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Mira-Cristiana Anisiu
Institutul de Matematica, Cluj-Napoca, Romania
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M.-C. Anisiu, On multivalued mappings satisfying the condition T(F_T)=F_T, Seminar on Fixed Point Theory, 1-8, Preprint, 85-3, Univ. Babeş-Bolyai, Cluj-Napoca, 1985 (pdf file here)
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[1] Purdea, I., Pic, Gh., Treatise of modern algebra, vol.1, Ed. Acad. R.S.R., 1977 (in Romanian)
[2] Rus, I.A., Fixed and strict fixed points for multivalued mappings, this Preprint
"BABES-BOLYAI" UNIVERSITIT
FACULTY OF MATHEMATICS
RESEARGH SEMINARIRS
SEMINAR ON FIXED POINT THEORY
Preprint ar. 3, 1985, pp. A-BA-B
Let XX be a nonvold set and T:X rarr XT: X \rightarrow X a multivalued mapping. Let
{:[F_(T)={x in X:x in T(x)}","" respectively "],[(SF)_(T)={x in X:T(x)={x}}]:}\begin{aligned}
& F_{T}=\{x \in X: x \in T(x)\}, \text { respectively } \\
& (S F)_{T}=\{x \in X: T(x)=\{x\}\}
\end{aligned}
denote the fixed point set, respectively the strict fixed point set of T.
For the multivalued mapping T:X rarr XT: X \rightarrow X we consider the graph
G(T)={(x,y)in X:X:y in T(x)}G(T)=\{(x, y) \in X: X: y \in T(x)\}
which may be regarded as a relation on XX. The properties of the relations are exposed, for example, in [1].
It is obvious that any function f:X rarr Xf: X \rightarrow X satisfies the equality f(F_(f))=F_(f)f\left(F_{f}\right)=F_{f}. In the paper [2], I.A. Rus shows that there are multivalued mappings which have not this property, but any multivalued mapping having only strict fixed points ( F_(T)=(SF)_(T)\mathrm{F}_{T}=(S F)_{T} ) verifies the condition T(F_(T))=F_(T)T\left(F_{T}\right)=F_{T} ( Lemma 4.1). The Problem 4.1[2] asks what are the conditions under which the set F_(T)F_{T} is fixed for a multivalued mapping T.
In the sequel we give sufficient conditions for the equality T(F_(T))=F_(T)T\left(F_{T}\right)=F_{T} holds and then we study some properties of a multivalued mapping tilde(T)\tilde{T} induced by the given TT.
For the set X!=O/X \neq \varnothing we denote
I_(X):X rarr X,I_(X)(x)={x}" for any "x in XI_{X}: X \rightarrow X, I_{X}(x)=\{x\} \text { for any } x \in X
Delta_(X)=G(I_(X))={(x,x)in X xx X:x in X}\Delta_{X}=G\left(I_{X}\right)=\{(x, x) \in X \times X: x \in X\},
and for the multivalued mapping T:X rarr XT: X \rightarrow X
dom T={x in X:T(x)!=O/}\operatorname{dom} T=\{x \in X: T(x) \neq \varnothing\}
Im T={y in XT=\{y \in X : there is x in Xx \in X such that y in T(x)}y \in T(x)\}
T^(-1):X rarr X,T^(-1)(y)={x in X:y in T(x)}T^{-1}: X \rightarrow X, T^{-1}(y)=\{x \in X: y \in T(x)\}.
For a multivalued mapping T:X rarr XT: X \rightarrow X the following theorem holds.
and for the multivalued mapping T:X rarr XT: X \rightarrow X
dom T={x in X:T(x)!=O/}\operatorname{dom} T=\{x \in X: T(x) \neq \varnothing\}
Im T={y in XT=\{y \in X : there is x in Xx \in X such that y in T(x)}y \in T(x)\}
T^(-1):X rarr X,T^(-1)(y)={x in X:y in T(x)}T^{-1}: X \rightarrow X, T^{-1}(y)=\{x \in X: y \in T(x)\}.
For a multivalued mapping T:X rarr XT: X \rightarrow X the following theorem holds.
