Survey of Metric fixed point theory in random functional analysis

Abstract: Based on the idea of randomizing the traditional space theory of functional analysis, random functional analysis has been developed as functional analysis over random metric spaces, random normed modules and random locally convex modules. Since these random frameworks have much more complicated algebraic, topological and geometric structures than their prototypes, the development of fixed point theory in random functional analysis had been almost stagnant before 2010. Unexpectedly, with the deep development of stable set theory fixed point theory in random functional analysis, including both its metric and topological fixed point theory, has made considerable progress in the recent 15 years. The purpose of this paper is to survey the important progress in metric fixed point theory in random functional analysis, including the random Banach contraction mapping principle and Caristi fixed point theorem on complete random metric spaces, and fixed point theorems for random nonexpansive and asymptotically nonexpansive mappings in complete random normed modules. Besides, the connections among the topics surveyed, random equations and random fixed point theorems for random operators are also briefly mentioned.

• The paper unifies 15 years of metric fixed point research in random metric spaces and random normed modules, replacing classical weak compactness with stability, random conjugate spaces, and L0-convex compactness.
• Metric fixed point methods extend contraction, Ekeland–Caristi, Browder–Göhde–Kirk, common fixed point, and asymptotically nonexpansive theorems under random completeness, convexity, and normal-structure assumptions.
• The lifting method converts strong random operators into mappings on sigma-stable random spaces, yielding random fixed points without measurable selection, separability, or completeness assumptions in important cases.

This survey by Guo, Tu, Mu, and Sun (2603.27963) synthesizes the past fifteen years of progress in metric fixed point theory within random functional analysis — the framework obtained by randomizing the space theory of functional analysis rather than its operator theory. The central objects are random metric spaces ($RM$ spaces), random normed modules ($RN$ modules), and random locally convex modules, developed under two topologies: the classical $(\varepsilon,\lambda)$-topology of convergence in probability and the stronger, generally non-linear locally $L^0$-convex topology. The authors argue that the development of fixed point theory in this framework was nearly stagnant before 2010, and that the obstacle was structural: $RN$ modules are non-locally convex under the $(\varepsilon,\lambda)$-topology, so the theory of dual spaces — the standard tool behind weak compactness arguments in the Browder–Göhde–Kirk theorem — universally fails. The resolution came through $\sigma$-stability, the theory of random conjugate spaces, $L^0$-convex compactness, and the geometry of $RN$ modules. The survey covers the random Banach contraction principle, the Ekeland variational principle and Caristi fixed point theorem, the Browder–Göhde–Kirk theorem, common fixed point theorems for families of nonexpansive and isometric mappings, and fixed point theorems for asymptotically nonexpansive mappings, together with their consequences for classical random fixed point theory of random operators.

The survey first unifies several notions of stability. A $B$-regular equivalence relation on $RN$0, for a complete Boolean algebra $RN$1, allows the definition of $RN$2-stable sets and the concatenation $RN$3 along a partition of unity. The authors record that a $RN$4-regular relation amounts to a $RN$5-valued metric, that $RN$6-stability generalizes $RN$7-stability of subsets of an $RN$8-module, and that when $RN$9 itself is $(\varepsilon,\lambda)$0-stable the notion recovers Drapeau–Jamneshan–Karliczek–Kupper's conditional sets — while being, in their view, more convenient for applications because it avoids a non-traditional set-theoretic formalism.

An $(\varepsilon,\lambda)$1 space $(\varepsilon,\lambda)$2 takes values in $(\varepsilon,\lambda)$3; an $(\varepsilon,\lambda)$4 module is an $(\varepsilon,\lambda)$5-module with an $(\varepsilon,\lambda)$6-norm satisfying $(\varepsilon,\lambda)$7. The $(\varepsilon,\lambda)$8-topology makes $(\varepsilon,\lambda)$9 a topological module over the topological algebra $L^0$0, while the $L^0$1-topology $L^0$2 makes $L^0$3 merely a topological ring. A pivotal fact is that the two completeness notions are equivalent for finitely stable spaces: $L^0$4 is $L^0$5-complete iff it is $L^0$6-stable and $L^0$7-complete, and closures of $L^0$8-stable sets coincide under both topologies. Consequently, for $L^0$9-stable functions the two notions of lower semicontinuity coincide — a technical fact that underlies the equivalence between the $RN$0- and $RN$1-versions of nearly every theorem surveyed.

