Vector Fixed Point Equations: Theory & Applications

• Vector fixed point equations are operator equations on vector spaces that use multivariate metrics and cone-induced orderings.
• They extend classical fixed point theories to address complex systems in optimization, nonlinear PDEs, and probabilistic models.
• Recent research establishes existence, uniqueness, and convergence using modular spaces, matrix contractivity, and iterative schemes.

A vector fixed point equation is an operator equation of the form $x = F(x)$ or, more generally, an inclusion or distributional equation involving a mapping $F$ from a subset of a vector space (often infinite-dimensional, possibly equipped with additional algebraic or topological structure) into itself, where $x$ is expected to belong to the same space or a related product/hierarchical structure. In contrast to scalar fixed point theories—which typically address operators in real or complex vector spaces measured with norms or metrics—vector fixed point equations may involve vector-valued distances, orderings induced by cones, modular or lattice structures, or deal with set-valued or multi-criteria operators.

1. Vector Fixed Point Equations: Contexts and General Frameworks

Vector fixed point equations naturally arise in analysis, operator theory, optimization, probabilistic models, and nonlinear partial differential equations whenever the solution structure is inherently vectorial, or when the error/residual is to be measured componentwise or in an ordered sense. Several modern frameworks extend classical fixed point principles to the vector setting:

• Vector ultrametric spaces: $d:X \times X \rightarrow A_p$, where $(A_p,p,P)$ is a complete modular space equipped with a convex modular $p$ and a cone $P$, and $d$ satisfies strong triangle/ultrametric properties generalized to the vectorial setting (Nourouzi, 2013).
• Vector S-metric spaces: Generalized “distance” maps $S:R \times R \times R \to V$ (with $V$ a linear lattice), satisfying S-metric axioms that encode vectorial order and symmetry (Yadav et al., 2022, Yadav et al., 2023, Yadav et al., 2022).
• Vector B-metric spaces: $d:X \times X \to \mathbb{R}^n$ (or an ordered vector space), satisfying a triangle inequality with a matrix constant $B$ rather than a scalar (Precup et al., 7 Feb 2025).
• Quasi-locally convex and quasi-point-separable spaces: Topological vector spaces structured by generalized norm-like objects (weaknorms, m-quasiconvex functionals) rather than classical norms or seminorms (Li, 2019, Li, 2020, Li, 2022).

Depending on the structure, fixed point equations may involve non-contractive mappings, set-valued correspondences, recursive distributional operations, or may be analyzed using order-theoretic/topological properties.

2. Main Results: Existence and Uniqueness Theorems

Several foundational existence and uniqueness results for vector fixed point equations have been developed, tailored to various settings:

2.1 Spherically Complete Vector Ultrametric Spaces

Let $(X,d,P)$ be a spherically complete, unital-normal vector ultrametric space, $p$ a convex modular satisfying the $\Delta_2$-condition, and $\varphi:X\to c(X)$ a correspondence (set-valued mapping with nonempty, compact images). The main theorem (Nourouzi, 2013) is:

For each $x,y\in X$, every $p\in\varphi(x)$, there exists $q\in\varphi(y)$ such that

$$p\bigl(d(p,q)\bigr) < \max\left\{p\bigl(d(x,p)\bigr),\; p\bigl(d(x,y)\bigr),\; p\bigl(d(y,q)\bigr)\right\}.$$

Then there exists $g\in X$ such that $g\in\varphi(g)$.

Key: The comparison in the contractive condition is performed on the images under the convex modular $p$, generalizing fixed point criteria from scalar-valued ultrametric spaces.

2.2 Vector S-metric Spaces and Common Fixed Points

For complete vector S-metric spaces $(\mathcal{R}, S, V)$ and self-maps $f, K$ commuting under certain conditions, the following contractive form yields unique fixed points (Yadav et al., 2022):

$S(fx, fx, fy) \preceq q\cdot U(x, x, y)$

for $q\in[0,1/3)$, with $U$ chosen from specific vector S-metric combinations.

More general theorems address common fixed points of three or four self-maps, under weak compatibility and point of coincidence, extending to cases with cyclic relations or weaker commutativity (Yadav et al., 2023, Yadav et al., 2022).

2.3 Vector B-metric and Perov Contractions

Given a complete vector $B$-metric space and $N:X\to X$ such that

$d(N(x), N(y)) \leq A d(x, y), \quad \text{with $A$a matrix satisfying$r(A)<1$},$

iteration $x_{k+1} = N(x_k)$ yields convergence to a unique fixed point $x^* = N(x^*)$ (Precup et al., 7 Feb 2025). Error estimates and variants (Maia-type, Caristi-type) are established for both unique and non-unique cases, with the vectorial (matrix) structure allowing system-wise error analysis and sharper convergence guarantees.

2.4 Topological Vector Spaces Beyond Local Convexity

In quasi-locally convex or quasi-point-separable Hausdorff topological vector spaces, every continuous self-map on a nonempty compact convex subset has a fixed point (Li, 2019, Li, 2020, Li, 2022). The underlying “distance” is measured by a family of norm-like (quasi-weaknorm, m-quasiconvex) functionals, with fixed points characterized as

$u_\alpha(x_0 - f(x_0)) = 0 \quad \forall\alpha,$

which, due to the separating property, implies $x_0 = f(x_0)$.

