lemma A.439 Existence, uniqueness, and the structure of the zeros

open in the book · appendices/A-long-proofs.tex:21538 · p. 3007

Rests on

depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
- depends_on definition 9.2 Linear equation; homogeneity ¶
  - depends_on definition 9.1 Ordinary differential equation ¶
- depends_on remark 9.3 Normal form and first-order systems ¶
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on definition 9.4 Lipschitz condition ¶
  - depends_on equation 9.3 eq:slt-ode-system ¶
  - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
- proves proof ch:07-odes-sturm-liouville@proof-3 ¶
depends_on definition A.438 Momentum variable and the accessory system ¶
- depends_on equation 16.49 eq:calcvar-jacobi ¶
- depends_on equation 9.3 eq:slt-ode-system ¶ ↺
proves proof app:A-long-proofs@proof-261 ¶

Supports

depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
- depends_on definition A.467 Green function of the shifted problem ¶
  - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
    - depends_on theorem A.470 $T$ is compact ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on lemma A.472 The pairing identity ¶
    - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶ ↺
- depends_on theorem A.443 Sturm separation theorem ¶
depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
- depends_on corollary A.445 Displacing the initial point ¶

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`	→	Linear equations: existence on the whole interval	declared	appendices/A-long-proofs.tex:21551
`depends_on`	→	Momentum variable and the accessory system	declared	appendices/A-long-proofs.tex:21551
`depends_on`	←	The Wronskian of the accessory equation is constant	declared	appendices/A-long-proofs.tex:21748
`depends_on`	←	Continuous dependence on the data and on the initial point	declared	appendices/A-long-proofs.tex:21656
`proves`	←	app:A-long-proofs@proof-261	declared	appendices/A-long-proofs.tex:21554