Solutions of an Initial Value Problem Are the Continuous Solutions of the Integral Equation

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b,c\in\mathbb{R}$ with $a<c\le b$, and let $n,p,q\in\mathbb{N}$. $\mathbb{R}\times\mathbb{R}^{n}$ carries the product metric of $d_{\mathbb{R}}$ and $d_{E}$, and $\mathbb{R}\times\mathcal{M}_{p\times q}(\mathbb{R})$ the product metric of $d_{\mathbb{R}}$ and $d_{p\times q}$, metrics by The Product Metric is a Metric (i). Continuity of maps on $[a,c]$ refers to $d_{\mathbb{R}}$ on $[a,c]$ and to $d_{E}$ on $\mathbb{R}^{n}$ and $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, as in Continuous Map Between Metric Spaces (ii). Integrals of maps $[a,c]\to\mathbb{R}^{n}$ and $[a,c]\to\mathcal{M}_{p\times q}(\mathbb{R})$, and their being defined, are as in Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv).

(i) Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be continuous on $[a,b]\times\mathbb{R}^{n}$, for the product metric of $d_{\mathbb{R}}$ and $d_{E}$ on $\mathbb{R}\times\mathbb{R}^{n}$ and for $d_{E}$ on $\mathbb{R}^{n}$, and let $x:[a,c]\to\mathbb{R}^{n}$ be continuous on $[a,c]$. Then the map $[a,c]\to\mathbb{R}^{n}$, $s\mapsto F(s,x(s))$, is continuous on $[a,c]$. The same holds for an $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ continuous on $[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})$, for the product metric of $d_{\mathbb{R}}$ and $d_{p\times q}$ on $\mathbb{R}\times\mathcal{M}_{p\times q}(\mathbb{R})$ and for $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, and a map $X:[a,c]\to\mathcal{M}_{p\times q}(\mathbb{R})$ continuous on $[a,c]$ in place of $F$ and $x$.

(ii) Let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ be continuous on $[a,b]\times\mathbb{R}^{n}$, for the product metric of $d_{\mathbb{R}}$ and $d_{E}$ on $\mathbb{R}\times\mathbb{R}^{n}$ and for $d_{E}$ on $\mathbb{R}^{n}$, let $\xi\in\mathbb{R}^{n}$ and let $x:[a,c]\to\mathbb{R}^{n}$. Then $x$ is a solution on $[a,c]$ of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (i) with right-hand side $F$ and initial value $\xi$ if and only if $x$ is continuous on $[a,c]$ and, for every $t\in[a,c]$, the integral $\int_{a}^{t}F(s,x(s))\,ds$ is defined and

$$x(t)=\xi+\int_{a}^{t}F\bigl(s,x(s)\bigr)\,ds .$$

(iii) Let $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ be continuous on $[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})$, for the product metric of $d_{\mathbb{R}}$ and $d_{p\times q}$ on $\mathbb{R}\times\mathcal{M}_{p\times q}(\mathbb{R})$ and for $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, let $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$ and let $X:[a,c]\to\mathcal{M}_{p\times q}(\mathbb{R})$. Then $X$ is a solution on $[a,c]$ of the initial value problem of The Initial Value Problem for a First-Order System of Ordinary Differential Equations (ii) with right-hand side $F$ and initial value $\Xi$ if and only if $X$ is continuous on $[a,c]$ and, for every $t\in[a,c]$, the integral $\int_{a}^{t}F(s,X(s))\,ds$ is defined and

$$X(t)=\Xi+\int_{a}^{t}F\bigl(s,X(s)\bigr)\,ds .$$