A Fixed Point Theorem for Random Asymptotically Pointwise Contractions

Abstract: This paper combines the decomposition technique ($σ$-stability) in random functional analysis with the deterministic theory of asymptotically pointwise contractions to provide a complete self-contained derivation of a fixed point theorem for random asymptotically pointwise contractions. We assume the contraction function is linear $ψ(t)=λt$ ($λ&lt;1$) and focus on the linear case under the assumption that $G$ is bounded. By choosing $p$ sufficiently large so that $5<sup>{1/p}λ&lt;1$, we apply the deterministic theorem in $L<sup>p(E)$. The paper gives detailed explanations of concepts such as random normed modules, the $(ε,λ)$-topology, and $σ$-stability, and reviews the historical development of fixed point theory in the introduction.

• The paper establishes a unique fixed point and convergence of every iterate for linear random asymptotically pointwise contractions on deterministically bounded complete random normed modules.
• The proof embeds the module into an L^p space, choosing p so that 5^(1/p)λ<1, and combines a tail-diameter argument with measurable σ-stable decomposition techniques.
• The theorem guarantees convergence in probability but remains limited to bounded domains and linear contraction functions, leaving unbounded and nonlinear extensions open.

This paper establishes a fixed point theorem for random asymptotically pointwise contractions on complete random normed modules (RN modules), combining the deterministic theory of asymptotic contractions with the decomposition ($\sigma$-stability) technique from random functional analysis. The author restricts attention to the linear case $\psi(t)=\lambda t$ with $\lambda<1$ and assumes the domain $G$ is deterministically bounded; under these hypotheses, existence and uniqueness of a fixed point are obtained, together with convergence of all iterates in the $(\epsilon,\lambda)$-topology.

Background and motivation

The paper situates itself within two converging lines of research. On the deterministic side, it traces the progression from Banach's contraction principle through Browder–Göhde–Kirk nonexpansive theory, Boyd–Wong nonlinear contractions, Ćirić quasi-contractions (with the maximum $M(x,y)$ over five distances), Goebel–Kirk asymptotically nonexpansive maps, and Kirk's 2003 asymptotic contractions. The modern refinement adopted here weakens Kirk's global uniform convergence of the comparison functions to local uniform convergence (uniform convergence on bounded sets) and allows the limiting estimate to involve the Ćirić-type maximum $M(x,y)$.

On the stochastic side, the paper builds on Guo's theory of RN modules, where the norm takes values in $L^0_+(F)$ rather than $\mathbb{R}_+$, and on the 2025 result of Sun, Guo and Tu extending the Goebel–Kirk theorem to random asymptotically nonexpansive mappings via decomposition into countably many deterministic pieces glued by $\sigma$-stability. The stated goal is to merge these two frameworks: a fixed point theorem for random asymptotically pointwise contractions.

Preliminaries

The paper reviews the required machinery in detail. An RN module $\psi(t)=\lambda t$0 is a left module over the algebra $\psi(t)=\lambda t$1 of equivalence classes of random variables, equipped with an $\psi(t)=\lambda t$2-norm satisfying the usual axioms with values in $\psi(t)=\lambda t$3. Convergence is taken in the $\psi(t)=\lambda t$4-topology, generated by neighborhoods $\psi(t)=\lambda t$5; this topology coincides with convergence in probability and is metrizable. A subset $\psi(t)=\lambda t$6 is $\psi(t)=\lambda t$7-stable if it is closed under gluing along countable measurable partitions: for any sequence $\psi(t)=\lambda t$8 and partition $\psi(t)=\lambda t$9 there is a unique $\lambda<1$0 with $\lambda<1$1. A map $\lambda<1$2 has the local property if $\lambda<1$3 for all events $\lambda<1$4; the author notes this follows automatically from $\lambda<1$5-Lipschitz continuity when $\lambda<1$6 is $\lambda<1$7-stable, citing Lemma 2.11 of Sun–Guo–Tu, but chooses to assume it directly. Finally, $\lambda<1$8 with norm $\lambda<1$9 is a classical Banach space whose convergence implies $G$0-convergence but not conversely — the key fact enabling the embedding argument.

Deterministic preparation

The paper first gives a self-contained proof of the deterministic result for linear asymptotically pointwise contractions. A continuous map $G$1 on a complete metric space satisfies (D1) $G$2, (D2) $G$3 pointwise and uniformly on bounded sets, and (D3) $G$4 with $G$5, where $G$6 is the five-term Ćirić maximum. For an orbit with tail diameters $G$7, the central lemma shows that for every $G$8 there exists $G$9 independent of $(\epsilon,\lambda)$0 such that

$(\epsilon,\lambda)$1

Since $(\epsilon,\lambda)$2 is nonincreasing with limit $(\epsilon,\lambda)$3, taking $(\epsilon,\lambda)$4 yields $(\epsilon,\lambda)$5, hence $(\epsilon,\lambda)$6: the orbit is Cauchy, converges to some $(\epsilon,\lambda)$7, and continuity gives $(\epsilon,\lambda)$8. Uniqueness follows by passing (D1)–(D3) to the limit at two fixed points, where $(\epsilon,\lambda)$9 collapses the estimate to $M(x,y)$0.

