[1] F.E. Browder, Fixed points theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sc. 53(1965), 1272-1276.
[2] R.F.Brown, Retraction methods in Nielsin fixed point theory, Pacific J.Math. 115(1984), 277-316.
[3] D.J.Dowhing, W.O.Ray, Some remarks on set-valued mappings, Nonlinear Analysis, T.M.A. 5(12)(1981), 1367-1377
[4] T.C.Lim, A fixed point theorm for multivalued nonexpresive mappings in a unformly convex Banach space, Bull. Amer. Math. Soc, 80(1974), 1123-1125.
[5] I.A. Rus, The fixed point structures and the retraction mappings principle, “Babes-Bolyai” Univ. Fac. of Math., Research Sem., Preprint nr.3, 1986, 175-184.
[6] R. Schoneberg, Some fixed point theorems for mappings of nonexpansive type, Comm. Math. Univ. Carolinae 17 (1976), 399-411.
[7] S.P.Singh, S. Massa, D. Roux, Approximation technique in fixed point theory, Rendiconti Sem. Math.Fis. Milano VIII(1983), 165-172.
[8] Gh.Siretchi, Analiză funcțională, Univ. din București, 1982 (mimeographed).
1988-Anisiu-FixedPointsOfRetractible
{:[" "GELS-EOLYRT" UNIVARSITY "],[" Dacilury of Dothesetics and Physics "],[" Research Seminars "],[" Senincr on Hathematical Analysis "],[" Preprint Hr. 7, 1988, PF • 87-96 "]:}\begin{aligned}
& \text { "GELS-EOLYRT" UNIVARSITY } \\
& \text { Dacilury of Dothesetics and Physics } \\
& \text { Research Seminars } \\
& \text { Senincr on Hathematical Analysis } \\
& \text { Preprint Hr. 7, 1988, PF • 87-96 }
\end{aligned}
{:[" FEND POTHS OF RE FRACTIBLE SIAPETIGG VITH RESPECT TO "],[" ART JUARIC FROJLCTION "],[" Hira-Oristiana Inisiu "]:}\begin{aligned}
& \text { FEND POTHS OF RE FRACTIBLE SIAPETIGG VITH RESPECT TO } \\
& \text { ART JUARIC FROJLCTION } \\
& \text { Hira-Oristiana Inisiu }
\end{aligned}
In the paper one presents the notions of retract, retractible amotion ana some properties following the papers //2,5/// 2,5 / in ordit to obtain generalizations of come fixed point theorens. The notrions are extended to potnt-to-set mapgings.
Let X!in O/X \notin \varnothing a set and O/!in A in X\emptyset \notin A \in X. AA function rr iX rarr\rightarrow A is a ratect of XX onto hh if F|_(A)=1d_(A)\left.F\right|_{A}=1 d_{A}. A function f:A rarr Xf: A \rightarrow X is retracEible onto 4 with respect to the retract rr if Fix ror =Fix,1=F i x, 1, Where Fir ff denotes the set of the fixed points of ff.
Femary 1. It is obvious that Fix f subef \subseteq Fix rof, because x=f(x)∈∈Ax=f(x) \in \in A implies x=F(x)in x(f(x))x=F(x) \in x(f(x)) i it follows that in the definition of the retractible function one may douand only Fix rof sube F1-f\subseteq F 1-f.
R. F. Brown gives in //2/// 2 / the following condition for ff to be retractible onto AA with respect to the retract rr;
(1) x in r(f(A)\\A)x \in r(f(A) \backslash A) impliss f(x)=xf(x)=x or f(x)!inf^(-1)(x)f(x) \notin f^{-1}(x).
Condition (1) way be reformulated as
(1') quad r(f(A)\\A)sube{x in A:f(x)=x:}\quad r(f(A) \backslash A) \subseteq\left\{x \in A: f(x)=x\right. or {:f(x)!inr^(-1)(x)}\left.f(x) \notin r^{-1}(x)\right\}.
The next proposition is obvious.
