f ( a ) | . = ( 1 − t ) | a − f ( a ) | < | a − f ( a ) | . =(1-t)|a-f(a)| < |a-f(a)|.=(1-t)|a-f(a)|<|a-f(a)| .=(1−t)|a−f(a)|<|a−f(a)|.
For a ∈ ∂ A a ∈ ∂ A a in del Aa \in \partial Aa∈∂A, zet us suppose ‖ b − f ( a ) ‖ ⩾ ‖ a − I ( a ) ‖ ‖ b − f ( a ) ‖ ⩾ ‖ a − I ( a ) ‖ ||b-f(a)|| >= ||a-I(a)||\|b-f(a)\| \geqslant\|a-I(a)\|‖b−f(a)‖⩾‖a−I(a)‖ for each b b bbb in I A ( a ) I A ( a ) I_(A)(a)I_{A}(a)IA(a). Then ‖ a − f ( a ) ‖ d ( f ( a ) , I A ( a ) ‖ a − f ( a ) ‖ d f ( a ) , I A ( a ) ||a-f(a)||d(f(a),I_(A)(a):}\|a-f(a)\| d\left(f(a), I_{A}(a)\right.‖a−f(a)‖d(f(a),IA(a) and applying 3 0 ⇒ 1 0 3 0 ⇒ 1 0 3^(0)=>1^(0)3^{0} \Rightarrow 1^{0}30⇒10 in Iema 2 it follows P 0 f ( a ) = a P 0 f ( a ) = a P_(0)f(a)=a\mathrm{P}_{0} \mathrm{f}(\mathrm{a})=aP0f(a)=a, contradicting (4). It follows that (2) is true in this case too.
(2) ⇒ ⇒ =>\Rightarrow⇒ (4). Let a ∈ ∂ A , a ≠ f ( a ) a ∈ ∂ A , a ≠ f ( a ) a in del A,a!=f(a)a \in \partial A, a \neq f(a)a∈∂A,a≠f(a). Condition (2) implies d ( I ( a ) , I A ( a ) ) < ‖ a − f ( a ) ‖ d I ( a ) , I A ( a ) < ‖ a − f ( a ) ‖ d(I(a),I_(A)(a)) < ||a-f(a)||d\left(I(a), I_{A}(a)\right)<\|a-f(a)\|d(I(a),IA(a))<‖a−f(a)‖; using now the implication 1 0 ⇒ 3 0 1 0 ⇒ 3 0 1^(0)=>3^(0)1^{0} \Rightarrow 3^{0}10⇒30
in lemma 2, Pof ( a ) f̸ ( a ) f̸ (a)f(a) \not f(a)f̸ a and (4) is proved.
All the implications which do not contain condition (2) are true in the complex case too. It remains to prove only that in
Tisis case（1＇） ⟹ ⟹ Longrightarrow\Longrightarrow⟹（4）．
Let a ∈ ∂ A , a ≠ f ( a ) . By ( 1 ′ ) , thare is λ ∈ C , | λ | < 1 such  Let  a ∈ ∂ A , a ≠ f ( a ) . By  1 ′ , thare is  λ ∈ C , | λ | < 1  such  " Let "a in del A,a!=f(a)". By "(1^('))", thare is "lambda in C,|lambda| < 1" such "\text { Let } a \in \partial A, a \neq f(a) \text {. By }\left(1^{\prime}\right) \text {, thare is } \lambda \in C,|\lambda|<1 \text { such } Let a∈∂A,a≠f(a). By (1′), thare is λ∈C,|λ|<1 such 
that a ′ = λ a + ( 1 − λ ) f ( a ) ∈ A a ′ = λ a + ( 1 − λ ) f ( a ) ∈ A a^(')=lambda a+(1-lambda)f(a)in Aa^{\prime}=\lambda a+(1-\lambda) f(a) \in Aa′=λa+(1−λ)f(a)∈A ．
Lieve ‖ a − f ( a ) ‖ > ‖ λ ‖ a − f ( a ) ‖ = ‖ λ a − λ f ( a ) ‖ = ‖ a − f ( a ) ‖ > ‖ λ ‖ a − f ( a ) ‖ = ‖ λ a − λ f ( a ) ‖ = ||a-f(a)|| > ||lambda||a-f(a)||=||lambda a-lambda f(a)||=\|a-f(a)\|>\|\lambda\| a-f(a)\|=\| \lambda a-\lambda f(a) \|=‖a−f(a)‖>‖λ‖a−f(a)‖=‖λa−λf(a)‖=
= ‖ λ a + ( 1 − λ ) f ( a ) − I ^ ( a ) ‖ = ‖ a ! − f ( a ) ‖ ≥ d ( f ( a ) , A ) = ‖ λ a + ( 1 − λ ) f ( a ) − I ^ ( a ) ‖ = a ! − f ( a ) ≥ d ( f ( a ) , A ) =||lambda a+(1-lambda)f(a)- hat(I)(a)||=||a^(!)-f(a)|| >= d(f(a),A)=\|\lambda a+(1-\lambda) f(a)-\hat{I}(a)\|=\left\|a^{!}-f(a)\right\| \geq d(f(a), A)=‖λa+(1−λ)f(a)−I^(a)‖=‖a!−f(a)‖≥d(f(a),A) ，hence
