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Existence and Uniqueness Theorems

In: Fundamentals of Ordinary Differential Equations

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  • Uri Elias

    (Technion – Israel Institute of Technology, Department of Mathematics)

Abstract

In this chapter we learn about existence and uniqueness theorems for initial value problems as a central pivot of the theory of differential equations. We start with examples which show that neither existence nor uniqueness are self-evident, and even if that is the case, the domain of existence of the solution may surprise us. The Picard-Lindelöf theorem is first outlined without a proof and it is shown that its assumptions are sufficient (but not necessary), its result is local (and not global). Section 3.3 presents a detailed proof of the existence, while uniqueness is proved in Sect. 3.5. Later we discuss applications of uniqueness and global solutions and explain where a local solution “terminates.” These techniques are applied to describe the behavior of solutions of some differential equations that we do not know to solve explicitly. Funnels and anti-funnels, as well as comparison of equations, are used. Finally, stability and asymptotic stability are explained and exemplified.

Suggested Citation

  • Uri Elias, 2025. "Existence and Uniqueness Theorems," Springer Books, in: Fundamentals of Ordinary Differential Equations, chapter 0, pages 47-93, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-86532-9_3
    DOI: 10.1007/978-3-031-86532-9_3
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