A Short Note on Fractal Interpolation in the Space of Convex Lipschitz Functions

Abstract

In this short note, we consider fractal interpolation in the Banach space $V^{\theta }\left(I\right)$ of convex Lipschitz functions defined on a compact interval $I\subset \mathbb{R}$. To this end, we define an appropriate iterated function system and exhibit the associated Read–Bajraktarević operator T. We derive conditions for which T becomes a Ratkotch contraction on a closed subspace of $V^{\theta }\left(I\right)$, thus establishing the existence of fractal functions of class $V^{\theta }\left(I\right)$. An example illustrates the theoretical findings.

1. Introduction

Over the last few decades, the theory of fractal interpolation has been employed successfully to describe and model highly non-smooth functions naturally found in numerous applied situations. One of the main purposes of fractal interpolation or approximation is to describe intrinsically occurring complex geometric self-referential structures and to employ approximants that are well suited to be adapted to these types of structures. Usually, the approximants or interpolants belong to certain Banach spaces; see, for instance, reference [1] for a discussion of these issues in a (slightly) more general setting.

Many results in fractal interpolation come from the use of the Banach fixed point theorem, but recently, more general contraction-type results yielding a unique fixed point have been studied. Cf., for instance, references [2,3,4,5,6] for an albeit incomplete list of recent references. In [7], a review of such results is given in a convenient overall framework.

In this short note, we consider fractal interpolation in the Banach space of convex Lipschitz function using Rakotch contractions. Convex Lipschitz functions were introduced in [8] and play an important role in, for instance, optimization theory [9].

The organization of this paper is as follows. Section 2, introduces the fixed point theorems that are used in the sequel. Fractal interpolation is briefly presented in Section 3, and in Section 4, convex Lipschitz functions are introduced. In the final Section 5, the main result, namely fractal interpolation in the Banach space of convex Lipschitz functions, is discussed and the main theorem, namely the existence of fractal functions of class $V^{\theta }\left(I\right)$, is proven.

Throughout this paper, the following notations are employed: $\mathbb{N}$ denotes the set of positive integers; ${\mathbb{N}}_0$ is the set of non-negative integers; and ${\mathbb{N}}_n:=\{1,\dots ,n\}$ and ${\mathbb{N}}_{0,n}:=\{0,1,\dots ,n\}$ are initial segments of $\mathbb{N}$ and ${\mathbb{N}}_0$, respectively.

2. Some Important Theorems in Fixed Point Theory

Undoubtedly, one of the most important theorems in fixed point theory is the Banach contraction principle proven by Stefen Banach in 1922. The immense applicability of this results lies in the existence of a unique fixed point of a contractive mapping $f:X\to X$, where $(X,d)$ is a complete metric space. Numerous other contractive mappings have been introduced and studied over the last few decades. For a comparison of these generalized contractions, the interested reader may consult [10].

In this section, we introduce the Banach, the Rakotch, and the Matkowski fixed point theorem, with the later two being a generalization of the first.

Definition 1. 

Let $(X,d)$ be a complete metric space and let $f:X\to X$ be a map. If there exists a $\beta \in [0,1)$ such that for all $x,y\in X$,

$$d\left(f\right(x),f(y\left)\right)\le \beta d(x,y),$$

(1)

then the map f is called a (d-)Banach contraction on X.

Theorem 1 (Banach Fixed Point Theorem). 

Assume that $(X,d)$ is a complete metric space and $f:X\to X$ is a contractive map in the sense of (1). Then, f has a unique fixed point.

A more general contraction mapping is the following introduced in [11].

Definition 2 (Rakotch Contraction). 

Let $\psi :[0,\infty )\to [0,\infty )$ be a map such that for all $t>0$, we have $0\le \psi \left(t\right)<1$ and $\psi \left(t\right)$ is non-increasing. If there exists a map $f:X\to X$ such thatthen f is called a ψ-Rakotch contraction or, when the map ψ is clear from the context, just a Rakotch contraction.

Note that setting $\psi =\mathrm{constant}$ shows that every Banach contraction is a Rakotch contraction.

In the following, we also use an equivalent definition of Rakotch contraction presented in the theorem below. For a reference, see [4], p. 963.

Theorem 2 (Equivalent Definitions for Rakotch Contraction). 

Let $(X,d)$ be a complete metric space and $f:X\to X$ a map. Then, the following two definitions of Rakotch contractions are equivalent:

1. 

