Brown A new fixed point theorem is proved by using the theorem of Miranda.. Introduction In 1940, Miranda published the following theorem [1].. Mi-randa proved his theorem using the Brou
Trang 1Volume 2007, Article ID 78706, 6 pages
doi:10.1155/2007/78706
Research Article
A Fixed Point Theorem Based on Miranda
Uwe Sch¨afer
Received 5 June 2007; Revised 17 August 2007; Accepted 1 October 2007
Recommended by Robert F Brown
A new fixed point theorem is proved by using the theorem of Miranda
Copyright © 2007 Uwe Sch¨afer This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
In 1940, Miranda published the following theorem ([1])
Theorem 1.1 LetΩ=x ∈ R n:x i ≤ L, i =1, , n
and let f : Ω →R n be continuous satisfying
f i
x1,x2, , xi −1,− L, x i+1, , xn
≥0,
f i
x1,x2, , xi −1, +L, xi+1, , xn
≤0, ∀ i {1, , n} (1.1)
Then, f (x) = 0 has a solution in Ω.
Forn =1,Theorem 1.1reduces to the well-known intermediate-value theorem Mi-randa proved his theorem using the Brouwer fixed point theorem Using the Brouwer degree of a mapping, Vrahatis gave another short proof ofTheorem 1.1(see [2]) Follow-ing this proof it is easy to see thatTheorem 1.1is also true, ifL is dependent of i; that is,
Ω can also be a rectangle and need not to be a cube Even some L ican be zero Very often, the theorem of Miranda is stated as in the following corollary (see also [3,4]), which is not the theorem of Miranda in its original form, but a consequence of it
Corollary 1.2 Letx ∈ R n , L =(li)∈ R n , i ≥ 0, for i =1, , n, let Ω be the rectangle
Ω := { x ∈ R n:| x i − x i | ≤ l i, =1, , n} and let f : Ω →R n be a continuous function on Ω.
Trang 2Also let
F+
i := { x ∈ Ω : x i = x i+l i }, F i −:= { x ∈ Ω : x i = x i − l i }, i =1, , n, (1.2)
be the pairs of parallel opposite faces of the rectangle Ω If for all i =1, , n
f i(x)· f i(y)≤0, ∀ x ∈ F+
i ,∀ y ∈ F i −, (1.3)
then there exists some x ∗ ∈ Ω satisfying f (x ∗)= 0.
In principle,Corollary 1.2says thatTheorem 1.1is also true if the≤-sign and the≥ -sign are exchanged with each other in (1.1).Corollary 1.2also says thatTheorem 1.1is not restricted to a rectangle with 0 as its center
Many generalizations have been given (see, e.g., [2,4–6] for the finite-dimensional case and see [7,8] for the infinite-dimensional case) In the presented paper we give a generalization ofCorollary 1.2 in the infinite-dimensional Hilbert spacel2 Finally, we prove a fixed point version ofTheorem 1.1inl2
2 The infinite-dimensional case
Letl2be the infinite-dimensional Hilbert space of all square summable sequences of real numbers equipped with the natural order
x ≤ y : ⇐⇒ x i ≤ y i, ∀ i ∈ N, (2.1) and equipped with the norm x := ∞
i =1x2
i Theorem 2.1 Let x = { x i } ∞ i =1∈ l2, L = { l i } ∞ i =1∈ l2, i ≥ 0, for all i ∈ N , Ω := { x ∈ l2:
| x i − x i | ≤ l i,f or all i ∈ N} and let f : Ω → l2be a continuous function on Ω Also let
F i+:= { x ∈ Ω : x i = x i+l i }, F i −:= { x ∈ Ω : x i = x i − l i }, ∀ i ∈ N (2.2)
If for all i ∈ N it holds that
f i(x)· f i(y)≤0, ∀ x ∈ F i+,∀ y ∈ F i −, (2.3)
then there exists some x ∗ ∈ Ω satisfying f (x ∗)= 0.
Proof For fixed n ∈ N, we consider the functionh(n):Ω→ l2defined by
h(n)(x) :=
⎛
⎜
⎜
⎜
⎜
⎝
f1
x1,x2, , xn −1,x n,x n+1, .
f n
x1,x2, , x n −1,xn,xn+1, .
0
⎞
⎟
⎟
⎟
⎟
⎠
SinceΩ is compact and since f is continuous, the set f (Ω) is compact Therefore, for
givenε > 0 there is a finite set of elements v(1), , v(p)∈ f (Ω) such that if f (x) ∈ f (Ω),
Trang 3then there is av ∈ { v(1), , v(p)}such that
and there existsn1= n1(ε)∈ Nsuch that for alln > n1it holds that
∞
j = n+1
v j
2
≤ ε, ∀ v ∈v(1), , v(p)
So, ifn > n1is valid, then for all f (x) ∈ f (Ω) we have some v ∈ { v(1), , v(p)}such that
f (x) h(n)(x) =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
0
f n+1(x)
f n+2(x)
⎞
⎟
⎟
⎟
⎟
⎟
⎠
f (x) − v +
⎛
⎜
⎜
⎜
⎜
⎜
⎝
0
0
v n+1
v n+2
⎞
⎟
⎟
⎟
⎟
⎟
⎠
≤2ε (2.7)
for allx ∈ Ω Now, for fixed n ∈ Nwe define
Ωn:=
⎛
⎜
⎝
x1− l1,x1+l1
x n − l n,x n+l n
⎞
⎟
andh(n):Ωn →R nby
h(n)(x) :=
⎛
⎜
⎝
f1
x1,x2, , xn −1,xn,xn+1,xn+2, .
f n
x1,x2, , xn −1,x n,x n+1,xn+2, .
