A homotopy-based existence method that connects a problem with known solvability to a target problem via a continuous family of problems parameterized by t in [0,1], relying on a priori estimates and topological or analytic continuation to carry solutions along the path.

Principle

Deform a solvable equation continuously into the target while maintaining uniform a priori bounds and invertibility properties (or index control) of the linearized operator so that solutions persist for all parameter values.

Demonstration

To solve a nonlinear elliptic PDE L(u)=N(u), consider L(u)=tN(u)+(1-t)G(u) where G is a linear operator with known solvability; prove uniform bounds on solutions and use openness/closedness arguments in t to extend solvability from t=0 to t=1.

Misapplication

Using the method without establishing uniform a priori estimates or without controlling loss of compactness can fail: the solution branch may blow up or cease to exist at some parameter value despite initial solvability.

Consequence

When applicable, one obtains existence (and sometimes continuity) of solutions for the target problem, often with continuation of regularity and sometimes uniqueness if invertibility is preserved along the path.

Reversal

If a parameter path crosses a bifurcation point where linearized invertibility fails, the continuity argument breaks down and one must use bifurcation theory or alternative global methods.

Boundary

Requires a connected parameter family, a starting problem with known solutions, and control (a priori estimates, index theory or compactness) along the path; does not replace local techniques when only local data are available.

Semantic Tension

Sits between local implicit-function techniques (which give local continuation) and topological degree or variational methods (which give existence without pathwise control); the method emphasizes deformation plus estimates.

Synthesis

The Method of Continuity leverages a continuous deformation from an easy problem to a hard one, using a priori bounds and analytic/topological continuation to propagate solvability across the parameter interval.