Abstract
Authors
Mira-Cristiana Anisiu
Institutul de Matematica, Cluj-Napoca, Romania
Otilia Mark
Cluj-Napoca, Romania
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M.-C. Alicu, O. Mark, Some properties of the fixed point set for multifunctions, Studia Univ. Babeş-Bolyai, Math. XXV (1980) no. 4, 77-79.
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Studia Univ. Babes-Bolyai, Mathematica
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Babes-Bolyai University
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[1] Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79.
[2] Rus, I. A., Some metrical fixed point theorems, Studia, Univ. Babes-Bolyai 24, 1 (1979), 73-77.
[3] Schirmer, H., Properties of the Öxed point set of contractive multifunction, Canad. Math. Bull.13, 2 (1970), 169-173.
SOME PROPERTIES OF THE FIXED POINTS SET FOR MULTIFUNCTIONS
Many fixed point theorems can be extended for multifunctions, elementary examples showing that in this case the unicity part is lost. We study the transfer of some properties of the sets that are values of a multifunction to the set of fixed points.
Let ( X,dX, d ) be a metric space and S(X)={A inP(X)∣A!=O/,A= bar(A)}S(X)=\{A \in \mathcal{P}(X) \mid A \neq \varnothing, A=\bar{A}\}. We denote by HH the Hausdorff metric on S(X)S(X), and for x in X,D(x,A)=i n f_(y in A)d(x,y)x \in X, D(x, A)= \inf _{y \in A} d(x, y).
Definition 1. The multifunction f:X rarr S(X)f: X \rightarrow S(X) is contractive if
H(f(x),f(y)) < d(x,y),AA x,y in X,x!=y.H(f(x), f(y))<d(x, y), \forall x, y \in X, x \neq y .
Definition 2. Let varphi:R_(+)^(5)rarrR_(+)\varphi: \mathbb{R}_{+}^{5} \rightarrow \mathbb{R}_{+}be a function satisfying the following properties:
a) varphi(r) <= varphi(s)\varphi(r) \leq \varphi(s), for r <= s,r,s inR_(+)^(5)r \leq s, r, s \in R_{+}^{5}
b) varphi\varphi is continuous
c) varphi(r,r,r,r,r) <\varphi(r, r, r, r, r)<,r,forr > 0r , for r>0
d) r-varphi(r,r,r,r,r,)rarr+oor-\varphi(r, r, r, r, r,) \rightarrow+\infty, for r rarr+oor \rightarrow+\infty.
a) varphi(r) <= varphi(s)\varphi(r) \leq \varphi(s), for r <= s,r,s inR_(+)^(5)r \leq s, r, s \in R_{+}^{5}
b) varphi\varphi is continuous
c) varphi(r,r,r,r,r) <\varphi(r, r, r, r, r)<,r,forr > 0r , for r>0
d) r-varphi(r,r,r,r,r,)rarr+oor-\varphi(r, r, r, r, r,) \rightarrow+\infty, for r rarr+oor \rightarrow+\infty.
The multifunction f:X rarr S(X)f: X \rightarrow S(X) is a varphi\varphi-contraction if
H(f(x)f(y)) <= varphi(d(x,y),D(x,f(x)),D(y,f(y)),D(x,f(y)),D(y,f(x)))H(f(x) f(y)) \leq \varphi(d(x, y), D(x, f(x)), D(y, f(y)), D(x, f(y)), D(y, f(x))), AA x,y in X\forall x, y \in X.
H(f(x)f(y)) <= varphi(d(x,y),D(x,f(x)),D(y,f(y)),D(x,f(y)),D(y,f(x)))H(f(x) f(y)) \leq \varphi(d(x, y), D(x, f(x)), D(y, f(y)), D(x, f(y)), D(y, f(x))), AA x,y in X\forall x, y \in X.
We consider the case of the multifunctions defined on the real axis R\mathbb{R} or on some subsets of R\mathbb{R}.
Theorem 1 Let f:[a,b]rarrP([a,b])f:[a, b] \rightarrow \mathcal{P}([a, b]) be a contractive multifunction with f(x)f(x) non-void, compact and convex, AA x in[a,b]\forall x \in[a, b]. Then the fixed points set is non-void, compact and convex.
Proof. The only non-void, compact and convex sets in R\mathbb{R} being the closed intervals, we have f(x)=[m(x),M(x)]f(x)=[m(x), M(x)], where mm and MM are functions which are defined on [a,b][a, b] with values in the same interval. We obtain H(f(x),f(y))=H(f(x), f(y))=
max{|M(x)-M(y)|,|m(X)-m(y)|}\max \{|M(x)-M(y)|,|m(X)-m(y)|\}. The multifunction ff being contractive, we have for x!=yx \neq y
max{|M(x)-M(y)|,|m(X)-m(y)|}\max \{|M(x)-M(y)|,|m(X)-m(y)|\}. The multifunction ff being contractive, we have for x!=yx \neq y
{:[|M(x)-M(y)| < |x-y|],[|m(x)-m(y)| < |x-y|]:}\begin{aligned}
|M(x)-M(y)| & <|x-y| \\
|m(x)-m(y)| & <|x-y|
\end{aligned}
We can apply the theorem of Edelstein [1], following which the functions mm and MM have a unique fixed point. Let x_(m)x_{m} and x_(M)x_{M} be the fixed points, which are also fixed points for ff. It is easy to show that x_(m) <= x_(M)x_{m} \leq x_{M} and that on the left of x_(m)x_{m} and on the right of x_(M)x_{M} there are not other fixed points. Any point xx in ]x_(m),x_(M)[] x_{m}, x_{M}[ is also in [m(x),M(x)][m(x), M(x)], so it is a fixed point.