∴ ( f ( a ) , A ) < ‖ a − f ( a ) ‖ ∴ ( f ( a ) , A ) < ‖ a − f ( a ) ‖ :.(f(a),A) < ||a-f(a)||\therefore(f(a), A)<\|a-f(a)\|∴(f(a),A)<‖a−f(a)‖ and a ≠ P o f ( a ) a ≠ P o f ( a ) a!=Pof(a)a \neq P o f(a)a≠Pof(a) ．
One may present the five classes of functions in tha following Yonn diagram（2 and 4 coincide）：
It follows a list of six examples el－e 6 showing that every set which appears in the diagram is nonvoid．
Example el satisfies（ 1 ），but not（5）．
Let X = B , A = [ 0 , 1 ] , f ( x ) = 3 − 3 x ; x 0 = 3 / 4 X = B , A = [ 0 , 1 ] , f ( x ) = 3 − 3 x ; x 0 = 3 / 4 X=B,A=[0,1],f(x)=3-3x;x_(0)=3//4X=B, A=[0,1], f(x)=3-3 x ; x_{0}=3 / 4X=B,A=[0,1],f(x)=3−3x;x0=3/4 is a fixed point． For each a in Λ Λ Lambda\LambdaΛ and λ = 2 / 3 λ = 2 / 3 lambda=2//3\lambda=2 / 3λ=2/3 one has λ a + ( 1 − λ ) r ( a ) = 1 − a / 3 ∈ Λ λ a + ( 1 − λ ) r ( a ) = 1 − a / 3 ∈ Λ lambda a+(1-lambda)r(a)=1-a//3in Lambda\lambda a+(1-\lambda) r(a)=1-a / 3 \in \Lambdaλa+(1−λ)r(a)=1−a/3∈Λ and（I）holds．But for a = 0 a = 0 a=0a=0a=0 one has f ( a ) ′ ′ = 3 f ( a ) ′ ′ = 3 f(a)^('')=3f(a)^{\prime \prime}=3f(a)′′=3 and | 3 − a ′ | > a ′ 3 − a ′ > a ′ |3-a^(')| > a^(')\left|3-a^{\prime}\right|>a^{\prime}|3−a′|>a′ for each a＇in Λ Λ Lambda\LambdaΛ ．
Example e2 satisfies（1）and（5）．
Let X X XXX be a normed spece， A ⊆ X A ⊆ X A sube XA \subseteq XA⊆X nonvede and I : A → K , f ( x ) == X I : A → K , f ( x ) == X I:A rarr K,f(x)==XI: A \rightarrow K, f(x)= =XI:A→K,f(x)==X（or any f : Λ → Λ f : Λ → Λ f:Lambda rarr Lambdaf: \Lambda \rightarrow \Lambdaf:Λ→Λ ，for A A AAA convers set）．
Graple e3 satisfies（3）and（5），but not（1）．
Let X = R 2 , Δ = { ( a 1 , a 2 ) ∈ R 2 ; a 1 ≥ 0 , a 2 ≥ 0 , a 1 2 + a 2 2 ≤ 1 } X = R 2 , Δ = a 1 , a 2 ∈ R 2 ; a 1 ≥ 0 , a 2 ≥ 0 , a 1 2 + a 2 2 ≤ 1 X=R^(2),Delta={(a_(1),a_(2))inR^(2);a_(1) >= 0,a_(2) >= 0,a_(1)^(2)+a_(2)^(2) <= 1}X=R^{2}, \Delta=\left\{\left(a_{1}, a_{2}\right) \in R^{2} ; a_{1} \geq 0, a_{2} \geq 0, a_{1}^{2}+a_{2}^{2} \leq 1\right\}X=R2,Δ={(a1,a2)∈R2;a1≥0,a2≥0,a12+a22≤1}. The Iunction f ( a ) = ( a 1 1 + 1 − a a 2 , 1 ) , a = ( a 1 , a 2 ) f ( a ) = a 1 1 + 1 − a a 2 , 1 , a = a 1 , a 2 f(a)=((a_(1))/(1+sqrt(1-a_(a)^(2))),1),a=(a_(1),a_(2))f(a)=\left(\frac{a_{1}}{1+\sqrt{1-a_{a}^{2}}}, 1\right), a=\left(a_{1}, a_{2}\right)f(a)=(a11+1−aa2,1),a=(a1,a2) bes ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) as
a fixed point．