The bridge to classical random fixed point theory is the lifting construction. For a sample-continuous strong random operator $RN$2, the lifted operator $RN$3 acts on $RN$4, the $RN$5-stable $RN$6 space of equivalence classes of strong random elements. Fixed points of $RN$7 are exactly equivalence classes of random fixed points of $RN$8. This converts random fixed point problems into ordinary fixed point problems on $RN$9 spaces, bypassing measurable selection theorems entirely — a methodological point the authors emphasize repeatedly and which, as discussed below, yields results where measurable selection fails.

The random Banach contraction principle holds verbatim: if $(\varepsilon,\lambda)$0 is $(\varepsilon,\lambda)$1-complete, $(\varepsilon,\lambda)$2 with $(\varepsilon,\lambda)$3 on $(\varepsilon,\lambda)$4, and $(\varepsilon,\lambda)$5, then $(\varepsilon,\lambda)$6 has a unique fixed point with the a.s. error estimate $(\varepsilon,\lambda)$7. Two refinements are highlighted. First, for $(\varepsilon,\lambda)$8-stable spaces and mappings, the contraction may hold only for a random iterate: if $(\varepsilon,\lambda)$9 for some $\sigma$0, uniqueness of the fixed point still follows. Second, the multivalued Nadler contraction principle extends to the random Hausdorff metric $\sigma$1 on $\sigma$2, the family of nonempty, a.s. bounded, $\sigma$3-stable, $\sigma$4-closed subsets. The authors note that Hanš's classical random fixed point theorem and Itoh's multivalued version are both special cases obtained via the lifting method.

The Ekeland variational principle and Caristi fixed point theorem are established on complete $\sigma$5 spaces with $\sigma$6-valued functions. The precise version states: for a proper, $\sigma$7-stable, lower semicontinuous, lower bounded $\sigma$8 on a $\sigma$9-stable complete $L^0$0 space, and any $L^0$1 with $L^0$2 and any $L^0$3, there exists $L^0$4 with $L^0$5, $L^0$6, and $L^0$7 for $L^0$8. The Caristi theorem, equivalent to the variational principle, gives fixed points of mappings satisfying $L^0$9. These results were developed in connection with random convex analysis and conditional convex risk measures, and the survey notes they have been applied to backward stochastic differential equations.

The main structural difficulty here is that weak compactness is meaningless for closed $RN$0-convex subsets of an $RN$1 module. The survey presents the replacement: $RN$2-convex compactness, defined by the finite intersection property for families of $RN$3-closed $RN$4-convex subsets. Its key characterization is duality-theoretic: a closed $RN$5-convex set $RN$6 is $RN$7-convexly compact iff for every $RN$8 the functional $RN$9 attains its maximum on $B$0 — an exact analogue of James-type characterizations, achieved through the theory of random conjugate spaces. Moreover, a.s. bounded closed $B$1-convex subsets are all $B$2-convexly compact iff $B$3 is random reflexive, and weak compactness of a convex $B$4 in a Banach space $B$5 is equivalent to $B$6-convex compactness of $B$7.

On the geometric side, random normal structure is defined via nondiametral points with respect to the random diameter $B$8, and the survey records two sufficient conditions: every closed $B$9-convex subset of a complete random uniformly convex $RN$00 module has random normal structure, and $RN$01 has random normal structure whenever $RN$02 is weakly compact with normal structure. Random uniform convexity itself is characterized exactly: $RN$03 is random uniformly convex iff $RN$04 is uniformly convex for some $RN$05.