(a)   There exists a $\psi :[0,\infty )\to [0,\infty )$;

(b) 

$0\le \psi \left(t\right)<1$, $\forall t>0$;

(c) 

ψ is non-increasing;

(d) 

$d\left(f\right(x),f(y\left)\right)\le \psi \left(d\right(x,y\left)\right)d(x,y)$, $\forall x,y\in X$.

2. 

(a)   There exists a non-decreasing $\tau :\left(0,\infty \right)\to \left(0,\infty \right)$;

(b) 

$\tau \left(t\right)<t$, $\forall t>0$;

(c) 

The map ${\displaystyle \frac{\tau \left(t\right)}{t}}$ is non-increasing, $\forall t>0$;

(d) 

$d\left(f\right(x),f(y\left)\right)\le \tau \left(d\right(x,y\left)\right)$, $\forall x,y\in X$.

The next theorem whose proof can be found in [11] shows that a $\psi$-Rakotch contraction has a unique fixed point.

Theorem 3. 

Let $(X,d)$ be a complete metric space and $f:X\to X$ a ψ-Rakotch contraction. Then, f has a unique fixed point.

Not every Banach contraction is a Rakotch contraction. The following example demonstrates this (see, also, reference [4]). Let $X:=[0,\infty )$ and $f:X\to X$ be given by $f\left(x\right):={(1+x)}^{-1}$. Then, using the usual metric d on $\mathbb{R}$ restricted to X, one has

$$d\left(f\right(x),f(y\left)\right)\le \psi \left(d\right(x,y\left)\right)d(x,y),$$

where $\psi \left(t\right):={(1+t)}^{-1}$. Hence, f is a Rakotch contraction but not a Banach (d-)contraction on $X\subset \mathbb{R}$.

Definition 3 

(Matkowski Contraction [12]). Let $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ be a non-decreasing map such that for all $t>0$, we have $\underset{n\to \infty }{\lim }{\varphi }^n\left(t\right)=0$. If the map $f:X\to X$ satisfies

$$d\left(f\right(x),f(y\left)\right)\le \varphi \left(d\right(x,y\left)\right),forallx,y\in X,$$

then f is called a ϕ-Matkowski contraction or, when the map ϕ is clear from the context, just a Matkowski contraction.

For Matkowski contractions, we have the following result (see, e.g., [13]).

Theorem 4. 

Let $(X,d)$ be a complete metric space and $f:X\to X$ be a ϕ-Matkowski contraction. Then, the following properties hold:

(i) 

For all $t>0$, $\varphi \left(t\right)<t$;

(ii) 

The map f is continuous;

(iii) 

The map f has a unique fixed point.

3. Fractal Interpolation

In this section, we give a very brief and compact introduction to iterated function systems, fractals, and fractal interpolation. The interested reader can find more details about these concepts in [14,15,16,17] and the references given therein.

Let $(X,d)$ be a complete metric. Let $N>1$ be an integer and, for $i\in {\mathbb{N}}_N$, consider Banach contractions $f_i:X\to X$. The collection $\left\{X;f_1,\dots ,f_N\right\}$ is called an iterated function system (IFS) on X. Further, the collection of all nonempty compact subsets of X is denoted by $\mathcal{H}\left(X\right)$ and the Hausdorff–Pompeiu metric on $\mathcal{H}\left(X\right)$ is denoted by $h_d$, which are defined by

$$h_d(A,B):=\max \left\{\underset{x\in A}{\max }\underset{y\in B}{\min }d(x,y),\underset{y\in B}{\max }\underset{x\in A}{\min }d(x,y)\right\},\forall A,B\in \mathcal{H}\left(X\right).$$

It is known that the completeness of $(X,d)$ implies the completeness of $(\mathcal{H}\left(X\right),h_d)$. Define a set-valued mapping $\mathcal{F}:\mathcal{H}\left(X\right)\to \mathcal{H}\left(X\right)$ by

$$\mathcal{F}\left(E\right):=\bigcup _{i=1}^N{f}_i\left(E\right).$$

Then, $\mathcal{F}$ is contractive on $\mathcal{H}\left(X\right)$ with Lipschitz constant $\mathrm{Lip}\mathcal{F}:=\max \{\mathrm{Lip}{f}_i:i\in {\mathbb{N}}_N\}$. Here, $\mathrm{Lip}f$ of a mapping $f:X\to X$ is defined by

$$\mathrm{Lip}f:=\underset{x,y\in X,x\ne y}{\sup }{\displaystyle \frac{d\left(f\right(x),f(y\left)\right)}{d(x,y)}}.$$

Hence, by the Banach fixed point theorem, there exists a unique $A\in \mathcal{H}\left(X\right)$, called the attractor of the IFS, such that $\mathcal{F}\left(A\right)=A$ or, equivalently,

$$A=\bigcup _{i=1}^N{f}_i\left(A\right).$$

This latter equation reflects the fact that the attractor A is self-referential, and thus, in general, a fractal set.