Propogtring 1. The following sers are equal M={x in A;f(x)=x:}\mathbb{M}=\left\{x \in A ; f(x)=x\right. or {:f(x)!inr^(-1)(x)}\left.f(x) \notin r^{-1}(x)\right\} N={N in A:x^(˙)^(')(x)!inI^(-1)(x)\\{x}}N=\left\{N \in A: \dot{x}^{\prime}(x) \notin I^{-1}(x) \backslash\{x\}\right\} P=P= Fix {uuC_(lambda) vec(Fix):}\left\{\cup C_{\lambda} \overrightarrow{F i x}\right. ror．
Proposition 2．Condivion（1）is equivalent to each of the fol－ lowing
（2）Fix rof sube\subseteq Fiz
（3）Fixs rol subeP^(-1)(A)\subseteq \mathrm{P}^{-1}(A) ．
Proof．（1）=>\Rightarrow（2）．Let x in Fi=x \in F i= rofi，hence x in Ax \in A and x=r(f(x))x=r(f(x)) ． If f(x)!=Af(x) \neq A ，we have y=f(x)in P(A)\\Ay=f(x) \in P(A) \backslash A ，hence x=r(y)in FixfU uuuC_(A)Fixx=r(y) \in F i x f U \bigcup C_{A} F i x rof．It follows x in FixPx \in F i x P ，which contradicts f(x)!in Af(x) \notin A ．It remains that f(x)in Af(x) \in A and x=f(f(x))=f(x)x=f(f(x))=f(x) ，so x in bar(F)ixfx \in \bar{F} i x f and（a） is proved．
（2）=>\Rightarrow（3）is obvious，since Fix f subef^(-1)(A)f \subseteq f^{-1}(A) ．
（3）=>(1)\Rightarrow(1) ，in Inciv（3）=>(A sube:}\Rightarrow\left(A \subseteq\right. Fix i uuC_(i)i \cup C_{i} Fix rcf）．Let x in Ax \in A ． Suppose that x(d)/(i)C_(A)x \frac{d}{i} C_{A} Fix rof，nence x in Hixx \in H i x ror -=x^(-1)(A)\equiv x^{-1}(A) and x^(')(z)=y <=x^{\prime}(z)=y \leqslant A．Then x=r(f(x))=f(x)in Ax=r(f(x))=f(x) \in A and x in Fixfx \in F i x f ，so A sube Fixf uuA \subseteq F i x f \cup UCAFix rof and（I）is proved．The last inclusion is in fact an equality，the reverse inclusion being obvious．
It follows that if a function if A longrightarrow XA \longrightarrow X admits a retract rr and Fix rof !=O/\neq \varnothing ，then Fix f!=O/f \neq \varnothing ．
In che following we give a general Iom of some fixed point theorems，using as a retract the acric projection．Wa recall some of the proporties of the metric projection in dilbert spaces which are mentioned and used in／6／to obtain fixed point theorems．
Let HH be a Hilbert space and phi!=0sube H\phi \neq 0 \subseteq H a closed convex Eomega^(')E \omega^{\prime} ． Then for ach x\operatorname{ach} x in AA there exists a unique y in Cy \in C such that ||x-y||=d(x,C)=i n f{||x-z||;z in C}\|x-y\|=d(x, C)=\inf \{\|x-z\| ; z \in C\}.
