**Connection 3**

Let *dn* : *I* ! be a continuously differentiable function such that

$$\max\_{\boldsymbol{\pi} \in I} |d\_n(\boldsymbol{\pi})| \le 1.$$

3. In Theorem 2.2, for all *x*, *y*<sup>0</sup> ð Þ, *x*, *y*<sup>00</sup> ð Þ∈ *I* � ,

*DOI: http://dx.doi.org/10.5772/intechopen.92662*

*How Are Fractal Interpolation Functions Related to Several Contractions?*

contraction with respect to the second variable.

**Connection 5**

**Connection 6**

**Connection 7**

are obtained as follows:

*n* ¼ 1, 2, … , *N*.

**127**

*<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>*

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

(cf. [20], p. 3), for all *t*> 0,

∣*Fn*ð Þ *x*, *y* ∣ ¼ ∣*cnx* þ *dn*ð Þ *x y* þ *f <sup>n</sup>*∣ ≤ *M* þ max

<sup>∣</sup>*Fn <sup>x</sup>*, *<sup>y</sup>*<sup>0</sup> ð Þ� *Fn <sup>x</sup>*, *<sup>y</sup>*<sup>00</sup> ð Þ<sup>∣</sup> <sup>¼</sup> <sup>∣</sup>*dn*ð Þk *<sup>x</sup> sn <sup>y</sup>*<sup>0</sup> ð Þ� *sn <sup>y</sup>*<sup>00</sup> ∣≤∣*sn <sup>y</sup>*<sup>0</sup> ð Þ� *sn <sup>y</sup>*<sup>00</sup> ð Þ∣ ≤*<sup>φ</sup>* <sup>j</sup>*y*<sup>0</sup> � *<sup>y</sup>*<sup>00</sup> ð Þj *:*

4.Even though *sn* : ! are Rakotch contractions, *wn* : *I* � are not in general Rakotch contractions on the metric space ð Þ *I* � , *d*<sup>0</sup> , and thus, the IFSs defined above are not IFSs of [14] (cf. second and third line in p. 215 of [2]).

In the case where the vertical scaling factors are constants, in [1], the existence of affine FIFs by using the Banach fixed point theorem was investigated, whereas in [20], a generalisation of affine FIFs by using vertical scaling factors as (continuous) 'contraction functions' and Banach's fixed point theorem was introduced. Theorem 2.2 gives the existence of fractal interpolation curves by using the Rakotch fixed point theorem and vertical scaling factors as (continuous) 'contraction functions'.

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

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).

In view of a *φ*-contraction, the connections between the coefficients of *y* variable

2. In the FIF with vertical scaling factors as (continuous) 'contraction functions'

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

max *<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*t*,

∣*dn*∣*t*,

*<sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> max *<sup>n</sup>*¼1, 2, … , *<sup>N</sup>*

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

*φ*ð Þ*t* ≔ max *i*¼1, 2, … , *N* . Then for all *y*∈½ � �*h*, *h* ,

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þk *<sup>x</sup> <sup>y</sup>*∣ ≤ *<sup>M</sup>* <sup>þ</sup> max

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*h*≤*h:*

1� max *<sup>x</sup>*∈*I*∣*dn*ð Þ *x* ∣

That is, each *wn*ð Þ *x*, *y* is chosen so that function *Fn*ð Þ *x*, *y* is Rakotch

Then, by the Differential Mean Value Theorem and the extreme value theorem, we can see that for some *Ldn* > 0,

$$|d\_n(\mathbf{x'}) - d\_n(\mathbf{x''})| \le L\_{d\_n} |\mathbf{x'} - \mathbf{x''}|,$$

where *x*<sup>0</sup> , *x*<sup>00</sup> ∈*I*. Hence, *dn* is Lipschitz continuous function defined on *I* satisfying max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤ 1, but the converse is not true in general.