Next, we consider a special class of IFSs, namely those whose attractors are graphs of continuous functions passing through a prescribed set of interpolation points. Such functions are termed fractal interpolation functions (FIFs) and were first introduced in [14,16].

For this purpose, let $I:=\left[x_0,x_N\right]\subset \mathbb{R}$, where N is an integer greater than one. Further, let $\Delta :=\left\{(x_i,y_i)\in I\times \mathbb{R}:i\in {\mathbb{N}}_{0,N}\right\}$ be a given set of interpolation points with $x_0<x_1<\dots <x_N$.

Define subintervals $I_i:=\left[x_{i-1},x_i\right]$ of I and contractive homeomorphisms $l_i:I\to I_i$, $i\in {\mathbb{N}}_N$, such that

$$\begin{array}{c}\hfill l_i\left(x_0\right)=x_{i-1},l_i\left(x_N\right)=x_i,\end{array}$$

(2)

and

$$\begin{array}{cc}\hfill & |l_i\left(x\right)-l_i\left(x^{\prime }\right)|\le a_i|x-x^{\prime }|,\mathrm{for}\mathrm{all}x,x^{\prime }\in I\mathrm{and}0\le a_i<1.\hfill \end{array}$$

Furthermore, let $K:=I\times \left[a,b\right]$ where $a<b$ are finite numbers with $y_0,y_1,\dots ,y_N\in \left[a,b\right]$. Thus, $\Delta =\{(x_0,y_0),\dots ,(x_N,y_N)\}\subset K$. In addition, we require continuous maps $F_i:K\to \left[a,b\right]$ with the property that

$$F_i(x_0,y_0)=y_{i-1}\mathrm{and}{F}_i(x_N,y_N)=y_i,i\in {\mathbb{N}}_N.$$

Finally, define maps

$$\begin{array}{cc}\hfill w_i:K& \to K,\hfill \\ \hfill w_i(x,y):=& (l_i\left(x\right),F_i(x,y)).\hfill \end{array}$$

If $G\in \mathcal{H}\left(K\right)$ is the (unique) attractor of the IFS $\left\{K;w_1,\dots ,w_n\right\}$ and also the graph of a continuous function $f:I\to \left[a,b\right]$ satisfying

$$f\left(x_i\right)=y_i,i\in {\mathbb{N}}_{0,N},$$

then f is called a fractal interpolation function (FIF) as it passes through the interpolation points $\Delta :=\left\{(x_i,y_i):i\in {\mathbb{N}}_{0,N}\right\}$. This is, for instance, the case when for some $M\ge 0$ and $s\in [0,1)$, each $F_i$ satisfies

$$|F_i(x,y)-F_i(x^{\prime },y^{\prime })|\le M|x-x^{\prime }|+s|y-y^{\prime }|,$$

(3)

for all $x,x^{\prime }\in I$ and $y,y^{\prime }\in [a,b]$.

A different approach to FIFs is given as follows. (Cf., for instance, [17].) To this end, define

$$\begin{array}{cc}\hfill C\left(I\right):=& \left\{g:I\to \left[a,b\right]:g\mathrm{continuous}\right\},\hfill \\ \hfill C^{*}\left(I\right):=& \left\{g\in C\left(I\right):g\left(x_0\right)=y_0,g\left(x_N\right)=y_N\right\},\hfill \\ \hfill C^{**}\left(I\right):=& \left\{g\in C^{*}\left(I\right):g\left(x_i\right)=y_i,i\in {\mathbb{N}}_{N-1}\right\}.\hfill \end{array}$$

When endowed with the norm ${\parallel g\parallel }_{\infty }^I:=\sup \left\{\right|g\left(x\right)|:x\in I\}$, the spaces $\left({C\left(I\right),\parallel ·\parallel }_{\infty }^I\right)$, $\left(C{\left(I\right)}^{*},{\parallel ·\parallel }_{\infty }^I\right)$, and $\left(C{\left(I\right)}^{**},{\parallel ·\parallel }_{\infty }^I\right)$ all become complete metric spaces.

