A N N A L E S SO C IE T A T IS M A TH EM A TIC A E PO LO N A E Series I : C OM M EN TATIO N E S M A TH EM A TIC A E X I X (1977) R O C Z N IK I P O L SK IE G O TO W A RZY STW A M ATEM ATYCZNEGO
Séria I : P R A C E M A TEM A TY CZ N E X I X (1977).
K. Zima (Bzeszôw)
On the Schauder’s fixed point theorem with respect to para-normed space
1. Introduction. Let E be a linear space over the real or complex number field. The function || ||* (E->[0, oo)) will be called para-norm if:
1. ||a?||* = 0 if and only if x = в, 2. Ц— x\\* — ||a?||* for each xeE,
3. ||a? + 2/||*< IN|* + |!y||* for each x ,y eE ,
4. if I K - ® 0||*->-0 and Лп->Л0, then ||Ana?n- A 0®0||*-»-0.
The function q ( ExE -> [0, + oo)) defined by g(x,y) = \\œ — y\\* is the distance function on E. If the system (E, q) is the complete metric space, then it will be called Fréchet space and will be denoted by F.
2. Schauder’s type lemma. Let К be a bounded subset of F and let 'ïtf(K-^-F) be a completely continuous operator. I f there exists a positive number С {C = such that:
(1) ||Аж||* < CÀ ||a?||* for 0 < A < 1 , xej<f(K) — s/(K ) (1),
then there exists a sequence {s#n} of continuous on К operators with the range lying in a finite dimensional space and \\s#n(x) — s#{x)\\*->0 uniformly on K.
P ro o f (see [1], Lemma 2). Let a (fc) = {y[k)yik), . . . , у™) be
“ e^-net” of s / ( K ) . We put:
nk ' nk
(y))
( 2 ) P k(y) = 2 , <4*) ( ÿ ) - # ,
i=l and
where a ffy ) = v{f ](y): ( ^ vf]
i = l
Л * if Wy-y\k)W*<eki
if \\y~y(i )\\*> 4- (x) Algebraic difference of sets.
422 K . Z i m a
Functions v[k) and a{k) are continuous on Thus finite dimen
sional operators P k are continuous on j/(K ) too.
Because of 0 < ctfHy) < 1 and condition (1), for each y es/(K ) we obtain
(3) ii» -p * ( ÿ )r = ! | ^ ’ « f)(ÿ )-(ÿ -ÿ ? ))||*nk
г = 1
nk
< C - Y a t\y )-\\y - y f> V * iC - ek.
i = l
Denoting by ^ k{x) = P k(s/(x)) for x eK we obtain the mentioned above sequence {
3. Schauder’s fixed point theorem. Let К be a bounded(2), closed and convex subset of F . Let s# (K -+F) be a completely continuous on К ope
rator. If:
(4 ) stf{K ) <= К ,
(5) there exists a number C > 0 such that:
\\Щ\* < <7*A||a>||* for 0 < Л < 1 and XeK, then there exists an element p e K with jaf(p) — p.
The proof of the above Schauder’s theorem is quite similar to this one contained in [1].
4. Example. Let functions fi(t,x 1, x 2, хщ), i — 1 , 2 , 3 , . . . , are defined for $<•[(), а], х ^ ( — oo, + oo), j = 1 , 2, ..., %, where sup{%} = +oo.
г
Suppose that:
(6) functions f {, i = 1 , 2 , 3 , . . . , are continuous with respect to xjr j = 1 , 2 , . . . , % and measurable in t for every ( a q , a?2 , ..., хщ) ‘, (7) there exist Lebesgue integrable on [0, a] functions mt such that:
\fi(t,x 1, x 2, . . . , х щ) \ ^ т {^), t €[0, a], i = 1 , 2 , 3 , . . . ,
(8) i = 1, 2, 3, . . . , j = 1, 2, ..., % are continuous transform
ations defined in space O(0>a) into C(0>a).
We consider the infinite system of integral equations t
(9) x{ = J f {(s, Ап (хг), A i2(x2), ..., Ain.{xn%))ds, г = 1 , 2 , 3 , . . . 0
(2) The boundedness of i f c J means: if xne M and An~>0, then ||Алагг*,||:*с—>0.
Schauder's fixed point theorem 423
Let E denote the set of all sequences y — (ylf y 2, y3,...) with coor
dinates yk continuous in interval [0, a]. We introduce a paranorm in E by defining
(10)
oo Ml* = J T l
¥
Ы \ M{ + no
where Mi = j mi(t)dt and ||^|| = max \q>i{t)\.
0 [0,a]
The space (E , || ||*) we will denote by 8*. It is complete metric space (see, for example, [2]).
Putting K 0 — {y: < Mt} and defining the operator T as follows:
t
(11) Ту = j J /i(sf ^-xi(9?i), • • • ? -^•1и1(97п1)) ds,
о t
/ л (s, A 21(ç’1), A 22( y 2), . . . , A 2n2( y n2)) d s, . . . j
о we see that
(12) the set K 0 is convex, closed and bounded in 8 * and T (K 0) c K Q.
Since
H t
---<3A- — --- for 0 < K 1 , ^ [ 0 ,2 i f ,] ,
Mi + M Mf + t ’
then for paranorm (10) we obtain:
(13) I M * < 3 - A | H * for 0 < Л < 1 and y e E 0- K 0 => T {E 0) - T { E 0).
It is easy to see that a subset X 8* is relatively compact in 8*
iff for each natural Tc the set P k(X) defined by Pk(y) — yk is equi-bounded and equi-continuous on [0, a]. Taking into consideration above and suppositions (6), (7), (8) we conclude, by virtue of Schauder’s theorem, that there exists a point y e E 0 such that Т(ф) = у. This proves the exist
ence in 8* of the solution of system (9).
References
[1] L. L u ste rn ik and W. Sobolew , Elementy analizy funîccjonalnej, PWN, War
szawa 1959.
[2] K. Y o sid a , Functional analysis, Springer Yerlag, 1965.