THHOREM 1. Let the below conditions be given
(a) T(F_(T))subeF_(T)T\left(F_{T}\right) \subseteq F_{T}
(a') T(F_(T))=F_(T)T\left(F_{T}\right)=F_{T}
(a") T(dom(I_(X)nn T))sube dom(I_(X)nn T)T\left(\operatorname{dom}\left(I_{X} \cap T\right)\right) \subseteq \operatorname{dom}\left(I_{X} \cap T\right)
(b) for any x inF_(T)T(x)subennn_(y in T(x))T(y)x \in \mathbb{F}_{T} T(x) \subseteq \bigcap_{y \in T(x)} T(y)
(b') for any x inF_(T)x \in F_{T} and z in T(x)z \in T(x), it follows T(x)sube T(z)T(x) \subseteq T(z)
(c) G(T)G(T) is a symmetrical and transitive relation
(d) G(T)G(T) is a reflexive relation
(e) F_(T)=(SF)_(T)F_{T}=(S F)_{T}
(a) T(F_(T))subeF_(T)T\left(F_{T}\right) \subseteq F_{T}
(a') T(F_(T))=F_(T)T\left(F_{T}\right)=F_{T}
(a") T(dom(I_(X)nn T))sube dom(I_(X)nn T)T\left(\operatorname{dom}\left(I_{X} \cap T\right)\right) \subseteq \operatorname{dom}\left(I_{X} \cap T\right)
(b) for any x inF_(T)T(x)subennn_(y in T(x))T(y)x \in \mathbb{F}_{T} T(x) \subseteq \bigcap_{y \in T(x)} T(y)
(b') for any x inF_(T)x \in F_{T} and z in T(x)z \in T(x), it follows T(x)sube T(z)T(x) \subseteq T(z)
(c) G(T)G(T) is a symmetrical and transitive relation
(d) G(T)G(T) is a reflexive relation
(e) F_(T)=(SF)_(T)F_{T}=(S F)_{T}
The following implications are true
(a)=>(a)<=>(a^('))<=>(a^(''))(\mathrm{a}) \Rightarrow(\mathrm{a}) \Leftrightarrow\left(\mathrm{a}^{\prime}\right) \Leftrightarrow\left(\mathrm{a}^{\prime \prime}\right)
hat(n)\hat{n}
(e) =>(b)Longleftrightarrow(b^('))\Rightarrow(b) \Longleftrightarrow\left(b^{\prime}\right)
hat(n)\hat{n}
(c)
(a)=>(a)<=>(a^('))<=>(a^(''))(\mathrm{a}) \Rightarrow(\mathrm{a}) \Leftrightarrow\left(\mathrm{a}^{\prime}\right) \Leftrightarrow\left(\mathrm{a}^{\prime \prime}\right)
hat(n)\hat{n}
(e) =>(b)Longleftrightarrow(b^('))\Rightarrow(b) \Longleftrightarrow\left(b^{\prime}\right)
hat(n)\hat{n}
(c)
Proof. (a) =>(a^('))\Rightarrow\left(a^{\prime}\right). It is obvious that (a^('))=>(a)\left(a^{\prime}\right) \Rightarrow(a); if (a)(a) is true, we obtain ( a^(')a^{\prime} ) because from x inF_(T)x \in F_{T} we deduce x in T(x)x \in T(x), so x∈∈T(F_(T))x \in \in T\left(F_{T}\right).
(a)<=>(a^(''))*dom(I_(X)nn T)={x in X:I_(X)(x)nn T(x)!=O/}=={x in X:x in T(x)}=F_(T)(\mathrm{a}) \Leftrightarrow\left(\mathrm{a}^{\prime \prime}\right) \cdot \operatorname{dom}\left(I_{\mathrm{X}} \cap T\right)=\left\{x \in X: I_{\mathrm{X}}(x) \cap T(x) \neq \emptyset\right\}= =\{x \in X: x \in T(x)\}=F_{T} and the equivalence holds.
(b) =>\Rightarrow (b'). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x). We prove that T(x)sube T(z)T(x) \subseteq T(z). For any y in T(x)y \in T(x), we obtain y in T(x)subennn_(y in T(x))T(y)subeTV_(3)(!in so T(x)y \in T(x) \subseteq \bigcap_{y \in T(x)} T(y) \subseteq \operatorname{TV}_{3}(\notin \operatorname{so} T(x)
sube T(z)\subseteq T(z).
(b') =>\Rightarrow (B). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x); by (b') we have T(x)sube T(B)T(x) \subseteq T(\mathbf{B}), hence T(x)subennn_(z in T(x))T(z)T(x) \subseteq \bigcap_{z \in T(x)} T(z).
