bai tap diem bat dong
TRANSCRIPT
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1. Exercises-Fixed points of c.c. maps
(1) Let K be a bounded, open, convex subset of E. Let F : K E be
completely continuous and be such that F(K) K. Then F has a fixedpoint in K.
Solution 1:(Berton Earnshaw) This solution reduces the exercise to anapplication of the Schauder Fixed Point Theorem.
Choose c K and R > 0, such that the open ball of radius R centeredat c, BR(c) K. For each b BR(c) define
r(b) = {tb + (1 t)c : t 0},
the ray starting at c and containing b Since K is bounded, it follows that
r(b) K = ,
and equals exactly one point, since K is convex. Call this point p(b). Foreach x r(b), define
F(x) =
F(x), x K
F(p(b)), x / K.
Notice that F is continuous and that F(E) F(K), implying that F iscompletely continuous.
Put
A = co
F(K) K
.
Then A is a bounded, closed, and convex set and
F : A A,
hence by the Schauder Fixed Point Theorem, F has a fixed point in the setA, call it y. Notice that ify / K, then F(y) = F(p(b)), for some b BR(c),hence, by hypothesis (F(K) K) y K. Thus the fixed point y must liein K, but then F(y) = F(y).
Solution 2: Choose c K and make the change of variables
v = u c,
then the fixed point equation
u = F(u)
is equivalent to the fixed point equation
v = F(v + c) c =: G(v),
where G is a completely continuous mapping. Define the set
K := {v = u c : u K}.
This set is a bounded open neighborhood of 0 E and
u K v = u c K
and
u K v = u c K.
We furthermore conclude that
G(K) K.1
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If we have that F has a fixed point u K, then the proof is complete andG has a fixed point in K. Thus, assume that G has no fixed points in K.Consider the family of mappings
g(t, v) := v tG(v), 0 t 1.
This is a family of completely continuous perturbations of the identity andfor v K, g(t, v) = 0, t [0, 1]. Thus by the homotopy invarianceprinciple of the Leray-Schauder degree, we have that
d(g(t, ), K, 0) = d(id, K, 0) = 1.
We therefore conclude that the equation
v G(v) = 0
has a solution in K.We remark that in the above proof the convexity of the set K may be
replaced by the weaker requirement that K be starlike with respect to apoint c K, i.e. that the ray emenating from c will intersect the boundaryof K in exactly one point.
(2) Let be a bounded open set in Ewith 0 . Let F : Ebe completelycontinuous and satisfy
x F(x)2 F(x)2 x2, x .
then F has a fixed point in .Solution: Let us assume that F has no fixed points in . Consider the
family of c.c. perturbations of the identity
f(t, x) := x tF(x), 0 t 1.
This family has no zeros on , for t = 0, 1. If, on the other hand
f(t, x) = 0,
for some t (0, 1), and some x , then
x = tF(x),
and the inequality in the exercise becomes
(1 t)2F(x)2 (1 t2)F(x)2.
But F(x) = 0, and thus
(1 t)2 1 t2,
i.e., t 1, contradicting that t (0, 1). We hence may conclude, by the
homotopy invariance principle of Leray-Schauder degree that
d(id, , 0) = 1 = d(id F, , 0).
Which implies that F has a fixed point in . Note a particular case, wherethe above condition hold is the following:
F(x) x, x .
As an example, where this condition holds, consider the following:
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LetF : E E
be a completely continuous mapping such there exist nonnegative constantsa and b, a < 1, such that
F(x) ax + b.
Choose R >> 1, so that b (1 a)R. Then for x R
F(x) ax + b ax + (1 a)x = x.
Thus, for such R, we may choose = BR(0) and apply the above result.