**Connection 8**

We refer to *f* of Theorem 2.2 as a nonlinear FIF. The reason is that the functions *Fn* take the form

we can see that max ð Þ *<sup>x</sup>*,*<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>*∣*dm*,*n*ð Þ *x*, *y* ∣< 1, and so each *wm*,*n*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*n*ð Þ *x*, *y*, *z* is a Banach contraction with respect to the third

∣*Fm*,*n*ð Þ� *x*, *y*, *z Fm*,*<sup>n</sup> x*, *y*, *z*<sup>0</sup> ð Þ∣ ¼ ∣*dm*,*n*ð Þk *x*, *y sm*,*n*ð Þ� *z sm*,*<sup>n</sup> z*<sup>0</sup> ð Þ∣

That is, each *Fm*,*n*ð Þ *x*, *y*, *z* is Rakotch contraction with respect to the third variable. So, each *wm*,*n*ð Þ *x*, *y*, *z* is chosen so that the function *Fm*,*n*ð Þ *x*, *y*, *z* is a

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

2. In the FIS with vertical scaling factors as function (cf. [21]), for all *t*> 0,

max *<sup>n</sup>*¼1, 2, … , *<sup>N</sup>*

where *dm*,*<sup>n</sup>*ð Þ *x* is Lipschitz function defined on *I* satisfying sup*<sup>x</sup>*∈*<sup>I</sup>*∣*dm*,*<sup>n</sup>*ð Þ *x* ∣<1 for

The continuity of bivariable FIFs differ from the continuity of univariable FIFs.

2.There are bivariable discontinuous functions that interpolate the given data;

3.Theorem 2.3 ensures that attractors of constructed IFSs are graphs of some bounded functions which interpolate the given data, but these graphs (i.e., the graphs of bivariable FIFs) are not always continuous surfaces. Some continuity conditions of bivariable FIFs are given explicitly by Corollary 2.1

The key difficulty in constructing fractal interpolation surfaces (or volumes) involves ensuring continuity. Another important element necessary in modelling complicated surfaces of this type is the existence of the contractivity, or vertical

1.The graphs of linear univariable FIFs are always continuous curves.

max *<sup>n</sup>*¼1, 2, … , *<sup>N</sup>*

max

∣*dm*,*<sup>n</sup>*∣*t*,

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dm*,*<sup>n</sup>*ð Þ *<sup>x</sup>* <sup>∣</sup>*t*,

,

≤ ∣*sm*,*n*ð Þ� *z sm*,*<sup>n</sup> z*<sup>0</sup> ð Þ∣ ≤*φ* j*z* � *z*<sup>0</sup> ð Þj *:*

2. In Theorem 2.3, for all ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*<sup>0</sup> ð Þ<sup>∈</sup> *<sup>K</sup>* <sup>⊂</sup> <sup>3</sup>

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

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

Rakotch contraction with respect to the third variable.

*<sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> max *<sup>m</sup>*¼1, 2, … , *<sup>M</sup>*

where ∣*dm*,*<sup>n</sup>*∣< 1 for all *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

*<sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> max *<sup>m</sup>*¼1, 2, … , *<sup>M</sup>*

1. In the affine FIS (cf. [22]), for all *t*>0,

all *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

(see for instance [23], p. 630, 631).

and Corollary 2.2.

**Connection 12**

scaling, factors.

**129**

variable.

**Connection 10**

**Connection 11**

are obtained as follows:

$$F\_n(\mathfrak{x}, \mathfrak{y}) = c\_n \mathfrak{x} + d\_n(\mathfrak{x})\mathfrak{s}\_n(\mathfrak{y}) + e\_n\mathfrak{y}$$

where max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤1 and each *sn* is Rakotch contraction. That is, each *Fn*, in general, is nonlinear with respect to the second variable (cf. [17]). In fact, in [2] or [20], since 0< ∣*dn*ð Þ *x* ∣ � ∣*dn*∣< 1 or 0< max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣< 1 and

$$d\_n(\mathbf{x})\mathbf{y} = \frac{d\_n(\mathbf{x})}{\max\_{\mathbf{x}\in I} |d\_n(\mathbf{x})|} \max\_{\mathbf{x}\in I} |d\_n(\mathbf{x})|\mathbf{y}, \mathbf{y}$$

