Multiple Fixed Point Theorems and Applications in the Theory of ODEs, FDEs, and PDEs

Fixed point theorems are fundamental tools in mathematical analysis. They provide conditions under which a function has a fixed point, that is, a point that is mapped to itself. Multiple fixed point theorems extend these results to the case where a function has multiple fixed points.

In this article, we will discuss three important multiple fixed point theorems: Banach's fixed point theorem, Schauder's fixed point theorem, and the Leray-Schauder fixed point theorem. We will also discuss applications of these theorems in the theory of ordinary differential equations (ODEs),functional differential equations (FDEs),and partial differential equations (PDEs).

Banach's fixed point theorem is one of the most basic and important fixed point theorems. It states that if a function $f$ is a contraction on a complete metric space $X$, then $f$ has a unique fixed point.

Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

by Lilian Darcy

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More formally, Banach's fixed point theorem can be stated as follows:

Theorem 1 (Banach's Fixed Point Theorem). Let $X$ be a complete metric space and let $f: X \to X$ be a contraction. Then $f$ has a unique fixed point $x^* \in X$.

Banach's fixed point theorem has many applications in analysis, including in the theory of ODEs, FDEs, and PDEs. For example, Banach's fixed point theorem can be used to prove the existence and uniqueness of solutions to initial value problems for ODEs and FDEs.

Schauder's fixed point theorem is a generalization of Banach's fixed point theorem to the case of compact metric spaces. It states that if a function $f$ is continuous on a compact metric space $X$, then $f$ has a fixed point.

More formally, Schauder's fixed point theorem can be stated as follows:

Theorem 2 (Schauder's Fixed Point Theorem). Let $X$ be a compact metric space and let $f: X \to X$ be continuous. Then $f$ has a fixed point $x^* \in X$.

Schauder's fixed point theorem has many applications in analysis, including in the theory of ODEs, FDEs, and PDEs. For example, Schauder's fixed point theorem can be used to prove the existence of solutions to boundary value problems for ODEs and FDEs.

The Leray-Schauder fixed point theorem is a generalization of Schauder's fixed point theorem to the case of non-compact metric spaces. It states that if a function $f$ is continuous and compact on a closed convex subset of a Banach space, then $f$ has a fixed point.

More formally, the Leray-Schauder fixed point theorem can be stated as follows:

Theorem 3 (Leray-Schauder Fixed Point Theorem). Let $X$ be a closed convex subset of a Banach space $E$ and let $f: X \to X$ be continuous and compact. Then $f$ has a fixed point $x^* \in X$.

The Leray-Schauder fixed point theorem has many applications in analysis, including in the theory of ODEs, FDEs, and PDEs. For example, the Leray-Schauder fixed point theorem can be used to prove the existence of solutions to nonlinear boundary value problems for PDEs.

Multiple fixed point theorems have many applications in the theory of ODEs. For example, Banach's fixed point theorem can be used to prove the existence and uniqueness of solutions to initial value problems for ODEs. Schauder's fixed point theorem can be used to prove the existence of solutions to boundary value problems for ODEs. The Leray-Schauder fixed point theorem can be used to prove the existence of solutions to nonlinear boundary value problems for ODEs.

Here is an example of how Banach's fixed point theorem can be used to prove the existence and uniqueness of solutions to initial value problems for ODEs:

Theorem 4. Consider the following initial value problem for an ODE:

$$y' = f(t, y),\quad y(t_0) = y_0.$$

Assume that $f$ is continuous on a rectangle $R = [t_0, t_0 + a] \times [b_1, b_2]$ and that there exists a constant $L$ such that

$$|f(t, y) - f(t, z)| \le L|y - z|$$

for all $(t, y),(t, z) \in R$. Then there exists a unique solution $y(t)$ to the initial value problem on the interval $[t_0, t_0 + a]$.

Proof. Let $X$ be the space of continuous functions on $[t_0, t_0 + a]$ with the norm

$$|y| = \max_{t \in [t_0, t_0 + a]}|y(t)|.$$

Define a function $F: X \to X$ by

$$F(y)(t) = y_0 + \int_{t_0}^t f(s, y(s)) ds.$$

Then $F$ is a contraction on $X$ with contraction constant $L$. Therefore, by Banach's fixed point theorem, $F$ has a unique fixed point $y^* \in X$. This fixed point $y^*$ is the unique solution to the initial value problem on the interval $[t_0, t_0 + a]$.

Multiple fixed point theorems also have many applications in the theory of FDEs. For example, Schauder's fixed point theorem can be used to prove the existence of solutions to boundary value problems for FDEs. The Leray-Schauder fixed point theorem can be used to prove the existence of solutions to nonlinear boundary value problems for FDEs.

Here is an example of how Schauder's fixed point theorem can be used to prove the existence of solutions to boundary value problems for FDEs:

Theorem 5. Consider the following boundary value problem for an FDE:

$$y

Multiple Fixed-Point Theorems and Applications in the Theory of ODEs, FDEs and PDEs (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

by Lilian Darcy

4.5 out of 5

Language	:	English
File size	:	4247 KB
Screen Reader	:	Supported
Print length	:	288 pages

FREE