Not checking the conditions of the theorem

Oct 13, 2025 by thesweetdesigns.com 41 views

Introduction

In the realm of functional analysis, the Peano theorem is a fundamental result that deals with the existence and uniqueness of solutions to certain types of differential equations. In this article, we will explore an example of the general Peano theorem in Banach spaces, which is a crucial concept in the study of differential equations. We will also examine the conditions of the Arzela-Ascoli theorem and determine which one does not hold to prove that a given function is not compact.

Background

The Peano theorem is a result in functional analysis that deals with the existence and uniqueness of solutions to certain types of differential equations. In its general form, the theorem states that if we have a differential equation of the form:

y'(t) = f(t, y(t))

where f is a continuous function from a Banach space X to a Banach space Y, and if we have an initial condition y(0) = y0, then there exists a unique solution to the differential equation.

The Arzela-Ascoli Theorem

The Arzela-Ascoli theorem is a fundamental result in functional analysis that deals with the compactness of sets of functions. In its general form, the theorem states that a set of functions is compact if and only if it is closed, bounded, and equicontinuous.

Conditions of the Arzela-Ascoli Theorem

The Arzela-Ascoli theorem has three main conditions that must be satisfied for a set of functions to be compact:

Closed: The set of functions must be closed, meaning that it contains all its limit points.
Bounded: The set of functions must be bounded, meaning that there exists a constant M such that ||f(x)|| ≤ M for all x in the domain.
Equicontinuous: The set of functions must be equicontinuous, meaning that for every ε > 0, there exists a δ > 0 such that ||f(x) - f(y)|| ≤ ε whenever ||x - y|| ≤ δ.

Example

Let's consider an example of a function f that is not compact in the sense of the Arzela-Ascoli theorem. Suppose we have a function f: [0, 1] → ℝ² defined by:

f(x) = (x, sin(1/x))

where x ∈ (0, 1].

Analysis

To determine which condition of the Arzela-Ascoli theorem does not hold for the function f, let's examine each condition separately.

Closed: The set of functions {f(x) = (x, sin(1/x))} is not closed, since it does not contain its limit point at x = 0.
Bounded: The set of functions {f(x) = (x, sin(1/x))} is bounded, since ||f(x)|| ≤ √(x² + 1) for all x ∈ (0, 1].
Equicontinuous: The set of functions {f(x) = (x, sin(1/x))} is not equicontinuous, since for every ε > 0, there exists a δ > 0 such that ||f(x) - f(y)|| > ε whenever ||x - y|| ≤ δ.

Conclusion

In conclusion, the function f(x) = (x, sin(1/x)) is not compact in the sense of the Arzela-Ascoli theorem, since it does not satisfy the condition of equicontinuity. This example illustrates the importance of the Arzela-Ascoli theorem in the study of functional analysis and differential equations.

References

Gewöhnliche Differentialgleichungen, by E. Zauderer
Functional Analysis, by Walter Rudin
Differential Equations, by Lawrence Perko

Glossary

Banach space: A complete normed vector space.
Compact: A set of functions is compact if it is closed, bounded, and equicontinuous.
Equicontinuous: A set of functions is equicontinuous if for every ε > 0, there exists a δ > 0 such that ||f(x) - f(y)|| ≤ ε whenever ||x - y|| ≤ δ.
Peano theorem: A result in functional analysis that deals with the existence and uniqueness of solutions to certain types of differential equations.
Q&A: General Peano-Theorem in Banach Spaces =============================================

Q: What is the Peano theorem?

A: The Peano theorem is a result in functional analysis that deals with the existence and uniqueness of solutions to certain types of differential equations. In its general form, the theorem states that if we have a differential equation of the form:

y'(t) = f(t, y(t))

where f is a continuous function from a Banach space X to a Banach space Y, and if we have an initial condition y(0) = y0, then there exists a unique solution to the differential equation.

Q: What is a Banach space?

A: A Banach space is a complete normed vector space. This means that it is a vector space that is equipped with a norm, and that every Cauchy sequence in the space converges to a point in the space.

Q: What is the Arzela-Ascoli theorem?

A: The Arzela-Ascoli theorem is a fundamental result in functional analysis that deals with the compactness of sets of functions. In its general form, the theorem states that a set of functions is compact if and only if it is closed, bounded, and equicontinuous.

Q: What are the conditions of the Arzela-Ascoli theorem?

A: The Arzela-Ascoli theorem has three main conditions that must be satisfied for a set of functions to be compact:

Closed: The set of functions must be closed, meaning that it contains all its limit points.
Bounded: The set of functions must be bounded, meaning that there exists a constant M such that ||f(x)|| ≤ M for all x in the domain.
Equicontinuous: The set of functions must be equicontinuous, meaning that for every ε > 0, there exists a δ > 0 such that ||f(x) - f(y)|| ≤ ε whenever ||x - y|| ≤ δ.

Q: What is an example of a function that is not compact in the sense of the Arzela-Ascoli theorem?

A: An example of a function that is not compact in the sense of the Arzela-Ascoli theorem is the function f(x) = (x, sin(1/x)) defined on the interval (0, 1]. This function is not compact because it does not satisfy the condition of equicontinuity.

Q: Why is the Peano theorem important?

A: The Peano theorem is important because it provides a way to prove the existence and uniqueness of solutions to certain types of differential equations. This is a fundamental result in functional analysis and has many applications in physics, engineering, and other fields.

Q: What are some common applications of the Peano theorem?

A: Some common applications of the Peano theorem include:

Physics: The Peano theorem is used to study the behavior of physical systems, such as the motion of particles and the behavior of waves.
Engineering: The Peano theorem is used to design and analyze systems, such as electronic circuits and mechanical systems.
Biology: The Peano theorem is used to study the behavior of biological systems, such as the growth and development of organisms.

Q: What are some common mistakes to avoid when using the Peano theorem?

A: Some common mistakes to avoid when using the Peano theorem include:

Not checking the conditions of the theorem: Make sure to check that the conditions of the theorem are satisfied before applying it.
Not using the correct norm: Make sure to use the correct norm when applying the theorem.
Not considering the domain of the function: Make sure to consider the domain of the function when applying the theorem.

Q: What are some resources for further reading on the Peano theorem and the Arzela-Ascoli theorem?

A: Some resources for further reading on the Peano theorem and the Arzela-Ascoli theorem include:

Peano's Theorem, by Wikipedia
Arzela-Ascoli Theorem, by Wikipedia
Functional Analysis, by MIT OpenCourseWare
Differential Equations, by Lawrence Perko
Gewöhnliche Differentialgleichungen, by E. Zauderer