### **Connection 4**

1.The function *dn*ð Þ *x sn*ð Þ*y* is a generalisation of the bivariable function *dn*ð Þ *x y* with vertical scaling factors as (continuous) 'contraction functions'. In fact, in the case when 0< max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣<1 (see [20], p. 3), obviously,

$$d\_n(\boldsymbol{\kappa})\boldsymbol{y} = \frac{d\_n(\boldsymbol{\kappa})}{\max\_{\boldsymbol{\kappa}\in I}|d\_n(\boldsymbol{\kappa})|} \max\_{\boldsymbol{\kappa}\in I} |d\_n(\boldsymbol{\kappa})| \boldsymbol{y} \dots$$

Let *sn*ð Þ¼ *<sup>y</sup>* max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*<sup>y</sup>* and *<sup>d</sup>*<sup>∗</sup> *<sup>n</sup>* ð Þ¼ *x dn*ð Þ *x* max *<sup>x</sup>*∈*I*∣*dn*ð Þ *x* ∣ . Then *dn*ð Þ *x y* ¼ *d*∗ *<sup>n</sup>* ð Þ *<sup>x</sup> sn*ð Þ*<sup>y</sup>* , max *<sup>x</sup>*∈*<sup>I</sup>*∣*d*<sup>∗</sup> *<sup>n</sup>* ð Þ *x* ∣ ¼ 1 and *sn* is a Banach (or Rakotch) contraction.

2.The functional condition max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤1 is essential in order to show the difference between Banach contractibility of *Fn*ð Þ �, *y* and Rakotch contractibility of *Fn*ð Þ �, *y* ; compare with [20]. In fact, since *φ*ð Þ*t* <*t* for any *t* >0,

$$\begin{aligned} |F\_n(\mathbf{x}, \mathbf{y}') - F\_n(\mathbf{x}, \mathbf{y}'')| &= |d\_n(\mathbf{x})| |s\_n(\mathbf{y}') - s\_n(\mathbf{y}'')| \\ &\le \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| |s\_n(\mathbf{y}') - s\_n(\mathbf{y}'')| \\ &\le \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| \rho(|\mathbf{y}' - \mathbf{y}''|) \\ &\le \max\_{\mathbf{x} \in I} |d\_n(\mathbf{x})| |\mathbf{y}' - \mathbf{y}''|, \end{aligned}$$

where *<sup>x</sup>*, *<sup>y</sup>*<sup>0</sup> ð Þ, *<sup>x</sup>*, *<sup>y</sup>*<sup>00</sup> ð Þ<sup>∈</sup> <sup>2</sup> . Hence, if max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣<1, as can be seen, notwithstanding each *sn* is a Rakotch contraction that is not a Banach contraction, each *Fn* is Banach contraction with respect to the second variable because

$$|F\_n(\mathbf{x}, \boldsymbol{\mathcal{y}}') - F\_n(\mathbf{x}, \boldsymbol{\mathcal{y}}'')| \le \max\_{\boldsymbol{\mathcal{x}} \in I} |d\_n(\boldsymbol{\mathcal{x}})| |\boldsymbol{\mathcal{y}}' - \boldsymbol{\mathcal{y}}''|.$$

On the other hand, if max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ¼ 1, then we conclude that each *Fn* is Rakotch contraction with respect to the second variable whenever each *sn* is a Rakotch contraction because

$$|F\_n(\mathfrak{x}, \mathfrak{y}') - F\_n(\mathfrak{x}, \mathfrak{y}'')| \le \max\_{\mathfrak{x} \in I} |d\_n(\mathfrak{x})| \rho(|\mathfrak{y}' - \mathfrak{y}''|) \dots$$

*How Are Fractal Interpolation Functions Related to Several Contractions? DOI: http://dx.doi.org/10.5772/intechopen.92662*

3. In Theorem 2.2, for all *x*, *y*<sup>0</sup> ð Þ, *x*, *y*<sup>00</sup> ð Þ∈ *I* � ,

$$|F\_n(\mathbf{x}, \mathbf{y'}) - F\_n(\mathbf{x}, \mathbf{y''})| = |d\_n(\mathbf{x})| |\mathbf{s}\_n(\mathbf{y'}) - \mathbf{s}\_n(\mathbf{y'}')| \le |\mathbf{s}\_n(\mathbf{y'}) - \mathbf{s}\_n(\mathbf{y''})| \le \rho(|\mathbf{y'} - \mathbf{y''}|).$$

That is, each *wn*ð Þ *x*, *y* is chosen so that function *Fn*ð Þ *x*, *y* is Rakotch contraction with respect to the second variable.

4.Even though *sn* : ! are Rakotch contractions, *wn* : *I* � are not in general Rakotch contractions on the metric space ð Þ *I* � , *d*<sup>0</sup> , and thus, the IFSs defined above are not IFSs of [14] (cf. second and third line in p. 215 of [2]).
