A Fixed Point Theorem Based on Miranda

A new ﬁxed point theorem is proved by using the theorem of Miranda.

For n = 1, Theorem 1.1 reduces to the well-known intermediate-value theorem.Miranda proved his theorem using the Brouwer fixed point theorem.Using the Brouwer degree of a mapping, Vrahatis gave another short proof of Theorem 1.1 (see [2]).Following this proof it is easy to see that Theorem 1.1 is also true, if L is dependent of i; that is, Ω can also be a rectangle and need not to be a cube.Even some L i can 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.
..,n} and let f : Ω→R n be a continuous function on Ω.

Also
be the pairs of parallel opposite faces of the rectangle Ω.If for all i = 1,...,n then there exists some x * ∈ Ω satisfying f (x * ) = 0.
In principle, Corollary 1.2 says that Theorem 1.1 is also true if the ≤ -sign and the ≥sign are exchanged with each other in (1.1).Corollary 1.2 also says that Theorem 1.1 is not restricted to a rectangle with 0 as its center.
Many generalizations have been given (see, e.g., [2,[4][5][6] for the finite-dimensional case and see [7,8] for the infinite-dimensional case).In the presented paper we give a generalization of Corollary 1.2 in the infinite-dimensional Hilbert space l 2 .Finally, we prove a fixed point version of Theorem 1.1 in l 2 .

The infinite-dimensional case
Let l 2 be the infinite-dimensional Hilbert space of all square summable sequences of real numbers equipped with the natural order and equipped with the norm x : or all i ∈ N} and let f : Ω→l 2 be a continuous function on Ω.Also let If for all i ∈ N it holds that then there exists some Proof.For fixed n ∈ N, we consider the function h (n) : Ω→l 2 defined by Since Ω is compact and since f is continuous, the set f (Ω) is compact.Therefore, for given ε > 0 there is a finite set of elements v (1) ,..., and there exists n 1 = n 1 (ε) ∈ N such that for all n > n 1 it holds that ∞ j=n+1 v j 2 ≤ ε, ∀v ∈ v (1) ,...,v (p) . (2.6) So, if n > n 1 is valid, then for all f (x) ∈ f (Ω) we have some v ∈ {v (1) ,...,v (p) } such that for all x ∈ Ω.Now, for fixed n ∈ N we define and h (n) : (2.9) Due to (2.3) and Corollary 1.2 there exists x (n) ∈ Ω n with .11) it holds that (2.12) Now, let n 1 .Then, Hence, lim n→∞ f ( x (n) ) = 0. Since Ω is compact, the sequence x (n) has an accumulation point in Ω, say x * .Without loss of generality, we assume that lim n→∞ x (n) = x * holds.On the one hand, it follows that lim n→∞ 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 of Theorem 1.1 in l 2 .
,∀i ∈ N} and suppose that the mapping g : Ω→l 2 is continuous satisfying Then, g(x) = x has a solution in Ω .
Then, the mapping .17) is (even) a compact mapping from l 2 to l 2 .Now, if A is some kind of diagonally dominant in the sense that there exists some  In both pictures the thick line is the graph of a function y = g(x), x ∈ Ω.In the left picture, Ω = [−L,L] and g(−L) < 0, g(L) > 0. According to Corollary 1.2 g(x) has a zero in Ω.However, g(x) has no fixed point in Ω, which is no contradiction to Theorem (2.2), since g(−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.
Remark 2.5.Theorem 2.2 is also valid in R n of 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 in