⎞
⎟
Due to (2.3) andCorollary 1.2there existsx(n)∈Ωnwith
h(n)
x(n)
Setting
x(n):=
⎛
⎜
⎜
⎝
x(n)
x n+1
x n+2
⎞
⎟
⎟
it holds that
x(n)∈Ω, h(n)
x(n)
Trang 4Now, letn > n1 Then,
f
x(n) = f
x(n)
h(n)
x(n) ≤2ε (2.13) Hence, limn →∞ f (x(n))=0 SinceΩ is compact, the sequencex(n)has an accumulation point inΩ, say x ∗ Without loss of generality, we assume that limn →∞ x(n)= x ∗holds On the one hand, it follows that limn →∞ f (x(n))= f (x ∗), since f is continuous On the other
hand, it follows that f (x ∗)=0, since the limit is unique
Next, we prove the fixed point version ofTheorem 1.1inl2
Theorem 2.2 Let L = { l i } ∞
i =1∈ l2, i ≥ 0, for all i ∈ N LetΩ= { x ∈ l2:| x i | ≤ l i,∀ i ∈ N} and suppose that the mapping g : Ω → l2is continuous satisfying
g i(x1,x2, , x i −1,− l i,xi+1, ) ≥0,
g i(x1,x2, , x i −1, +li,xi+1, )≤0, ∀ i ∈ N (2.14)
Then, g(x) = x has a solution in Ω
Proof We consider the continuous function
Since for alli ∈ N
f i
x1, , xi −1,− l i,xi+1, .
= g i
x1, , xi −1,− l i,x i+1, .
+l i ≥0,
f i
x1, , xi −1, +li,x i+1, .
= g i
x1, , x i −1, +li,xi+1, .
− l i ≤0, (2.16) due toTheorem 2.1there existsx ∈ Ω satisfying f (x) =0; that is,g(x) = x.
Example 2.3 Let b ∈ l2andA =(aik) satisfying ∞
i,k =1| a ik |2< ∞ Then, the mapping
g(x) : =
b1−
∞
k =1
a1kx k,b2−
∞
k =1
a2kx k,
(2.17)
is (even) a compact mapping froml2tol2 Now, ifA is some kind of diagonally dominant
in the sense that there exists someL = { l i } ∞ i =1∈ l2such that for alli ∈ N
a ii · l i ≥b i+ ∞
k =1,k = i
a ik · l k, (2.18)
then byTheorem 2.1there exists someξ ∈Ω= { x ∈ l2:| x i | ≤ l i,∀ i ∈ N}withAξ = b.
ByTheorem 2.2it follows that there existsη ∈ Ω satisfying η = b − Aη.
Remark 2.4 Note that inTheorem 2.2it is not necessary thatg is a self-mapping as it is
assumed in many other fixed point theorems
Remark 2.5. Theorem 2.2is also valid inRnof course Note, however, that the conditions (2.14) cannot be changed analogously as the conditions (1.1) have been changed to (1.3)
We demonstrate this inFigure 2.1forn =1
Trang 5(a)
y
(b)
Figure 2.1 In both pictures the thick line is the graph of a functiony = g(x), x ∈Ω In the left pic-ture, Ω=[− L, L] and g( − L) < 0, g(L) > 0 According toCorollary 1.2g(x) has a zero in Ω However, g(x) has no fixed point in Ω, which is no contradiction to Theorem (2.2 ), sinceg( − L) ≥0,g(L) ≤0
is not valid, here In the right picture, Ω=[ x − L, x + L] and g( x − L) > 0, g( x + L) < 0 According to Corollary 1.2 ,g(x) has a zero in Ω However, g(x) has no fixed point in Ω.
Acknowledgments
The author would like to thank the anonymous referee(s) for many suggestions and com-ments that helped to improve the paper Furthermore, he would like to thank Professor Mitsuhiro Nakao for his invitation to the Kyushu University in Fukuoka, where this work was started
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Uwe Sch¨afer: Institut f¨ur Angewandte und Numerische Mathematik, Fakult¨at f¨ur Mathematik, Universit¨at Karlsruhe (TH), D-76128 Karlsruhe, Germany
Email address:Uwe.Schaefer@math.uni-karlsruhe.de
... Trang 5(a)
y
(b)
Figure... 1978.
[4] J Mayer, ? ?A generalized theorem of Miranda and the theorem of Newton-Kantorovich,”
Numer-ical Functional Analysis and Optimization,... Universitatii Al I Cuza din Iasi Serie Noua Matematica, vol 37, no 2, pp 161–164,
1991.
[7] C Avramescu, ? ?A generalization of Miranda? ??s theorem, ” Seminar