Linked from

The 24 pages that link to Banach fixed-point theorem, each with the reason it gives.

• Metric spaceRelated: Completeness and distance contraction together guarantee convergence to a solution.

• Banach spaceRelated: Completeness ensures the successive approximations converge to the fixed point.

• Brouwer fixed-point theoremCompared with: Unlike Brouwer's existence-only result, it requires contraction and provides uniqueness and an iterative method.

• Fixed pointBroader topic: It guarantees both a unique fixed point and convergence of iteration under precise conditions.

• Fixed-point theoremBroader topic: It guarantees existence and uniqueness by repeatedly applying a distance-shrinking map.

• Lipschitz continuityRelated: Its contraction hypothesis is a strict Lipschitz bound.

• Picard–Lindelöf theoremRelated: Its fixed point is the function that satisfies the integral form of the initial-value problem.

• Complete metric spaceRelated: Completeness ensures the successive approximations converge to the fixed point.

• Hahn–Banach theoremCompared with: It also bears Banach's name but concerns iterative existence, not linear-functional extension.

• Inverse function theoremCompared with: It can prove local solvability by iteration rather than by the inverse theorem's differential criterion.

• Fixed-point iterationRelated: Its completeness and contraction assumptions explain one rigorous convergence guarantee.

• Stefan BanachBroader topic: It exemplifies how completeness and a quantitative contraction condition guarantee existence.

• Lax–Milgram theoremCompared with: It is another route to well-posedness, but requires a contraction rather than coercivity.

• Schauder fixed-point theoremCompared with: Unlike Schauder’s theorem, it requires contraction but gives uniqueness and iterative convergence.

• Uniqueness of solutionsRelated: A contraction forces any two fixed points to coincide, giving uniqueness alongside existence.

• Lefschetz fixed-point theoremCompared with: It uses metric contraction rather than homology to guarantee a unique fixed point.

• Kleene's recursion theoremCompared with: Unlike Banach's theorem, this result concerns computable program behavior rather than metric convergence.

• Kirszbraun theoremCompared with: It concerns maps that shrink distances, rather than extending maps while preserving distances.

• Kleene fixed-point theoremCompared with: It obtains fixed points through metric contraction, rather than order and directed continuity.

• Knaster–Tarski theoremCompared with: It uses metric contraction rather than order preservation and gives uniqueness rather than a lattice of solutions.

• Cantor's intersection theoremRelated: The iterates generate a Cauchy sequence, reflecting the shrinking-set logic behind the theorem.

• Caristi fixed-point theoremCompared with: Banach requires uniform distance contraction; Caristi instead uses a decreasing potential.

• Fredholm's theoremCompared with: It proves uniqueness by contraction, not by the Fredholm alternative's range conditions.

• Lions–Lax–Milgram theoremCompared with: It offers a general alternative route to existence and uniqueness through contraction.