we can see that

$$F\_n(\mathfrak{x}, \mathfrak{y}) = c\_n \mathfrak{x} + d\_n(\mathfrak{x})\mathfrak{y} + e\_n = c\_n \mathfrak{x} + d\_n^\*(\mathfrak{x})\mathfrak{s}\_n(\mathfrak{y}) + e\_n,$$

where *d*<sup>∗</sup> *<sup>n</sup>* ð Þ *<sup>x</sup>* <sup>≔</sup> *dn*ð Þ *<sup>x</sup>* max *<sup>x</sup>*∈*I*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup> and *sn*ð Þ*<sup>y</sup>* <sup>≔</sup> max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*y*, and thus, each *sn* is a special Banach contraction and linear with respect to the second variable. Obviously, we can say that nonlinear FIFs may have more flexibility and applicability.

### **Connection 9**

1.The well-known FIS in theory and applications is generated by an IFS of the form f g *K*, *wm*,*<sup>n</sup>* : *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N* under some conditions, where the maps are transformations of the special structure

$$w\_{m,n}\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}u\_m(\mathbf{x})\\v\_n(\mathbf{y})\\F\_{m,n}(\mathbf{x},\mathbf{y},\mathbf{z})\end{pmatrix} = \begin{pmatrix}a\_mx+b\_m\\c\_ny+d\_n\\e\_{m,n}\mathbf{x}+f\_{m,n}\mathbf{y}+g\_{m,n}\mathbf{x}\mathbf{y}+d\_{m,n}(\mathbf{x},\mathbf{y})\mathbf{z}+h\_{m,n}\end{pmatrix},$$

where <sup>∣</sup>*dm*,*<sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∣</sup><sup>&</sup>lt; 1 for all ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>* <sup>⊂</sup> <sup>2</sup> . Then for all ð Þ *x*, *y*, *z* , *x*, *y*, *z*<sup>0</sup> ð Þ∈*K*,

$$\begin{aligned} |F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}) - F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}')| &= |d\_{m,n}(\mathbf{x}, \mathbf{y})\mathbf{z} - d\_{m,n}(\mathbf{x}, \mathbf{y})\mathbf{z}'|\\ &\leq \max\_{(\mathbf{x}, \mathbf{y}) \in D} |d\_{m,n}(\mathbf{x}, \mathbf{y})| |\mathbf{z} - \mathbf{z}'|.\end{aligned}$$

That is, each *wm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is a Banach contraction with respect to the third variable. So, the existence of bivariable FIFs follows from Banach's fixed point theorem. In fact, in [22], since for all ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>* <sup>⊂</sup> <sup>2</sup> , *dm*,*<sup>n</sup>*ð Þ� *x*, *y sm*,*<sup>n</sup>* and 0≤ ∣*sm*,*<sup>n</sup>*∣< 1, we can see that each *wm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is Banach contraction with respect to the third variable. Also in [21], since

$$d\_{m,n}(\boldsymbol{\kappa}, \boldsymbol{\jmath}) = \lambda\_{m,n} (\boldsymbol{\kappa} - \boldsymbol{\varkappa}\_0) (\boldsymbol{\kappa}\_M - \boldsymbol{\varkappa}) \left(\boldsymbol{\jmath} - \boldsymbol{\jmath}\_0\right) (\boldsymbol{\jmath}\_N - \boldsymbol{\jmath})^2$$

and

$$|\lambda\_{m,n}| < \frac{16}{\left(\varkappa\_M - \varkappa\_0\right)^2 \left(\wp\_N - \wp\_0\right)^2},$$

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

we can see that max ð Þ *<sup>x</sup>*,*<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>*∣*dm*,*n*ð Þ *x*, *y* ∣< 1, and so each *wm*,*n*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*n*ð Þ *x*, *y*, *z* is a Banach contraction with respect to the third variable.

2. In Theorem 2.3, for all ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*<sup>0</sup> ð Þ<sup>∈</sup> *<sup>K</sup>* <sup>⊂</sup> <sup>3</sup> ,

$$\begin{aligned} |F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}) - F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}')| &= |d\_{m,n}(\mathbf{x}, \mathbf{y})| |s\_{m,n}(\mathbf{z}) - s\_{m,n}(\mathbf{z}')| \\ &\le |s\_{m,n}(\mathbf{z}) - s\_{m,n}(\mathbf{z}')| \le q(|\mathbf{z} - \mathbf{z}'|). \end{aligned}$$

That is, each *Fm*,*n*ð Þ *x*, *y*, *z* is Rakotch contraction with respect to the third variable. So, each *wm*,*n*ð Þ *x*, *y*, *z* is chosen so that the function *Fm*,*n*ð Þ *x*, *y*, *z* is a Rakotch contraction with respect to the third variable.

