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. By Extreme Value Theorem on a Compact Interval the continuous entries of $A$ are bounded: $|A_{ij}(t)|\le\alpha$ for all $i,j,t$. Throughout, matrix products are manipulated with Associativity of the Matrix Product, and sums and products of continuous functions of $t$ are continuous by Sum and Product Rules for One-Dimensional Derivatives and Continuity.

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 Fundamental Theorem of Calculus, Part I in One Dimension, each entry of $\Phi$ and of $\Psi$ is differentiable at every point of $(a,b)$ with $\Phi'=A\Phi$ and $\Psi'=-\Psi A$ (entrywise; the integrands are continuous). 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 $M_{il}$: $M_{il}$ is continuous on $[a,t]$ and differentiable on $(a,t)$ with vanishing derivative, so Mean Value Theorem in One Dimension 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 interior points, differentiable with derivative the corresponding entry of $AN-NA$, which is continuous on $[a,b]$; thus each entry of $N$ is an antiderivative of the corresponding entry of $AN-NA$ on $[a,b]$ (that definition requires the derivative only at interior points), and $N$ is continuous on the closed interval. Applying Fundamental Theorem of Calculus, Part II in One Dimension 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 interior points with $y'=\Psi g$ by Fundamental Theorem of Calculus, Part I in One Dimension. 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 $[a,b]$, so each component of $x$ is an antiderivative of the corresponding component of $Ax+g$ in the sense of Antiderivative on an Interval, and Fundamental Theorem of Calculus, Part II in One Dimension, applied on $[a,t]$ for each $t$, 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. $\blacksquare$