Existence theorem

Existence theorem . An existence theorem is a theorem that asserts that each of a wide range of problems has a solution of a particular type. A case could be this way: when can a value for x be found as a function of y an equation of the form f (x) = y? In this case x can be defined by a formula such as x 2 – log (2x-1), established for the real numbers x of some interval, such as [2, 6] and y represents, for example, 27/4. The problem is stated in this way: is there any value of x between 2 and 6 such that x 2 – log (2x-1) = 27/4?


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  • 1 Definitions of the theorem
  • 2 System of equations in R
  • 3 Two-year history lesson
  • 4 Differential equations
  • 5 Source
  • 6 See also

Definitions of the theorem

The answer to this question should not resort to the method of finding the value or values ​​of x in particular cases. The issue is to find a criterion to determine whether or not there is a solution, but it is usable in an extensive class of problems. Already with the criterion that assures us that the specific problem has a solution, we can embark on the effort to find it, but with the certainty that our claim will be successful.

What is required is that the function be continuous in the closed interval; take different signs of the function at the ends of the interval. If all this happens it is certain that the solution exists.

If we want to know if for a second degree equation in an unknown, with rational coefficients, there are real roots, it will be enough to prove that the discriminant D = b 2 -4ac> = 0.

In which cases can the system f (x, y) = m, g (x, y) = n be solved for x , e y as a function of a and b? A simple example is the system of linear equations

5x + 3y = 9 and x- 3y = 3. Its resolvability can be discerned by the criterion of the determinant of the coefficients of the unknowns.

System of equations in R

It is about finding a para of real numbers x and y that satisfy the two equations:

x log y / (1 + y 2 ) = -174, 2x 2 = 8

Wondering if there is a solution for x between 2/5 and 2, and y is between -0.75 and 1.5.

The criterion of the possible existence of the solution involves the concept related to the number of turns that a plane curve surrounds a point. The concepts of compactness, connection and continuity will also have to be used. The main theorems are statements regarding the existence of zeros of polynomials, fixed points of transformations, and singularities of vector fields .

Two-thousand-year history lesson

Keep in mind the story of the famous problems of angle resection , squaring the circle, and doubling the cube , using only the ruler and the compass. They made an immutable effort to solve the problem with the implicit admission that the solution existed; the hard part was finding her. Until the end of the century of enlightenment, it never occurred to anyone to ponder the possibility that the solution did not exist. In other words, the evidence of non-existence was close, very close. Once the issue of existence was clearly stated, it was quickly resolved. In current research, existence issues are tackled first. Your answer is essential for theories to have a firm foundation.

Differential equations

Consider the following initial value problem, in ordinary differential equations:

y ‘ x = f (x, y), y (x 0 ) = y 0

This problem has a positive outcome with the Cauchy existence theorem . If f (x, y) is continuous in a region, in the neighborhood of the point (x 0 , y 0 ) that is to say for | xx 0 | <ay | yy 0 | <b, then there is at least one solution to the equation

y ‘ x = f (x, y)

which is defined and is continuous on an interval around x 0 and takes the value y 0 for x 0 .


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