Fuzzy-Valued Fractal Interpolation

• The paper establishes the construction of fuzzy-valued fractal interpolation functions as unique fixed points in fuzzy-number spaces, generating self-referential attractors.
• It employs iterated function system methods with contractive homeomorphisms and Lipschitz fuzzy mappings to ensure convergence and Hölder continuity of the interpolant.
• The framework decomposes the fuzzy interpolant into real-valued fractal functions via level sets, distinguishing it from ordinary fuzzy or fractal interpolation.

A fuzzy-valued fractal interpolation function is an interpolation function whose values are fuzzy numbers and whose graph is generated by a self-referential fractal mechanism. In the formulation developed in "Construction and properties of fuzzy-valued fractal interpolation function by using iterated function system" (Yun et al., 17 Jul 2025), the object is designed to interpolate a prescribed data set $P=\{(x_i,u_i)\}$ with $u_i\in\mathbb{R}^F$, to realize its graph as the attractor of an iterated function system (IFS), and to admit a regularity theory in the form of Hölder continuity. The construction is motivated by phenomena that exhibit both irregularity and uncertainty: irregularity is represented through fractal interpolation, while uncertainty is represented through fuzzy numbers. In this sense, fuzzy-valued fractal interpolation is positioned between ordinary fuzzy interpolation and ordinary fractal interpolation rather than as a minor variant of either (Yun et al., 17 Jul 2025).

1. Conceptual setting and mathematical object

The underlying data are fuzzy-number-valued samples

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$

Here $\mathbb{R}^F$ denotes the space of fuzzy numbers over $\mathbb{R}$. The fuzzy numbers used in the construction are assumed to be normal, upper semicontinuous, convex, and compactly supported. For each $u\in\mathbb{R}^F$, the $\alpha$-level set is an interval

$$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$

A fuzzy-valued fractal interpolation function is a continuous mapping $f:I\to\mathbb{R}^F$, with $I=[x_0,x_n]$, such that $u_i\in\mathbb{R}^F$0 for all nodes and such that $u_i\in\mathbb{R}^F$1 satisfies a self-referential equation on each subinterval $u_i\in\mathbb{R}^F$2. The function is not introduced merely as a pointwise fuzzy extension of a crisp interpolant; rather, it is defined through an IFS whose attractor is exactly the graph

$u_i\in\mathbb{R}^F$3

This framework addresses a common distinction. Ordinary fuzzy interpolation captures uncertainty but does not encode self-similar or fractal geometry. Ordinary fractal interpolation captures fractal structure but does not directly interpolate fuzzy-number-valued data. The fuzzy-valued fractal interpolation function combines both requirements in a single fixed-point construction (Yun et al., 17 Jul 2025).

2. Metric structure, level sets, and IFS data

The paper equips $u_i\in\mathbb{R}^F$4 with the supreme metric

$u_i\in\mathbb{R}^F$5

With this metric, $u_i\in\mathbb{R}^F$6 is a complete metric space. For fuzzy-valued continuous functions $u_i\in\mathbb{R}^F$7, the induced metric is

$u_i\in\mathbb{R}^F$8

and $u_i\in\mathbb{R}^F$9 is also complete (Yun et al., 17 Jul 2025).

The interval domain is decomposed as

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$0

For each $$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$1, one chooses a contractive homeomorphism

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$2

satisfying the endpoint conditions

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$3

These maps compress the whole interval onto its constituent subintervals.

The vertical component of the construction is provided by fuzzy-valued maps

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$4

where $$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$5 is a vertical scaling factor and $$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$6 is Lipschitz. The interpolation conditions are

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$7

An example given in the paper is

$$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$8

with the auxiliary terms chosen so that the endpoint conditions hold.

On the product space $$P=\{(x_i,u_i)\in\mathbb{R}\times\mathbb{R}^F\mid i=0,1,\dots,n\},\qquad x_0<x_1<\cdots<x_n.$$9, the paper first considers

$\mathbb{R}^F$0

and then introduces the maps

$\mathbb{R}^F$1

A key structural step is the proof that there exists a metric $\mathbb{R}^F$2 equivalent to $\mathbb{R}^F$3 such that each $\mathbb{R}^F$4 is contractive. Consequently,

$\mathbb{R}^F$5

forms a hyperbolic iterated function system (Yun et al., 17 Jul 2025).

