How can Banach's Fixed Point Theorem be applied to prove the existence and uniqueness of solutions to differential equations with boundary conditions?

Dieudonne Ndong Ovono

Hi Hamid, you can check these link:

https://wiki.math.ntnu.no/_media/tma4145/2020h/banach.pdf

or this book

Book Applied Mathematics: Theories, Methods and Practices, From s...

part IV or page 341, etc... Another interresting reference is

Book Implementation of Advanced Finite Element Solution to Comple...

Another interresting literature on wellposedness, coercivity, etc is:

Alexander Ern, Jean-Luc Guermond, "Theory and Practices of Finite Elements".

This reference addressed similar issues.

To your question I canadvise the following step-by-step outline of how to proceed:

Formulate the Problem: Consider a differential equation with boundary conditions. For example, let’s take a boundary value problem (BVP) of the form:⎩⎨⎧u′′(x)=f(x,u(x)),u(a)=α,u(b)=β.x∈(a,b)

Transform into an Integral Equation: Convert the differential equation into an equivalent integral equation using the Green’s function or an appropriate integral operator. For instance, the above BVP can be transformed into:u(x)=α+b−ax−a(β−α)+∫abG(x,s)f(s,u(s))ds,where ( G(x, s) ) is the Green’s function associated with the boundary conditions.

Define the Operator: Define an operator ( T ) on a suitable function space (e.g., the space of continuous functions on ([a, b])) such that finding a fixed point of ( T ) corresponds to solving the integral equation. For the example above, the operator ( T ) can be defined as:(Tu)(x)=α+b−ax−a(β−α)+∫abG(x,s)f(s,u(s))ds.

Verify Contraction Mapping: Show that the operator ( T ) is a contraction mapping. This involves proving that there exists a constant ( 0 < k < 1 ) such that for any two functions ( u ) and ( v ),∥Tu−Tv∥≤k∥u−v∥.This step often requires conditions on the function ( f ), such as Lipschitz continuity.

Apply Banach’s Fixed Point Theorem: Since ( T ) is a contraction mapping on a complete metric space (e.g., the space of continuous functions with the supremum norm), Banach’s Fixed Point Theorem guarantees that ( T ) has a unique fixed point. This fixed point is the unique solution to the original boundary value problem.

Hope this will be of some help.

Truman Prevatt

Fixed point theorems have been a useful tool for proving existence of solutions in differential equations. Below is an example.

Article Application of exponential dichotomies to asymptotic integra...

This uses the Tychonoff which is more general than the Contraction principle which requires a Banach space.

The key process is to define a mapping that associates with a function y a solution x. This mapping is constructed as such that the fixed point is the solution one seeks.

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