Define the following operator, called a Read–Bajractarević (RB) operator:

$$\begin{array}{c}T:C^{*}\left(I\right)\to C\left(I\right),\\ Tf\left(x\right):=F_i(l_i^{-1}\left(x\right),f\left(l_i^{-1}\left(x\right)\right)),\mathrm{for}x\in \left[x_{i-1},x_i\right]\mathrm{and}i\in {\mathbb{N}}_N.\end{array}$$

(4)

Lemma 1 

([18]). For all $f\in C^{*}\left(I\right)$, $Tf\in C^{**}\left(I\right)$. Consequently, $T:C^{*}\left(I\right)\to C^{**}\left(I\right)$ and $T^n:=\underset{n-times}{\underbrace{T\circ \cdots \circ T}}:C^{**}\left(I\right)\to C^{**}\left(I\right)$, for all integers $n\ge 2$.

If the maps $F_i$ in (4) satisfy condition (3), then T has a unique fixed point $f^{*}$ in $C^{*}\left(I\right)$ and by Lemma (1), $f^{*}=T{f}^{*}\in C^{**}\left(I\right)$. Hence, $f^{*}$ interpolates the data set $\Delta$. Moreover, the fixed point $f^{*}$ satisfies the self-referential equation

$$f^{*}\left(x\right)=F_i(l_i^{-1}\left(x\right),f^{*}\left(l_i^{-1}\left(x\right)\right)),\mathrm{for}x\in l_i\left(I\right)=\left[x_{i-1},x_i\right],i\in {\mathbb{N}}_N.$$

It is worthwhile to point out that such fractal functions can also be constructed using more general contractivity conditions than (3); see, for instance, [4] for one of the first such constructions.

4. Convex Lipschitz Functions

In this section, we consider convex Lipschitz functions and prove that under a certain norm, they form a Banach space.

Definition 4 

([8]). Let $\theta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ and let $f:[x_0,x_N]\to \mathbb{R}$. If there exists a constant M such that for $x_0\le x<x+y\le x_N$ and $0\le \delta \le 1$, inequality (5) holds, then f is called a convex Lipschitz of order θ on the interval $[x_0,x_N]$.

$$|\Delta (x,y,\delta \left)\right|:=\left|f\right(x+\delta y)-(\delta f(x+y)+(1-\delta )f\left(x\right)\left)\right|\le M\theta \left(y\right)$$

(5)

By a change in variables, $z=x+y$, rearranging, and renaming variables again, the expression for $\left|\Delta (x,y,\delta )\right|$ in inequality (5) can be rewritten in the more geometrical form

$$|f(\delta x+(1-\delta )y)-\left(\delta f\left(x\right)+(1-\delta )f\left(y\right)\right)|\le M\theta (x-y),$$

(6)

where the difference appearing in the left-hand side expresses the difference between the line through $(x,f(x\left)\right)$ and $(y,f(y\left)\right)$ and the function f.

It is worth mentioning that if f belongs the the Zygmund class ${\mathsf{\Lambda }}_{\alpha }$ ([19], Chapter 2, §3), then f is a convex Lipschitz of order $\theta \left(x\right)=x^{\alpha }$, $x>0$.

Following [20], we denote by $V^{\theta }\left(I\right)$ the set of convex Lipschitz functions of order $\theta$ on the interval $I:=[x_0,x_N]$. Clearly, $V^{\theta }\left(I\right)$ is an $\mathbb{R}$-vector space. A norm on $V^{\theta }\left(I\right)$ is defined by setting

$${\left[f\right]}^{*}:=\underset{x_0\le x<x+y\le x_N}{\sup }{\displaystyle \frac{\left|f\right(x+\delta y)-(\delta f(x+y)+(1-\delta )f\left(x\right)\left)\right|}{\theta \left(y\right)}}$$

and then ${\parallel f\parallel }_{V^{\theta }}:={\parallel f\parallel }_{\infty }^I+{\left[f\right]}^{*}$.

The proof for the next result can be found in [20].

Theorem 5. 

The space $\left(V^{\theta }\left(I\right),{\parallel ·\parallel }_{V^{\theta }}\right)$ is a Banach space.

Now, let

$$V_{*}^{\theta }\left(I\right):=\left\{f\in V^{\theta }\left(I\right):f\left(x_0\right)=y_0,f\left(x_N\right)=y_N\right\},$$

and

$$V_{**}^{\theta }\left(I\right):=\left\{f\in V_{*}^{\theta }\left(I\right):f\left(x_i\right)=y_i,\mathrm{for}i\in {\mathbb{N}}_{0,N}\right\}.$$

Then, the above theorem implies the following corollary.

Corollary 1. 

