the fixed-point formulation
The fixed-point formulation is the change of viewpoint that makes the existence theory work: instead of looking for a function whose derivative obeys a rule, you look for a function that equals a certain transformation of itself. Recasting 'solve the differential equation' as 'find a fixed point of an operator' is the pivot that lets the contraction-mapping principle take over and deliver existence, uniqueness, and a construction all at once.
The recasting is concrete. Start from y' = f(x, y), y(x0) = y0. Integrate both sides from x0 to x and use the initial condition to get rid of the constant: y(x) = y0 + the integral from x0 to x of f(t, y(t)) dt. A continuous function satisfies this integral equation if and only if it solves the original initial value problem — the two are equivalent, but the integral form is far friendlier, because integration smooths things out where differentiation makes demands. Now define an operator T that takes a candidate function y and returns the right-hand side, T(y)(x) = y0 + the integral from x0 to x of f(t, y(t)) dt. Solving the ODE is now exactly the same as finding a y with T(y) = y — a fixed point of T.
Why bother trading a differential equation for a fixed-point hunt? Because the fixed-point question has a powerful general answer. On a short enough interval the Lipschitz condition on f makes T a contraction, and then the contraction-mapping principle hands you a unique fixed point reached by iteration. The integral form also builds the initial condition right into the equation (no separate step) and tolerates merely continuous data, which is why it, not the raw derivative form, is the natural starting line for both Peano's and Picard-Lindelof's theorems.
The problem y' = x + y, y(0) = 0 becomes the fixed-point equation y(x) = the integral from 0 to x of (t + y(t)) dt. A function solves the ODE exactly when feeding it through this integral returns the same function — that is the fixed point the contraction principle then locates and Picard iteration approaches.
Integrate the ODE into y = T(y); solving the equation is now hunting a fixed point of the integral operator T.
The integral form is genuinely equivalent to the ODE plus its initial condition, not an approximation — and it is preferred precisely because integration is forgiving (it needs only continuity) where differentiation is not.