Main theorem and proof strategy

The main theorem states that if $M(x,y)$1 is a random asymptotically pointwise contraction — meaning $M(x,y)$2 is $M(x,y)$3-stable with the local property, continuous in the $M(x,y)$4-topology, the random variables $M(x,y)$5 converge almost surely to $M(x,y)$6 locally uniformly in probability, and $M(x,y)$7 a.s. — then $M(x,y)$8 has a unique fixed point and every iterate sequence converges to it in probability.

The proof proceeds by embedding into $M(x,y)$9. Because $M(x,y)$0 is deterministically bounded ($M(x,y)$1 a.s.), $M(x,y)$2 lies in $M(x,y)$3 for every $M(x,y)$4, and $M(x,y)$5-closedness plus $M(x,y)$6-stability imply $M(x,y)$7 is closed in the $M(x,y)$8-norm, hence complete. The exponent $M(x,y)$9 is chosen so that

$L^0_+(F)$0

e.g., $L^0_+(F)$1. Verifying (D1) is immediate. For (D2), the local uniform convergence in probability combined with the uniform bound $L^0_+(F)$2 a.s. yields, via splitting the expectation at level $L^0_+(F)$3 and applying the mean value theorem to $L^0_+(F)$4,

$L^0_+(F)$5

so $L^0_+(F)$6 uniformly on $L^0_+(F)$7. For (D3), the pointwise maximum inequality $L^0_+(F)$8 produces the effective contraction constant $L^0_+(F)$9. This factor $\mathbb{R}_+$0 is the price paid for replacing the pointwise maximum of five distances by its $\mathbb{R}_+$1 analogue; choosing $\mathbb{R}_+$2 large absorbs it since $\mathbb{R}_+$3 strictly.

Applying the deterministic theorem on $\mathbb{R}_+$4 yields an $\mathbb{R}_+$5-convergent iterate sequence with limit $\mathbb{R}_+$6. Since $\mathbb{R}_+$7-convergence implies convergence in probability and $\mathbb{R}_+$8 is assumed continuous in the $\mathbb{R}_+$9-topology, $\sigma$0. Uniqueness is established directly in the almost sure sense without recourse to the $\sigma$1 embedding: at two fixed points, $\sigma$2 a.s., forcing $\sigma$3 a.s. Global convergence of iterates from any initial point follows from the $\sigma$4 conclusion.

Scope and limitations

Several restrictions are explicit and should be weighed when assessing the result's generality. First, the analysis is confined to the linear contraction function $\sigma$5; the author concedes that general nonlinear Boyd–Wong functions cannot be unconditionally reduced to this case because one cannot guarantee $\sigma$6. Second, the deterministic boundedness of $\sigma$7 is essential throughout: it places $\sigma$8 inside every $\sigma$9, supplies the uniform bound $\psi(t)=\lambda t$00 used in the uniform convergence argument, and allows the constant radius $\psi(t)=\lambda t$01 in the local uniform convergence condition. Whether the theorem extends to merely $\psi(t)=\lambda t$02-bounded or unbounded domains is not addressed. Third, the proof requires $\psi(t)=\lambda t$03 to be continuous in the $\psi(t)=\lambda t$04-topology while the contraction estimates are verified only in the $\psi(t)=\lambda t$05 metric; the paper does not establish $\psi(t)=\lambda t$06-continuity of $\psi(t)=\lambda t$07 itself, instead routing the fixed-point identification through the weaker topology. Fourth, the local property is assumed directly rather than derived, although the author notes it follows from $\psi(t)=\lambda t$08-Lipschitz conditions under $\psi(t)=\lambda t$09-stability. Finally, the local uniform convergence condition in Definition 3.4 is formulated with matching parameters ($\psi(t)=\lambda t$10 chosen so that both the threshold and the probability bound equal $\psi(t)=\lambda t$11); whether this can be relaxed to independent parameters is not discussed.

Conclusion

The paper delivers a complete, self-contained fixed point theorem for random asymptotically pointwise contractions in the linear case: uniqueness of the fixed point and convergence of all iterates in the $\psi(t)=\lambda t$12-topology follow from a tail-diameter argument lifted to $\psi(t)=\lambda t$13 via the choice $\psi(t)=\lambda t$14. The result recovers random asymptotically nonexpansive mappings as the regime $\psi(t)=\lambda t$15 and subsumes the deterministic theory as the degenerate case. Its principal open questions concern extension beyond the linear contraction function — where the ratio $\psi(t)=\lambda t$16 need not be bounded away from 1 — and removal of the deterministic boundedness assumption on $\psi(t)=\lambda t$17, both of which would require techniques beyond the $\psi(t)=\lambda t$18-embedding employed here.