Non-Stationary Fractal Functions on the Sierpiński Gasket

Abstract

Following the work on non-stationary fractal interpolation (Mathematics 7, 666 (2019)), we study non-stationary or statistically self-similar fractal interpolation on the Sierpiński gasket (SG). This article provides an upper bound of box dimension of the proposed interpolants in certain spaces under suitable assumption on the corresponding Iterated Function System. Along the way, we also prove that the proposed non-stationary fractal interpolation functions have finite energy.

1. Introduction and Background

The concept of fractals naturally emerges in the examination of non-linear functions. The fractal theory offers different approaches to describe, quantify, and investigate natural phenomena, such as lightning, rivers, trees, mountains, heart sounds, and so on. One effective approach to creating fractals involves identifying the fixed points of contractive operators within a specific class of Iterated Function Systems (IFSs) [1].

Constructive approximation theory is based on classical interpolation techniques, which can be substituted by fractal interpolation. This approach is utilized in various domains, including computer graphics, picture compression, and multiwavelets. A fractal set is defined as the graph of a continuous function, and which interpolates a given collection of data, which is known as a fractal interpolation function (FIF). Barnsley presented this concept in 1986 [2], and it gained popularity through the work of Navascués, Chand, and their respective research groups [3,4,5,6].

The attractor of a single Iterated Function System is a self-similar fractal, meaning that its local shape is in line with regard to the appropriate contraction maps. When approximating complex curves with non-uniform self-similarity (e.g., simulating stochastic processes such as Brownian motion), it is logical to consider creating fractals with varying levels of structure. Barnsley et al. [7] examined a collection of infinitely many Iterated Function Systems (IFSs) in order to generate fractals with a variable T. A T-variable fractal is a set that, depending on the magnification level, produces at most T-distinct local patterns at each level. A uniform contractive and uniformly bounded sequence of IFSs is postulated.

By employing the concept of a non-stationary subdivision technique, David and colleagues [8,9] recently developed a more generic class of sequences of operators, and investigated their convergence features. The IFS system sequence has been applied to non-stationary subdivision schemes by the authors of [8]. In doing so, the uniform contractivity constraint that is imposed in the production of T-variable fractals is relaxed. Furthermore, the freedom to obtain attractors (fractals) with distinct patterns at various dimensions is provided by the employment of various contractive operators.

The investigation of non-stationary fractal functions on support graphs is the primary subject of this paper. As a result of the research conducted by Barnsley [2] and Levin et al. [9], Massopust conducted research on non-stationary FIFs in 2019 [10]. On the one hand, despite the fact that it appears to be natural, and has an extensive range of applications, the utilization of a sequence of IFSs in the aforementioned subjects has received a significantly lower amount of attention. On the other hand, Celik et al. developed the construction of fractal functions on SG in 2008 [11], and thereafter, its analytic properties are examined by numerous researchers [12,13,14,15]. Using the Sierpiński gasket, Sahu and Priyadarshi [15] determined the upper bounds for the box-counting dimension of graphs of harmonic functions and fractal interpolation functions. The study of vector-valued FIFs on SG was conducted by Navascués et al. [16] in recent times. In addition to demonstrating the smoothness quality of vector-valued FIF on SG, they also launched a systematic approximation theory by making use of the vector-valued FIF on SG.

Our current study is dedicated to the development of the non-stationary FIF on the Sierpiński gasket. There is a significant impact that the parameters of the IFS have on the characteristics and shapes of the FIFs that correspond to them [5]. The identification of the IFS parameters and the determination of the fractal dimension of the non-stationary FIFs in SG is, therefore, a question that could be considered intriguing. A challenge like the one described above is among the most difficult questions in the field of fractal theory. Determining the exact upper limit of the box-counting and Hausdorff dimension presents an additional challenge that adds to the complexity of the problem that was presented earlier. As part of our recent research [17], we have conducted an investigation into the dimensional analysis of fractal functions. Additionally, the creation of a non-stationary fractal function in oscillation spaces on SG are covered in this article.

According to David et al. [9] and Celik et al. [11], our construction here is based on non-stationary IFS. On the other hand, it is important to mention that the purpose of our work is different. The analysis on SG is the primary focus of our study and, in particular, it connects the theory of analysis on fractals from a theoretical perspective.

The article is structured as follows. Section 2 begins with a review of the concept of a system of function systems and trajectories. Section 3 describes how to construct a non-stationary fractal function on SG. Section 4 presents the results of a thorough examination of the suggested architecture in oscillation spaces, as well as the dimension findings. Finally, we demonstrate in Section 5 that the suggested fractal functions on SG have finite energy.

2. Preliminaries

Let us begin by examining the basic principles of function systems sequences and the backward trajectories of sequences of contraction maps. For a more comprehensive explanation, we recommend referring to [8,18,19].

2.1. Sequences of Transformations and Trajectories

Let $(E,d)$ be a complete metric space. A map $f:E\to E$ is known as Lipschitz in case a constant $c\ge 0$, such that $d(f\left({\upsilon }_1\right),f\left({\upsilon }_2\right))\le cd({\upsilon }_1,{\upsilon }_2)$ for all ${\upsilon }_1,{\upsilon }_2\in E$. If the constant $0\le c<1$, the map is known as contraction.

Let $H\left(E\right)$ be the collection of all non-empty compact subsets of E. Let Y, Z ∈ $H\left(E\right)$ and Hausdorff distance is defined between Y and Z as

$$\begin{array}{c}\hfill h(Y,Z)=max\{d(Y,Z),d(Z,Y)\},\mathrm{where}d(Y,Z)=\underset{\upsilon \in Y}{sup}\underset{\omega \in Z}{inf}d(\upsilon ,\omega ).\end{array}$$

Denote ${\mathbb{N}}_k=\{1,2,3,\cdots ,k\}$. The set $\{E;{\omega }_i:i\in {\mathbb{N}}_k\}$ is called an IFS on E, where ${\omega }_i$ are continuous functions on E. If ${\omega }_i$’s are contractions, the above IFS is contractive. For this contractive IFS, the Hutchinson map $B:H\left(E\right)\to H\left(E\right)$ is defined as

$$B\left(Y\right)=\underset{i=1}{\stackrel{N}{\cup }}{\omega }_i\left(Y\right).$$

Note that B is a contraction on $H\left(E\right)$.

The Banach fixed point theorem asserts a unique $P\in H\left(E\right)$ such that $P=B\left(P\right)$. This P is referred as the attractor of the IFS (see [2,11] for more information). The attractor P can be derived as the limit of the iterative process $P_k=B\left(P_{k-1}\right),k\in \mathbb{N}$.