(b) =>\Rightarrow (a). Let y in T(F_(T))y \in T\left(F_{T}\right), i.e. there exists x inF_(T)x \in F_{T} such that y∈∈T(x)y \in \in T(x). The condition (b) implies T(x)subennn_(z in T(x))T(z)sube T(y)T(x) \subseteq \bigcap_{z \in T(x)} T(z) \subseteq T(y); but y∈∈T(x)y \in \in T(x) and it follows y in T(y)y \in T(y), hence y inF_(T)y \in F_{T}.
(c) =>\Rightarrow (B). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x). We prove that for any y in Xy \in X such that y in T(x)y \in T(x) we have y in T(z)y \in T(z). The symmetry of G(T)G(T) implies x in T(y)x \in T(y); but z in T(x)z \in T(x) and from the transitivity of G(T)G(T) we obtain z in f(y)z \in \boldsymbol{f}(y). Applying again the symmetry, we have y in T(z)y \in T(z) and T(x)⊆⊆nnn_(z in T(x))T(z)T(x) \subseteq \subseteq \bigcap_{z \in T(x)} T(z).
(e)=>(b)(e) \Rightarrow(b). Let x inF_(T)=(SF)_(T)x \in F_{T}=(S F)_{T} so T(x)={x}=nnn_(y in T(x))T(y)T(x)=\{x\}=\bigcap_{y \in T(x)} T(y) and (b) holds.
(d)=>(a).G(T)(d) \Rightarrow(a) . G(T) being reflexive, we have Delta_(X)sube G(T)\Delta_{X} \subseteq G(T) and it follows F_(T)=XF_{T}=X; it is obvious that T(F_(T))subeF_(T)T\left(F_{T}\right) \subseteq F_{T}.
(a)<=>(a^(''))*dom(I_(X)nn T)={x in X:I_(X)(x)nn T(x)!=O/}=={x in X:x in T(x)}=F_(T)(\mathrm{a}) \Leftrightarrow\left(\mathrm{a}^{\prime \prime}\right) \cdot \operatorname{dom}\left(I_{\mathrm{X}} \cap T\right)=\left\{x \in X: I_{\mathrm{X}}(x) \cap T(x) \neq \emptyset\right\}= =\{x \in X: x \in T(x)\}=F_{T} and the equivalence holds.
(b) =>\Rightarrow (b'). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x). We prove that T(x)sube T(z)T(x) \subseteq T(z). For any y in T(x)y \in T(x), we obtain y in T(x)subennn_(y in T(x))T(y)subeTV_(3)(!in so T(x)y \in T(x) \subseteq \bigcap_{y \in T(x)} T(y) \subseteq \operatorname{TV}_{3}(\notin \operatorname{so} T(x)
sube T(z)\subseteq T(z).
(b') =>\Rightarrow (B). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x); by (b') we have T(x)sube T(B)T(x) \subseteq T(\mathbf{B}), hence T(x)subennn_(z in T(x))T(z)T(x) \subseteq \bigcap_{z \in T(x)} T(z).
(b) =>\Rightarrow (a). Let y in T(F_(T))y \in T\left(F_{T}\right), i.e. there exists x inF_(T)x \in F_{T} such that y∈∈T(x)y \in \in T(x). The condition (b) implies T(x)subennn_(z in T(x))T(z)sube T(y)T(x) \subseteq \bigcap_{z \in T(x)} T(z) \subseteq T(y); but y∈∈T(x)y \in \in T(x) and it follows y in T(y)y \in T(y), hence y inF_(T)y \in F_{T}.
(c) =>\Rightarrow (B). Let x inF_(T)x \in F_{T} and z in T(x)z \in T(x). We prove that for any y in Xy \in X such that y in T(x)y \in T(x) we have y in T(z)y \in T(z). The symmetry of G(T)G(T) implies x in T(y)x \in T(y); but z in T(x)z \in T(x) and from the transitivity of G(T)G(T) we obtain z in f(y)z \in \boldsymbol{f}(y). Applying again the symmetry, we have y in T(z)y \in T(z) and T(x)⊆⊆nnn_(z in T(x))T(z)T(x) \subseteq \subseteq \bigcap_{z \in T(x)} T(z).
