Proof of the Picard–Lindelöf theorem in integral form

Builds on Picard–Lindelöf hypotheses Picard operator An iterate of the Picard operator is a contraction Banach fixed point theorem Completeness of the space of bounded continuous functions Finite-dimensional normed spaces are complete Continuous functions are Riemann integrable Composition of continuous functions is continuous Linearity of the integral Monotonicity of the integral Cauchy–Schwarz inequality

Proof

We find the curve as the unique fixed point of the Picard operator, with the Banach fixed point theorem.

If $t_{\min} = t_{\max}$ , then $J = \set{t_0}$ . The equation at $t_0$ says $\alpha(t_0) = x_0$ , so exactly one curve works. From now on let $t_{\min} < t_{\max}$ , and let $T = \max(t_{\max} - t_0, t_0 - t_{\min})$ .
$f$ is continuous on $J \times B$ . Indeed, $\norm{f(s, x) - f(s', x')} \le K \norm{x - x'} + \norm{f(s, x') - f(s', x')}$ , by condition (1) of the Picard–Lindelöf hypotheses. The last term tends to $0$ as $s \to s'$ , by condition (2).
Let $\pi(x) = x$ if $\norm{x - x_0} \le a$ , and $\pi(x) = x_0 + a (x - x_0)/\norm{x - x_0}$ otherwise. Then $\pi : \R^n \to B$ is continuous, since both formulas are continuous and agree where $\norm{x - x_0} = a$ . If $a = 0$ , then $\pi$ is the constant $x_0$ .
Let $\tilde f(t, x) = f(t, \pi(x))$ . By steps 2 and 3, $\tilde f$ is continuous on $J \times \R^n$ . It equals $f$ on $J \times B$ , so it satisfies the hypotheses with the same constants $a, L, K$ .
For each continuous $\gamma : J \to \R^n$ , the map $s \mapsto \tilde f(s, \gamma(s))$ is continuous, so it is integrable. So the Picard operator $P$ of $\tilde f$ at $(t_0, x_0)$ is defined.
Let $X$ be the set of continuous $\gamma : J \to B$ , with $d(\gamma, \delta) = \sup_{t \in J} \norm{\gamma(t) - \delta(t)}$ . It is nonempty, since it contains the constant curve $x_0$ .
$X$ is complete. The curves in $X$ are bounded, and the space of bounded continuous maps $J \to \R^n$ is complete, since $\R^n$ is complete. A Cauchy sequence $(\gamma_k)$ in $X$ converges uniformly to a continuous $\gamma$ .
For $t \in J$ , $\norm{\gamma(t) - x_0} \le \norm{\gamma(t) - \gamma_k(t)} + a$ , and the first term tends to $0$ . So $\gamma \in X$ , and $\gamma_k \to \gamma$ in $X$ .
Lemma. If $h : [c, e] \to \R^n$ and $\phi : [c, e] \to \R$ are integrable with $\norm{h} \le \phi$ , then $\norm{\int_c^e h} \le \int_c^e \phi$ .
Proof of the lemma: for $v = \int_c^e h \ne 0$ , linearity gives $\norm{v}^2 = \int_c^e v \cdot h$ . By Cauchy–Schwarz and monotonicity, this is at most $\norm{v} \int_c^e \phi$ .
$P$ maps $X$ into $X$ . Let $\gamma \in X$ . By condition (3), $\norm{\tilde f(s, \gamma(s))} \le L$ . The lemma gives $\norm{(P\gamma)(t) - (P\gamma)(t')} \le L \abs{t - t'}$ , so $P\gamma$ is continuous.
The lemma also gives $\norm{(P\gamma)(t) - x_0} \le L \abs{t - t_0} \le L T \le a$ , by condition (4). So $P\gamma \in X$ .
By the contraction lemma for the Picard operator, applied to $\tilde f$ , there is $m \ge 1$ such that $P^m$ is a contraction of $X$ .
By the Banach fixed point theorem, $P^m$ has exactly one fixed point $\alpha \in X$ .
Then $P^m(P\alpha) = P(P^m \alpha) = P\alpha$ , and $P\alpha \in X$ . So $P\alpha$ is a fixed point of $P^m$ , and $P\alpha = \alpha$ by uniqueness.
If $\beta \in X$ and $P\beta = \beta$ , then $P^m \beta = \beta$ , so $\beta = \alpha$ . Thus $\alpha$ is the only fixed point of $P$ in $X$ .
For $\gamma \in X$ , the values $\gamma(s)$ lie in $B$ , where $\tilde f = f$ . So $(P\gamma)(t) = x_0 + \int_{t_0}^{t} f(s, \gamma(s))\, ds$ .
Hence the continuous curves $J \to B$ that satisfy the integral equation are exactly the fixed points of $P$ in $X$ . By step 16, there is exactly one, namely $\alpha$ .

Idea

The integral equation says that $\alpha$ is a fixed point of the Picard operator. On continuous curves with values in $B$ , the operator itself may not contract, but a high iterate does. The Banach fixed point theorem then gives one fixed point of the iterate, and it is also the only fixed point of $P$ .

Reading path

To understand this page from the ground up, read these 310 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the diameter characterisation of Cauchy sequences, Proof of the completeness of the space of bounded continuous functions, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of Riemann's criterion for integrability, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that limits preserve non-strict inequalities, Proof that closed sets contain their limits, Proof of the linearity of the integral, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that an iterate of the Picard operator is a contraction, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof that a composition of continuous functions is continuous, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that open balls are open, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that a set is open if and only if its complement is closed, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable.

