ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)
O. HadZic (Novi Sad, Yugoslavia)
On Sadovski’s fixed point theorem in topological vector spaces
In [7] Zima has proved a fixed point theorem in a non-locally convex topological vector space. Rzepecki proved in [5] a generalization of Zima’s fixed point theorem for compact mappings and a theorem on the fixed point of Danes type. Here we shall give a further generalization of Zima’s result.
Le m m a. Let E be a Hausdorff topological vector space, К a convex subset of Ё such that 0 e K , let % be the fundamental system of balanced neigh
bourhoods of zero in E and for every V e°U there exists U e âÜ such that
(1) c o ( U n ( K - K ) ) c z V .
I f a subset A of К is precompact, then со A is precompact,
P roof. The proof is similar to the proof from [2] (p. 240). Let Vetff.
We shall prove that there exists (yl5 y 2, ..., ym} <=: со A such that
m
(2) CO A cz ( J { y f + V ].
i= 1
Suppose that U e W is such that U + U cz V and U ' e att such that
(3) c o ( U ' n ( K - K ) ) c z U.
Since the set A is precompact, it is also bounded and so there exists 3 > 0 such that З А cz U’. Since V is balanced, we can suppose that 3 < 1. From the fact that К is convex and 0e K it follows that A cz К implies that à A cz K , for 3 < 1 and so:
ЗА cz U’ r \ K cz U' r\(K — K) and со (ЗА) cz со (U' n ( K - K ) ) . Since the set A is precompact, there exists a finite set x2, ..., x N} c: A such that
W) A c (J \x, + U' } .
i = I
Key words: Fixed point theorem, topological vector spaces.
Let S = {(s1, s2, sN), Y si = 1» si ^ 0, i = 1, N] c R*. Then there i = 1
exists a finite set of points j80) = , f$p, /1$) c S such that for each («i, s2, 5jv)e5 there exists pU)eS so that
X |s , - S '1i « s . i = 1
N
Let us prove that со A a (J ( Y PŸ'xî+V)- If yeco A, then j /=1
r
Y *кУк, Ук^А (к = 1, 2, г), к = 1
г
1 ^ = 1, ^ 0 (к = 1, 2, ..., г).
к= 1
From (4) it follows that yk = xik + zk, where zkeU' (к — 1, 2, . . r) and so
N r 1V
y =
Z
a**k+Z
ak2k>where Z
«к = 1, «к ^ 0 (к = 1, 2, N).к - 1 к = 1
к = 1
Since (a'j, a'2, a)y)eS, there exists such that Y lak“ It remains to prove that
(5) X К - $ " ) * * + I « л е и
k = 1
since then y e Y № >л:к + F Let k= l
*k, <*k-PiP>0
Xt « 4 - д » < о * 1,2>
Since l/' is balanced, it follows that (3) implies
n N la' —
X (« ;- # > )* , = X - - - g— К = с о ( i/ 'n ( K - K ) ) = и
к = 1
and similarly
к = 1
^ atzk с со ((/' Г\{К — К)) с I / .
к = 1
Since U + U с Ц the lemma is proved because (5) implies (2).
R em ark . It is obvious that the lemma remains true if ОфК since in this case the set K' — K — x {xeK) satisfies all the conditions of the lemma.
In [3] Clemens Krauthausen remarked that the following theorem can be proved similarly as in [6]:
Let E be a Hausdorff complete topological vector space, let К be a closed convex and locally convex subset of E such that for every A a К which is compact, со A is also compact and f: К -> К so that
(i) / is continuous.
(ii) I f 0 ф А с К , f (A) c: A and A \co f (A) is compact, then A is compact.
Then there exists x e K such that x = f (x).
R em ark . A set К c= E is locally convex if for every V e°U and x e K there exists U e JU such that со ((x + U) n K) ci V + x. It is easy to prove that from (1) it follows that К is locally convex. The open question is the following: Let E be a Hausdorff topological vector space and let К be a convex locally convex subset in E. Is it true that A is precompact implies со A is precompact, where А с К ?
Now, we can formulate the following fixed point theorem.
Th e o r e m. Let E be a complete Hausdorff topological vector space, К be a closed and convex subset of E such that (1) holds and f: К К satisfies (i) and (ii). Then there exists x e K such that x = f (x).
Now, we shall give two examples of subset К such that (1) holds.
Let E be a linear space over the real or complex number field. The function || U*: £ — [0, oc) will be called quasinorm iff:
1° ||x||*=0<t>x = 0,
2° J| — x||* = ||x||* for every x e E ,
3° ||x + y||* ^ 1М1* + 1Ы1* for every x , y e E ,
4° if ||x „ -x 0||*-> 0 and A„-*A0, then p „x n-A 0x0||* -» 0.
The function q: £ x £ - > [ 0 , x ) defined by g(x, y) = ||x —y||* is the distance function on £, and if (£, q) is a complete metric space, then it is a Fréchet space. Further (£, || ||*) is a topological vector space.
In [7] is given an example of E and К a E such that there exists a number C > 0 with
(6) ||Ax||* < CA||x||* for every 0 ^ X ^ 1 and every x cr K — K.
