Differential Equations and Linear Algebra, 4th Edition

A.4 Uniqueness of Solutions

It is possible to establish the existence of solutions of the initial value problem in (35) under the much weaker hypothesis that f(x, t) is merely continuous; techniques other than those used in this section are required. By contrast, the Lipschitz condition that we used in proving Theorem 1 is the key to uniqueness of solutions. In particular, the solution provided by Theorem 3 is unique near the point $t=a.$

THEOREM 1 Uniqueness of Solutions

Suppose that on some region R in $(m+1)$-space, the function f in (35) is continuous and satisfies the Lipschitz condition

$$|\mathbf{\text{f}}({\mathbf{\text{x}}}_1,t)-\mathbf{\text{f}}({\mathbf{\text{x}}}_2,t)|\leqq k\cdot |{\mathbf{\text{x}}}_1-{\mathbf{\text{x}}}_2|.$$ (20)

If ${\mathbf{\text{x}}}_1(t)$ and ${\mathbf{\text{x}}}_2(t)$ are two solutions of the initial problem in (35) on some open interval I containing $x=a,$ such that the ...