For the concept of trajectories, we look at an order of transformations ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ on the set E. A subset $\mathcal{A}$ of E is called an invariant set of the sequence ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ if for all $k\in \mathbb{N}$ and $\forall \upsilon \in \mathcal{A}$, $G_k\left(\upsilon \right)\in \mathcal{A}$. To obtain an invariant set from a sequence of transformations ${\left\{G_k\right\}}_{k\in \mathbb{N}}$, one may consult ([9], Lemma 3.7).

For a given complete metric space $(E,d)$, we have considered the sequence of set valued maps $B_k:H\left(E\right)\to H\left(E\right)$ connected with the Systems of Function Systems (SFSs) as

$$B_k\left(Y\right)=\bigcup _{i=1}^{{\eta }_k}{\omega }_{i,k}\left(Y\right),Y\in H\left(E\right),$$

(1)

where $B_k=\{{\omega }_{1,k},{\omega }_{2,k},\cdots ,{\omega }_{{\eta }_k,k}\}$ is a family of contractions form an IFS on $(E,d)$.

Definition 1

(Forward and Backward Trajectories). Let E be a metric space and ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ be a sequence of Lipschitz maps on E. The forward and backward procedures are defined procedures,

$${\mathrm{\Phi }}_k:=G_k\circ G_{k-1}\circ \dots G_1\mathit{and}{\mathsf{\Psi }}_k:=G_1\circ G_2\circ \dots G_k.$$

The convergence of both type trajectories was studied in [9]. It is discerned that the limits of forward trajectories do not lead to a new class of fractals. However, it is discovered that mild conditions are required for convergence of backward trajectories.

For detailed investigations of the trajectories and their convergence results, the reader is asked to see [9].

2.2. Non-Stationary or Statistically Self-Similar FIFs

In this section, we revisit the construction of non-stationary FIFs. For more discussions, we refer the reader to [10]. Let $I=[a,b]$ and $f:I\to \mathbb{V}$ be a continuous function, where $\mathbb{V}$ may be $\mathbb{R}$ or ${\mathbb{R}}^n$ or $\mathbb{C}$ with their standard norm. Consider $I_i=[{\upsilon }_i,{\upsilon }_{i+1}]$ and the partition of I as $\{{\upsilon }_1,{\upsilon }_2,\cdots ,{\upsilon }_{\eta }:a={\upsilon }_1<{\upsilon }_2<\cdots <{\upsilon }_{\eta }=b\}$. For each $i\in {\mathbb{N}}_{\eta -1}:=\{1,2,\cdots ,\eta -1\}$, let $A_i:I\to I_i$ be affine maps such that

$$A_i\left({\upsilon }_1\right)={\upsilon }_i,A_i\left({\upsilon }_{\eta }\right)={\upsilon }_{i+1}.$$

(2)

Let $K=I\times \mathbb{V}$. For $k\in \mathbb{N},$ let $b_k,\alpha :I\to \mathbb{V}$ be continuous functions satisfying $b_k\ne {f,\parallel b\parallel }_{\infty }=\underset{k\in \mathbb{N}}{sup}{\parallel b_k\parallel }_{\infty }<\infty ,b_k\left({\upsilon }_1\right)=f\left({\upsilon }_1\right),b_k\left({\upsilon }_{\eta }\right)=f\left({\upsilon }_{\eta }\right)$ and

$${\parallel \alpha \parallel }_{\infty }=\underset{k\in \mathbb{N}}{sup}{\parallel {\alpha }_k\parallel }_{\infty }<1.$$

(3)

We define $F_{i,k}:K\to \mathbb{V}$ as $F_{i,k}(\upsilon ,\omega )={\alpha }_k\left(A_i\left(\upsilon \right)\right)\omega +f\left(A_i\left(\upsilon \right)\right)-{\alpha }_k\left(A_i\left(\upsilon \right)\right)b_k\left(\upsilon \right)$. Write $B_{i,k}(\upsilon ,\omega )=\left(A_i\left(\upsilon \right),F_{i,k}(\upsilon ,\omega )\right)$. With these settings, we obtain a sequence of IFSs $\{K;B_{i,k}:i\in {\mathbb{N}}_{\eta }\}$. For $f\in \mathcal{C}[a,b]$, define a subset

$${\mathcal{C}}_f[a,b]=\{w\in \mathcal{C}[a,b]:w\left(a\right)=f\left(a\right),w\left(b\right)=f\left(b\right)\},$$

and ${\parallel w\parallel }_{\infty }:=\underset{\upsilon \in [a,b]}{sup}\left|w\left(\upsilon \right)\right|$. Then, ${\mathcal{C}}_f[a,b]$ becomes a Polish space for being a closed subset of the Banach space $\mathcal{C}[a,b]$ with respect to the sup norm. For $k\in \mathbb{N}$, define a sequence of RB-operators $G_k:{\mathcal{C}}_f[a,b]\to {\mathcal{C}}_f[a,b]$ by

$$\left(G_{k}w\right)\left(\upsilon \right)=F_{i,k}\left(A_i^{-1}\left(\upsilon \right),w\left(A_i^{-1}\left(\upsilon \right)\right)\right)\forall \upsilon \in [{\upsilon }_i,{\upsilon }_{i-1}].$$

(4)

Proposition 1.

A non-empty closed invariant set for ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ in ${\mathcal{C}}_f[0,1]$ can be considered

$$\mathcal{A}=\left\{w\in {\mathcal{C}}_f[0,1]:{\parallel w\parallel }_{\infty }\le {\displaystyle \frac{{\parallel f\parallel }_{\infty }+{\parallel \alpha \parallel }_{\infty }{\parallel b\parallel }_{\infty }}{1-{\parallel \alpha \parallel }_{\infty }}}\right\},$$

where ${\parallel \alpha \parallel }_{\infty }$ is defined in (3).

The next result from [10] is used to provide the existence of a non-stationary fractal function.

Theorem 1.

Let $\left\{G_k\right\}$ be a sequence of RB-operators of the form (4). Then, the backward trajectories $\left\{{\mathsf{\Psi }}_k\left(f_0\right)\right\}$ converge to a function $f^{*}\in \mathcal{A}$, for any $f_0\in \mathcal{A}$.

The function $f^{*}$ is called a non-stationary (or statistically self-similar) FIF.

Definition 2.