In this cass P=P_(C):H rarr C,P(x)=vP=P_{C}: H \rightarrow C, P(x)=v is a function named Metric projection．
The function P;H rarr CP ; H \rightarrow C is a retract ofHo f H on CC ，because P|_(C)==1d_(C)\left.P\right|_{C}= =1 d_{C} f it satisfies the well－known relations
（4） se(x-ix,Px-y) >= 0,AA x,y in C\operatorname{se}(x-i x, P x-y) \geqslant 0, \forall x, y \in C
（5）quad||Fx-Fy|| <= ||x-y||,AA x,y in E\quad\|F x-F y\| \leqslant\|x-y\|, \forall x, y \in E（ PP is nonexpansive）．
Fropsition j．Let EE be a Hilbert syace，phi!in C sube H\phi \notin C \subseteq H a closed conver set and f:C rarr Hf: C \rightarrow H a given function．If for each x∈∈P(f(C)\\C)x \in \in P(f(C) \backslash C) which is not a fixed point for II it follows that there exists y in Cy \in C such that
（6） Re(f(x)-x,x-y) < 0\operatorname{Re}(f(x)-x, x-y)<0 ，
Then ff is retractible on CC with respect to the retraction PP ．
Proof．Lat x in P(x(C)\\C)x \in P(x(C) \backslash C) and x!=ixf. bar(I) vec(I)P(x(x))=xx \neq i x f . \bar{I} \vec{I} P(x(x))=x ，then for each y in Cy \in C Re (f(x)-x,x-y) >= 0(f(x)-x, x-y) \geqslant 0 ，contradiction．It follows f(+-)!inP^(-1)(x)f( \pm) \notin P^{-1}(x) ，hence the condition（1）teres place and ff is retrac－ tib2e onto CC with respect to the retract PP ．
how we can prove
敋联形 1．Lay AA be AA Hilbert space，A!in C sube EA \notin C \subseteq E a closed bounded conver set．Let f:c rarr Hf: c \rightarrow H sixh that for each x in P(f(c)\\c),xqx \in P(f(c) \backslash c), x q ｜Six ff inere exists y in Cy \in C such that Re (f(+-)-x,x-y) < 0(f( \pm)-x, x-y)<0 and Pof ：C rarr CC \rightarrow C is nonezoansive．Then ff has in CC at least a fixed peimt．
Eroof．Accordinaly to Enoposition zz ，f is retractible on CC with respect to the reiract FF and Fix Fof =Fix^(1)=F i x{ }^{1} ．But Pof being nonempansive，the wheorem of Browder／ 1//1 / implies Fix Pof !in D\notin D ．
Corollary 1//6//1 / 6 / ．Let HH be a Hilbert space，O/!in C sube H\varnothing \notin C \subseteq H a closed bounded convex set．Lat f:C rarr Hf: C \rightarrow H a nonexpansive mapping such that for each x in JCx \in J C there exists y in Cy \in C such that
（7）quad||f(x)-y|| <= ||x-y||\quad\|f(x)-y\| \leq\|x-y\| ．
Then ff has at least a fixed point．
Froof．Let x in P(P(C)uu C)⊆⊃C,x!inx \in P(P(C) \cup C) \subseteq \supset C, x \notin Fix ff ．There exists y in Cy \in C such that（7）takes place and 0 >= ||f(x)-y||^(2)-||x-y||^(2)=(f(x)-x+x-y,f(x)-x+x--y)-||x-y||^(2)=||f(x)-x||^(2)+2Re(f(x)-x,x-y)0 \geq\|f(x)-y\|^{2}-\|x-y\|^{2}=(f(x)-x+x-y, f(x)-x+x- -y)-\|x-y\|^{2}=\|f(x)-x\|^{2}+2 \operatorname{Re}(f(x)-x, x-y),
fle hothetheses of Thecrem 1 are savisfied, since PP and PP are nonexpansive.
Ronsas 2. Fhere Gre functions f:C rarr Hf: C \rightarrow H which are not nonexpansive and fulfil the condition is heroren 1 , out ouv vhowe in Corcllary 1.
Let f:[0,1)rarrR\mathrm{f}:[0,1) \rightarrow \mathrm{R},
f(x)={[-4x+3",",x in[0","1//2)],[3//2-x",",x in[1//2","1]]:}f(x)= \begin{cases}-4 x+3, & x \in[0,1 / 2) \\ 3 / 2-x, & x \in[1 / 2,1]\end{cases}
a function which verifies the hypotheses of Ibeorem 1.
Let x=0in JCx=0 \in J C; if yy varifies (7), it follows j-y <= yj-y \leqslant y, hence y >= 3//2y \geqslant 3 / 2, contradiction to y in Cy \in C.
Applying Browder's theorem in uniformly convex soaces one obtains
THEOREM 2. Let XX be a uniformly convez Engen spuce, O/!=G sube X\emptyset \neq G \subseteq X a closed bounded corvex set. If r:C rarr Xr: C \rightarrow X is retractitle grivo CC Pith raspect to the metric projection P=P_(C)P=P_{C} and For is noncyansive, then ff nes in CC at lessi a fixed point.
Proof. Because ff is retractible onto CC with respect to PP, we have Fix f=f= Fix Pof. But Pof is nonexpansive, hence it has a fixed point by the Browder's theorem.
