A counterexample in nonlinear boundary value problems

Abstract:

For the boundary value problem $(1.1), (1.2)$ below where initial value problems of $(1.1)$ are unique and exist on $(a,b)$ it is known that global uniqueness on $(a,b)$ implies global existence on $(a,b)$ if $\beta \delta = 0$. It is also known that this is false if $\beta \delta \ne 0$ and $\alpha \delta - \beta \gamma = 0$. It is shown here by example that this is also false if $\beta \delta \ne 0$ and $\alpha \delta - \beta \gamma \ne 0$.