Two sequences ${\left\{{\upsilon }_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{{\omega }_k\right\}}_{k\in \mathbb{N}}$ in a metric space $(E,d)$ are said to be asymptotically similar if $d({\upsilon }_k,{\omega }_k)\to 0$ as $k\to \infty$.

Example 1.

Let $E=\mathbb{V}$ and ${\upsilon }_k={\displaystyle \frac{1}{k}},{\omega }_k={\displaystyle \frac{1}{2k}}$. Then, ${\left\{{\upsilon }_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{{\omega }_k\right\}}_{k\in \mathbb{N}}$ are asymptotically similar.

Remark 1.

If ${\left\{{\upsilon }_k\right\}}_{k\in \mathbb{N}}$ and ${\left\{{\omega }_k\right\}}_{k\in \mathbb{N}}$ are asymptotically similar then

$$\underset{k\to \infty }{lim}{\upsilon }_k=\upsilon \iff \underset{k\to \infty }{lim}{\omega }_k=\upsilon .$$

Proposition 2

([9], Proposition $3.4$). Let ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ be a sequence of Lipschitz maps on a Banach space E such that $G_k$ has Lipschitz constant $c_k$. If ${lim}_{k\to \infty }{\coprod }_{i=1}^k{c}_i=0$, then $\left\{{\mathrm{\Phi }}_k\left(\upsilon \right)\right\}$, $\left\{{\mathrm{\Phi }}_k\left(\omega \right)\right\}$ are asymptotically similar for all $\upsilon ,\omega \in E$, and so are $\left\{{\mathsf{\Psi }}_k\left(\upsilon \right)\right\}$, $\left\{{\mathsf{\Psi }}_k\left(\omega \right)\right\}$ for all $\upsilon ,\omega \in E$.

Proposition 3

([20]). Let $(E,d)$ be a Banach space and ${\left\{G_k\right\}}_{k\in \mathbb{N}}$ be a sequence of Lipschitz maps on E. If there exists ${\upsilon }_{*}\in E$ such that the sequence $\left\{d({\upsilon }_{*},G_k\left({\upsilon }_{*}\right))\right\}$ is bounded, and ${\sum }_{k=1}^{\infty }{\coprod }_{i=1}^k{c}_i<\infty$ then the sequence $\left\{{\mathsf{\Psi }}_k\left(\upsilon \right)\right\}$ converges for all $\upsilon \in E$ to a unique limit $\overline{\upsilon }$.

3. Non-Stationary Fractal Interpolation Functions on SG

Let $T_0=\{p_1,p_2,p_3\}$ be the vertices of an equilateral triangle on ${\mathbb{V}}^2$ and $l_i\left(\upsilon \right)={\displaystyle \frac{1}{2}}(\upsilon +p_i)$, where $i=1,2,3$, three contractions of the plane which constitutes an IFS. The Sierpiński gasket (abbreviated as SG) is the attractor of this IFS,

$$SG=l_1\left(SG\right)\cup l_2\left(SG\right)\cup l_3\left(SG\right).$$

For fix $\eta \in \mathbb{N}$, consider the iterations $l_i=l_{i_1}{l}_{i_2}\cdots l_{i_{\eta }}$ for any sequence $i=(i_1,i_2,\cdots ,i_{\eta })\in I^{\eta }:={\{1,2,3\}}^{\eta }$. The union of images of $T_0$ under these iterations constitutes the set of $\eta$-th stage vertex $T_{\eta }$ of $SG$. Let $Z:T_{\eta }\to \mathbb{V}$ be a given function. We find an IFS whose attractor is the graph of a continuous function on $SG$, such that ${f|}_{T_{\eta }}=Z$. For $k\in \mathbb{N}$, define maps $B_{\omega ,k}:SG\times \mathbb{V}\to SG\times \mathbb{V}$ by

$$B_{\omega ,k}(\upsilon ,z)=\left(l_{\omega }\left(\upsilon \right),F_{\omega ,k}(\upsilon ,z)\right),\omega \in I^{\eta },$$

where $F_{\omega ,k}(\upsilon ,z):SG\times \mathbb{V}\to \mathbb{V}$ are required to satisfy the following conditions:

$$\parallel F_{\omega ,k}(.,z_1)-F_{\omega ,k}(.,z_2)\parallel \le c_{\omega ,k}|z_1-z_2|$$

and $F_{\omega ,k}(p_j,Z\left(p_j\right))=Z\left(l_i\left(p_j\right)\right)$ for every $\omega \in I^{\eta },j\in I$, where $c:=\underset{k\in \mathbb{N}}{sup}\underset{\omega \in I^{\eta }}{max}{c}_{\omega ,k}<1$. For the aforementioned objective, we consider $F_{\omega ,k}(\upsilon ,z)={\alpha }_{\omega ,k}\left(\upsilon \right)z+q_{\omega ,k}\left(\upsilon \right)$, where ${\alpha }_{\omega ,k}:SG\to \mathbb{V}$ and $q_{\omega ,k}:SG\to \mathbb{V}$ are continuous functions with ${\parallel \alpha \parallel }_{\infty }:=\underset{k\in \mathbb{N}}{sup}max\{\parallel {\alpha }_{\omega ,k}{\parallel }_{\infty }:\omega \in I^{\eta }\}<1$ and ${\parallel q\parallel }_{\infty }:={sup}_{k\in \mathbb{N}}max\{\parallel q_{\omega ,k}{\parallel }_{\infty }:\omega \in I^{\eta }\}<\infty$. Let $K=SG\times \mathbb{V}$. We obtain a sequence of IFSs ${\mathcal{I}}_k:=\{K;B_{\omega ,k}:\omega \in I^{\eta }\}$.

Theorem 2.

Let $\eta \in \mathbb{N}$ and $Z:T_{\eta }\to \mathbb{V}$ be given. The sequence of IFSs $\{K;B_{\omega ,k}:\omega \in I^{\eta }\}$ defined above produces a continuous function $w_{*}:SG\to \mathbb{V}$ which fulfills $w_{*}{|}_{T_{\eta }}=Z$.

Proof. 

