Quasi-Nonexpansive Operators: Theory & Applications

• Quasi-nonexpansive operators are mappings in metric, Banach, and Hilbert spaces that do not increase the distance to any fixed point, ensuring convergence properties.
• They generalize nonexpansive and firmly nonexpansive operators and underpin key fixed-point, projection, and splitting algorithms in convex optimization and nonlinear analysis.
• Their generalizations extend to Bregman, Wasserstein, and geometric settings, enriching convergence analysis in diverse mathematical and algorithmic frameworks.

A quasi-nonexpansive operator is a mapping acting on a metric, Banach, or Hilbert space that preserves or decreases the distance to its fixed point set, generalizing both nonexpansive and firmly nonexpansive operators. This notion underpins the analysis of many splitting, projection, and fixed-point algorithms that are central in convex optimization and nonlinear analysis, and has robust generalizations to Bregman, Wasserstein, and geometric settings beyond linear spaces.

1. Definitions and Basic Properties

Let $(X, \|\cdot\|)$ be a Banach or Hilbert space, or more generally a metric space $(X,d)$. Let $T: X \to X$ (or, more generally, $T: X \rightrightarrows X$ in the multivalued case). The set of fixed points is $\Fix(T) := \{ x \in X \mid T(x) = x \}$.

Quasi-nonexpansive mappings:

• Single-valued: $T$ is quasi-nonexpansive (QNE) if $\Fix(T) \neq \emptyset$ and

$\|T(x) - p\| \le \|x - p\|, \quad \forall x \in X,\, \forall p \in \Fix(T).$

In metric spaces, replace norm with ambient metric.

• Multivalued (Hausdorff): For $T: D(T) \to CB(D(T))$, take

$H(Tx, Tp) \le \|x - p\|, \quad \forall x \in D(T),\, p \in \Fix(T),$

where $(X,d)$0 is the Hausdorff metric (Mendy et al., 11 Jan 2025).

• Strict and strong variants: T is strictly QNE (sQNE) if the inequality is strict for $(X,d)$1; T is strongly QNE if vanishing of the Fejér gap implies vanishing of the residual (Kostecki, 18 Nov 2025, Worapitpong et al., 30 Apr 2025).

Relations: Every nonexpansive map is QNE; every firmly nonexpansive map is QNE and, often, strongly QNE. The quasi-firmly nonexpansive property adds a term penalizing the residual (Bërdëllima et al., 2021, Bërdëllima et al., 2022). The class of QNE operators is strictly larger than nonexpansive or firmly nonexpansive maps (Mohammed et al., 2016).

2. Geometric and Functional-Analytic Generalizations

Quasi-nonexpansiveness extends beyond normed linear spaces:

• Hilbert geometry: $(X,d)$2 is QNE if for all $(X,d)$3 and all fixed points $(X,d)$4, the norm decreases to $(X,d)$5 (Berinde, 2024). QNE maps arise in the context of demicontractive and pseudocontractive mappings, with averaging providing embeddings between these classes.
• Banach spaces: In uniformly convex Banach spaces, quasi-nonexpansiveness is defined via Lyapunov functionals (as in $(X,d)$6) or using Bregman/Vaĭnberg–Brègman divergences $(X,d)$7 (Anh et al., 2015, Kostecki, 18 Nov 2025), and is intrinsically asymmetric. Calculus rules (closure under convex combinations, compositions) remain, though with adjustments to the modulus (Bërdëllima et al., 2021).
• Geodesic spaces / CAT(0) and Hadamard spaces: QNE maps are key in fixed-point theory for nonpositively curved spaces, with Δ-convergence (asymptotic center convergence) replacing weak convergence (Worapitpong et al., 30 Apr 2025, Berdellima, 2020). Operators are classified as strongly QNE if their residuals vanish when the Fejér gap closes.
• Wasserstein and metric measure spaces: Quasi-nonexpansive and quasi-α-firmly nonexpansive mappings admit analogous definitions based on transportation cost, enabling convergence results for measure-valued proximal and splitting algorithms (Bërdëllima et al., 2022).
• Information/Bregman geometry: Quasi-nonexpansiveness arises for operators that are nonexpansive relative to Bregman or Csiszár–Morimoto-type divergences, e.g., left DΨ-quasinonexpansiveness (Kostecki, 18 Nov 2025). Here, Chebyshev (entropic) projections and generalized resolvents are central.