(e)=>(b)(e) \Rightarrow(b). Let x inF_(T)=(SF)_(T)x \in F_{T}=(S F)_{T} so T(x)={x}=nnn_(y in T(x))T(y)T(x)=\{x\}=\bigcap_{y \in T(x)} T(y) and (b) holds.
(d)=>(a).G(T)(d) \Rightarrow(a) . G(T) being reflexive, we have Delta_(X)sube G(T)\Delta_{X} \subseteq G(T) and it follows F_(T)=XF_{T}=X; it is obvious that T(F_(T))subeF_(T)T\left(F_{T}\right) \subseteq F_{T}.
We mention now some connections between the classes of multivalued mappings satisfying the conditions in Theorem. 1.
THEOREM 2. If the graph of the multivalued mapping TT satisfying (b) is a reflexive relation, it is also symmetrical and transitive.
Proof. G(T)G(T) being reflexive, the condition (b) is satisfied for any x in Xx \in X. Let bar(x)in X\bar{x} \in X be arbitrary and y in T(x)y \in T(x). Using (b),x in T(x)sube(b), x \in T(x) \subseteq 里 sube T(y)\subseteq T(y), so x in T(y)x \in T(y) and the symmetry is proved.
For the transitivity, we consider x in X,y in T(x)x \in X, y \in T(x) and z in T(y)z \in T(y). It follows by (b) that x in T(x)sube T(y)sube T(z)x \in T(x) \subseteq T(y) \subseteq T(z), so x in T(z)x \in T(z). Applying the symmetry of G(T)G(T) we obtain z in T(x)z \in T(x) and the proof is over.
REMARK 1. There is only one mailhytrod mapping which has only strict fixed points and a reflexive graph, namely I_(X)I_{X}, whose graph Delta_(X)\Delta_{X} is also symmetrical and transitive.
We are able now to present the relative position of the classes
of multivalued mappings involved in Theorem 1 using the diagram in Fig. 1; rectangles having the basas on the same line and the top vertexes marked with a letter stand for the classes denoted by that letter. All the regions marked by a number are nonvoid, as the following examples show.
of multivalued mappings involved in Theorem 1 using the diagram in Fig. 1; rectangles having the basas on the same line and the top vertexes marked with a letter stand for the classes denoted by that letter. All the regions marked by a number are nonvoid, as the following examples show.
| aa | 1 | bb | cc | ||
| dd | cc | b^(2)b^{2} | 8 | cc | |
| 2 | ee | 3 | |||
| 5 | 7 | 4 |
Fig. 1
EXAMPLIES.
- I satisfies (a), but none of (b) and (d).
X=R,T(x)={[{0","x}","x in[-1","1]],[{1}","x in R\\[-1","1]]:}X=R, T(x)=\left\{\begin{array}{l}
\{0, x\}, x \in[-1,1] \\
\{1\}, x \in R \backslash[-1,1]
\end{array}\right.
- TT satisfies (d), but it does not satisfy (b).
X=R,T(x)={[{x}",",x!=0],[{0","1}",",x=0]:}X=R, T(x)= \begin{cases}\{x\}, & x \neq 0 \\ \{0,1\}, & x=0\end{cases}
- I satisfies (b), but none of (c), (e) and (d).
X=R,T(x)={[{2}",",x=0],[{-x","x}",",x in[-1","1]\\{0}],[{0}",",x in R\\[-1","1]]:}X=R, T(x)= \begin{cases}\{2\}, & x=0 \\ \{-x, x\}, & x \in[-1,1] \backslash\{0\} \\ \{0\}, & x \in R \backslash[-1,1]\end{cases}
- T satisfies (e), but none of (c) and (d).
X=R,T(x)={0,-x}X=R, T(x)=\{0,-x\}
- T satisfies (c), (d) and (e).
By Remark 1, the only multivalued mapping satisfying these conditions is I_(X)I_{X}.
6. I satisfies (c) and (d), but it does not satisfy (e).
6. I satisfies (c) and (d), but it does not satisfy (e).
X=R,T(x)={[{x}",",x in R\\{1","2}],[{1","2}",",x in{1","2}]:}X=R, T(x)= \begin{cases}\{x\}, & x \in R \backslash\{1,2\} \\ \{1,2\}, & x \in\{1,2\}\end{cases}
- I satisfies (c) and (e), but it does not satisfy (d).
X=RU{oo},I(x)={[{x}","x in R],[phi;x=oo]:}X=R U\{\infty\}, I(x)=\left\{\begin{array}{c}
\{x\}, x \in R \\
\phi ; x=\infty
\end{array}\right.
