4
votes
3answers
75 views

generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
3
votes
1answer
83 views

Cone metric spaces and fixed point theory

These days cone metric spaces, as a generalization of metric spaces, started to be very interesting... A special interest for this space is its application to the fixed point theory. On the last ...
0
votes
1answer
47 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
1
vote
2answers
84 views

Applying the Banach's Contraction Principle

I have a problem: For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if $$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then ...
1
vote
1answer
57 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one ...
2
votes
1answer
60 views

Application of fixed point theorem in $R^n$

Let $A=(a_{ij}) \in \mathbb R^{n \times n}$ a matrix such that $|a_{ij}|<\frac{1}{n}$ for every $i,j$. Prove that $I-A$ is invertible. My attempt at a solution: $I-A$ is invertible $\iff$ ...
3
votes
1answer
73 views

Getting a root of a continuously differentiable function by Banach's Fixed Point Theorem.

Banach Fixed Point Theorem: Consider a metric space $X = (X, d)$, where $X\neq \varnothing$. Suppose that $X$ is complete and let $T: X \to X$ be a contraction on $X$. Then $T$ has precisely one ...
29
votes
2answers
609 views

Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
0
votes
1answer
37 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
4
votes
2answers
109 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
1
vote
1answer
93 views

upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
2
votes
3answers
158 views

Is this function necessarily a contraction?

If $(X, d)$ is a compact metric space satisfying $d(f(x), f(y)) < d(x, y)$ for all $x, y \in X$ such that $x \ne y$, is $f$ necessarily a contraction? I know an analogue of the Banach Fixed Point ...
1
vote
0answers
59 views

Fixed point of continuous function on compact metric space [duplicate]

Possible Duplicate: Prove the map has a fixed point Let $X$ be a compact metric space and $f:X\rightarrow X$ be a continuous function such that $d(f(x),f(y))<d(x,y)$ for every $x,y\in X$ ...
0
votes
1answer
186 views

Schauder's fixed point theorem for metric linear space

Is there an analogue of Schauder type fixed point theorems that can be used over a metric linear space. So, here $(X,d)$ is a complete vector space with metric $d$. If $C\subseteq X$ and ...
2
votes
2answers
434 views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
5
votes
1answer
1k views

Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$,$\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$.Prove A have a unique fixed point in K. The uniqueness is easy.My ...
1
vote
1answer
225 views

A corollary of Banach's fixed-point theorem

Can someone give me an idea how to generalize Banach's fixed-point theorem for complete metric spaces such that the constant contraction coefficient $c$ (as in $d(Tx,Ty)\leq c \ d(x,y)$ ) may be ...
5
votes
2answers
591 views

Opposite of a contraction mapping

I am taking Real Analysis and we recently went over the Banach Fixed-point Theorem, also commonly known as the Contraction Mapping Theorem which states: If $(X,d)$ is a complete metric space, and ...
3
votes
2answers
856 views

Contraction mapping does not hold in metric space

Let $X=\mathbb{Q}\cap [1,2]$, i.e $X$ is the set of rational number between 1 and 2 inclusive. We can consider $X$ to be a metric space by endowing it with the usual distance function, i.e for $x,y ...
10
votes
2answers
574 views

Contraction mapping in an incomplete metric space

Let us consider a contraction mapping $f$ acting on metric space $(X,~\rho)$ ($f:X\to X$ and for any $x,y\in X:\rho(f(x),f(y))\leq k~\rho(x,y),~ 0 < k < 1$). If $X$ is complete, then there ...
1
vote
3answers
394 views

Finding the fixed points of a contraction

Banach's fixed point theorem gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction. I'm interested in how to calculate the ...