3. Composition, Averaging, and Calculus Rules

Critical algorithmic and structural properties stem from the stability of QNE mappings under functional calculus:

Operation	Condition	Result	Source
Convex combination	QNE, weights in Δ	Combination is QNE	(Bërdëllima et al., 2021, Berdellima, 2020)
Composition	QNE, nonempty common fixed set	Composition is QNE or qα-firmly NE, with parameters updated	(Berdellima, 2020, Bërdëllima et al., 2021, Worapitpong et al., 30 Apr 2025)
Averaging/relaxation	QNE and λ-in [0,1)	Averaged map is QNE and preserves fixed points	(Berinde, 2024, Alaviani, 2019)
Generalized relaxation	sQNE and σ(x)>0	Map remains sQNE for appropriate σ	(Nikazad et al., 2021)

A central fact is the embedding of demicontractive maps into the QNE class by averaging (Berinde, 2024). All standard convergence theorems derived for QNE iterates transfer to demicontractive maps via this embedding. Variants such as quasi-firm nonexpansive and quasi-φ-nonexpansive operators are stable under similar operations in uniformly convex or information-geometric contexts (Anh et al., 2015, Kostecki, 18 Nov 2025).

4. Fixed-Point Iteration, Convergence Properties, and Regularity

Fejér monotonicity is inherent to QNE operators: the iterative sequence reduces (or at least does not increase) the distance to $(X,d)$8. Convergence analysis is stratified by space geometry and operator properties:

• Hilbert/Banach:
  • Mann and Halpern iterations with QNE maps converge weakly under mild control on step sizes, with strong convergence possible under enhanced regularity (demiclosedness, uniform convexity, Opial property) (Combettes et al., 2017, Dogan et al., 2015, Mohammed et al., 2016).
  • Strong convergence is ensured in uniformly convex Banach spaces for Halpern-type schemes over (possibly infinite) families of QNE operators (Dogan et al., 2015).
  • Demiclosedness at 0 is often needed for strong/weak convergence, but in some schemes may be relaxed to the fixed-point closedness property (Bauschke et al., 2012).
• Geodesic spaces (CAT(0)/Hadamard):
  • Δ-convergence to a fixed point is established for (finite or infinite) compositions/products of strongly QNE, Δ-demiclosed maps (Worapitpong et al., 30 Apr 2025, Berdellima, 2020).
  • Weak convergence is replaced by convergence in the sense of asymptotic centers.
• Stochastic and random settings:
  • Iterative schemes over random QNE operators yield almost sure and mean-square convergence to the solution of convex feasibility or minimization problems under mild diameter and step decays (Alaviani, 2019).
• Bregman/entropic geometry:
  • QNE maps are characterized through generalized Pythagorean inequalities for Bregman divergences; entropic projections and resolvents satisfy norm/Hölder regularity estimates matching those of the underlying Banach geometry (Kostecki, 18 Nov 2025).

5. Principal Applications and Algorithmic Contexts

Quasi-nonexpansive operators feature as fundamental components in iterative methods for addressing fixed-point, feasibility, and equilibrium problems:

• Convex feasibility and splitting: Projection, block-iterative, string-averaging, and hybrid methods utilize QNE maps and their generalizations for rapid and robust solution of large-scale feasibility systems (Anh et al., 2015, Nikazad et al., 2021, Combettes et al., 2017).
• Operator-splitting frameworks: Krasnoselskii–Mann, forward–backward, and adaptive Douglas–Rachford schemes rely on QNE forms to extend classical contraction-mapping arguments to broader operator classes (Berinde, 2024, Zhu et al., 2020).
• Wasserstein proximal and cyclic splitting: Proximal maps and compositions in Wasserstein geometry are shown to be quasi-α-firmly nonexpansive, enabling convergence of measure-valued splitting algorithms (Bërdëllima et al., 2022).
• Bregman projection and noncommutative state geometry: Entropic projections and relative-entropy-based procedures leverage left/right QNE properties to realize robust projection algorithms over general state spaces (Kostecki, 18 Nov 2025).

Concrete domains exploited in applications include infinite-dimensional neural network architectures (stabilized by QNE/averaged layer maps), variational inequalities, distributed optimization over random networks, and image reconstruction from projections (Bërdëllima et al., 2021, Maulén et al., 2022, Alaviani, 2019, Nikazad et al., 2021).