Remary 3. In the conditions of Meorem 2, f:0rarr Xf: 0 \rightarrow X is retractible onto CC with rospect to tho metric projection PP if and only if for ouch x in P(f(C)\\C),f(x)!=xx \in P(f(C) \backslash C), f(x) \neq x there exists y in Cy \in C such that ||f(x)-x|| > ||f(x)-y||\|f(x)-x\|>\|f(x)-y\|.
Indeed, let x in P(f(C)\\C)sube C,f(x)!inx_("; in this case we have ")f(x)!inP^(-1)(x)<=>P(f(x))!=x<=>EE y in C,||f(x)-x|| > ||f(x)-y||x \in P(f(C) \backslash C) \subseteq C, f(x) \notin x_{\text {; in this case we have }} f(x) \notin P^{-1}(x) \Leftrightarrow P(f(x)) \neq x \Leftrightarrow \exists y \in C,\|f(x)-x\|>\|f(x)-y\|
iie obtain
Conollary 2. In the conditions of Theorem 2, if I:C rarr KI: C \rightarrow K has the property that for each x in P(f(C)\\C)⊆⊃C,x!=f(x)x \in P(f(C) \backslash C) \subseteq \supset C, x \neq f(x) there
exists y in Cy \in C such that
(8) ||f(x)-x|| > ||f(x)-y||\|f(x)-x\|>\|f(x)-y\|
and PofP o f is monexpansive, then ff has in CC at least a fixed point.
Using the condition (7) one obtains
Gorollazy 3. In the hypotheses of Theorem 2, if f:C longrightarrow Xf: C \longrightarrow X has the property that for each x in P(f(C)\\C)⊆⊃Cx \in P(f(C) \backslash C) \subseteq \supset C there exists y in Cy \in C such that (7) takes place and Pof is nonerpansive, then ff has in cc at least a firod point.
Proof. Let x in P(f(C)uu C)x \in P(f(C) \cup C); if x**f(x)x * f(x) the theorem is proved. If x dots f(x)x \ldots f(x), it follows that there exists y in Cy \in C such that ||f(x)-y||⩽≤||x-y||\|f(x)-y\| \leqslant \leq\|x-y\|. If we suppose x=Pof(x)x=\operatorname{Pof}(x), then ||f(x)-x||==i n f{||f(x)-y||z in C} <= i n f{||x-y||z,y in C}=0\|f(x)-x\|= =\inf \{\|f(x)-y\| z \in C\} \leqslant \inf \{\|x-y\| z, y \in C\}=0, which is a contradiction. It follows that ff is retractible onto CC with respect to PP, hence Fix F∤F \nmid.
Now we shall generalize the notion of retractible function to point-to-set mappings (shortly, mappings).
Let X!in aX \notin a set, O/ in A sube X\emptyset \in A \subseteq X. AA mapping R:X rarr2^(A)\\{O/}R: X \rightarrow 2^{A} \backslash\{\varnothing\} is a retrest of XX onto AA if R|_(A)=id_(A)\left.R\right|_{A}=i d_{A}. Therefore RR restricted on the set AA is a function which coincides to the identical function. AA maping F:A rarr2^(X)\\{O/}F: A \rightarrow 2^{X} \backslash\{\emptyset\} is retractible onto AA with respect to the retract RR if Fix RoF == Fix PP, where Fix F={x in A;x in F(x)}F=\{x \in A ; x \in F(x)\}.
The analogrus of condition (1) is
(9) x in R(F(A)\\A)x \in R(F(A) \backslash A) implies x in F(x)x \in F(x) or F(x)nnR^(-1)(x)=O/F(x) \cap R^{-1}(x)=\varnothing, where R^(-1)(x)={z in X:x in R(z)}R^{-1}(x)=\{z \in X: x \in R(z)\}. The condition (9) aay be reformulate' as
(9') quad R(F(A)\\A)sube{x inA:x in F(x):}\quad R(F(\mathbb{A}) \backslash A) \subseteq\left\{x \in \mathbb{A}: x \in F(x)\right. or {:F((x^(˙)))nnR^(-1)(x)=O/}\left.F(\dot{x}) \cap R^{-1}(x)=\varnothing\right\}.
We obtain some results analogous to those for functions.