$(V_{*}^{\theta }\left(I\right),\parallel ·{\parallel }_{V^{\theta }})$ and $(V_{**}^{\theta }\left(I\right),\parallel ·{\parallel }_{V^{\theta }})$ are complete metric spaces.

Proof. 

Let $\left\{f_n\right\}$ be a convergent sequence in $V_{*}^{\theta }\left(I\right)$ and assume that the limit of the sequence is $f\in V^{\theta }\left(I\right)$. Thus,

$$\forall ϵ>0\exists n_0\forall n\ge n_0:{\parallel f_n-f\parallel }_{V^{\theta }}<ϵ.$$

As ${\parallel f\parallel }_{V^{\theta }}={\parallel f\parallel }_{\infty }^I+{\left[f\right]}^{*}$, this in particular means that

$$\forall ϵ>0\exists n_0\forall n\ge n_0:{\parallel f_n-f\parallel }_{\infty }<ϵ$$

If $f\in V^{\theta }\left(I\right)\setminus V_{*}^{\theta }\left(I\right)$, then either $f\left(x_0\right)\ne y_0$ or $f\left(x_N\right)\ne y_N$. Without loss of generality, assume that $f\left(x_0\right)\ne y_0$ and set ${ϵ}_0|f\left(x_0\right)-y_0|>0$. Since $\left\{f_n\right\}\subset V_{*}^{\theta }\left(I\right)$, for all n, we have $f_n\left(x_0\right)=y_0$. Therefore, for all n, we must have $\parallel f_n{-f\parallel }_{\infty }\ge {ϵ}_0$, which contradicts the fact that $\underset{n\to \infty }{\lim }{f}_n=f$. So, we must have $f\in V_{*}^{\theta }\left(I\right)$. As a result, $V_{*}^{\theta }\left(I\right)$ is a closed subset of $V^{\theta }\left(I\right)$. Therefore, $(V_{*}^{\theta }\left(I\right),\parallel ·{\parallel }_{V^{\theta }})$ is a complete metric space. Similarly, it can be shown that $(V_{**}^{\theta }\left(I\right),\parallel ·{\parallel }_{V^{\theta }})$ is a complete metric space. □

5. Fractal Interpolation in the Space ${\mathit{V}}^{\mathit{\theta }}\left(\mathit{I}\right)$

In this section, besides the assumptions made in the previous section, we will also assume that $F_i$ are $\rho$-Matkowski contractions (with the same function $\rho$) with respect to the second variable, i.e., for some non-decreasing function $\rho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, where, for all $t>0$, we have $\underset{n\to \infty }{\lim }{\rho }^n\left(t\right)=0$, and every $F_i$ satisfies the following condition:

$$\forall x\in I\forall y,y^{\prime }\in \left[a,b\right]|F_i(x,y)-F_i(x,y^{\prime })|\le \rho (|y-y^{\prime }\left|\right).$$

It is worth pointing out the following theorem from [4].

Theorem 6. 

Under the given conditions on $l_i$, $F_i$, and K, the operator $T:C^{*}\left(I\right)\to C^{*}\left(I\right)$ has a unique fixed point $f^{*}\in C^{**}\left(I\right)$. Furthermore, the graph $G:=\left\{(x,f^{*}\left(x\right)):x\in I\right\}$ of $f^{*}$ is invariant with respect to the IFS $\left\{K;w_1,\dots ,w_N\right\}$, i.e., $G={\displaystyle \bigcup _{i=1}^N}{w}_i\left(G\right)$.

From now on, let us furthermore assume that

$$l_i\left(x\right)=a_{i}x+b_i\mathrm{and}{F}_i(x,y)={\alpha }_i\left(x\right)y+q_i\left(x\right),$$

where ${\alpha }_i:I\to \mathbb{R}$ is a multiplier in $V^{\theta }\left(I\right)$ and $q_i\in V^{\theta }\left(I\right)$. The $a_i$ and $b_i$ are determined by the conditions (2) imposed on $l_i$. We also set $|{\alpha }_i{|}_{\infty }:=\sup \left\{|{\alpha }_i\left(x\right)|:x\in I\right\}$ and ${\left|\alpha \right|}_{\infty }:=\underset{i\in {\mathbb{N}}_N}{\max }\sup \left\{|{\alpha }_i\left(x\right)|:x\in I\right\}$. Note that due to the specific structure of the mappings $F_i$, we have that $T:V^{\theta }\left(I\right)\to V^{\theta }\left(I\right)$.

Theorem 7. 