- T satisfies (c), but none of (d) and (e).
X=R uu{oo},T(x)={[{-x","x}","x in R],[phi","x=oo]:}X=R \cup\{\infty\}, T(x)=\left\{\begin{array}{l}
\{-x, x\}, x \in R \\
\phi, x=\infty
\end{array}\right.
REMARK 2. If we consider only multivalued mappings T:X rarr XT: X \rightarrow X such that X=dom TX=\operatorname{dom} T (all the values of TT are nonvidid), the condition (c) implies (d.).
Indeed, for any x in Xx \in X we have T(x)!=phiT(x) \neq \phi and we can choose y in T(x)y \in T(x); from the symmetry we obtain x in T(y)x \in T(y) and the transitivity of G(x)G(x) imp plies x in T(x)x \in T(x), i.e. G(T)G(T) is reflexive.
If this is the case, the regions denoted by 7 and 8 in Fig. 1 an are void and the diagram looks like this
[[a,,b,,a","b],[,1,,,],[a,,c,c","d,3],[,2,(6)/(e),,],[,,,5,4]:}\left[\begin{array}{cccc|c}a & & b & & a, b \\ & 1 & & & \\ \hline a & & c & c, d & 3 \\ & 2 & \frac{6}{e} & & \\ & & & 5 & 4\end{array}\right.
Fig. 2
It follows that a multivalued mapping Psi:X rarr X\Psi: X \rightarrow X with X=dom XX=\operatorname{dom} X has a symmetrical and transitive graph ( satisfies the condition (c) ) if and only if TT satisfies the condition (b) and G(T)G(T) is a reflexive relation.
The condition (b) leeds us to the definition of a multivalued mapping attached to TT.
Let XX be a nonvoid set and T:X rarr XT: X \rightarrow X a multivalued mapping. We define tilde(T):X rarr X\tilde{T}: X \rightarrow X given by
tilde(T)(x)={[nnn in T(x)],[y","" for "x in X\\dom T]:}\tilde{T}(x)=\left\{\begin{array}{l}
\bigcap \in T(x) \\
y, \text { for } x \in X \backslash \operatorname{dom} T
\end{array}\right.
The the termes of tilde(T)\tilde{T}, the conditione (6) becomes T|_(F_(T))sube T|_(F_(T))\left.\left.T\right|_{F_{T}} \subseteq T\right|_{F_{T}}.
If we consider a new condition
(P) T sube tilde(T)T \subseteq \tilde{T}
we obtain obviously that (f) implies (b).
If T=G:X rarr XT=G: X \rightarrow X is a function, we have g^(N)(x)=(g@g)(x)g^{N}(x)=(g \circ g)(x) for any x in Xx \in X; the condition (f)(f) is equivalent to g(x)=(g@g)(x)g(x)=(g \circ g)(x) for an any x in Xx \in X, i.e. to Im g=F_(g)\operatorname{Im} g=F_{g}.
(P) T sube tilde(T)T \subseteq \tilde{T}
we obtain obviously that (f) implies (b).
If T=G:X rarr XT=G: X \rightarrow X is a function, we have g^(N)(x)=(g@g)(x)g^{N}(x)=(g \circ g)(x) for any x in Xx \in X; the condition (f)(f) is equivalent to g(x)=(g@g)(x)g(x)=(g \circ g)(x) for an any x in Xx \in X, i.e. to Im g=F_(g)\operatorname{Im} g=F_{g}.
RMARK 3. If TT is a multivalued mapping, we have only T sube T@TT \subseteq T \circ T on dom TT, the inclusion being generally strict, as the following example shows. Let T:R rarr RT: R \rightarrow R be given by T(x)={0,x}T(x)=\{0, x\}; then T*T(x)=={0,x}⇌ tilde(T)(x)={0}T \cdot T(x)= =\{0, x\} \rightleftharpoons \tilde{T}(x)=\{0\}, for any x!=0x \neq 0.