First, let $\mathcal{C}(SG,\mathbb{V})$ denote the Banach space of real-valued continuous functions $w:SG\to \mathbb{V}$ with norm ${\parallel w\parallel }_{\infty }=max\left\{\right|w\left(\upsilon \right)|:\upsilon \in SG\}$. Let ${\mathcal{C}}^{*}(SG,\mathbb{V})=\left\{{w\in \mathcal{C}(SG,\mathbb{V}):w|}_{T_0}{=Z|}_{T_0}\right\}$; it is obvious that ${\mathcal{C}}^{*}(SG,\mathbb{V})$ is a complete space with respect to the metric induced by norm ${\parallel ·\parallel }_{\infty }$. For $k\in \mathbb{N}$, a mapping $G_k:{\mathcal{C}}^{*}(SG,\mathbb{V})\to {\mathcal{C}}^{*}(SG,\mathbb{V})$ is defined by

$$\left(G_{k}w\right)\left(\upsilon \right)=F_{\omega ,k}(l_{\omega }^{-1}\left(\upsilon \right),w\left(l_{\omega }^{-1}\left(\upsilon \right)\right))\forall \upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }.$$

We see that $G_k$ is a contraction map through the following lines:

$$\begin{array}{cc}\hfill & \parallel G_k{w}_1-G_k{w}_2{\parallel }_{\infty }\hfill \\ \hfill =& max\left\{|\left(G_k{w}_1\right)\left(\upsilon \right)-\left(G_k{w}_2\right)\left(\upsilon \right)|:\upsilon \in SG\right\}\hfill \\ \hfill =& max\left\{|F_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right),w_1\left(l_{\omega }^{-1}\left(\upsilon \right)\right)\right)-F_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right),w_2\left(l_{\omega }^{-1}\left(\upsilon \right)\right)\right)|:\upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }\right\}\hfill \\ \hfill \le & max\left\{\left|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)\right||w_1\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-w_2\left(l_{\omega }^{-1}\left(\upsilon \right)\right)|:\upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }\right\}\hfill \\ \hfill \le & {\parallel \alpha \parallel }_{\infty }max\left\{|w_1\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-w_2\left(l_{\omega }^{-1}\left(\upsilon \right)\right)|:\upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }\right\}\hfill \\ \hfill =& {\parallel \alpha \parallel }_{\infty }max\left\{|w_1\left(\upsilon \right)-w_2\left(\upsilon \right)|:\upsilon \in SG\right\}\hfill \\ \hfill =& {\parallel \alpha \parallel }_{\infty }{\parallel w_1-w_2\parallel }_{\infty }.\hfill \end{array}$$

It may be observed that the sequence $\{\parallel G_{k}w-w{\parallel }_{\infty }\}$ is bounded. By applying Proposition 3, the backward trajectories ${\mathsf{\Psi }}_k:=G_1\circ G_2\circ \cdots \circ G_k$ of $\left\{G_k\right\}$ converge for every $w\in {\mathcal{C}}^{*}\left(SG\right)$ to a unique attractor $w_{*}\in {\mathcal{C}}^{*}\left(SG\right)$. This concludes the proof. □

Remark 2.

The aforementioned outcome must be examined with [10], Theorem 4. More specifically, [10] Theorem 4 uses the assumption of closed invariant set (see [10], Proposition 3); nevertheless, our proof does not required that assumption. Therefore, our result may be treated as a generalization of [10] Theorem 4.

4. Oscillation Spaces

For $w:SG\to \mathbb{V}$, we define total oscillation of order $\eta$ by

$$V(\eta ,w)=\sum _{\omega \in {\{1,2,3\}}^{\eta }}{V}_w\left[l_{\omega }\left(SG\right)\right],$$

where $V_w\left[l_{\omega }\left(SG\right)\right]=sup\left\{\right|w\left({\upsilon }_1\right)-w\left({\upsilon }_2\right)|:{\upsilon }_1,{\upsilon }_2\in l_{\omega }\left(SG\right)\}$. For $0\le \lambda \le {\displaystyle \frac{log3}{log2}}$, we consider a new class as

$${\mathcal{C}}^{\lambda }\left(SG\right):=\left\{w:SG\to \mathbb{V}:w\mathrm{is}\mathrm{continuous}\mathrm{and}\underset{\eta \in \mathbb{N}}{sup}{\displaystyle \frac{V(\eta ,w)}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}<\infty \right\}.$$

For details related to oscillation spaces, the reader is recommended to refer to [21].

Let ${\overline{dim}}_B\left(A\right)$ denotes the upper box dimension of A. Then, we have the following result.

Theorem 3.

Let $f,b_k,{\alpha }_{\omega ,k}(\omega \in I^{\eta },k\in \mathbb{N})\in {\mathcal{C}}^{\lambda }\left(SG\right)$ be such that $b_k{{|}_{T_0}=f|}_{T_0}$. Suppose that ${\parallel b\parallel }_{\infty }={sup}_{k\in \mathbb{N}}{\parallel b_k\parallel }_{\infty }<\infty$. Then, for

$$max\left\{{\parallel \alpha \parallel }_{\infty }+{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\underset{\omega \in I^{\eta },k\in \mathbb{N}}{sup}\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,{\alpha }_{\omega ,k})}{2^{m\left(\frac{log3}{log2}-\lambda \right)}}},{\displaystyle \frac{3^{\eta }{\parallel \alpha \parallel }_{\infty }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\right\}<1,$$

there exists a non-stationary fractal function $f_{*}^{\alpha }\in {\mathcal{C}}^{\lambda }\left(SG\right)$. Furthermore,

${\overline{dim}}_B(\mathrm{Gr}\left(f_{*}^{\alpha }\right)\le 1-\lambda +{\displaystyle \frac{log3}{log2}}$.

Proof. 

Let ${\mathcal{C}}_f^{\lambda }\left(SG\right)=\{w\in {\mathcal{C}}^{\lambda }{\left(SG\right):w|}_{T_0}=f{|}_{T_0}\}$. We identify that the space ${\mathcal{C}}_f^{\lambda }\left(SG\right)$ is a closed subspace of ${\mathcal{C}}^{\lambda }\left(SG\right)$. Since $({\mathcal{C}}^{\lambda }\left(SG\right),\parallel .{\parallel }_{{\mathcal{C}}^{\lambda }})$ is a complete metric space [22] (Theorem 4) with respect to the metric induced by ${\parallel w\parallel }_{{\mathcal{C}}^{\lambda }}:={\parallel w\parallel }_{\infty }+{sup}_{\eta \in \mathbb{N}}{\displaystyle \frac{V(\eta ,w)}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}$, it follows that ${\mathcal{C}}_f^{\lambda }\left(SG\right)$ is a complete metric space with respect to the metric induced by norm ${\parallel .\parallel }_{{\mathcal{C}}^{\lambda }}$. A sequence of mappings $G_k:{\mathcal{C}}_f^{\lambda }\left(SG\right)\to {\mathcal{C}}_f^{\lambda }\left(SG\right)$ is defined by

