Solution of an Initial Value Problem on $[a,c]$

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 $a<c\le b$. By Basic Facts about Intervals of the Real Line (v) and Basic Facts about Intervals of the Real Line (ii), $[a,c]$ is an interval containing the two distinct points $a$ and $c$, and $[a,c]\subseteq[a,b]$. Differentiability and derivatives of maps on $[a,c]$ are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii), taken relative to $[a,c]$.

(i) Let $n\in\mathbb{N}$, let $F:[a,b]\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ and let $\xi\in\mathbb{R}^{n}$. 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 $\xi$ is a map $x:[a,c]\to\mathbb{R}^{n}$ that is differentiable and satisfies

$$x'(t)=F\bigl(t,x(t)\bigr)\quad\bigl(t\in[a,c]\bigr),\qquad x(a)=\xi ;$$

here $F(t,x(t))$ is defined because $t\in[a,b]$, and $a\in[a,c]$.

(ii) Let $p,q\in\mathbb{N}$, let $F:[a,b]\times\mathcal{M}_{p\times q}(\mathbb{R})\to\mathcal{M}_{p\times q}(\mathbb{R})$ and let $\Xi\in\mathcal{M}_{p\times q}(\mathbb{R})$. 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 $\Xi$ is a map $X:[a,c]\to\mathcal{M}_{p\times q}(\mathbb{R})$ that is differentiable and satisfies

$$X'(t)=F\bigl(t,X(t)\bigr)\quad\bigl(t\in[a,c]\bigr),\qquad X(a)=\Xi ;$$

here $F(t,X(t))$ is defined because $t\in[a,b]$, and $a\in[a,c]$.

(iii) A solution on $[a,b]$ of the initial value problem with right-hand side $F$ and initial value $\xi$ (respectively $\Xi$), in the sense of (i) (respectively (ii)) with $c=b$, is also called a solution of that problem.