6. Structural Extensions and Open Directions

Recent research extends quasi-nonexpansiveness to diverse analytic, geometric, and probabilistic frameworks:

• Divergence-based and asymmetric geometries: The theory of QNE operators on spaces with Bregman, Tsallis–Petz, and related divergences unifies convex analysis with information geometry, providing a compositional and functorial language for entropic projections and resolvents (Kostecki, 18 Nov 2025).
• Nonlinear metric spaces: The Δ-convergence paradigm in Hadamard and general CAT(0) spaces adapts classical weak-convergence results, supporting convex optimization in singular or geodesically convex contexts (Worapitpong et al., 30 Apr 2025, Berdellima, 2020).
• Strong and linear rates: When QNE maps become strictly or strongly QNE (e.g., quasi-contractive), convergence accelerates to strong or even linear (Maulén et al., 2022, Cegielski et al., 2017).
• Stochastic and adaptive frameworks: Quasi-nonexpansive theory underpins adaptive and distributed stochastic optimization algorithms, providing both mean-square and almost sure convergence guarantees (Alaviani, 2019).

Challenges and open problems include the characterization of QNE operator classes under nonconvexity, extension to nonmetric settings, robust and adaptive parameter selection, and analysis of operator compositions in infinite and dynamically changing environments.

7. Examples and Illustrative Constructions

A representative selection:

Setting	Operator Construction	QNE Property and Role	Reference
Hilbert/Euclidean	Subgradient/projectors, Halpern/KM	QNE/fQNE/strong QNE iterations, strong convergence	(Mohammed et al., 2016, Dogan et al., 2015)
Operator Splitting	Demicontractive $(X,d)$9, averaging $T: X \to X$0	Averaged map QNE; convergence results port to $T: X \to X$1	(Berinde, 2024)
CAT(0)/Hadamard	Projections in geodesic spaces	Products/compositions strongly QNE, Δ-convergence	(Berdellima, 2020, Worapitpong et al., 30 Apr 2025)
Bregman geometry	Entropic projection, Bregman resolvent	QNE w.r.t. $T: X \to X$2, generalized Pythagoras, Hölder	(Kostecki, 18 Nov 2025)
Stochastic systems	Random QNE block/coordinate/consensus map	A.s. and mean-square convergence in distributed systems	(Alaviani, 2019)
Wasserstein geometry	Pushforward and cycle of proximity maps	Quasi-α-firm NE, narrow convergence in $T: X \to X$3	(Bërdëllima et al., 2022)

These setups enable robust fixed-point algorithms across a vast spectrum of mathematical, statistical, and computational problems.

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References:

• "On a useful lemma that relates quasi-nonexpansive and demicontractive mappings in Hilbert spaces" (Berinde, 2024)
• "On a notion of averaged operators in CAT(0) spaces" (Berdellima, 2020)
• "Quasi $T: X \to X$4-Firmly Nonexpansive Mappings in Wasserstein Spaces" (Bërdëllima et al., 2022)
• "Vaĭnberg--Brègman geometry and quasinonexpansive operators" (Kostecki, 18 Nov 2025)
• "A product of strongly quasi-nonexpansive mappings in Hadamard spaces" (Worapitpong et al., 30 Apr 2025)
• "On $T: X \to X$5-Firmly Nonexpansive Operators in $T: X \to X$6-Uniformly Convex Spaces" (Bërdëllima et al., 2021)
• "Asymptotic behavior of a nonautonomous evolution equation governed by a quasi-nonexpansive operator" (Zhu et al., 2020)
• "Inertial Krasnoselskii-Mann Iterations" (Maulén et al., 2022)
• "The split feasibility and fixed point equality problems for quasi-nonexpansive mappings in Hilbert spaces" (Mohammed et al., 2016)
• "On some strong convergence results of a new Halpern-type iterative process for quasi-nonexpansive mappings and accretive operators in Banach spaces" (Dogan et al., 2015)
• "A string averaging method based on strictly quasi-nonexpansive operators with generalized relaxation" (Nikazad et al., 2021)
• "Regular Sequences of Quasi-Nonexpansive Operators and Their Applications" (Cegielski et al., 2017)
• "Convex Optimization over Fixed Value Point Set of Quasi-Nonexpansive Random Operators on Hilbert Spaces" (Alaviani, 2019)
• "Parallel and sequential hybrid methods for a finite family of asymptotically quasi $T: X \to X$7-nonexpansive mappings" (Anh et al., 2015)
• "A projection method for approximating fixed points of quasi nonexpansive mappings without the usual demiclosedness condition" (Bauschke et al., 2012)
• "Viscosity Iterative algorithm for solving Variational Inclusion and Fixed point problems involving Multivalued Quasi-Nonexpansive and Demicontractive Operators in real Hilbert Space" (Mendy et al., 11 Jan 2025)