3. Fixed-point construction and attractor characterization

The interpolation space is

$\mathbb{R}^F$6

Within this space, the construction is driven by the Read–Bajraktarević operator

$\mathbb{R}^F$7

Using the specific form of $\mathbb{R}^F$8, this becomes

$\mathbb{R}^F$9

The operator $\mathbb{R}$0 is contractive in the metric $\mathbb{R}$1, with contraction constant

$\mathbb{R}$2

By Banach’s fixed point theorem, there exists a unique fixed point $\mathbb{R}$3 satisfying

$\mathbb{R}$4

This fixed point is the fuzzy-valued fractal interpolation function (Yun et al., 17 Jul 2025).

The same object can be described geometrically through the Hutchinson–Barnsley operator

$\mathbb{R}$5

where $\mathbb{R}$6 denotes the family of nonempty compact subsets endowed with the Hausdorff metric. Since the $\mathbb{R}$7 are contractive, $\mathbb{R}$8 has a unique fixed point $\mathbb{R}$9, the attractor of the IFS.

The paper proves that the graph of the fixed point is invariant: $u\in\mathbb{R}^F$0 By uniqueness of the attractor,

$u\in\mathbb{R}^F$1

Thus the fuzzy-valued fractal interpolation function is simultaneously a fixed point in function space and an attractor in the hyperspace of compact subsets. This equivalence is central: the interpolation property is encoded analytically by equation $u\in\mathbb{R}^F$2 and geometrically by graph invariance under the IFS.

4. Level-set decomposition into ordinary fractal interpolation functions

A fundamental structural theorem of the construction is that the fuzzy-valued interpolant decomposes levelwise into ordinary real-valued fractal interpolation functions. For each $u\in\mathbb{R}^F$3, one defines the real-valued data sets

$u\in\mathbb{R}^F$4

$u\in\mathbb{R}^F$5

The paper then introduces real-valued maps on $u\in\mathbb{R}^F$6 associated with the lower and upper endpoints of the level sets, and denotes the resulting ordinary fractal interpolation functions by $u\in\mathbb{R}^F$7 and $u\in\mathbb{R}^F$8. Their fixed-point equations are

$u\in\mathbb{R}^F$9

$\alpha$0

Taking $\alpha$1-level sets in the fuzzy fixed-point equation yields

$\alpha$2

which is equivalent, at the endpoint level, to

$\alpha$3

$\alpha$4

By uniqueness of the fixed point in the corresponding contraction scheme, the lower and upper endpoint functions of the fuzzy-valued interpolant coincide with the ordinary fractal interpolation functions built from the $\alpha$5-level data: $\alpha$6 This is formulated in the paper as Theorem 4 (Yun et al., 17 Jul 2025).

The result has a precise interpretive significance. The fuzzy-valued fractal interpolation function is not merely associated with level sets; it is exactly a family of ordinary fractal interpolation functions indexed by $\alpha$7, one for the lower endpoint and one for the upper endpoint of each level interval. A plausible implication is that many analytical properties of the fuzzy-valued object can be studied by reducing them to the corresponding properties of these real-valued endpoint interpolants.

5. Hölder continuity and regularity theory

The regularity analysis proceeds in two steps. First, the paper studies ordinary real-valued fractal interpolation functions $\alpha$8 associated with data $\alpha$9 satisfying

$$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$0

The assumptions include $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$1, Lipschitz continuity of each $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$2, a uniform bound $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$3, endpoint conditions

$$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$4

and the requirement that each $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$5 is a similitude: $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$6

With

$$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$7

Theorem 5 states that if $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$8 for all $$[u]^\alpha=[u^-(\alpha),u^+(\alpha)],\qquad \alpha\in[0,1].$$9, then there exists a constant $f:I\to\mathbb{R}^F$0, independent of $f:I\to\mathbb{R}^F$1, such that

$f:I\to\mathbb{R}^F$2

and $f:I\to\mathbb{R}^F$3 is Hölder continuous: $f:I\to\mathbb{R}^F$4 for some constants $f:I\to\mathbb{R}^F$5 and $f:I\to\mathbb{R}^F$6 independent of $f:I\to\mathbb{R}^F$7 (Yun et al., 17 Jul 2025).