THEOREM 3. If T:X rarr XT: X \rightarrow X satisfles the condition (f)(f) we have Im T=F_(T)^(**)T=F_{T}{ }^{*}
Procif. We have obviously F_(T)sube Im TF_{T} \subseteq \operatorname{Im} T. Let y in Im Ty \in \operatorname{Im} T and x in Xx \in X such that y in T(x)y \in T(x). Applying (f)(f), we get y in T(x)sube tilde(T)(x) <= T(y)y \in T(x) \subseteq \tilde{T}(x) \leq T(y) and y inF_(T)y \in F_{T}. It follows that In T subeF_(T)T \subseteq \mathbb{F}_{T}, so Im T=F_(T)\operatorname{Im} T=\mathbb{F}_{T} holds.
REMARK 4. The reverse implication is not true. For the multivalued mapping TT from Example 1 we have Im T=F_(T)=[-1,1]\operatorname{Im} T=F_{T}=[-1,1], but T(x)={[{0}","x in[-1","1]],[{0","1}","x in R\\[-1","1]]:}T(x)=\left\{\begin{array}{l}\{0\}, x \in[-1,1] \\ \{0,1\}, x \in R \backslash[-1,1]\end{array}\right., so T⊈ tilde(T)T \nsubseteq \tilde{T}.
The next theorem gives a condition for a point xinX\mathbf{x} \in \mathbf{X} be a fixed point for T.
THEOREM 4. The element x in Xx \in X is a fixed point for TT if and only if T(x)subeT^(-1)(x)T(x) \subseteq T^{-1}(x).
Proof. Let x in Xx \in X be a fixed point for tilde(T)\tilde{T}; if x in dom Tx \in \operatorname{dom} T, we have T(x)=phi subeT^(-1)(x)T(x)=\phi \subseteq T^{-1}(x). In the case that x inx \in dom T,T(x)T, T(x) is a nonvoid set; let y in T(x)y \in T(x). It follows that x in tilde(T)(x)sube T(y)x \in \tilde{T}(x) \subseteq T(y), i.e. y inT^(-1)(x)y \in T^{-1}(x) and we obtain again T(x)subeT^(-1)(x)T(x) \subseteq T^{-1}(x).
Let now x in Xx \in X be a point such that T(x)subeT^(-1)(x)T(x) \subseteq T^{-1}(x). If T(x)=O/T(x)=\emptyset, we have T(x)=XT(x)=X and xx is obviously a fixed point for tilde(T)\tilde{T}. But yy was
arbitrary in T(x)T(x), so x innnn_(y in T(x))T(y)=T(x)x \in \underset{y \in T(x)}{\bigcap} T(y)=T(x), hence x inF_(T)x \in F_{T}.
COROILARY. The fixed point set for tilde(T)\tilde{T} is the largest subset of XX on which G(T)G(T) is symmetrical.
arbitrary in T(x)T(x), so x innnn_(y in T(x))T(y)=T(x)x \in \underset{y \in T(x)}{\bigcap} T(y)=T(x), hence x inF_(T)x \in F_{T}.
COROILARY. The fixed point set for tilde(T)\tilde{T} is the largest subset of XX on which G(T)G(T) is symmetrical.
Propf. F_(T)={x in X:T(x)subeT^(-1)(x)}F_{T}=\left\{x \in X: T(x) \subseteq T^{-1}(x)\right\}.
REMARK 5. Any strict fixed point for TT is also a strict fixed point for widetilde(T)\widetilde{T}. The reverse implication is not true. Indeed, for T:R rarr RT: R \rightarrow R given by
REMARK 5. Any strict fixed point for TT is also a strict fixed point for widetilde(T)\widetilde{T}. The reverse implication is not true. Indeed, for T:R rarr RT: R \rightarrow R given by
T(x)={[{-1","1}","x=0],[{0","-2}","x=-1],[{0","2}","x=1],[{x}","x in R\\{0","-1","1}]:}T(x)=\left\{\begin{array}{l}
\{-1,1\}, x=0 \\
\{0,-2\}, x=-1 \\
\{0,2\}, x=1 \\
\{x\}, x \in R \backslash\{0,-1,1\}
\end{array}\right.
we have tilde(T)(0)=T(-1)nn T(1)={0}\tilde{T}(0)=T(-1) \cap T(1)=\{0\}, so 0in(SF)_( tilde(T))0 \in(S F)_{\tilde{T}}, but 0!in(SF)_(T)0 \notin(S F)_{T}.
REFWRENOES
- Purdea, I., Pic,Gh., Treatise of modern algebra, Vol. I, ED. Acad. R.S.R. 1977 (in Romanian)
- Rus, I.A., Fixed and strict fixed points for multivalued mappings, this Preprint