**Connection 6**

The boundedness of *sn* is the essential condition to establish a unique invariant set of an iterated function system. In the fractal interpolation curve with vertical scaling factors as 'contraction function', 0< max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣< 1 (see [20]). Let *<sup>M</sup>* <sup>≔</sup> max *<sup>x</sup>*∈*<sup>I</sup>*∣*cnx* <sup>þ</sup> *<sup>f</sup> <sup>n</sup>*<sup>∣</sup> and *<sup>h</sup>*<sup>≥</sup> *<sup>M</sup>* 1� max *<sup>x</sup>*∈*I*∣*dn*ð Þ *x* ∣ . Then for all *y*∈½ � �*h*, *h* ,

$$|F\_n(\mathbf{x}, \mathbf{y})| = |\mathbf{c}\_n \mathbf{x} + d\_n(\mathbf{x})\mathbf{y} + f\_n| \le M + \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| |\mathbf{y}| \le M + \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| h \le h.$$

So, for all ð Þ *x*, *y* ∈*I* � �½ � *h*, *h* , we can see that *Fn*ð Þ *x*, *y* ∈½ � �*h*, *h* . That is, an IFS of the form f g *I* � �½ � *h*, *h* ; *w*<sup>1</sup>�*<sup>N</sup>* has been constructed (cf. [21], p. 1897). Thus *D s*ð Þ¼ � *<sup>n</sup>* ½ � *h*, *h* and *sn*ð Þ*y* ≔ max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣*y* is bounded in *D s*ð Þ*<sup>n</sup>* . Hence the boundedness of *sn* in *D s*ð Þ*<sup>n</sup>* is the essential condition to establish a unique invariant set of an IFS (cf. [21], p. 1897).

### **Connection 7**

In view of a *φ*-contraction, the connections between the coefficients of *y* variable are obtained as follows:

1. In the affine FIF (cf. [1], p. 308, Example 1), for all *t*>0,

$$\boldsymbol{\rho}(t) := \max\_{n=1,2,\dots,N} |d\_n|t,$$

where ∣*dn*∣<1 for all *i* ¼ 1, 2, … , *N*.

2. In the FIF with vertical scaling factors as (continuous) 'contraction functions' (cf. [20], p. 3), for all *t*> 0,

$$\rho(t) \coloneqq \max\_{i=1,2,\ldots,N} \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| t,$$

where *dn*ð Þ *x* is Lipschitz function defined on *I* satisfying sup*<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣< 1 for all *n* ¼ 1, 2, … , *N*.