Combining these ingredients yields the random Browder–Göhde–Kirk theorem: every nonexpansive self-mapping of a closed, $RN$06-convexly compact set with random normal structure in a complete $RN$07 module has a fixed point. Via lifting, this immediately implies that a nonexpansive strong random operator on a weakly compact convex set with normal structure has a strong random fixed point — without the completeness of $RN$08, separability of $RN$09, or measurable selection arguments required in earlier approaches such as Xu's.

Extending from single mappings to families required a random analogue of complete normal structure, a problem with a demanding classical history (Belluce–Kirk, Lim, and others). The notion involves random Chebyshev radii and centers and a condition on decreasing consistent nets of $RN$10-stable subsets — consistency being precisely where the abstract $RN$11-stable set theory of the prerequisites section enters. The central result is the equivalence: an $RN$12-convexly compact set has random complete normal structure iff it has random normal structure. This is described as solving a central problem in metric fixed point theory in random functional analysis.

The consequences are the random versions of Lim's theorem for commutative families of nonexpansive mappings, of Lim–Lin–Petalas–Vidalis for isometries (fixed point in the Chebyshev center $RN$13), and of Brodskii–Milman for families of surjective isometries (common fixed point in $RN$14, with commutativity unnecessary). Lifting again produces corresponding results for random operators, including common fixed points of commutative families of strong random nonexpansive operators. The survey is explicit about a limitation of the classical approach here: the available measurable selection theorems fail for uncountable families $RN$15, and likely fail even for a single operator when $RN$16 is not separable — cases fully covered by the lifting method.

The final technical section treats random asymptotically nonexpansive mappings — those with $RN$17 for $RN$18 a.s. — in complete random uniformly convex $RN$19 modules. The proof strategy is notable: rather than mimicking Goebel–Kirk's techniques in uniformly convex Banach spaces, which the authors state is impossible given the less developed theory of random uniform convexity, they use the equivalence between random uniform convexity of $RN$20 and uniform convexity of $RN$21 to decompose a random asymptotically nonexpansive mapping into a family of classical ones on uniformly convex Banach spaces. Under these hypotheses, existence of fixed points, closedness and $RN$22-convexity of the fixed point set, and an "eventual" version all hold. The random demiclosedness principle — $RN$23 is demiclosed at $RN$24 with respect to the random weak topology $RN$25 — is obtained analogously, using the duality $RN$26, since the theory of random weak topologies is itself still under development.

Several caveats are stated plainly in the survey. The theory of stable compactness, more general than random sequential compactness and needed for a complete topological fixed point theory, is still under development; consequently the topological results (random Brouwer, noncompact Schauder, random Kakutani and Markov–Kakutani theorems) are deferred to a future survey. The theory of random weak topologies for $RN$27 modules is likewise incomplete, which is why the demiclosedness principle had to be obtained indirectly through the $RN$28-duality rather than directly. The claim that measurable selection theorems fail for a single nonexpansive random operator on a nonseparable $RN$29 is offered as the authors' assessment ("it seems to us") rather than a proved statement. Finally, whether the fixed point theory surveyed here will find further applications in random equations, dynamic mathematical finance, and nonsmooth differential geometry on metric measure spaces is posed as an expectation grounded in existing connections, not an established result.

The survey documents a coherent fifteen-year program in which metric fixed point theory has been extended to random metric spaces and random normed modules by replacing the unavailable tools of classical local convexity — dual spaces, weak compactness, weak topologies — with random conjugate spaces, $RN$30-convex compactness, random normal structure, and stability theory. The recurring lifting argument shows that classical random fixed point theorems for random operators are recoverable as corollaries of this space-theoretic framework, including in situations where measurable selection fails. Together with the topological fixed point results obtained in parallel, the metric theory surveyed here constitutes, by the authors' account, an essentially complete analogue of the classical metric fixed point theory of Banach, Caristi, Browder–Göhde–Kirk, Lim, Brodskii–Milman, Goebel–Kirk, and Xu.