3. Structural Properties and Methodologies

The theoretical analysis in vector settings employs a combination of:

• Partial ordering via cones or lattices: Essential for defining positivity and enabling order-theoretic arguments, especially in Banach spaces with cone-induced orders or when considering operator systems localized on cones (Nourouzi, 2013, Rodríguez-López, 2022).
• Generalized notion of “balls” and completeness: For spaces with modulars or lattice-valued metrics, balls and Cauchy sequences are defined in the context of the modular or the lattice ordering (Nourouzi, 2013, Yadav et al., 2022).
• Zorn’s Lemma and KKM methods: Used in proofs requiring spherically completeness or the intersection property for families of closed balls or KKM maps, to guarantee fixed points or common intersection points (Nourouzi, 2013, Li, 2019, Li, 2022).

4. Applications and Examples

Vector fixed point equations have diverse applications:

• Set-valued correspondences: The general framework allows for treatment of operator inclusions or systems where the solution must belong to a set-valued mapping, important in logic programming, domain theory, and differential inclusions (Nourouzi, 2013).
• Systems of integral and differential equations: Vector versions of Krasnosel’skii–Precup and related fixed point index theorems are applied to Hammerstein integral equations, $p$-Laplacian systems, and other operator systems modeling coexistence or positivity phenomena (Rodríguez-López, 2022).
• Random recursive structures: In probability, recursive distributional equations for random objects with structural complexity (e.g., continuum trees) are framed as vector fixed point problems, with characterizations via the law of the process or tree (Albenque et al., 2015).
• Operator equations with nonlinear dependence on vector norms: Nonlinear vector equations (such as $Ax - \|x\|_1 x = b$) with “state-dependent” nonlinearities are solved using auxiliary scalar–vector fixed point iterations and accelerated algorithms, with rigorous convergence and computational guarantees (Wang et al., 7 Jul 2025).
• Functional equations for ODEs with discontinuous vector fields: Hybrid fixed point theorems incorporating both contraction and order structures enable generalizations of Carathéodory’s existence theorem beyond classical continuity frameworks (Zubelevich, 2023).

5. Generalizations and Connections to Variational Principles

Several results extend classical variational and minimization principles:

• Vector Ekeland’s Variational Principle: Extended to vector $B$-metric spaces, with ‘distance’ and variational increments measured in $\mathbb{R}^n$ and the triangle constant replaced by a matrix, yielding both weak and strong forms (Precup et al., 7 Feb 2025).
• Caristi-type Fixed Point Theorems: Derived in vector settings, where the “potential” $f:X\to\mathbb{R}^n$ replaces the scalar functional, and the mapping is shown to be non-expansive up to a B-matrix “potential drop” (Precup et al., 7 Feb 2025).
• Kuratowski–Ryll-Nardzewski and Schauder theorems for $p$-vector spaces: Generalized to upper semicontinuous set-valued mappings in non-locally convex vector spaces, confirming the fixed point property in settings where convexity is given in a $p$-convex or $s$-convex sense $0 < p, s \leq 1$ (Yuan, 2022).

6. Iterative Schemes and Computational Methods

For practical solution of vector fixed point equations, several iterative and structure-preserving schemes have been analyzed:

Scheme	\ LaTeX Formula	Convergence Rate
Relaxed fixed-point	$\mu_{k+1} = \tau f(\mu_k) + (1-\tau)\mu_k$	Linear (tuned by $\tau$)
Newton iteration	$$\mu_{k+1} = \mu_k - \frac{f(\mu_k) - \mu_k}{f'(\mu_k)-1}$$	Quadratic (at root)
Doubling algorithm	Recursive block-structured updates on auxiliary system	At least linear ($1/2^k$)

The robustness and convergence of these methods depend critically on the spectral properties of the associated matrices, the contractivity of the operator, and the structural features of the vector space (such as the M-matrix property or invertibility in the case of nonlinear operator equations) (Wang et al., 7 Jul 2025, Precup et al., 7 Feb 2025).

7. Outlook and Open Problems

The vector fixed point paradigm unifies and extends fixed point theory in several directions: handling multivariate errors, incorporating natural partial orders, dealing with set-valued and nonlinear correspondences, and generalizing to non-locally convex, modular, or $p$-convex structures. Open directions include:

• Nonexpansive and multivalued maps in vector S-metric and modular spaces (connections to selection theory and measure-valued operator equations).
• Further sharpness and computable bounds for convergence rates, especially in non-classical settings.
• Generalized stochastic vector fixed point equations and their tail behavior in probabilistic models (Burdzy et al., 2020).
• Understanding the structural requirements necessary for extension of classical fixed point properties (such as those guaranteed by Ekeland’s principle) to even more abstract algebraic/topological vector frameworks.

The interplay between function space geometry, partial ordering, and operator theory in the vector fixed point setting offers a rich ground for both theoretical advancement and practical computational algorithms.