The Gronwall inequality
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915218906415104 |
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| author | Howard, Ralph |
| author_facet | Howard, Ralph |
| contents | We prove the following version generalization of the Gronwall inequality:
Let $\mathbf X$ be a Banach space and $U\subset \mathbf X$ an open convex set in $\mathbf X$. Let $f,g\colon [a,b]\times U\to \mathbf X$ be continuous functions and let $y,z\colon [a,b]\to U$ satisfy the initial value problems \begin{align*} y'(t)&=f(t,y(t)),\quad y(a)=y_0,\\ z'(t)&=g(t,z(t)),\quad z(a)=z_0. \end{align*} Also assume there is a constant $C\ge 0$ so that $$ \|g(t,x_2)-g(t,x_1)\|\le C\|x_2-x_1\| $$ and a continuous function $ϕ\colon [a,b]\to [0,\infty)$ so that $$ \|f(t,y(t))-g(t,y(t))\|\le ϕ(t). $$ Then for $t\in [a,b]$ $$ \|y(t)-z(t)\|
\le e^{C|t-a|}\|y_0-z_0\|+e^{C|t-a|}\int_a^te^{-C|s-a|}ϕ(s)\,ds.
$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Gronwall inequality Howard, Ralph Functional Analysis Classical Analysis and ODEs 34A12, 34A34 We prove the following version generalization of the Gronwall inequality: Let $\mathbf X$ be a Banach space and $U\subset \mathbf X$ an open convex set in $\mathbf X$. Let $f,g\colon [a,b]\times U\to \mathbf X$ be continuous functions and let $y,z\colon [a,b]\to U$ satisfy the initial value problems \begin{align*} y'(t)&=f(t,y(t)),\quad y(a)=y_0,\\ z'(t)&=g(t,z(t)),\quad z(a)=z_0. \end{align*} Also assume there is a constant $C\ge 0$ so that $$ \|g(t,x_2)-g(t,x_1)\|\le C\|x_2-x_1\| $$ and a continuous function $ϕ\colon [a,b]\to [0,\infty)$ so that $$ \|f(t,y(t))-g(t,y(t))\|\le ϕ(t). $$ Then for $t\in [a,b]$ $$ \|y(t)-z(t)\| \le e^{C|t-a|}\|y_0-z_0\|+e^{C|t-a|}\int_a^te^{-C|s-a|}ϕ(s)\,ds. $$ |
| title | The Gronwall inequality |
| topic | Functional Analysis Classical Analysis and ODEs 34A12, 34A34 |
| url | https://arxiv.org/abs/2503.23639 |