The proof employs the recursive structure of the interpolation equation and a geometric-series argument based on

$f:I\to\mathbb{R}^F$8

Three regimes are distinguished. If $f:I\to\mathbb{R}^F$9, equivalently $I=[x_0,x_n]$0, the function is Lipschitz, so $I=[x_0,x_n]$1. If $I=[x_0,x_n]$2, a logarithmic estimate still yields Hölder continuity with some $I=[x_0,x_n]$3. If $I=[x_0,x_n]$4, the paper derives Hölder continuity with an exponent given in terms of the displayed formula in the derivation. The exact exponent is therefore governed by the balance between vertical scaling and horizontal contraction.

The fuzzy-valued case follows by applying this real-valued theory to the lower and upper endpoint functions $I=[x_0,x_n]$5 and $I=[x_0,x_n]$6. Because each fuzzy number $I=[x_0,x_n]$7 has compact support, the endpoint data are bounded; because the $I=[x_0,x_n]$8 are Lipschitz, their level-set endpoint functions inherit the same Lipschitz control. Theorem 6 then states that the fuzzy-valued fractal interpolation function $I=[x_0,x_n]$9 is Hölder continuous: $u_i\in\mathbb{R}^F$00 for some constant $u_i\in\mathbb{R}^F$01 and some $u_i\in\mathbb{R}^F$02, with

$u_i\in\mathbb{R}^F$03

This regularity result places the construction within the standard analytical regime expected of fractal interpolation while preserving fuzzy-number-valued uncertainty (Yun et al., 17 Jul 2025).

6. Related developments and scope of the framework

A closely related development is the construction of fuzzy valued recurrent fractal interpolation functions by means of a recurrent iterated function system (RIFS) in "Construction of fuzzy valued recurrent fractal interpolation functions and their properties" (Choe et al., 17 Jul 2025). That paper starts from the same basic motivation—data with both local self-similarity and uncertainty—but replaces the ordinary IFS architecture by a recurrent structure determined by a row-stochastic matrix $u_i\in\mathbb{R}^F$04. The resulting fuzzy-valued RFIF is again the unique fixed point of an operator on $u_i\in\mathbb{R}^F$05, and its graph is the attractor of the associated hyperbolic RIFS.

The recurrent construction differs from the non-recurrent fuzzy-valued fractal interpolation function in one essential respect: the horizontal dynamics are organized through domains $u_i\in\mathbb{R}^F$06 that may consist of unions of several subintervals, thereby encoding local recurrence rather than a single global-to-local interval mapping. The RFIF satisfies a self-referential equation

$u_i\in\mathbb{R}^F$07

and the paper proves Hölder continuity for the resulting fuzzy-valued interpolant. It additionally establishes stability under perturbations of the nodes $u_i\in\mathbb{R}^F$08, the fuzzy values $u_i\in\mathbb{R}^F$09, and the vertical scaling factors $u_i\in\mathbb{R}^F$10, with explicit estimates involving the contraction parameter $u_i\in\mathbb{R}^F$11, where $u_i\in\mathbb{R}^F$12.

Within the immediate research trajectory, the non-recurrent fuzzy-valued fractal interpolation function (Yun et al., 17 Jul 2025) provides the foundational IFS-based model, while the recurrent version (Choe et al., 17 Jul 2025) extends the same basic synthesis of fuzziness and fractality to a locally recursive setting and adds perturbation theory. This suggests a broader program in which fuzzy-valued interpolation is studied not only as an uncertainty-aware generalization of classical interpolation, but also as a family of self-referential constructions whose analytical properties can be tracked through IFS or RIFS machinery.

A common misconception is to identify fuzzy-valued fractal interpolation with either ordinary fuzzy interpolation or ordinary fractal interpolation applied componentwise. The published construction does not support that simplification. Its defining features are the fixed-point equation in a fuzzy-valued function space, the attractor characterization of the graph, and the theorem that each $u_i\in\mathbb{R}^F$13-level endpoint is itself an ordinary fractal interpolation function. Those three ingredients together distinguish the subject as a specific interpolation theory rather than an ad hoc combination of two unrelated methods (Yun et al., 17 Jul 2025).