Caristi fixed-point theorem

A fixed-point theorem for self-maps of complete metric spaces: a lower semicontinuous potential that bounds each point’s displacement guarantees a fixed point.

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Caristi's inequality: An inequality of the form d(x,Tx) ≤ φ(x) − φ(Tx), linking displacement to the decrease of a potential. This pointwise descent condition is the theorem’s central hypothesis.
Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. The theorem measures each point’s displacement using a metric.
Banach fixed-point theorem: A theorem stating that a contraction on a complete metric space has a unique fixed point. Banach requires uniform distance contraction; Caristi instead uses a decreasing potential.
Optimization: The study of selecting decision variables to minimize or maximize an objective under specified constraints. A potential function can encode an objective whose descent forces a stationary point.
Lower semicontinuity: A function is lower semicontinuous when its value at a limit is no greater than the limit inferior of nearby values. It preserves the potential’s lower bound when the constructed sequence converges.
Self-map: A function from a set to itself. The theorem applies to maps whose inputs and outputs lie in the same space.
Kirk fixed-point theorem: A fixed-point theorem for nonexpansive mappings on weakly compact convex subsets of Banach spaces with normal structure. Kirk’s theorem extends fixed-point methods to nonexpansive maps under geometric assumptions.
Variational inequality: A problem seeking a point whose relation to a feasible set and operator satisfies a specified inequality. Caristi-type arguments establish existence for problems expressed through inequalities.
Complete metric space: A metric space in which every Cauchy sequence converges to a point in the space. Completeness supplies the limit of the sequence generated by repeated application of the map.
Potential function: A scalar-valued function used to measure state, cost, or progress in a system. Its decrease along the map’s action pays for the distance traveled.