Froposition 4. The next two sets are equal U={x in A:x in F(x):}U=\left\{x \in A: x \in F(x)\right. or {:F(x)nnR^(-1)(x)=O/}\left.F(x) \cap R^{-1}(x)=\emptyset\right\}
Procosition 5. The condition (9) is equivalent to (10) Fix RoF sube\subseteq Fix FF.
Repark 4. It is obvious that Fix F <= FixF \leq F i x Rof, since x in FixFx \in F i x F implies x in F(x)x \in F(x) and x in Ax \in A, hence x=R(x)sube R(F(x))x=R(x) \subseteq R(F(x)) and x in Fixx \in F i x RoF. In fact, (10) ueans that Fix RoF =FixF=F i x F.
Proof of Proposition 5.
(9') =>\Rightarrow (10). Let x inx \in Fix RoF, so x in RoF(x)x \in \operatorname{RoF}(x); there exists y in F(x)y \in F(x) such that x in R(y)x \in R(y). If y in A,R(y)={y}y \in A, R(y)=\{y\} and x=yx=y, hence x in FixFx \in F i x F. If y!in A,x in R(y)sube R(F(A)\\A)sube FixF uuC_(A)Fixy \notin A, x \in R(y) \subseteq R(F(A) \backslash A) \subseteq F i x F \cup C_{A} F i x RoF. But x in RoF(x)x \in \operatorname{RoF}(x) and again x in FixPx \in F i x P.
Conversely, Wo show that (10) implies A sube Fix in UC_(A)A \subseteq F i x \in U C_{A} Fix Ror, the inclusion meaning in fact equality. Indeod, Fix RoF sube\subseteq Fix FF implies C_(A)FdxC_{A} F d x RoF supeC_(A)FixF\supseteq C_{A} F i x F, hence A=FixF uuC_(A)FixA=F i x F \cup C_{A} F i x Rof.
In the following we obtain for point-to-set mappings some results which are analogous to those in the first part of this paper. Wo shall use agatin as a retract tho metric projection on closed convex sets in uniformly convex spaces, which is in fact a function.
In uniformly convex spaces, the metric profection is no more a nonexpansive mapping, but it is a continuous one.
Indeed, let XX be a uniformly convex Banach space, O/quad!in C sube X\emptyset \quad \notin C \subseteq X a closed conver : We prove that P_(C)=PP_{C}=P is continuous.
Let x_(n)rarr"n"xx_{n} \xrightarrow{n} x. He have d(x,C) <= ||x-Px_(n)|| <= ||x-x_(n)||+||x_(n)-Px_(n)||=||x-x_(n)||++d(x_(n),C) <= 2||x-x_(n)||+d(x,C)d(x, C) \leq\left\|x-P x_{n}\right\| \leq\left\|x-x_{n}\right\|+\left\|x_{n}-P x_{n}\right\|=\left\|x-x_{n}\right\|+ +d\left(x_{n}, C\right) \leq 2\left\|x-x_{n}\right\|+d(x, C).
It follows that ||x-Px_(n)||rarr"n"d(x,C)\left\|x-P x_{n}\right\| \xrightarrow{n} d(x, C), hence (Ex_(n))_(n in N)\left(E x_{n}\right)_{n \in N} is a minimizing sequence. If x in Cx \in C x than means prectsely ||Dx-Dx_(n)||rarr"n"0\left\|D x-D x_{n}\right\| \xrightarrow{n} 0 and the continuity of PP is proved.
