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  2. Picard–Lindelöf theorem - Wikipedia

    en.wikipedia.org/wiki/Picard–Lindelöf_theorem

    A standard proof relies on transforming the differential equation into an integral equation, then applying the Banach fixed-point theorem to prove the existence of a solution, and then applying Grönwall's lemma to prove the uniqueness of the solution.

  3. Uniqueness quantification - Wikipedia

    en.wikipedia.org/wiki/Uniqueness_quantification

    which completes the proof that 3 is the unique solution of + =. In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.

  4. Uniqueness theorem - Wikipedia

    en.wikipedia.org/wiki/Uniqueness_theorem

    A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition). [3]

  5. Cauchy–Kovalevskaya theorem - Wikipedia

    en.wikipedia.org/wiki/Cauchy–Kovalevskaya_theorem

    In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy , and the full result by Sofya Kovalevskaya .

  6. Disjunction and existence properties - Wikipedia

    en.wikipedia.org/wiki/Disjunction_and_existence...

    The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem.; The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term t such that the theory proves A(t).

  7. Existence theorem - Wikipedia

    en.wikipedia.org/wiki/Existence_theorem

    Despite that, the purely theoretical existence results are nevertheless ubiquitous in contemporary mathematics. For example, John Nash's original proof of the existence of a Nash equilibrium in 1951 was such an existence theorem. An approach which is constructive was also later found in 1962. [6]

  8. Haar measure - Wikipedia

    en.wikipedia.org/wiki/Haar_measure

    The existence and uniqueness (up to scaling) of a left Haar measure was first proven in full generality by André Weil. [4] Weil's proof used the axiom of choice and Henri Cartan furnished a proof that avoided its use. [5] Cartan's proof also establishes the existence and the uniqueness simultaneously.

  9. Banach fixed-point theorem - Wikipedia

    en.wikipedia.org/wiki/Banach_fixed-point_theorem

    A standard application is the proof of the Picard–Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator on the space of continuous functions under the uniform norm. The Banach ...