An Overview of Existence and Uniqueness Theorem

•  Existence Theorem

An existence theorem is said to be entirely theoretical in the field of mathematics if the proof that is provided for it does not imply the construction of the object whose existence is being stated. 

A proof of this kind is not constructive because the overall strategy might not lend itself to construction in the first place. 

When it comes to locating something that is declared to exist, purely theoretical existence theorems are able to circumvent any and all algorithms. 

These are to be contrasted with the so-called “constructive” existence theorems, which many constructivist mathematicians working in extended logics (such as intuitionistic logic) claim to be inherently more powerful than the non-constructive versions of the same theorems.

• Uniqueness Theorem

A theorem, also known as a unicity theorem, states the uniqueness of a mathematical object, which often means that there is only one item fulfilling specific properties or that all objects of a particular class are equivalent.

 This type of theorem is sometimes referred to as a uniqueness theorem (i.e., they can be represented by the same model).

Properties of the existence theorem

In the field of mathematics, a theorem that states the existence of a particular object is referred to as a “existence theorem.”

 It might be a statement that starts with the phrase “there exist(s)”, or it could be a universal statement whose last quantifier is existential (for example, “for every x, y,… there exist(s)…”).

 Either way, it could be considered an existential statement. 

Although in practice, existence theorems are typically stated in standard mathematical language, in the formal terms of symbolic logic, an existence theorem is a theorem with a prenex normal form involving the existential quantifier. 

This is despite the fact that in the formal terms of symbolic logic, an existence theorem is a theorem. 

For instance, the assertion that the sine function is continuous everywhere, or any theorem is written in big O notation, can both be thought of as examples of theorems that are existential by their very nature

This is due to the fact that quantification can be found in the definitions of the concepts that are being used.

Properties of Uniqueness Theorem

In the field of mathematics, a uniqueness theorem is a theorem that asserts either the one-of-a-kind quality of an object that satisfies a set of requirements or the equivalence of all things that meet the set of conditions in question.

The following are some examples of uniqueness theorems:

The uniqueness of three-dimensional polyhedra according to Alexandrov’s theorem.

Theorem on the singularity of black holes.

The Cauchy–Kowalevski theorem is the most important local existence and uniqueness theorem for analytic partial differential equations that are linked with Cauchy initial value problems.

The Cauchy–Kowalevski–Kashiwara theorem is a generalisation of the Cauchy–Kowalevski theorem for linear partial differential equations with analytic coefficients. 

It applies to systems of partial differential equations.

The Division Theorem proves that every quotient and remainder resulting from an application of Euclidean division is unique.

The uniqueness of the factorization of prime numbers is a fundamental theorem in mathematics.

The Existence and Uniqueness Theorem: Some Illustrative Examples

Example 1:

Investigate whether or not there is a solution to the starting value problem and whether or not it is unique.

y’ = x – y + 1, y(1) = 2.

Solution:

Given the issue with the starting value

y’ = x – y + 1, y(1) = 2.

where f(x, y) = x – y + 1 and its partial derivative with respect to y, 

fy = -1, which is continuous in every real interval. 

Where f(x, y) = x – y + 1 

and its partial derivative with respect to y. As a result, the existence and uniqueness theorem guarantees that a solution to the specified ODE does, in fact, exist in some open interval with the origin at 1.

Now,

The linear differential equation y’ + y = x + 1

 has the form y’ + P(x)y = Q(x), 

where P = 1 and Q(x) is a function of x.

Q = x + 1.

Hence, I.F. = e∫P dx = e∫ 1 . dx = ex

The answer is as follows:

y ex = ∫ (x + 1). ex dx 

⇒ y ex  = ∫ x.ex dx + ∫ ex  dx + C

 ⇒ y = x + C e–x

At x = 1 ⇒

 y = 2, 

we get 2 = 1 + C e–1 

⇒ C = e

Therefore, the answer to the given ODE is y = x + e 1 – x, which is true for every value of x within the range R.

Example 2:

Investigate whether or not there is a solution to the starting value problem and whether or not it is unique.

 

y’ = y2 , y(0) = 1.

 

Solution:

Given the issue with the starting value

y’ = y2 , y(0) = 1.

where f(x, y) = y2 and its partial derivative with respect to y, fy = 2y,

 which is continuous in every real interval where y can be measured.

 As a result, the existence and uniqueness theorem guarantees that a solution to the specified ODE does, in fact, exist in some open interval with the origin at 0.

Now, let’s break out the factors individually.

y – 2 dy = dx

By bringing together both perspectives, we gain

∫ y – 2 dy = ∫ dx + C1 

⇒ – 2/ y = x + C1 

⇒ y = − 2/ (x + C1)

At x = 0,

 y = 1 1 = −2/C1

 or C = 1 {Let – 2/C1 = C}

Therefore, the answer to the given ordinary differential equation (ODE) is y = 1/ (1 – x), which is true for any value of x that falls between – ∞ and 1.

Conclusion

Existence and uniqueness the tool that allows us to get to the conclusion that there is only one solution to a first-order differential equation that satisfies a particular initial condition is called a theorem. 

 

This tool makes it possible for us to reach this conclusion.

 

In the event that the two curves did meet, we would be able to use the point where they met as the starting point for the differential equation.