If x!in Cx \notin C, then d(x,C) > 0d(x, C)>0. The set CC being convex it rollotys that (Px_(n)+Px_(a))//2in C\left(P x_{n}+P x_{a}\right) / 2 \in C and 2a(x_(1)0) <= 2hx-(P_(2)+P_(2))//2h <= ||x-P_(n)||+||x-P_(n)||rarr"x_(n)"rarr"m_(x)"2d(x,C)2 a\left(x_{1} 0\right) \leq 2 h x-\left(P_{2}+P_{2}\right) / 2 h \leq\left\|x-P_{n}\right\|+\left\|x-P_{n}\right\| \xrightarrow{x_{n}} \xrightarrow{m_{x}} 2 d(x, C), honce H(-Px_(y))//d(x,C)+(y-Px_(m))//d(x,C)||rarr"m,n"2H\left(-P x_{y}\right) / d(x, C)+\left(y-P x_{m}\right) / d(x, C) \| \xrightarrow{m, n} 2. Donoting z_(n)=(x-P_(n))//d(x;C)z_{n}=\left(x-\mathrm{P}_{n}\right) / \mathrm{d}(x ; \mathrm{C}), we have ||z_(n)||rarr"^^"1\left\|z_{n}\right\| \xrightarrow{\wedge} 1, ||(z_(n)+z_(n))//2||^(m)=1\left\|\left(z_{n}+z_{n}\right) / 2\right\|^{m}=1 and using the uniformly converity of xx it follows that (z_(n))_(n in T)\left(z_{n}\right)_{n \in T} is a Cauchy sequence //8/ 8, Lilo.2.2, p. 379//379 / hence a convergent one. Let y=lim_(2)Fx_(2)y=\lim _{2} \mathrm{Fx}_{2}.
USERE d(x,C) <= ||x-Px_(n)|| <= 2||x-x_(n)||+d(x,C)d(x, C) \leq\left\|x-P x_{n}\right\| \leq 2\left\|x-x_{n}\right\|+d(x, C) we obtain ||x-y||=d(x,C)\|x-y\|=d(x, C), hence y=Pxy=P x and the continuity of PP is proved in this vaso too.
Nic: we prove a theorem which extends to mappings whose range is not nocossarily in CC a theorem of Inm//4//\operatorname{Inm} / 4 /. We denote by rho_(c)(C)\rho_{c}(C) the samily of the compact nonvold subsets of CC. !=0sube X\neq 0 \subseteq X a clange bourged conver set, F:C longrightarrowP_(c)(c)F: C \longrightarrow \mathcal{P}_{c}(c) a nonespan-立吅 provins ( tr rho_(c)(c)\rho_{c}(c) one considers the Hausdorff-Ponpeiu metric) 저술 서숭으 어서 x in Cx \in C surfy that x in F(x)x \in F(x).
Enozal 4. Tot xx be a pitamely conver Ranach space, O/!in C sube X\emptyset \notin C \subseteq X a glaca gopuled conver got, F:C longrightarrowuuu_(c)(X)F: C \longrightarrow \bigcup_{c}(X) a mapping which is Juch text Pow is gonerganelve, Then the napping PP has in ^(C){ }^{C} at least a pinga point.
Hrgos. Because F(I)F(I) is a compact set for each xx in CC and PP is a continuous function, we have For : c longrightarrowP_(c)(c)c \longrightarrow P_{c}(c). The mapping FF being ratractible on CC with respect to PP, it follows FixF=F i x F= = Pix For and using Theorem 4 one obtains fix bar(i)!=c\bar{i} \neq c.
Ore remaiks that FF is retractible onto CC with respect to PP if and only if for each x in P(F(C)\\C)x \in P(F(C) \backslash C) which is not a fixed point for FF and Por each z in F(=)z \in F(=) thare exists z in Cz \in C such thet iz-x|| > ||z-y||i z-x\|>\| z-y \|. Equivalenting, for each z in Z(x)z \in Z(x) we have ||z-x|| > d(z,0)\|z-x\|>d(z, 0).
第 obtain
Corollarr 4. Let XX and CC bo as in Nheoren 4; if F:C rarrS_(c)(X)F: C \rightarrow S_{c}(X) has theoroperty that for each x in P(F(C)\\C)x \in P(F(C) \backslash C) which is not a fired point for ff and for each z inf^(')(x)z \in f^{\prime}(x) there axists y in Cy \in C such that ||z-x||>>||\|z-x\|> >\| 2-y ||\| and PoF is nonexpansive, then FF has in CC at last a fired point.
Theoren 3 is extended in several papers to mappings whose range is not contained in the convex set CC, but F(x)SJ_(0)(x)=F(x) S J_{0}(x)= x{(1-a)x+ay:y quad Cx\{(1-a) x+a y: y \quad C, Ro a > 1//2}a>1 / 2\} for each xx in CC.
Remark 5. If XX is a real Banacin space, then J_(C)(x)=J_{C}(x)= =quad(1-b)x+by=\quad(1-b) x+b y a y in C,b >= 0}y \in C, b \geqslant 0\} for xx in CC.