$$\left(G_{k}w\right)\left(\upsilon \right)=f\left(\upsilon \right)+{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)(w-b_k)\left(l_{\omega }^{-1}\left(\upsilon \right)\right)$$

for all $\upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }$. It is visible that the mapping $G_k$ is clearly defined. Applying Remark 5 in [22], for $w,g\in {\mathcal{C}}_f^{\lambda }\left(SG\right)$, we have

$$\begin{array}{cc}& {\parallel G_{k}w-G_{k}g\parallel }_{{\mathcal{C}}^{\lambda }}\hfill \\ \hfill =& \parallel G_{k}w-G_k{g\parallel }_{\infty }+\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,G_{k}w-G_{k}g)}{2^{m\left(\frac{log3}{log2}-\lambda \right)}}}\hfill \\ \hfill \le & {\parallel \alpha \parallel }_{\infty }{\parallel w-g\parallel }_{\infty }+{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}{\parallel \alpha \parallel }_{\infty }\underset{m>\eta \in \mathbb{N}}{sup}{\displaystyle \frac{V(m-\eta ,w-g)}{2^{(m-\eta )\left(\frac{log3}{log2}-\lambda \right)}}}\hfill \\ & +{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}{\parallel w-g\parallel }_{\infty }\underset{m>\eta \in \mathbb{N}}{sup}{\displaystyle \frac{V(m-\eta ,{\alpha }_{\omega ,k})}{2^{(m-\eta )\left(\frac{log3}{log2}-\lambda \right)}}}\hfill \\ \hfill \le & \left({\parallel \alpha \parallel }_{\infty }+{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\underset{\omega \in I^{\eta },k\in \mathbb{N}}{sup}\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,{\alpha }_{\omega ,k})}{2^{m\left(\frac{log3}{log2}-\lambda \right)}}}\right){\parallel w-g\parallel }_{\infty }\hfill \\ & +{\displaystyle \frac{3^{\eta }{\parallel \alpha \parallel }_{\infty }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,w-g)}{2^{m\left(\frac{log3}{log2}-\lambda \right)}}}\hfill \\ \hfill \le & max\left\{{\parallel \alpha \parallel }_{\infty }+{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\underset{\omega \in I^{\eta },k\in \mathbb{N}}{sup}\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,{\alpha }_{\omega ,k})}{2^{m\left(\frac{log3}{log2}-\lambda \right)}}},{\displaystyle \frac{3^{\eta }{\parallel \alpha \parallel }_{\infty }}{2^{\eta \left(\frac{log3}{log2}-\lambda \right)}}}\right\}{\parallel w-g\parallel }_{{\mathcal{C}}^{\lambda }}.\hfill \end{array}$$

Using the hypothesis, we see that $G_k$ is a contraction map on ${\mathcal{C}}_f^{\lambda }\left(SG\right)$. It is visible that the sequence $\{\parallel G_{k}w-w{\parallel }_{{\mathcal{C}}^{\lambda }}\}$ is bounded. By employing Proposition 3, the backward trajectories ${\mathsf{\Psi }}_k\left(w\right):=G_1\circ G_2\circ \cdots \circ G_k\left(w\right)$ of $\left\{G_k\right\}$ converge for every $w\in {\mathcal{C}}_f^{\lambda }\left(SG\right)$ to a unique attractor $f_{*}^{\alpha }\in {\mathcal{C}}_f^{\lambda }\left(SG\right)$. Since $f_{*}^{\alpha }\in {\mathcal{C}}^{\lambda }\left(SG\right)$, we have ${\overline{dim}}_B(\mathrm{Gr}\left(f_{*}^{\alpha }\right)\le 1-\lambda +{\displaystyle \frac{log3}{log2}}$ by [23], Theorem $3.7$. □

Remark 3.

Based on the aforementioned proof, it is evident that each $G_k$ is a contraction and, hence, has a unique fixed point $f_k^{\alpha }$ termed as (stationary) FIF, as in [15]. It also fulfills that

$$f_k^{\alpha }\left(\upsilon \right)=f\left(\upsilon \right)+{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right).(f_k^{\alpha }-b_k)\left(l_{\omega }^{-1}\left(\upsilon \right)\right)\forall \upsilon \in l_{\omega }\left(SG\right),\omega \in I^{\eta }.$$

Now, the functions $B_{\omega ,k}:SG\times \mathbb{V}\to SG\times \mathbb{V}$ for $\omega \in I^{\eta }$ are defined by

$$B_{\omega ,k}(\upsilon ,\omega )=\left(l_{\omega }\left(\upsilon \right),{\alpha }_{\omega ,k}\left(\upsilon \right)\omega +f\left(l_{\omega }\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(\upsilon \right)b_k\left(\upsilon \right)\right).$$

Now, we demonstrate that the graph of the associated stationary fractal function $f_k^{\alpha }$ is an attractor of the backward trajectories of IFSs $\{SG\times \mathbb{V};B_{\omega ,k}:\omega \in I^{\eta }\}$.

$$\begin{array}{cc}\hfill & {\cup }_{\omega \in I^{\eta }}{B}_{\omega ,k}\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)\hfill \\ \hfill =& {\cup }_{\omega \in I^{\eta }}\left\{B_{\omega ,k}(\upsilon ,f_k^{\alpha }\left(\upsilon \right)):\upsilon \in SG\right\}\hfill \\ \hfill =& {\cup }_{\omega \in I^{\eta }}\left\{\left(l_{\omega }\left(\upsilon \right),{\alpha }_{\omega ,k}\left(\upsilon \right)f_k^{\alpha }\left(\upsilon \right)+f\left(l_{\omega }\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(\upsilon \right)b_k\left(\upsilon \right)\right):\upsilon \in SG\right\}\hfill \\ \hfill =& {\cup }_{\omega \in I^{\eta }}\left\{\left(l_{\omega }\left(\upsilon \right),f_k^{\alpha }\left(l_{\omega }\left(\upsilon \right)\right)\right):\upsilon \in SG\right\}\hfill \\ \hfill =& {\cup }_{\omega \in I^{\eta }}\left\{(\upsilon ,f_k^{\alpha }\left(\upsilon \right)):\upsilon \in l_{\omega }\left(SG\right)\right\}\hfill \\ \hfill =& \mathrm{Gr}\left(f_k^{\alpha }\right).\hfill \end{array}$$

In the below note, we have mentioned the dimension outcome of a stationary FIF $f_k^{\alpha }$ in the settings of a sequence of IFSs. The proof of this may be obtained using an idea similar to ref. [2], Theorem 4.