We have proved that the fixed points set is [x_(m),x_(M)]\left[x_{m}, x_{M}\right], so it is non-void, compact and convex.
Remark. The statement of Theorem 1 is not true for f:Rrarr S(R)f: \mathbb{R} \rightarrow S(\mathbb{R}), as the following example shows.
Let g:RrarrRg: \mathbb{R} \rightarrow \mathbb{R} be a function defined by g(x)={[{:(5)/(2)," for "x in]-oo","2[],[{:x+(1)/(x)," for "x in]2","+oo[]:}g(x)=\left\{\begin{array}{l}\left.\frac{5}{2}, \text { for } x \in\right]-\infty, 2[ \\ \left.x+\frac{1}{x}, \text { for } x \in\right] 2,+\infty[ \end{array}\right. and f:R rarr S(R),f(x)=[0,g(x)]f: R \rightarrow S(\mathbb{R}), f(x)=[0, g(x)].
ff satisfies the condition
ff satisfies the condition
H(f(x),f(y)) < |x-y|," for "x!=y,H(f(x), f(y))<|x-y|, \text { for } x \neq y,
so it is contractive. It also has as values non-void, compact and convex sets, but the fixed points set is [0,+oo[[0,+\infty[ and it is not compact.
Theorem 2 Let varphi:R_(+)^(5)rarrR_(+)\varphi: \mathbb{R}_{+}^{5} \rightarrow \mathbb{R}_{+}be a function as in Definition 2, and f:RrarrP(R)f: \mathbb{R} \rightarrow \mathcal{P}(\mathbb{R}) a varphi\varphi-contraction with f(x)f(x) non-void, compact and convex, AA x inR\forall x \in \mathbb{R}. In this case the fixed points set is non-void, compact and convex.
Proof. We have f(x)=[m(x),M(x)]f(x)=[m(x), M(x)] and H(f(x),f(x,y))=max{|m(x)-m(y)|H(f(x), f(x, y))=\max \{|m(x)-m(y)|, |M(x)-M(y)|}|M(x)-M(y)|\}. Because ff is a varphi\varphi-contraction, and varphi\varphi satisfies the condition a), it follows that
{:[|m(x)-m(y)| <= varphi(d(x","y)","D(x","f(x))","D(y","f(y))","D(x","f(y))","D(y","f(x))) <= ],[varphi(d(x","y)","d(x","m(x))","d(y","m(y))","d(x","m(y))","d(y","m(x)))]:}\begin{gathered}
|m(x)-m(y)| \leq \varphi(d(x, y), D(x, f(x)), D(y, f(y)), D(x, f(y)), D(y, f(x))) \leq \\
\varphi(d(x, y), d(x, m(x)), d(y, m(y)), d(x, m(y)), d(y, m(x)))
\end{gathered}
We obtain similarly
|M(x)-M(y)| <= varphi(d(x,y),d(x,M(x)),d(y,M(y)),d(x,M(y)),d(y,M(x)))|M(x)-M(y)| \leq \varphi(d(x, y), d(x, M(x)), d(y, M(y)), d(x, M(y)), d(y, M(x)))
By theorem 1 in [2] it follows that m:RrarrRm: \mathbb{R} \rightarrow \mathbb{R} has a unique fixed point x_(m)x_{m} and M:RrarrRM: \mathbb{R} \rightarrow \mathbb{R} a unique fixed point x_(M)x_{M}. There are not other fixed points on the left on x_(m)x_{m} and on the right of x_(M)x_{M}; any xx in [x_(m),x_(M)]\left[x_{m}, x_{M}\right] is a fixed point, so the fixed points set is compact and convex.
If we consider some other metric spaces, the properties mentioned above may not be true. In [3] there are given examples of multifunctions in R^(2)\mathbb{R}^{2} having as values convex and compact sets, whose fixed points set is however not convex; those multifunctions satisfy conditions of the type H(f(x),f(y)) <= kd(x,y)H(f(x), f(y)) \leq k d(x, y), with 0 < k < 10<k<1, so they are a special case of the multifunctions considered here.
(Received June 10, 1980)
(Received June 10, 1980)
References
[1] Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79.
[2] Rus, I. A., Some metrical fixed point theorems, Studia, Univ. Babeş-Bolyai 24, 1 (1979), 73-77.
[3] Schirmer, H., Properties of the fixed point set of contractive multifunction, Canad. Math. Bull.13, 2 (1970), 169-173.
[2] Rus, I. A., Some metrical fixed point theorems, Studia, Univ. Babeş-Bolyai 24, 1 (1979), 73-77.
[3] Schirmer, H., Properties of the fixed point set of contractive multifunction, Canad. Math. Bull.13, 2 (1970), 169-173.
UNELE PROPRIETĂŢI ALE MULŢIMII PUNCTELOR FIXE PENTRU APLICAŢII MULTIVOCE
(Rezumat)
În lucrare se studiază cazul în care proprietăţi ale imaginilor unei aplicaţii multivoce (convexitatea şi compactitatea) se transmit la mulţimea punctelor fixe. Rezultatele le generalizează pe cele obţinute de H. Schirmer pentru contracţii, aplicaţiile considerate aici fiind varphi\varphi-contracţii sau contractive, definite pe R\mathbb{R}, respectiv pe intervale compacte.