We obtain nom
Corollary 5. Let XX and CC be as in Theorem 4; if g:C rarrrho_(e)(X)g: C \rightarrow \rho_{e}(X) has the property that F(x)subeJ_(C)(x)={(1-a)x+ay,yF(x) \subseteq J_{C}(x)=\{(1-a) x+a y, y C, Re a > 1//2}a>1 / 2\} for each x in P(F(C)\\C)x \in P(F(C) \backslash C) which is not a fixed point for II and PoPP o P is nongrpansite, than vec(Fix)i!=0\overrightarrow{F i x} i \neq 0.
Eroof. Fet x in F(F(C)\\C),x!in TixFx \in F(F(C) \backslash C), x \notin T i x F and z in F(x)subeJ_(C)(x)z \in F(x) \subseteq J_{C}(x). It follows that there exists y in Cy \in C and a in Ca \in C, R_(a)a > 1//2R_{a} a>1 / 2 such that z==(l-a)x+ayz= =(l-a) x+a y.
We have y!in xy \notin x, since y=xy=x implias z=x,F(x)z=x, F(x), contradiction. men ||z-x||=|a|||y-x||\|z-x\|=|a|\|y-x\| and ||z-y||=|1-a|||y-x||\|z-y\|=|1-a|\|y-x\|. But Re a >>1//2> >1 / 2 implies |a| > |1-a||a|>|1-a|, hence ||z-x|| > ||z-y||\|z-x\|>\|z-y\| and Corollary 4 applies.
Theorem 4 has as a Corollary Theorem 2.3//3//2.3 / 3 /, wheme one imposes mother condition or inwardness than f(z)subeJ_(C)(x)f(z) \subseteq J_{C}(x).
Corollary 6/3/. Let H^(¨)\ddot{H} be a rilbert space, O/!=C sube H\varnothing \neq C \subseteq H a closed hounded convex set, N:C rarrP_(c)(H)N: C \rightarrow P_{c}(H) nonerpansite and A:C rarr[0,1)A: C \rightarrow[0,1) an aroitrary function. In addition one suppose that for each x in Cx \in C and y in bar(H)(x)y \in \bar{H}(x) one has
(11) quadl i m i n f_(n rarrc_(v))h^(-1)d((1-h)x+hy,x) <= A(x)d(x,Tx)\quad \liminf _{n \rightarrow c_{v}} h^{-1} d((1-h) x+h y, x) \leqslant A(x) d(x, T x). Gnen FF has in CC at least a fixed point.
TOOS. In the conditions of the corollary, Pos is nonexpaneive. :.\therefore Lure elso Fix Por sube\subseteq Fix F. Inderd; let x in PoF(x)x \in \operatorname{PoF}(x), hence there exists y <= F(x)y \leq F(x) such that x=P(y)x=P(y). Tie show that for h in(0,1)h \in(0,1) one has
(12) quad d((1-h)x+hy,0)=||(1-n)x+hy-x||\quad d((1-h) x+h y, 0)=\|(1-n) x+h y-x\|.
We suppose that these exists h in(0,1)h \in(0,1) such that z in C,z!in xz \in C, z \notin x and ||(1-h)x+hy-z|| < ||(1-h)x+hy-x||\|(1-h) x+h y-z\|<\|(1-h) x+h y-x\|.
But ||y-z|| <= ||(1-n)x+hy-z||+||(2-h)(x-y)|| <\|y-z\| \leq\|(1-n) x+h y-z\|+\|(2-h)(x-y)\|<
which contradicts x in P(y)x \in P(y). It follows that the relaticn (12) nolds, and consequently, d(x,F(x)) <= ||x-y||=limineh^(-1)a((1-h)x+by,0) <=d(x, F(x)) \leq\|x-y\|=\operatorname{limine} h^{-1} a((1-h) x+b y, 0) \leq <= A(x)d(x,F(x))\leq A(x) d(x, F(x)).
But A(x) < 1A(x)<1, hence bar(a)(x,F(x))=0\bar{a}(x, F(x))=0 and x in F(x)x \in F(x).
Applying insoren 4 one obtains the conclusion.
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