Let us prove that (6) implies (1). Suppose that £ > 0 and
V = Jx| x e E , ||x||* <" £ J and U — Jx| x e E , ||xj|* < £./C|.
Г
Then from weco (U n ( K — K)) it follows that и — where sfe[0, 1]
i = 1 r
(i = 1, 2, ..., r), Y, s i — 1 and ще1) n ( K — K). Now, we have /= i
||U ||*<C £ Sj ’ z/C — £ i = 1
and so и e V.
Let us remark that Zima applied his fixed point theorem to the infinite system of integral equations
t
Xi = ffi(s, А(1(Хi), Ai2(x2), ..., A/n (x„.))ds (i = 1, 2, ...).
о '
Now, let us give another example of a subset К <= F such that (1) holds.
First, we shall give some notations and definitions which will be used in the sequel [1]. By R we shall denote the set of all real numbers. Further, let E be a vector space over J f (real or complex number field), RA the set of all mappings from A into R with the Tikhonov product topology and the operations + and scalar multiplication as usual. If/, g e R A we say that f < g iff/ ( 0 ^ g (t) for every te A and / Ф g . By Рл we shall denote the cone of non-negative elements in R A. In [1] S. Kasahara introduced the notion of paranormed space.
Definition [1]. The triplet (F, || ||, Ф) is a paranormed space iff || ||: E -* PA, Ф is a linear, continuous, positive mapping from RA into RA such that the following conditions are satisfied:
1° ||x|| = 0 <=> x = 0,
2° ||Ax|| =A||x|| for every x e E and every Хе Ж, 3° ||х + у |К Ф (||х ||) + Ф(|Ы|) for every x , y e E .
Let us denote by °U the family of neighbourhoods of zero in RA and for every U e üU we shall denote the set {x| x e E , ||x ||el/} by Vv . Then E is a topological vector space in which {Vv}Ue% is the family of neighbourhoods of zero in E.
In [1] is proved that every Hausdorff topological vector space F is a paranormed space (F, || ||, Ф) over a topological semifield RA.
Suppose now that К is a subset of F and (F, || ||, Ф) is a paranormed space over a topological semifield R A. If for every n e N , every щ е К ~ К (i — 1, 2, ..., n) and (sl5 s2, ..., s„)eH" such that s,e[0, 1] (i = 1, 2, . . ., n)
П and Y, si ~ 1
i = 1
(7) || Y s«M«ll ^ Ê 5/ф (11м«11)
i= l i ~ 1
we shall say that the set К is of Ф type.
Relation (7) was used in the proof of fixed point theorem from [4]. We shall prove that from (7) follows (I).
If % is the family of neighbourhoods of zero in RA and Ue %, then {x| x e F , ||x||e U] is a neighbourhood of zero in F [1]. Let W be the fundamental system of neighbourhoods of zero in F and suppose that V e aU'.
Then there exist e > 0 and p = {tu t2, ..., tn] a A such that {x| x e E , ||x|| (t)
< s, t e g I с К Suppose that U is a symmetric neighbourhood of zero in R A
such that \\u\\gU implies Ф(||м||)(0 < e for every teft (Ф is a continuous mapping) and that u e U' ( V is a neighbourhood of zero in E) implies
||м||е1/. Let us show that со (IT n ( K — Kj) <= V П
In fact x e со (U1 n ( K — Kj) implies that x = ]T a w h e r e u . e l / 'n i = 1
n
n ( K — K ) for every i = 1, 2, n ( £ a, = 1, a, ^ 0, i = 1, 2, . . n) and
i= 1
so we have
П П
!MI(0 = |l Z ° w ||( rK Z * /ф (Н«1-11)(0 < £ for every
/=1 i = 1
which means that x e V
Using the theorem, it is easy to obtain a generalization of Matusov’s fixed point theorem for generalized condensing mapping, i.e. for the mapping which satisfies (i) and (ii).
References
[1] S. K a s a h a r a , On formulations of topological linear spaces by topological semifield, Math.
Japan. 19 (1974), 121-134.
[2] G. К б th e, Topologische lineare Rtiume, Springer-Verlag, Berlin Gtfttingen-Heidelberg 1960.
[3] C. K r a u t h a u s e n , Der Fixpunktsatz von Schauder in nicht notwendig konvexen Rdumen sowie Anwendungen auf Hammerstain'sche Gleichungen, Dokt. Diss., Aachen 1976.
[4] S. M at u so v , A generalization o f a Tikhonov's fixed point theorem (Russian), Dokl. Akad.
Nauk SSR 2 (1970), 12-14.
[5] B. R z e p e c k i, Remarks on Schauder's Fixed Point Principle and its applications, Bull. Acad.
Polon. Sci. Sér. Sci. Math. 27 (1979), 273-280.
[6] V. S t a llb o h m , Fixpunkte nicht expansiver Abbildungen, Fixpunkte kondensierender Abbildungen, Fredholm'sche Satze linearer kondensierender Abbildungen, Dokt. Diss., Aachen 1973.
[7] K. Z im a, On the Schauder's fixed point theorem with respect to para-normed space, Comment. Math. 19 (1977), 421-423.
DEPARTMENT O F MATHEMATICS UNIVERSITY OF NOVI SAD, YUGOSLAVIA