If the sequence of IFSs ${\mathcal{I}}_k=\{SG\times \mathbb{V};B_{\omega ,k}:\omega \in I^{\eta }\}$ as defined earlier fulfills the below given condition

$$c_{\omega ,k}{\parallel (\upsilon ,z)-(\omega ,l)\parallel }_2\le \parallel B_{\omega ,k}(\upsilon ,z)-B_{\omega ,k}{(\omega ,l)\parallel }_2\le C_{\omega ,k}{\parallel (\upsilon ,z)-(\omega ,l)\parallel }_2,$$

for every $(\upsilon ,z),(\omega ,l)\in SG\times \mathbb{V}$, where $0<c_{\omega ,k}\le C_{\omega ,k}<1\forall \omega \in I^{\eta }$, then $s_{*}\le {dim}_H\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)\le \overline{{dim}_B}\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)\le s^{*}$, where $s_{*}$ and $s^{*}$ are such that ${\sum }_{\omega \in I^{\eta }}{c}_{\omega ,k}^{s_{*}}=1$ and ${\sum }_{\omega \in I^{\eta }}{C}_{\omega ,k}^{s^{*}}=1$, and $f_k^{\alpha }$ is the stationary fractal interpolation function corresponding to the IFS ${\mathcal{I}}_k$.

Remark 4.

For the sequence of IFSs ${\mathcal{I}}_k=\{SG\times \mathbb{V};B_{\omega ,k}:\omega \in I^{\eta }\}$ satisfying the same hypothesis as in Note 1, we believe that the corresponding non-stationary FIF $f_{*}^{\alpha }$ will also have the same bound for the box and Hausdorff dimensions.

To calculate the exact bound for the Hausdorff and box dimensions of the non-stationary FIF, first we can define an open set $U=\mathrm{int}\left(▵\right)\times \mathbb{V}$, where $\mathrm{int}(▵)$ denotes interior of the filled triangle ▵ containing SG. It is easy to see that

$$l_{\omega }\left(\mathrm{int}(▵)\right)\cap l_{{\upsilon }_i}\left(\mathrm{int}(▵)\right)=\varnothing ,\omega \ne {\upsilon }_i.$$

This, in turn, yields that $B_{\omega ,k}\left(U\right)\cap B_{{\upsilon }_i,k}\left(U\right)=\varnothing$ for each $\omega \ne {\upsilon }_i$. Since $U\cap \mathrm{Gr}\left(f^{\alpha }\right)\ne \varnothing$, the IFS ${\mathcal{I}}_k$ will satisfy the strong open set condition. The proof may be completed using an idea form Theorem $2.4$ and Theorem $2.5$ of the work by Graf [24]. We leave the problem open for the further work in this direction.

Remark 5.

Let us write this remark on (stationary) fractal interpolation function. If $c_{\omega ,k}=C_{\omega ,k}$ for all $\omega ,k$, i.e., all maps $B_{\omega ,k}$ are similarity transformation, then by a result of Hutchinson [1], we obtain ${dim}_H\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)={dim}_B\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)=s$, where s is a unique solution of ${\sum }_{\omega \in I^{\eta }}{c}_{\omega ,k}^s=1$. In addition, we have $0<{\mathcal{H}}^s\left(\mathrm{Gr}\left(f_k^{\alpha }\right)\right)<\infty$.

Using the work of [25], for $\lambda \in \left[{\displaystyle \frac{log3}{log2}},1+{\displaystyle \frac{log3}{log2}}\right)$, we consider the space defined by

$${\mathcal{E}}_{\lambda }\left(SG\right):=\{f\in \mathcal{C}\left(SG\right):{\overline{dim}}_B(\mathrm{Gr}\left(f\right)\le \lambda \}.$$

Then, Verma et al. [22] Proposition 6 proved that ${\mathcal{E}}_{\lambda }\left(SG\right)$ is a vector space.

Moreover, we have the following result as per [25], Proposition $3.4$.

Lemma 1.

Let $\lambda \in \left[{\displaystyle \frac{log3}{log2}},1+{\displaystyle \frac{log3}{log2}}\right)$. Then,

$${\mathcal{E}}_{\lambda }\left(SG\right)={\cap }_{k\in \mathbb{N}}{\mathcal{C}}^{\lambda +{\displaystyle \frac{1}{k}}}\left(SG\right).$$

Moreover, $({\mathcal{E}}_{\lambda }\left(SG\right),d)$ is a complete metric space, where

$$d(f,w)=\sum _{k\in \mathbb{N}}min\left\{2^{-k},{\parallel f-w\parallel }_{{\mathcal{C}}^{\lambda +{\displaystyle \frac{1}{k}}}}\right\}.$$

Theorem 4.

Let $f,b_k,{\alpha }_{\omega ,k}(\omega \in I^{\eta },k\in \mathbb{N})\in {\mathcal{E}}_{\lambda }\left(SG\right)$ be such that $b_k{{|}_{T_0}=f|}_{T_0}$. Suppose that ${\parallel b\parallel }_{\infty }={sup}_{k\in \mathbb{N}}{\parallel b_k\parallel }_{\infty }<\infty$. Then, for

$$\underset{l\in \mathbb{N}}{sup}\left\{{\parallel \alpha \parallel }_{\infty }+{\displaystyle \frac{3^{\eta }}{2^{\eta \left(\frac{log3}{log2}-\lambda -\frac{1}{l}\right)}}}\underset{\omega \in I^{\eta },k\in \mathbb{N}}{sup}\underset{m\in \mathbb{N}}{sup}{\displaystyle \frac{V(m,{\alpha }_{\omega ,k})}{2^{m\left(\frac{log3}{log2}-\lambda -\frac{1}{l}\right)}}},{\displaystyle \frac{3^{\eta }{\parallel \alpha \parallel }_{\infty }}{2^{\eta \left(\frac{log3}{log2}-\lambda -\frac{1}{l}\right)}}}\right\}<1,$$

there exists a non-stationary fractal function $f_{*}^{\alpha }\in {\mathcal{E}}_{\lambda }\left(SG\right)$. Furthermore, ${\overline{dim}}_B(\mathrm{Gr}\left(f_{*}^{\alpha }\right)\le 1-\lambda +{\displaystyle \frac{log3}{log2}}$.

Proof. 

Proof follows from Theorem 3. □

Remark 6.

Our above work may be treated as an addendum to the similar works performed in univariate [26] and bivariate [27] settings.