For all $f\in V_{*}^{\theta }\left(I\right)$, we have that $Tf\in V_{**}^{\theta }\left(I\right)$. Therefore, $T:V_{*}^{\theta }\left(I\right)\to V_{**}^{\theta }\left(I\right)$.

Proof. 

We already know that $T:V^{\theta }\left(I\right)\to V^{\theta }\left(I\right)$. Let $f\in V_{*}^{\theta }\left(I\right)$, which means that $f\left(x_0\right)=y_0$ and $f\left(x_N\right)=y_N$. On the other hand, we know that for all $x\in \left[x_{i-1},x_i\right]$, we have $Tf\left(x\right)=F_i(l_i^{-1}\left(x\right),f\left(l_i^{-1}\left(x\right)\right))$. Therefore,

$$\begin{array}{cc}\hfill Tf\left(x_i\right)& =F_i(l_i^{-1}\left(x_i\right),f\left(l_i^{-1}\left(x_i\right)\right))\hfill \\ \hfill & =F_i(x_N,f\left(x_N\right))\left(\mathrm{since}{l}_i:I\to I_i\mathrm{is}\mathrm{a}\mathrm{homeomorphism}\right)\hfill \\ \hfill & =F_i(x_N,y_N)\left(\mathrm{since}f\in V_{*}^{\theta }\left(I\right)\right)\hfill \\ \hfill & =y_i\left(\mathrm{property}\mathrm{of}{F}_i\right).\hfill \end{array}$$

Hence, $Tf\left(x_i\right)=y_i$ for $i\in {\mathbb{N}}_N$. Similarly, it can be seen that $Tf\left(x_0\right)=y_0$. We conclude that $Tf\in V_{**}^{\theta }\left(I\right)$. □

Theorem 8. 

If $\max \left\{{\left|\alpha \right|}_{\infty },\underset{i\in {\mathbb{N}}_N}{\max }\left\{\right|{\alpha }_i{|}_{\infty }\sup \{{\displaystyle \frac{\theta \left(y\right)}{\theta \left(a_{i}y\right)}}:y\in I\}\}\right\}<1$, then $T:V_{*}^{\theta }\left(I\right)\to V_{**}^{\theta }\left(I\right)\subseteq V_{*}^{\theta }\left(I\right)$ is a Banach contraction.

This theorem was proven indirectly in [20] and the result appears in Theorem 2.7 under slightly different conditions. The proof provided there remains the same under the current setting including the extension to non-constant scaling factors ${\alpha }_i:I\to \mathbb{R}$. In addition, we corrected the statement in ([20], Theorem 2.7) as the term with the sup over $y\in I$ is missing, which would make the contractivity condition presented there dependent on y.

The following Theorem (9) provides conditions under which T becomes a Rakotch contraction.

Theorem 9. 

Assume that

$${|\beta |}_{\infty }:=\underset{i\in {\mathbb{N}}_N}{\max }\left\{|{\alpha }_i{|}_{\infty }\underset{x_0\le y\le x_N}{\sup }{\displaystyle \frac{\theta \left(y\right)}{\theta \left(a_{i}y\right)}}\right\}<1.$$

Let the maps $F_i$ be Rakotch contractions with respect to the second variable for the same function τ, i.e., for some non-decreasing function $\tau :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\tau \left(t\right)<t$ and ${\displaystyle \frac{\tau \left(t\right)}{t}}$ non-increasing for all $t>0$, we have that

$$\forall x\in I\forall y,y^{\prime }\in \left[a,b\right]:|F_i(x,y)-F_i(x,y^{\prime })|\le \tau (|y-y^{\prime }\left|\right).$$

Then, the operator $T:V_{*}^{\theta }\left(I\right)\to V_{**}^{\theta }\left(I\right)\subseteq V_{*}^{\theta }\left(I\right)$ is a Rakotch contraction.

Proof. 

First, we show that for all $f,g\in V^{\theta }\left(I\right)$, ${\parallel Tf-Tg\parallel }_{\infty }^I\le {\tau (\parallel f-g\parallel }_{\infty }^I)$. To this end, let $f,g\in V^{\theta }\left(I\right)$. Then,

