Proof of Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations

Conventions. Identify real $k\times k$ matrices with points of $\mathbb{R}^{k^{2}}$ by listing entries in a fixed order, so that Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form applies with $k^{2}$ unknowns; the Euclidean norm of a matrix $X$, written $|X|$, its largest absolute entry $|X|_{e}$, the inequalities $|X_{ij}|\le|X|\le k^{2}|X|_{e}$, and the product entry bound $|UV|_{e}\le k\,|U|_{e}|V|_{e}$ are those of claims 1 and 2 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals. The interval $[a,b]$ is nonempty, since $a<b$, and compact by Closed Interval $[a,b]$ is Compact in $\mathbb{R}$, so by Extreme Value Theorem on a Compact Subset of a Metric Space every real-valued continuous function on $[a,b]$ attains a maximum and a minimum and is in particular bounded; fix a real $\alpha\ge0$ with $|A_{ij}(t)|\le\alpha$ for all $i,j$ and all $t\in[a,b]$. Throughout, matrix products are manipulated with Associativity of the Matrix Product, and sums and products of continuous real-valued functions of $t$ are continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space.

Part 1. Consider $F(t,X)=A(t)X$ on $[a,b]\times\mathbb{R}^{k^{2}}$. Composition continuity: entries of $A(t)h(t)$ are finite sums of products of continuous functions. Lipschitz: each entry of $A(t)(X-Y)$ is bounded by $k\alpha\,|X-Y|_{e}\le k\alpha\,|X-Y|$, so $|A(t)X-A(t)Y|\le k^{2}\cdot k\alpha\,|X-Y|$; take $L=k^{3}\alpha$. By Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form (initial value $I_k$) there is exactly one continuous $\Phi$ with $\Phi(t)=I_k+\int_a^tA(r)\Phi(r)\,dr$.

Part 2. Similarly, $F(t,X)=-XA(t)$ satisfies the hypotheses of Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form, giving a unique continuous $\Psi$ with $\Psi(t)=I_k-\int_a^t\Psi(r)A(r)\,dr$.

By claim 3 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval, applied entrywise (the integrands are continuous), each entry of $\Phi$ and of $\Psi$ is differentiable at every point of $(a,b)$ with $\Phi'=A\Phi$ and $\Psi'=-\Psi A$. Let $M=\Psi\Phi$; its entries are continuous on $[a,b]$ (sums of products of continuous functions), and on $(a,b)$, by the sum and product rules of Sum and Product Rules for One-Dimensional Derivatives and Continuity, entrywise,

Fix $t\in(a,b]$ and an entry $M_{il}$: by Restriction Stability of Continuity and of the Derivative its restriction to $[a,t]$ is continuous on $[a,t]$ and differentiable at every point of $(a,t)\subseteq(a,b)$ with vanishing derivative, so A Continuous Function with Vanishing Derivative is Constant applied on $[a,t]$ gives $M_{il}(t)-M_{il}(a)=0$. Hence $M\equiv M(a)=I_kI_k=I_k$: $\Psi(t)\Phi(t)=I_k$ for all $t$.

Let $N=\Phi\Psi$, with continuous entries on $[a,b]$; on $(a,b)$, $N'=A\Phi\Psi-\Phi\Psi A=A N-NA$, and $N(a)=I_k$. Each entry of $N$ is continuous on $[a,b]$ and, at every point of $(a,b)$, differentiable with derivative the corresponding entry of $AN-NA$, which is continuous on $[a,b]$ and hence, its restriction to $[a,t]$ being continuous there by claim 1 of Restriction Stability of Continuity and of the Derivative, Riemann integrable on $[a,t]$ for every $t\in(a,b]$ by claim 3 of the integral toolkit on a compact interval; by the same lemma the continuity and the differentiability of the entries of $N$ likewise pass to the restrictions to $[a,t]$. Applying Fundamental Theorem of Calculus, Part II, on a Closed Real Interval on $[a,t]$ for each $t\in(a,b]$ (degenerate $t=a$ by the convention of Mean-Square Riemann Integral of a Family of Random Variables) yields the integral form $N(t)=I_k+\int_a^t(A(r)N(r)-N(r)A(r))\,dr$. The constant assignment $N_0(t)=I_k$ satisfies the same equation, because $A(r)I_k-I_kA(r)=0$. The map $F(t,X)=A(t)X-XA(t)$ is composition continuous and Lipschitz (constant $2k^{3}\alpha$ by the entry bounds above), so the uniqueness in Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form forces $N\equiv I_k$: $\Phi(t)\Psi(t)=I_k$.

Hence each $\Phi(t)$ is invertible with inverse $\Psi(t)$, the inverse being unique by Uniqueness of the Matrix Inverse; and $\Psi$ is the unique continuous solution of its equation by Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form.

Part 3. Set $y(t)=\int_a^t\Psi(r)g(r)\,dr$; its components are continuous on all of $[a,b]$ by claim 4 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals (the integrand is continuous), and differentiable at every point of $(a,b)$ with $y'=\Psi g$ by claim 3 of Fundamental Theorem of Calculus, Part I, on a Closed Real Interval, applied componentwise. Then $x=\Phi\xi+\Phi y$ has continuous components on $[a,b]$, $x(a)=\xi$, and on $(a,b)$, by the sum and product rules,

using $\Phi\Psi=I_k$ and associativity. The components of $Ax+g$ are continuous on $[a,b]$, hence continuous on $[a,t]$ for every $t\in(a,b]$ by claim 1 of Restriction Stability of Continuity and of the Derivative and Riemann integrable there by claim 3 of the integral toolkit on a compact interval; each component of $x$ is continuous on $[a,b]$ and differentiable at every point of $(a,b)$ with derivative the corresponding component of $Ax+g$, and by Restriction Stability of Continuity and of the Derivative these properties pass to the restrictions to $[a,t]$. Hence Fundamental Theorem of Calculus, Part II, on a Closed Real Interval, applied on $[a,t]$ for each $t\in(a,b]$ (degenerate $t=a$ by the convention of Mean-Square Riemann Integral of a Family of Random Variables), gives $x(t)=x(a)+\int_a^t(A(r)x(r)+g(r))\,dr=\xi+\int_a^t(A(r)x(r)+g(r))\,dr$.

Uniqueness: $F(t,v)=A(t)v+g(t)$ on $[a,b]\times\mathbb{R}^{k}$ is composition continuous and Lipschitz in $v$ (constant $k^{2}\alpha$: each component of $A(t)(v-w)$ is bounded by $k\alpha\max_l|v^{l}-w^{l}|\le k\alpha\,d(v,w)$, and claim 1 of Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals converts to the Euclidean bound), so Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form admits at most one continuous solution; $x$ is one. $\blacksquare$