5. Energy

Consider the complete graph ${\mathcal{G}}_0$ defined on the vertex set $T_0$. After constructing graph ${\mathcal{G}}_{\eta -1}$ with vertex set $T_{\eta -1}$ for some $\eta \ge 1$, the graph ${\mathcal{G}}_{\eta }$ on $T_{\eta }$ is defined as follows: for any $\upsilon ,\omega \in T_{\eta }$, $\upsilon {\sim }_{\eta }\omega$ holds if and only if $\upsilon =l_i\left({\upsilon }_1\right),\omega =l_i\left({\omega }_1\right)$ with ${\upsilon }_1{\sim }_{\eta -1}{\omega }_1$ and $i\in I$. Equivalently, $\upsilon {\sim }_{\eta }\omega$ if and only if there exists $i\in I^{\eta }$ such that $\upsilon ,\omega \in l_i\left(T_0\right)$. For $\eta =0,1,2,3,4\cdots$, the graph energies ${\mathcal{Q}}_{\eta }$ on ${\mathcal{G}}_{\eta }$ are defined by

$${\mathcal{Q}}_{\eta }\left(f\right):={\left({\displaystyle \frac{5}{3}}\right)}^{\eta }\sum _{\upsilon {\sim }_m\omega }{|f\left(\upsilon \right)-f\left(\omega \right)|}^2.$$

Note that the sequence of graph energies $\left\{{\mathcal{Q}}_{\eta }\right\}$ satisfies ${\mathcal{Q}}_{\eta -1}\left(f\right)=min{\mathcal{Q}}_{\eta }\left(\tilde{f}\right)$, where the minimum is taken over all $\tilde{f}$, satisfying $\tilde{f}{|}_{T_{\eta -1}}=f$ for any $f:T_{*}(:={\cup }_{m=0}^{\infty }{T}_m)\to \mathbb{V}$, and for any $\eta \ge 1$. After that, for each function f on $T_{*}$, we identify that ${\left\{{\mathcal{Q}}_{\eta }\left(f\right)\right\}}_{\eta =0}^{\infty }$ is an increasing sequence. The energy of f on $T_{*}$ is defined as

$$\mathcal{Q}\left(f\right):=\underset{\eta \to \infty }{lim}{\mathcal{Q}}_{\eta }\left(f\right).$$

A function $f\in \mathcal{C}\left(SG\right)$ is considered to have finite energy if $\mathcal{Q}\left(f\right)<+\infty$.

Let us define $dom\left(\mathcal{Q}\right)=\{w\in \mathcal{C}(SG):\mathcal{Q}(w)<\infty \}$. Then, the space $(dom\left(\mathcal{Q}\right),\parallel .{\parallel }_{\mathcal{Q}})$ is a Banach space ([28], Theorem $1.4.2$), where ${\parallel w\parallel }_{\mathcal{Q}}:={\parallel w\parallel }_{\infty }+\sqrt{\mathcal{Q}\left(w\right)}$.

Theorem 5.

Let $\eta \in \mathbb{N}$. Let germ function $f\in \mathrm{dom}\left(\mathcal{Q}\right)$ and $b_k\in \mathrm{dom}\left(\mathcal{Q}\right)$ with $b_k{{|}_{T_0}=f|}_{T_0}$. Suppose that $\mathcal{Q}\left(b\right):={sup}_{k\in \mathbb{N}}\mathcal{Q}\left(b_k\right)<\infty \mathrm{and}\mathcal{Q}\left(\alpha \right):=\underset{\omega \in I^{\eta },k\in \mathbb{N}}{sup}\mathcal{Q}\left({\alpha }_{\omega ,k}\right)<\infty$. If ${\parallel \alpha \parallel }_{\mathcal{Q}}<{\displaystyle \frac{1}{2\sqrt{5^{\eta }}}}$ then $f_{*}^{\alpha }\in \mathrm{dom}\left(\mathcal{Q}\right)$.

Proof. 

Let $k\in \mathbb{N}$. By applying the definition of RB operator $G_k:{\mathrm{dom}}_f\left(\mathcal{Q}\right):={\{w\in \mathrm{dom}\left(\mathcal{Q}\right):w|}_{T_0}=f{|}_{T_0}\}\to {\mathrm{dom}}_f\left(\mathcal{Q}\right)$. We have

$$\begin{array}{cc}\hfill {|G_{k}w\left(\upsilon \right)-G_{k}w\left(\omega \right)|}^2=& |f\left(\upsilon \right)-f\left(\omega \right)+{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)w\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)w\left(l_{\omega }^{-1}\left(\omega \right)\right)\hfill \\ & +{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)b_k\left(l_{\omega }^{-1}\left(\omega \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)b_k\left(l_{\omega }^{-1}\left(\upsilon \right)\right){|}^2\hfill \\ \hfill \le & {4|f\left(\upsilon \right)-f\left(\omega \right)|}^2+4|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right){|}^2{|w\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-w\left(l_{\omega ,k}^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +4|w\left(l_{\omega }^{-1}\left(\omega \right)\right){|}^2{|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +8|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right){|}^2{|b_k\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-b_k\left(l_i^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +8|b_k\left(l_{\omega }^{-1}\left(\upsilon \right)\right){|}^2{|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ \hfill \le & {4|f\left(\upsilon \right)-f\left(\omega \right)|}^2+{4\parallel \alpha \parallel }_{\infty }^2{|w\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-w\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +{4\parallel w\parallel }_{\infty }^2{|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +{8\parallel \alpha \parallel }_{\infty }^2{|b_k\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-b_k\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +{8\parallel b\parallel }_{\infty }^2{|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ \hfill \le & {4|f\left(\upsilon \right)-f\left(\omega \right)|}^2+{4\parallel \alpha \parallel }_{\infty }^2{|w\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-w\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2\hfill \\ & +{4(\parallel w\parallel }_{\infty }^2+2\parallel b_k{\parallel }_{\infty }^2)|{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-{\alpha }_{\omega ,k}\left(l_{\omega }^{-1}\left(\omega \right)\right){|}^2\hfill \\ & +{8\parallel \alpha \parallel }_{\infty }^2{|b_k\left(l_{\omega }^{-1}\left(\upsilon \right)\right)-b_k\left(l_{\omega }^{-1}\left(\omega \right)\right)|}^2.\hfill \end{array}$$

