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MATH
[1251]

Theorems from Existence, uniqueness and linear operators

All theorems from Existence, uniqueness and linear operators.

Back to Existence, uniqueness and linear operators

Local existence and uniqueness

For the IVP

y′=f(x,y),y(x0)=y0,y'=f(x,y),\qquad y(x_0)=y_0,

if ff is continuous and satisfies a Lipschitz condition in yy on a rectangle contained in its domain and containing (x0,y0)(x_0,y_0), then the IVP has a unique solution on some interval around x0x_0. The conclusion is local; a global solution needs the hypotheses to remain valid along the solution.

The Lipschitz condition is sufficient, not necessary: if it fails, uniqueness may still hold.

  1. [1]:Lipschitz condition in the dependent variable
  2. [2]:Reading the hypotheses

All forced solutions are an affine translate

If T[y]=fT[y]=f and ypy_p is one particular solution, then

{all solutions of T[y]=f}=yp+ker⁡(T).\{\text{all solutions of }T[y]=f\} =y_p+\ker(T).

Equivalently, every solution is yp+yhy_p+y_h for some homogeneous solution yhy_h.

The solution set is generally not a vector space when f≠0f\neq0, because it need not contain the zero function. It is a translate of the homogeneous solution space, as for solution classes of linear systems.

  1. [1]:Linear differential operator
  2. [2]:Kernel of an operator
  3. [3]:Classes of solutions of Ax=bAx=b
  4. [4]:Proof
  5. [5]:A particular solution translated by the kernel

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Local existence and uniquenessAll forced solutions are an affine translate