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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.
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]
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.
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 .
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).
The existence proof proceeds in imitation of the existence proof for Fraïssé limits. The uniqueness proof comes from an easy back and forth argument.
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]
As an important result, the inverse function theorem has been given numerous proofs. The proof most commonly seen in textbooks relies on the contraction mapping principle, also known as the Banach fixed-point theorem (which can also be used as the key step in the proof of existence and uniqueness of solutions to ordinary differential equations ...