Basis concept
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Differential equation concept
Element definition
Euclidean metric example
Extended real number line object
Function concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Interval concept
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Mathematical induction concept
Natural numbers object
Norm concept
Norm induced by an inner product concept
Ordered field definition
Partition of an interval definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Uniform metric on bounded functions example
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Ordinary differential equation definition
Lipschitz condition definition
Ring of integers statement
Limit of a function definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Complex conjugate definition
Empty set object
Family of sets definition
Ordered pair definition
Continuity at a point definition
Intersection definition
Union definition
Riemann integrable function definition
Solution of an ordinary differential equation definition
Addition of natural numbers definition
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Proof of the Archimedean property in an Archimedean ordered field proof
The successor function is injective statement
Zero is not a successor statement
The rationals form a field statement
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Inner product space definition
Linear independence definition
Linear isomorphism definition
Linear span definition
Completeness axiom statement
Cartesian product definition
Proof of the equality of complex numbers proof
Continuous function definition
Initial value problem definition
Riemann integral definition
Union of a family of sets definition
Subset definition
Archimedean property statement
Proof of the commutativity of addition by induction proof
Mathematical proof concept
Uniqueness quantifier definition
Propositional logic concept
Order on the natural numbers definition
Proof of the sum of a finite geometric series proof
The rationals form an ordered field statement
Convergent sequence definition
Sequential continuity definition
Upper and lower Darboux sums definition
Norm induced by an inner product definition
Spanning set definition
Equality of complex numbers statement
Binary relation definition
Picard–Lindelöf hypotheses definition
Proof that a composition of continuous functions is continuous proof
Universal set definition
Definite integral definition
Topological space definition
Order on the reals definition
Tautology definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Sum of a finite geometric series statement
Upper and lower Darboux integrals definition
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Picard operator definition
Function definition
Subspace topology definition
Complement definition
Sequential compactness definition
Interval definition
Logical equivalence definition
Coordinate vector definition
Strong induction statement
Preorder definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Choice function definition
Image of a set definition
Preimage of a set definition
Proof of the epsilon-delta form of continuity proof
Proof of the well-ordering principle from strong induction proof
Metric space definition
Proof of the double negation law by a truth table proof
Partial order definition
Left inverse definition
Right inverse definition
Surjective function definition
Axiom of choice statement
Epsilon-delta definition of continuity statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Double negation law statement
Diameter of a set definition
Well-ordering principle statement
Maximal element definition
Total order definition
Upper bound definition
Composition of continuous functions is continuous statement
Bijective function definition
Uniform continuity definition
Bounded set definition
Metric topology definition
Contraction mapping definition
Proof by contradiction concept
Proof of the diameter characterisation of Cauchy sequences proof
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that a function is bijective iff it has a two-sided inverse proof
Uniform continuity on a set definition
Open set definition
Epsilon characterisation of Cauchy sequences statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Open cover definition
Hausdorff space definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Closed set definition
Continuous map of topological spaces definition
Zorn's lemma statement
Equivalent metrics definition
Complete metric space definition
Proof of the description of the open sets of a metric space proof
The reals form an ordered field statement
Compact space definition
Open sets characterised by neighbourhoods statement
Proof that a set is open if and only if its complement is closed proof
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Compact set definition
Complement of an open set is closed statement
Continuity at a point of a topological space definition
Banach space definition
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Closed sets contain their limits statement
Proof of continuity via closed sets proof
Triangle inequality statement
Absolute value is multiplicative statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof of the limit of a sum of sequences proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the linearity of the integral proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of continuity via open sets in metric spaces proof
Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof that metric spaces are Hausdorff proof
Proof that limits preserve non-strict inequalities proof
Proof of the completeness of the space of bounded continuous functions proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Linearity of the integral statement
Proof that compactness and sequential compactness agree in metric spaces proof
Metric spaces are Hausdorff statement
Proof of the monotonicity of the integral proof
Limit of a sum of sequences statement
Proof of the triangle inequality for norms proof
Uniqueness of limits statement
Proof that Darboux and Riemann integrals agree proof
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Completeness of the space of bounded continuous functions statement
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
The sequence r^n tends to zero for |r| < 1 example
Proof of the Banach fixed point theorem by iteration proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Proof of Riemann's criterion for integrability proof
Decreasing sequence bounded below converges statement
Banach fixed point theorem statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Cauchy–Schwarz inequality statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof that an iterate of the Picard operator is a contraction proof
Bolzano–Weierstrass theorem statement
An iterate of the Picard operator is a contraction statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Completeness of the real numbers statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Boundedness theorem statement
Heine–Borel theorem statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Continuous real function on a compact set attains its bounds statement
Proof that continuous functions are Riemann integrable proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Continuous functions are Riemann integrable statement
Equivalence of norms in finite dimension statement
Proof that finite-dimensional normed spaces are complete proof
Finite-dimensional normed spaces are complete statement
Proof of the Picard–Lindelöf theorem in integral form proof

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Picard–Lindelöf theorem in integral form

Tag	`WR4W9EK4JV`
URI	`m2://WR4W9EK4JV`
Kind	proof
Writing	W0 stub
MSC	34A12
Proves	Picard–Lindelöf theorem in integral form
Uses	Picard–Lindelöf hypotheses, Picard operator, An iterate of the Picard operator is a contraction, Banach fixed point theorem, Completeness of the space of bounded continuous functions, Finite-dimensional normed spaces are complete, Continuous functions are Riemann integrable, Composition of continuous functions is continuous, Linearity of the integral, Monotonicity of the integral, Cauchy–Schwarz inequality
Changed	4 November 2026

Proof of the Picard–Lindelöf theorem in integral form

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