By employing the definition of energy at m-th level, for $m\ge \eta$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{Q}}_m\left(G_{k}w\right)\le & 4{\mathcal{Q}}_m\left(f\right)+{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }4{\parallel \alpha \parallel }_{\infty }^2{\mathcal{Q}}_{m-\eta }\left(w\right)\hfill \\ & +{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }{8\parallel \alpha \parallel }_{\infty }^2{\mathcal{Q}}_{m-\eta }\left(b_k\right)+{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }{4(\parallel w\parallel }_{\infty }^2+2\parallel b_k{\parallel }_{\infty }^2){\mathcal{Q}}_{m-\eta }\left({\alpha }_{\omega ,k}\right).\hfill \end{array}$$

This now produces

$$\begin{array}{cc}\hfill \mathcal{Q}\left(G_{k}w\right)\le & 4\mathcal{Q}\left(f\right)+5^{\eta }{8\parallel \alpha \parallel }_{\infty }^2\mathcal{Q}\left(b_k\right)+5^{\eta }4{\parallel \alpha \parallel }_{\infty }^2\mathcal{Q}\left(w\right)\hfill \\ & +5^{\eta }{4(\parallel w\parallel }_{\infty }^2+2\parallel b_k{\parallel }_{\infty }^2)\mathcal{Q}\left({\alpha }_{\omega ,k}\right).\hfill \end{array}$$

Further, we write

$$\begin{array}{cc}\hfill \mathcal{Q}\left(G_{k}w\right)\le & 4\mathcal{Q}\left(f\right)+5^{\eta }{8\parallel \alpha \parallel }_{\infty }\mathcal{Q}\left(b\right)+5^{\eta }4{\parallel \alpha \parallel }_{\infty }^2\mathcal{Q}\left(w\right)\hfill \\ & +5^{\eta }{4(\parallel w\parallel }_{\infty }^2+{2\parallel b\parallel }_{\infty }^2)\mathcal{Q}\left(\alpha \right),\hfill \end{array}$$

which shows that each $G_k$ is coherent under the specified criteria mentioned in the theorem. Now,

$$\begin{array}{cc}\hfill & \parallel G_{k}w-G_k{g\parallel }_{\mathcal{Q}}\hfill \\ \hfill =& \parallel G_{k}w-G_k{g\parallel }_{\infty }+\sqrt{\mathcal{Q}(G_{k}w-G_{k}g)}\hfill \\ \hfill \le & {\parallel \alpha \parallel }_{\infty }{\parallel w-g\parallel }_{\infty }\hfill \\ & +\sqrt{{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }{4\parallel \alpha \parallel }_{\infty }^2{lim}_{m\to \infty }{\mathcal{Q}}_{m-\eta }(w-g)+{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }4{\parallel w-g\parallel }_{\infty }^2{lim}_{m\to \infty }{\mathcal{Q}}_{m-\eta }\left(\alpha \right)}\hfill \\ \hfill =& {\parallel \alpha \parallel }_{\infty }{\parallel w-g\parallel }_{\infty }+\sqrt{{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }{4\parallel \alpha \parallel }_{\infty }^2\mathcal{Q}(w-g)+{\left({\displaystyle \frac{5}{3}}\right)}^{\eta }{3}^{\eta }4{\parallel w-g\parallel }_{\infty }^2\mathcal{Q}\left(\alpha \right)}\hfill \\ \hfill \le & {\parallel \alpha \parallel }_{\infty }{\parallel w-g\parallel }_{\infty }+{2\parallel \alpha \parallel }_{\infty }\sqrt{5^{\eta }}\sqrt{\mathcal{Q}(w-g)}+2{\parallel w-g\parallel }_{\infty }\sqrt{5^{\eta }}\sqrt{\mathcal{Q}\left(\alpha \right)}\hfill \\ \hfill \le & {2\parallel \alpha \parallel }_{\infty }\sqrt{5^{\eta }}\left({\parallel w-g\parallel }_{\infty }+\sqrt{\mathcal{Q}(w-g)}\right)+2{\parallel w-g\parallel }_{\infty }\sqrt{5^{\eta }}\sqrt{\mathcal{Q}\left(\alpha \right)}\hfill \\ \hfill \le & {2\parallel \alpha \parallel }_{\infty }\sqrt{5^{\eta }}{\parallel w-g\parallel }_{\mathcal{Q}}+2{\parallel w-g\parallel }_{\mathcal{Q}}\sqrt{5^{\eta }}\sqrt{\mathcal{Q}\left(\alpha \right)}\hfill \\ \hfill =& 2\sqrt{5^{\eta }}\left({\parallel \alpha \parallel }_{\infty }+\sqrt{\mathcal{Q}\left(\alpha \right)}\right){\parallel w-g\parallel }_{\mathcal{Q}}\hfill \\ \hfill =& 2\sqrt{5^{\eta }}{\parallel \alpha \parallel }_{\mathcal{Q}}{\parallel w-g\parallel }_{\mathcal{Q}}.\hfill \end{array}$$

Since ${\parallel \alpha \parallel }_{\mathcal{Q}}<{\displaystyle \frac{1}{2\sqrt{5^{\eta }}}}$, each $G_k$ is a contraction mapping.

It is easy to check that the sequence $\{\parallel G_{k}w-w{\parallel }_{\mathcal{Q}}\}$ is bounded. Thus, by using Proposition 3, the backward trajectories ${\mathsf{\Psi }}_k:=G_1\circ G_2\circ \cdots \circ G_k$ of $\left\{G_k\right\}$ converge for every $w\in {\mathrm{dom}}_f\left(\mathcal{Q}\right)$ to a unique attractor $f_{*}^{\alpha }\in {\mathrm{dom}}_f\left(\mathcal{Q}\right)$. Therefore, we have effectively demonstrated the validity of the outcome. □

Remark 7.

Since $G_k\left(f_k^{\alpha }\right)=f_k^{\alpha }$, from the first part of the above proof, we have $f_k^{\alpha }\in \mathrm{dom}\left(\mathcal{Q}\right)$, provided that ${\parallel \alpha \parallel }_{\infty }<{\displaystyle \frac{1}{2\sqrt{5^{\eta }}}}$ and $b_k,{\alpha }_{\omega ,k}\in \mathrm{dom}\left(\mathcal{Q}\right)$ for all $\omega \in I^{\eta }$.

6. Conclusions

In this paper, we have introduced the non-stationary fractal interpolation functions on the Sierpiński gasket (SG), which extends the existing results of stationary FIFs on SG. Under suitable assumptions on the IFS parameters, we calculated an upper bound of fractal dimension of the proposed interpolants for different function spaces with the underlying domain as SG. In the theory of analysis on fractals, the concept of energy plays a fundamental role. In our work, we have constructed the non-stationary FIF in the space of functions having finite energy. We believe that the current work presented in this paper may find applications in PDE on fractals.