### **Connection 10**

**Connection 8**

*Fn* take the form

we can see that

*<sup>n</sup>* ð Þ *<sup>x</sup>* <sup>≔</sup> *dn*ð Þ *<sup>x</sup>*

where *d*<sup>∗</sup>

**Connection 9**

*x y z* 1

ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>* <sup>⊂</sup> <sup>2</sup>

and

**128**

CA <sup>¼</sup>

0

B@

0

B@

*wm*,*<sup>n</sup>*

We refer to *f* of Theorem 2.2 as a nonlinear FIF. The reason is that the functions

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

max *<sup>x</sup>*∈*I*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup> and *sn*ð Þ*<sup>y</sup>* <sup>≔</sup> max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>*y*, and thus, each *sn* is a special

*<sup>n</sup>* ð Þ *x sn*ð Þþ *y en*,

*amx* þ *bm cny* þ *dn em*,*nx* þ *f <sup>m</sup>*,*<sup>n</sup>y* þ *gm*,*<sup>n</sup>xy* þ *dm*,*<sup>n</sup>*ð Þ *x*, *y z* þ *hm*,*<sup>n</sup>*

∣*dm*,*<sup>n</sup>*ð Þk *x*, *y z* � *z*<sup>0</sup>

� � *yN* � *<sup>y</sup>* � �

. Then for all ð Þ *x*, *y*, *z* , *x*, *y*, *z*<sup>0</sup> ð Þ∈*K*,

∣

∣*:*

1

CA,

*Fn*ð Þ¼ *x*, *y cnx* þ *dn*ð Þ *x sn*ð Þþ *y en*,

where max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣ ≤1 and each *sn* is Rakotch contraction. That is, each *Fn*, in general, is nonlinear with respect to the second variable (cf. [17]). In fact, in [2] or

[20], since 0< ∣*dn*ð Þ *x* ∣ � ∣*dn*∣< 1 or 0< max *<sup>x</sup>*∈*<sup>I</sup>*∣*dn*ð Þ *x* ∣< 1 and

*Mathematical Theorems - Boundary Value Problems and Approximations*

*dn*ð Þ *<sup>x</sup> <sup>y</sup>* <sup>¼</sup> *dn*ð Þ *<sup>x</sup>* max *<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* <sup>∣</sup>

*Fn*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> cnx* <sup>þ</sup> *dn*ð Þ *<sup>x</sup> <sup>y</sup>* <sup>þ</sup> *en* <sup>¼</sup> *cnx* <sup>þ</sup> *<sup>d</sup>*<sup>∗</sup>

can say that nonlinear FIFs may have more flexibility and applicability.

0

B@

the maps are transformations of the special structure

CA <sup>¼</sup>

1

*um*ð Þ *x vn*ð Þ*y Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z*

where <sup>∣</sup>*dm*,*<sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∣</sup><sup>&</sup>lt; 1 for all ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>* <sup>⊂</sup> <sup>2</sup>

respect to the third variable. Also in [21], since

∣*λ<sup>m</sup>*,*<sup>n</sup>*∣<

Banach contraction and linear with respect to the second variable. Obviously, we

1.The well-known FIS in theory and applications is generated by an IFS of the form f g *K*, *wm*,*<sup>n</sup>* : *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N* under some conditions, where

∣*Fm*,*<sup>n</sup>*ð Þ� *x*, *y*, *z Fm*,*<sup>n</sup> x*, *y*, *z*<sup>0</sup> ð Þ∣ ¼ ∣*dm*,*<sup>n</sup>*ð Þ *x*, *y z* � *dm*,*<sup>n</sup>*ð Þ *x*, *y z*<sup>0</sup>

follows from Banach's fixed point theorem. In fact, in [22], since for all

*dm*,*<sup>n</sup>*ð Þ¼ *x*, *y λ<sup>m</sup>*,*<sup>n</sup>*ð Þ *x* � *x*<sup>0</sup> ð Þ *xM* � *x y* � *y*<sup>0</sup>

ð Þ *xM* � *x*<sup>0</sup>

≤ max ð Þ *x*, *y* ∈ *D*

That is, each *wm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is a Banach contraction with respect to the third variable. So, the existence of bivariable FIFs

