Proof of continuous dependence on initial data by Grönwall's inequality

Builds on Solution of an ordinary differential equation Lipschitz condition Second fundamental theorem of calculus Linearity of the integral Cauchy–Schwarz inequality Monotonicity of the integral Triangle inequality for norms Grönwall's inequality

Proof

We write the solutions as integrals and apply the integral form of Grönwall's inequality. Here $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^n$ .

As solutions on $[a, b]$ , the curves $\gamma_1, \gamma_2$ are differentiable, hence continuous, on $[a, b]$ . For $t \in (a, b)$ they satisfy $\gamma_i'(t) = f(t, \gamma_i(t))$ . Only these facts are used.
For $i = 1, 2$ let $g_i(s) = f(s, \gamma_i(s))$ for $s \in [a, b]$ . As a composition of continuous maps, $g_i$ is continuous on $[a, b]$ .
Fix $t \in (a, b]$ . Apply the second fundamental theorem of calculus to each coordinate of $\gamma_i$ on $[a, t]$ . The derivative extends to the continuous function $g_i$ , so $\gamma_i(t) = \gamma_i(a) + \int_a^t g_i(s)\, ds$ , with coordinatewise integrals.
Let $e(t) = \gamma_1(t) - \gamma_2(t)$ . By step 3 and linearity of the integral, $e(t) = e(a) + \int_a^t \left( g_1(s) - g_2(s) \right) ds$ .
Let $v : [a, t] \to \mathbb{R}^n$ be continuous and let $w = \int_a^t v(s)\, ds$ . We show $\|w\| \le \int_a^t \|v(s)\|\, ds$ .
By linearity, $\|w\|^2 = \sum_{j} w_j \int_a^t v_j(s)\, ds = \int_a^t w \cdot v(s)\, ds$ .
By the Cauchy–Schwarz inequality, $w \cdot v(s) \le \|w\|\, \|v(s)\|$ . By monotonicity of the integral, $\|w\|^2 \le \|w\| \int_a^t \|v(s)\|\, ds$ .
If $w = 0$ , then $\|w\| = 0$ , and the integral of the non-negative function $\|v\|$ is non-negative. Otherwise divide by $\|w\| > 0$ .
By the Lipschitz condition in the hypothesis, $\|g_1(s) - g_2(s)\| \le K \|e(s)\|$ for $s \in [a, b]$ .
Let $u(s) = \|e(s)\|$ , a continuous function on $[a, b]$ . By step 4, the triangle inequality, step 5 and step 9, $u(t) \le u(a) + \int_a^t K\, u(s)\, ds$ .
This also holds for $t = a$ , where the integral is $0$ . So it holds for each $t \in [a, b]$ .
Apply the integral form of Grönwall's inequality on $I = [a, b]$ with $\beta = K \ge 0$ and the constant, hence non-decreasing, function $\alpha = u(a)$ .
It gives $u(t) \le u(a) \exp\left( \int_a^t K\, ds \right) = u(a)\, e^{K(t - a)}$ . This is $\|\gamma_1(t) - \gamma_2(t)\| \le \|\gamma_1(a) - \gamma_2(a)\|\, e^{K(t - a)}$ for each $t \in [a, b]$ .

Idea

The difference of two solutions changes at a rate bounded by $K$ times its own size, because of the Lipschitz condition. In integral form this is exactly the hypothesis of Grönwall's inequality, which bounds the size by exponential growth with rate $K$ .

Reading path

To understand this page from the ground up, read these 302 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Cauchy–Hadamard theorem, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that a function with nonnegative derivative is increasing, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that the exponential function is positive, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, 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 of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof of the comparison test, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of Grönwall's inequality, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Binomial coefficient definition
Complex numbers object
Complex numbers as pairs of reals concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Differential equation concept
Element definition
Factorial definition
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Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Interval concept
Limit point of a set definition
Local maximum definition
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
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Set definition
Ordinary differential equation definition
Lipschitz condition definition
Ring definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field definition
Mesh of a partition definition
Refinement of a partition definition
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Implication definition
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Addition of natural numbers definition
Riemann integrable function definition
Biconditional definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
Truth table concept
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Zero is not a successor statement
Universal quantifier definition
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Cauchy sequence definition
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Proof of the factorial formula for binomial coefficients by induction proof
Proof of the equality of complex numbers proof
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Archimedean property statement
Proof of the commutativity of addition by induction proof
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Mathematical proof concept
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The rationals form an ordered field statement
Limit superior definition
Proof of the sum of a finite geometric series proof
Upper and lower Darboux sums definition
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The sequence r^n tends to infinity for r > 1 example
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Equality of complex numbers statement
Factorial formula for binomial coefficients statement
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Proof that a composition of continuous functions is continuous proof
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Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Upper and lower Darboux integrals definition
Sum of a finite geometric series statement
Uniform convergence definition
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Convergent series definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
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Interval definition
Logical equivalence definition
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Strong induction statement
Sum of a series definition
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Absolute convergence definition
Binomial theorem statement
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Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Double negation law statement
Well-ordering principle statement
Total order definition
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Composition of continuous functions is continuous statement
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Modulus of a complex number definition
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof of the linearity of the integral proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
The sequence 1/n tends to zero example
Proof that Cauchy sequences are bounded proof
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Proof that a complex number times its conjugate is the square of its modulus proof
Differentiable functions are continuous statement
Linearity of the integral statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the triangle inequality for norms proof
Proof of the monotonicity of the integral proof
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Squeeze theorem statement
Proof of the ratio formula for the radius of convergence proof
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
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
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Triangle inequality for norms statement
The sequence r^n tends to zero for |r| < 1 example
Proof that a decreasing sequence bounded below converges proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Limit of a product of sequences statement
Geometric series example
Decreasing sequence bounded below converges statement
Proof of Riemann's criterion for integrability proof
A series of nonnegative terms converges iff its partial sums are bounded statement
Continuity is equivalent to sequential continuity statement
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the ratio test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Proof of the additivity of the integral over intervals proof
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Additivity of the integral over intervals statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Proof of the power rule by induction proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the first fundamental theorem of calculus proof
Boundedness theorem statement
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Proof of the formula for the derivative of a polynomial proof
Extreme value theorem statement
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with nonnegative derivative is increasing proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof of the second fundamental theorem of calculus proof
A function with nonnegative derivative is increasing statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Second fundamental theorem of calculus statement
Proof that the exponential series has infinite radius of convergence proof
Proof of term-by-term differentiation proof
The exponential series has infinite radius of convergence statement
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is positive proof
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
The exponential function is positive statement
The exponential function is its own derivative statement
Proof of Grönwall's inequality proof
Grönwall's inequality statement
Proof of continuous dependence on initial data by Grönwall's inequality proof

Where this leads

58 pages build on this one.

Tag	`RRBCGTD2SB`
URI	`m2://RRBCGTD2SB`
Kind	proof
Writing	W0 stub
MSC	34A12
Proves	Continuous dependence on initial data
Uses	Solution of an ordinary differential equation, Lipschitz condition, Second fundamental theorem of calculus, Linearity of the integral, Cauchy–Schwarz inequality, Monotonicity of the integral, Triangle inequality for norms, Grönwall's inequality
Changed	4 November 2026

Proof of continuous dependence on initial data by Grönwall's inequality

Proof

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