Existence and Uniqueness Theorem

Paper Summary:

This paper aims to provide information on generalization of existence and uniqueness theorem for ordinary differential equations (ODE). This contains mathematical definitions, notations, symbolisms, formulas, graphics, and equations of the theorems mentioned. A few examples were also provided to illustrate and prove the definitions of the theorems.

From the Paper:

The fundamental theorem of Existence and Uniqueness answer the questions does a solution exists and is it unique. If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of solutions (by means of a so-called existence theorem) and then investigate their uniqueness (by means of a so-called uniqueness theorem). The solutions to an ordinary differential equation (frequently abbreviated as ODE) satisfy the existence and uniqueness properties.

The foundation of the theory of differential equations is the theorem on the existence of solutions for the initial value problem
x = f ( x ), x ( t o ) = x o
for a function x: R - Rn and f: Rn - Rn The main tool that we will use in developing our theory is the reformation of the differential equation as an integral equation. Suppose that both x(t) and f(x) are continuous functions. Then we can formally integrate both sides with respect to t to obtain:
x ( t ) = x o + ....

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