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## What is real analysis?

In this article, we answer the question: what is real analysis about? Apparently the word real comes from the set of real numbers while analysis is what we study with…

## The optimal step gradient method

In this article, we propose an exercise that describes the optimal method of the step gradient for coercive functions on spaces of finite dimension. These specific problems are used in…

## Translation operator

We discuss the properties of the translation operator defined in a Lebesgue space. The latter appears in the study of differential equations, mainly equations with negative memories called also delay…

## Lyapunov stability for nonlinear systems

We discuss the Lyapunov stability for nonlinear systems. Indeed, having a reference solution, a stationary solution, one wonders if the ODE solution is closed to this reference when the time…

## Instability of solutions to nonlinear systems

In this post, we propose two results in the instability of solutions to nonlinear systems. Here, we study ordinary differential equations. Well-know stability theorems are due to Liapunov, based on…

## Open mapping theorem in functional analysis

In this article, we give an application of the open mapping theorem in functional analysis. This fundamental theorem in functional analysis plays a key role in the study of evolution…

## Differential calculus in Banach spaces

Differential calculus in Banach spaces is a very important part of mathematics. In fact, the treatment of partial differential equations strongly depends on this theory. Think about the heat equation….

## Heat equation using the Fourier series

In this article, we learn how to solve the heat equation using the Fourier series. The heat equation belongs to the class of partial differential equations widely used in physics….

## Gronwall lemma

In this article, we state and prove Gronwall lemma and give some of its applications. In fact, we will use this lemma for the stability of the solution to the…

## Stability analysis of differential equations

The stability analysis of solutions of differential equations is one of the most important axes in ODE. Here w gives a concise course on the stability of equilibrium (critical) points…

## Proof of peano existence theorem

We propose a nice proof of Peano existence theorem. This theorem shows that the continuity of the vector field suffices for the existence of solutions to the ODE; ordinary nonlinear…