$$\begin{array}{cc}\hfill {\parallel Tf-Tg\parallel }_{\infty }^I& =\underset{x\in I}{\sup }|Tf\left(x\right)-Tg\left(x\right)|\hfill \\ \hfill & =\underset{i\in {\mathbb{N}}_N}{\max }\underset{x\in I_i}{\sup }|Tf\left(x\right)-Tg\left(x\right)|\hfill \\ \hfill & =\underset{i\in {\mathbb{N}}_N}{\max }\underset{x\in I_i}{\sup }|F_i(l_i^{-1}\left(x\right),f\left(l_i^{-1}\left(x\right)\right))-F_i(l_i^{-1}\left(x\right),g\left(l_i^{-1}\left(x\right)\right))|\hfill \\ \hfill & \le \underset{i\in {\mathbb{N}}_N}{\max }\underset{x\in I_i}{\sup }\tau \left(\right|f\left(l_i^{-1}\left(x\right)\right)-g\left(l_i^{-1}\left(x\right)\right)\left|\right)\hfill \\ \hfill & \le \underset{i\in {\mathbb{N}}_N}{\max }\tau (\underset{x\in I_i}{\sup }|f\left(l_i^{-1}\left(x\right)\right)-g\left(l_i^{-1}\left(x\right)\right)\left|\right)\left(\mathrm{since}\tau \mathrm{is}\mathrm{non}-\mathrm{decreasing}\right)\hfill \\ \hfill & =\underset{i\in {\mathbb{N}}_N}{\max }{\tau (\parallel f-g\parallel }_{\infty }^I{)=\tau (\parallel f-g\parallel }_{\infty }^I).\hfill \end{array}$$

where in the penultimate equality, we used $\underset{x\in I_i}{\sup }|f\left(l_i^{-1}\left(x\right)\right)-g\left(l_i^{-1}\left(x\right)\right){|=\parallel f-g\parallel }_{\infty }^I)$ for $i=1,\dots ,N$.

Next, we show that for all $f,g\in V^{\theta }\left(I\right)$, ${\left[Tf-Tg\right]}^{*}\le {\left|\beta \right|}_{\infty }{\left[f-g\right]}^{*}$ where ${\left|\beta \right|}_{\infty }:=\underset{i\in {\mathbb{N}}_N}{\max }\left\{|{\alpha }_i{|}_{\infty }\underset{x_0\le y\le x_N}{\sup }{\displaystyle \frac{\theta \left(y\right)}{\theta \left(a_{i}y\right)}}\right\}$. Let $f,g\in V^{\theta }\left(I\right)$ and $0\le \delta \le 1$. Then, setting $h:=f-g$, we have for $i\in {\mathbb{N}}_N$,

$$\begin{array}{c}\hfill {\left[Th\right]}^{*}=\underset{x_{i-1}\le \tilde{x}<\tilde{x}+\tilde{y}\le x_i}{\sup }{\displaystyle \frac{|\left(Th\right)(\tilde{x}+\delta \tilde{y})-(\delta \left(Th\right)(\tilde{x}+\tilde{y})+(1-\delta )\left(Th\right)\left(\tilde{x}\right))|}{\theta \left(y\right)}}.\end{array}$$

Substituting the expression for the RB operator T into the above equation and using the fact that each $l_i$ is bijective with $x:=l_i^{-1}\left(\tilde{x}\right)\in I$ and $l_i^{-1}(\tilde{x}+\tilde{y})=x+y\in I$ with $y:=\tilde{y}/a_i$, yields

$$\begin{array}{cc}\hfill {\left[Th\right]}^{*}& \le \underset{i\in {\mathbb{N}}_N}{\max }\underset{x_0\le x<x+y\le x_N}{\sup }\hfill \\ \hfill & \left\{|{\alpha }_i\left(x\right)|{\displaystyle \frac{\left|h\right(x+\delta y)-(\delta h(x+y)+(1-\delta )h\left(x\right)\left)\right|}{\theta \left(y\right)}}·{\displaystyle \frac{\theta \left(y\right)}{\theta \left(a_{i}y\right)}}\right\}\hfill \\ \hfill & \le {\left|\beta \right|}_{\infty }{\left[f-g\right]}^{*}.\hfill \end{array}$$

Finally, we establish that T is a Rakotch contraction. To this end, let $f,g\in V^{\theta }\left(I\right)$. Then,

$$\begin{array}{cc}\hfill {\parallel Tf-Tg\parallel }_{V^{\theta }}& ={\parallel Tf-Tg\parallel }_{\infty }^I+{\left[Tf-Tg\right]}^{*}\hfill \\ \hfill & \le {\tau (\parallel f-g\parallel }_{\infty }^I)+{\left[Tf-Tg\right]}^{*}\hfill \\ \hfill & \le {\tau (\parallel f-g\parallel }_{V^{\theta }})+{\left[Tf-Tg\right]}^{*}\hfill \\ \hfill & \le {\tau (\parallel f-g\parallel }_{V^{\theta }}{)+|\beta |}_{\infty }{\left[f-g\right]}^{*}\hfill \\ \hfill & \le {\tau (\parallel f-g\parallel }_{V^{\theta }}{)+|\beta |}_{\infty }{\parallel f-g\parallel }_{V^{\theta }}\left({\left[f-g\right]}^{*}\le {\parallel f-g\parallel }_{V^{\theta }}\right),\hfill \end{array}$$

where proceeding from the first inequality to the second inequality above, we used the above since $\tau$ is non-decreasing and ${\parallel f-g\parallel }_{\infty }^I\le {\parallel f-g\parallel }_{V^{\theta }}$.

