Picard iteration is both a proof of local well-posedness and a constructive successive-approximation scheme. The contraction is obtained by controlling the state range and shortening the time interval.
The Picard operator
Assume on \[ R=[t_0-a,t_0+a]\times\overline{B(x_0,b)} \] that \[ \|f(t,x)\|\le M,\qquad \|f(t,x)-f(t,y)\|\le L\|x-y\|. \] Let \(h\le\min(a,b/M)\) and define \[ (Tx)(t)=x_0+\int_{t_0}^tf(s,x(s))\,\mathrm{d}s. \]
Theorem. If \(Lh<1\), \(T\) is a contraction on \[ X=\{x\in C([t_0-h,t_0+h]):\|x-x_0\|_\infty\le b\}. \] It has a unique fixed point, and Picard iterates \(x^{(n+1)}=Tx^{(n)}\) converge uniformly to it.
Proof. First, \[ \|Tx(t)-x_0\| \le M|t-t_0|\le Mh\le b, \] so \(T(X)\subset X\). Also, \[ \|Tx-Ty\|_\infty \le Lh\|x-y\|_\infty. \] Banach's fixed-point theorem gives a unique fixed point and geometric convergence.
Factorial error bounds
Proposition. Starting with \(x^{(0)}(t)=x_0\), \[ \|x^{(n+1)}(t)-x^{(n)}(t)\| \le \frac{M L^n|t-t_0|^{n+1}}{(n+1)!}. \] Consequently, the Picard sequence converges whenever the iterates stay in the rectangle, even without imposing \(Lh<1\) for the final interval.
Proof. The first difference is bounded by \(M|t-t_0|\). If the claim holds at \(n-1\), then \[ \|x^{(n+1)}(t)-x^{(n)}(t)\| \le L\int_{t_0}^t \frac{ML^{n-1}|s-t_0|^n}{n!}\,\mathrm{d}s, \] which gives the next bound. The exponential series makes the differences summable uniformly.
Uniqueness without a short interval
Proof. If \(x,y\) are two solutions with the same initial value, repeated substitution gives \[ \|x(t)-y(t)\| \le\frac{L^n|t-t_0|^n}{n!} \|x-y\|_\infty. \] For fixed \(t\), the factor tends to zero as \(n\to\infty\), proving equality. Equivalently, Gronwall gives the result immediately.
Weighted norms
On a longer interval \([t_0,t_0+T]\), use the Bielecki norm \[ \|x\|_\alpha =\sup_{t_0\le t\le t_0+T} e^{-\alpha(t-t_0)}\|x(t)\|. \] Then \[ \|Tx-Ty\|_\alpha \le\frac L\alpha\|x-y\|_\alpha. \] Choosing \(\alpha>L\) makes \(T\) contractive without shortening \(T\), provided the operator maps the chosen complete set into itself.
Picard series for linear systems
Example. For \(\dot x=Ax\), \(x(0)=x_0\), Picard iteration yields \[ x^{(n)}(t) =\sum_{k=0}^n\frac{t^kA^k}{k!}x_0. \] The limit is \[ x(t)=e^{tA}x_0, \qquad e^{tA}=\sum_{k\ge0}\frac{t^kA^k}{k!}. \] Thus the matrix exponential emerges directly from successive substitution.
Beyond Lipschitz contraction
One-sided Lipschitz conditions can give uniqueness even when ordinary Lipschitz bounds are poor. Osgood's criterion replaces \(L\|x-y\|\) by a modulus \(\omega(\|x-y\|)\) with \(\int_{0+}\mathrm{d}s/\omega(s)=\infty\). Compact fixed-point methods give existence under continuity but cannot by themselves supply uniqueness.