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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the
Banach fixed-point theorem In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqu ...
for maps of a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
into itself. Caristi's
fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. In mathematica ...
modifies the \varepsilon- variational principle of Ekeland (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk. Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
.


Statement of the theorem

Let (X, d) be a complete metric space. Let T : X \to X and f : X \to , +\infty) be a lower semicontinuous function from X into the non-negative real numbers">lower_semicontinuous.html" ;"title=", +\infty) be a lower semicontinuous">, +\infty) be a lower semicontinuous function from X into the non-negative real numbers. Suppose that, for all points x in X, d(x, T(x)) \leq f(x) - f(T(x)). Then T has a fixed point in X; that is, a point x_0 such that T(x_0) = x_0. The proof of this result utilizes Zorn's lemma to guarantee the existence of a Maximal and minimal elements, minimal element which turns out to be a desired fixed point.


References

{{Convex analysis and variational analysis Fixed-point theorems Metric geometry Theorems in real analysis