Hence, for ${\parallel f-g\parallel }_{V^{\theta }}\ne 0$, we have

$$\begin{array}{cc}\hfill {\parallel Tf-Tg\parallel }_{V^{\theta }}& \le {\displaystyle \frac{\tau (\parallel f-g{\parallel }_{V^{\theta }})}{{\parallel f-g\parallel }_{V^{\theta }}}}{\parallel f-g\parallel }_{V^{\theta }}+{\left|\beta \right|}_{\infty }{\parallel f-g\parallel }_{V^{\theta }}\hfill \\ \hfill & \le \max \left\{{\displaystyle \frac{\tau (\parallel f-g{\parallel }_{V^{\theta }})}{{\parallel f-g\parallel }_{V^{\theta }}}},{\left|\beta \right|}_{\infty }\right\}{\parallel f-g\parallel }_{V^{\theta }}\hfill \end{array}$$

Now, for all $t>0$, ${\displaystyle \frac{\tau \left(t\right)}{t}}<1$ is non-increasing and ${\left|\beta \right|}_{\infty }<1$. Define $\sigma \left(t\right):=t\max \left\{{\displaystyle \frac{\tau \left(t\right)}{t}},{\left|\beta \right|}_{\infty }\right\}$. Then, for all $t>0$, the map ${\displaystyle \frac{\sigma \left(t\right)}{t}}<1$ and is non-increasing. Therefore, T is a Rakotch contraction. □

Example 1. 

The result in the Theorem 9 establishes the existence of fractal functions of class $V^{\theta }\left(I\right)$. Here, we provide an example for such functions. For illustrative purposes, we choose convex Lipschitz functions of order $0<\alpha \le 1$, i.e., elements of the Zygmund class ${\mathsf{\Lambda }}_{\alpha }$. Let $I:=[0,1]$ and suppose $\Delta :=\{(0,0),({\displaystyle \frac{1}{2}},1),(1,0)\}$. Further, we assume that the scaling factors ${\alpha }_i$, $i=1,2$, are given by the two Weierstrass functions

$${\alpha }_1:I\to \mathbb{R},x\mapsto \gamma \sum _{n=0}^{\infty }{2}^{-\alpha n}\sin \left(2^n\pi x\right)$$

and

$${\alpha }_2:I\to \mathbb{R},x\mapsto \gamma \sum _{n=0}^{\infty }{2}^{-\alpha n}\sin (2^n\pi x+\pi ),$$

respectively, for some positive constant $\gamma \le 2^{-\alpha -1}$. It is known that ${\alpha }_1$ and ${\alpha }_2$ are convex Lipschitz of order α [8]. This choice ensures that for an $f\in {\mathsf{\Lambda }}_{\alpha }$, the product ${\alpha }_{i}f$ is also in ${\mathsf{\Lambda }}_{\alpha }$. Moreover, set $q_1:I\to \mathbb{R}$, $x\mapsto x^{\alpha }$, and $q_2:I\to \mathbb{R}$, $x\mapsto {(1-x)}^{\alpha }$. It is straight-forward to verify that all joined-up conditions are satisfied and that the RB operator T as defined above maps $V_{*}^{\theta }\left(I\right)\to V_{**}^{\theta }\left(I\right)$ with $\beta ={\displaystyle \frac{1}{2}}$.

6. Conclusions

In this paper, we introduced the concept of fractal interpolation on the Banach space $V^{\theta }\left(I\right)$ of convex Lipschitz functions of order $\theta$ defined on a compact interval $I\subset \mathbb{R}$. To achieve fractal interpolation, we introduced a Read–Bajrakterić operator T on a closed subspace of $V^{\theta }\left(I\right)$ and—in order to establish a more general result—derived conditions such that T becomes a Rakotch contraction. This includes and also corrects the case of Banach contractions considered in [20]. Our result then proves the existence of fractal functions of class $V^{\theta }\left(I\right)$. A class of examples is also provided.