lemma 9.10 Grönwall's inequality

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:390 · p. 277

Rests on

depends_on corollary 7.36 cor:ana-mvt-consequences ¶
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on proposition 7.29 Linearity ¶
    - depends_on definition 7.11 Sum of functions ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.14 eq:ana-derivh ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-13 ¶
  - depends_on theorem 7.34 Rolle ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - depends_on theorem 7.24 Extreme value theorem ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-19 ¶
- proves proof ch:05-real-analysis@proof-20 ¶
depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
- depends_on definition 7.26 Derivative of a function at a point ¶ ↺
- depends_on theorem 7.40 Continuous functions are integrable ¶
  - depends_on definition 7.39 Darboux sums and the definite integral ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
  - depends_on theorem 7.24 Extreme value theorem ¶ ↺
  - depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
    - depends_on proposition 7.22 Sequential characterization ¶ ↺
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶ ↺
    - proves proof ch:05-real-analysis@proof-10 ¶
  - proves proof ch:05-real-analysis@proof-23 ¶
- depends_on theorem 7.24 Extreme value theorem ¶ ↺
- proves proof ch:05-real-analysis@proof-24 ¶
proves proof ch:07-odes-sturm-liouville@proof-4 ¶

Supports

depends_on proposition 9.11 Continuous dependence on the initial data ¶
depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	cor:ana-mvt-consequences	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:402
`depends_on`	→	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:402
`depends_on`	←	Continuous dependence on the initial data	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:437
`depends_on`	←	Lyapunov's first method: asymptotic stability	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1170
`proves`	←	ch:07-odes-sturm-liouville@proof-4	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:405