Tor a = ( 1 , 0 ) a = ( 1 , 0 ) a=(1,0)a=(1,0)a=(1,0) one has λ λ lambda\lambdaλ \＆ μ ( 1 − λ ) f ( a ) = ( 1 , 1 − λ ) ≠ λ , 20 = | λ | < 1 μ ( 1 − λ ) f ( a ) = ( 1 , 1 − λ ) ≠ λ , 20 = | λ | < 1 mu(1-lambda)f(a)=(1,1-lambda)!=lambda,quad20=|lambda| < 1\mu(1-\lambda) f(a)=(1,1-\lambda) \neq \lambda, \quad 20=|\lambda|<1μ(1−λ)f(a)=(1,1−λ)≠λ,20=|λ|<1 ．
致品兰 e4 satistios（5），but not（3）．
Let X = Z 2 X = Z 2 X=Z^(2)X=Z^{2}X=Z2 ， A A AAA the same as in mamble ej and f : A → I f : A → I f:A rarr If: A \rightarrow If:A→I be siven by f ( a ) = ( a 1 2 ( 1 + 1 1 + 1 − a t 2 ; 1 , 1 ) Ior a = ⟨ a 1 , a 2 ; ∈ A f ( a ) = a 1 2 1 + 1 1 + 1 − a t 2 ; 1 , 1 Ior a = a 1 , a 2 ; ∈ A f(a)=((a_(1))/(2)(1+(1)/(1+sqrt(1-a_(t)^(2)));1,1)Ior a=(:a_(1),a_(2);in A:}f(a)=\left(\frac{a_{1}}{2}\left(1+\frac{1}{1+\sqrt{1-a_{t}^{2}}} ; 1,1\right) \operatorname{Ior} a=\left\langle a_{1}, a_{2} ; \in A\right.\right.f(a)=(a12(1+11+1−at2;1,1)Iora=⟨a1,a2;∈A, Which aas ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) as a pixed point．
For a = ( 2 2 , 2 2 ) a = 2 2 , 2 2 a=((sqrt2)/(2),(sqrt2)/(2))a=\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)a=(22,22) ，one has f ( a ) ≠ f ( a ) ≠ f(a)!=f(a) \neqf(a)≠ cl I d ( a ) I d ( a ) I_(d)(a)I_{d}(a)Id(a) ．
Frample es satisfies（3），but not（i）and（5）．
Iet Y = R 2 , α = arcsin ⁡ 1 / 3 , A = { ( a 1 , a 2 ) : a 1 ≥ 0 , a 2 ≥ ⩾ a 1 Y = R 2 , α = arcsin ⁡ 1 / 3 , A = a 1 , a 2 : a 1 ≥ 0 , a 2 ≥ ⩾ a 1 Y=R^(2),alpha=arcsin 1//3,A={(a_(1),a_(2)):a_(1) >= 0,a_(2) >= :} >= a_(1)Y=R^{2}, \alpha=\arcsin 1 / 3, A=\left\{\left(a_{1}, a_{2}\right): a_{1} \geq 0, a_{2} \geq\right. \geqslant a_{1}Y=R2,α=arcsin⁡1/3,A={(a1,a2):a1≥0,a2≥⩾a1 繁 α , a 1 2 + a 2 2 ≤ 1 } α , a 1 2 + a 2 2 ≤ 1 {: alpha,a_(1)^(2)+a_(2)^(2) <= 1}\left.\alpha, a_{1}^{2}+a_{2}^{2} \leq 1\right\}α,a12+a22≤1} and f : Δ → π f : Δ → π f:Delta rarr pif: \Delta \rightarrow \pif:Δ→π be given b J f ( a ) = b J f ( a ) = b_(J)f(a)=b_{J} f(a)=bJf(a)=
= ( 0 , 1 1 − a i 2 ) = 0 , 1 1 − a i 2 =(0,(1)/(sqrt(1-a_(i)^(2))))=\left(0, \frac{1}{\sqrt{1-a_{i}^{2}}}\right)=(0,11−ai2) ，for which ( 0 , 1 ) ( 0 , 1 ) (0,1)(0,1)(0,1) is a fixed potnt．
For a = ( cos ⁡ α , sin ⁡ α ) , f ( a ) = ( 0 , 3 ) a = ( cos ⁡ α , sin ⁡ α ) , f ( a ) = ( 0 , 3 ) a=(cos alpha,sin alpha),f(a)=(0,3)a=(\cos \alpha, \sin \alpha), f(a)=(0,3)a=(cos⁡α,sin⁡α),f(a)=(0,3) and（1）and（5）do not bold．
Example e6 satisfies（4），but not（3）and（5）．
Let X = R 2 , Δ = { ( a 1 , a 2 ) : a 1 2 + a 2 2 ≤ 1 } X = R 2 , Δ = a 1 , a 2 : a 1 2 + a 2 2 ≤ 1 X=R^(2),Delta={(a_(1),a_(2)):a_(1)^(2)+a_(2)^(2) <= 1}X=R^{2}, \Delta=\left\{\left(a_{1}, a_{2}\right): a_{1}^{2}+a_{2}^{2} \leq 1\right\}X=R2,Δ={(a1,a2):a12+a22≤1} and I : A → I I : A → I I:A rarr II: A \rightarrow II:A→I ，
