mathematics

Schauder fixed-point theorem

Schauder fixed-point theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Schauder fixed-point theorem rather than just read about it. In short: The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension. It asserts that if K {\displaystyle K} is a nonempty convex closed subset of a Hausdorff locally convex topological vector space V {\displaystyle V} and f {\displaystyle f} is a continuous mapping of K {\displaystyle K} into itself such that f ( K ) {\di…

Key takeaways

Schauder fixed-point theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Schauder fixed-point theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Schauder fixed-point theorem from memory before moving on to harder problems.

Reference excerpt

The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension. It asserts that if K {\displaystyle K} is a nonempty convex closed subset of a Hausdorff locally convex topological vector space V {\displaystyle V} and f {\displaystyle f} is a continuous mapping of K {\displaystyle K} into itself such that f ( K ) {\displaystyle f(K)} is contained in a compact subset of K {\displaystyle K} , then f {\displaystyle f} has a fixed point. A consequence, called Schaefer's fixed-point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was proved earlier by Juliusz Schauder and Jean Leray. The statement is as follows: Let f {\displaystyle f} be a continuous and compact mapping of a Banach space X {\displaystyle X} into itself, such that the set

{ x ∈ X : x = λ f ( x ) for some 0 ≤ λ ≤ 1 } {\displaystyle \{x\in X:x=\lambda f(x){\mbox{ for some }}0\leq \lambda \leq 1\}}

is bounded. Then f {\displaystyle f} has a fixed point. (A compact mapping in this context is one for which the image of every bounded set is relatively compact.)

History The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. His conjecture for the general case was published in the Scottish book. In 1934, Tychonoff proved the theorem for the case when K is a compact convex subset of a locally convex space. This version is known as the Schauder–Tychonoff fixed-point theorem. B. V. Singbal proved the theorem for the more general case where K may be non-compact; the proof can be found in the appendix of Bonsall's book (see references).

See also Fixed-point theorems Banach fixed-point theorem Kakutani fixed-point theorem

References F. F. Bonsall, Lectures on some fixed point theorems of functional analysis, Bombay 1962 D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. ISBN 3-540-41160-7. H. Schaefer, Über die Methode der a priori-Schranken, Math. Ann. 129, 415–416 (1955), doi:10.1007/BF01362380 J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180 A. Tychonoff, Ein Fixpunktsatz, Mathematische Annalen 111 (1935), 767–776 E. Zeidler, Nonlinear Functional Analysis and its Applications, I - Fixed-Point Theorems

External links "Schauder theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Schauder fixed point theorem". PlanetMath. "proof of Schauder Fixed Point Theorem". PlanetMath..

Worked examples

Example 1 — a first encounter with Schauder fixed-point theorem

Start with the simplest possible case. Write down what Schauder fixed-point theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Schauder fixed-point theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Schauder fixed-point theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Schauder fixed-point theorem

In research: Schauder fixed-point theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Schauder fixed-point theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Schauder fixed-point theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fixed-point theorems, Theorems in functional analysis, Topological vector spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Schauder fixed-point theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Schauder fixed-point theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Schauder fixed-point theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Schauder fixed-point theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Schauder fixed-point theorem in simple terms?

Why does Schauder fixed-point theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Schauder fixed-point theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Schauder fixed-point theorem.