, *dm*,*<sup>n</sup>*ð Þ� *x*, *y sm*,*<sup>n</sup>* and 0≤ ∣*sm*,*<sup>n</sup>*∣< 1, we can see that each *wm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is chosen so that function *Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* is Banach contraction with

16

2

*yN* � *y*<sup>0</sup> � �<sup>2</sup> ,

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

1. In the affine FIS (cf. [22]), for all *t*>0,

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

where ∣*dm*,*<sup>n</sup>*∣< 1 for all *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

2. In the FIS with vertical scaling factors as function (cf. [21]), for all *t*> 0,

$$\rho(t) := \max\_{m=1,2,\dots,M} \max\_{n=1,2,\dots,N} \max\_{\mathbf{x}\in I} |d\_{m,n}(\mathbf{x})|t,\mathbf{x}$$

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

### **Connection 11**

The continuity of bivariable FIFs differ from the continuity of univariable FIFs.


#### **Connection 12**

The key difficulty in constructing fractal interpolation surfaces (or volumes) involves ensuring continuity. Another important element necessary in modelling complicated surfaces of this type is the existence of the contractivity, or vertical scaling, factors.

1. In order to ensure continuity of a fractal interpolation surface, in [22], the interpolation points on the boundary was assumed collinear, whereas in [21], vertical scaling factors as (continuous) 'contraction functions' were used.

2. In Corollary 2.1, we can see that

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

, *bm* <sup>¼</sup> *xMxm*�<sup>1</sup> � *<sup>x</sup>*0*xm*

*yN* � *y*<sup>0</sup>

, *dn* <sup>¼</sup> *yNyn*�<sup>1</sup> � *<sup>y</sup>*0*yn*

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

*gm*,*<sup>n</sup>* <sup>¼</sup> <sup>ð</sup>*zm*,*<sup>n</sup>* � *zm*�1,*n*Þ � ð Þ *zm*,*n*�<sup>1</sup> � *zm*�1,*n*�<sup>1</sup> ð Þ *xM* � *x*<sup>0</sup> *yN* � *y*<sup>0</sup>

*em*,*<sup>n</sup>* <sup>¼</sup> *yN*ð*zm*,*n*�<sup>1</sup> � *zm*�1,*n*�1Þ � *<sup>y</sup>*0ð Þ *zm*,*<sup>n</sup>* � *zm*�1,*<sup>n</sup>* ð Þ *xM* � *x*<sup>0</sup> *yN* � *y*<sup>0</sup>

*<sup>f</sup> <sup>m</sup>*,*<sup>n</sup>* <sup>¼</sup> *xM*ð*zm*�1,*<sup>n</sup>* � *zm*�1,*n*�1Þ � *<sup>x</sup>*0ð Þ *zm*,*<sup>n</sup>* � *zm*,*n*�<sup>1</sup> ð Þ *xM* � *x*<sup>0</sup> *yN* � *y*<sup>0</sup>

*hm*,*<sup>n</sup>* <sup>¼</sup> *<sup>x</sup>*0*y*0*zm*,*<sup>n</sup>* � *<sup>x</sup>*0*yNzm*,*n*�<sup>1</sup> � *xMy*0*zm*�1,*<sup>n</sup>* <sup>þ</sup> *xMyNzm*�1,*n*�<sup>1</sup> ð Þ *xM* � *x*<sup>0</sup> *yN* � *y*<sup>0</sup>

*A fractal interpolation surface (a) that is associated with Banach contractions, (b) that is not necessarily*

*xM* � *x*<sup>0</sup>

,

,

,

,

,

� *sm*,*<sup>n</sup>*ð Þ *zM*,*<sup>N</sup> :*

*am* <sup>¼</sup> *xm* � *xm*�<sup>1</sup> *xM* � *x*<sup>0</sup>

*cn* <sup>¼</sup> *yn* � *yn*�<sup>1</sup> *yN* � *y*<sup>0</sup>

**Figure 2.**

**131**

*associated with Banach contractions.*

2.A new bivariable fractal interpolation function by using the Matkowski fixed point theorem and the Rakotch contraction is presented in [18]. In order to ensure the continuity of nonlinear FIS, the coplanarity of all the interpolation points on the boundaries instead of collinearity of interpolation points on the boundary was assumed in [18], whereas in [12], vertical scaling factors as (continuous) 'contraction functions' were used.