fixed point．
For a = ( 2 / 2 , 2 / 2 ) , ( 3 ) a = ( 2 / 2 , 2 / 2 ) , ( 3 ) a=(sqrt2//2,sqrt2//2),(3)a=(\sqrt{2} / 2, \sqrt{2} / 2),(3)a=(2/2,2/2),(3) and（5）do not hold．
All the maps in these sxamplos have fixed points，as a conse－ quence of the results in／4／，for example of Corollary 4．There are maps with Fix I Fiff which do not satisfy condition（4），as I → ( x ) = 2 x , I : [ 0 , 1 ] → R ． I → ( x ) = 2 x , I : [ 0 , 1 ] → R ． vec(I)(x)=2x,I:[0,1]rarrR_("．")\vec{I}(x)=2 x, I:[0,1] \rightarrow R_{\text {．}}I→(x)=2x,I:[0,1]→R．
Bemark．From the Venn diagram one can see that in the real case all the implications in the theorem are irreversible，finis is also true in the complex cane，because the examples related to the conditions（3）and（4），respectively（5）and（4）can bo considered in the space C C CCC with the scalar field C．It remains to sive an example for（1＇） ⇒ ⇒ =>\Rightarrow⇒（4）．
Itrapple el setisties（4），but not（1＇）．
Let X = C 2 , A = { ( a 1 , a 2 ) ∈ C 2 : | a 1 | 2 + | a 2 | 2 ≤ 1 } , f : : A ⋯ X , f ( a ) = ( a 1 + a 2 , a 2 ) , for a = ( a 1 , a 2 ) ∈ C 2 , which has a = = ( a 1 , 0 ) ∈ Δ as fined points. Me point a = ( 0 , 1 ) does not satisfy condition (I').  Let  X = C 2 , A = a 1 , a 2 ∈ C 2 : a 1 2 + a 2 2 ≤ 1 , f : : A ⋯ X , f ( a ) = a 1 + a 2 , a 2 ,  for  a = a 1 , a 2 ∈ C 2 ,  which has  a = = a 1 , 0 ∈ Δ  as fined points.   Me point  a = ( 0 , 1 )  does not satisfy condition (I').  {:[" Let "X=C^(2)","A={(a_(1),a_(2))inC^(2):|a_(1)|^(2)+|a_(2)|^(2) <= 1}","f:],[:A cdots X","f(a)=(a_(1)+a_(2),a_(2))","" for "a=(a_(1),a_(2))inC^(2)","" which has "a=],[=(a_(1),0)in Delta" as fined points. "],[quad" Me point "a=(0","1)" does not satisfy condition (I'). "]:}\begin{aligned} & \text { Let } X=C^{2}, A=\left\{\left(a_{1}, a_{2}\right) \in C^{2}:\left|a_{1}\right|^{2}+\left|a_{2}\right|^{2} \leq 1\right\}, f: \\ & : A \cdots X, f(a)=\left(a_{1}+a_{2}, a_{2}\right), \text { for } a=\left(a_{1}, a_{2}\right) \in C^{2}, \text { which has } a= \\ & =\left(a_{1}, 0\right) \in \Delta \text { as fined points. } \\ & \quad \text { Me point } a=(0,1) \text { does not satisfy condition (I'). } \end{aligned} Let X=C2,A={(a1,a2)∈C2:|a1|2+|a2|2≤1},f::A⋯X,f(a)=(a1+a2,a2), for a=(a1,a2)∈C2, which has a==(a1,0)∈Δ as fined points.  Me point a=(0,1) does not satisfy condition (I'). 

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1989

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