**2.2 Fractal interpolation in <sup>2</sup>**

Let *M*, *N* be two positive integers greater than 1. Let us represent the given set of *interpolation points* as *xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � �<sup>∈</sup> *<sup>K</sup>* : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>*; *<sup>j</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* n o, where *x*<sup>0</sup> < *x*<sup>1</sup> < ⋯ <*xM*, *y*<sup>0</sup> <*y*<sup>1</sup> < ⋯ <*yN* and *zi*,*<sup>j</sup>* ∈ ½ � *a*, *b* for all *i* ¼ 0, 1, … , *M* and *j* ¼ 0, 1, … , *N*. Set *I* ¼ ½ � *x*0, *xM* ⊂ and *J* ¼ *y*0, *yN* � �⊂ . Throughout this section, we will work in the complete metric space *K* ¼ *D* � , where *D* ¼ *I* � *J*, with respect to the Euclidean, or to some other equivalent, metric.

Set *Im* <sup>¼</sup> ½ � *xm*�1, *xm* , *Jn* <sup>¼</sup> *yn*�<sup>1</sup>, *yn* � �, *Dm*,*<sup>n</sup>* <sup>¼</sup> *Im* � *Jn* and let *um* : *<sup>I</sup>* ! *Im*, *vn* : *<sup>J</sup>* ! *Jn*, *Lm*,*<sup>n</sup>* : *D* ! *Dm*,*<sup>n</sup>* be defined for *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*, by

$$L\_{m,n}(\mathfrak{x}, \mathfrak{y}) = (u\_m(\mathfrak{x}), v\_n(\mathfrak{y})) = (a\_m \mathfrak{x} + b\_m, c\_n \mathfrak{y} + d\_n).$$

Thus, for *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*,

$$\begin{aligned} a\_m &= \frac{\varkappa\_m - \varkappa\_{m-1}}{\varkappa\_M - \varkappa\_0}, \quad b\_m = \varkappa\_{m-1} - \frac{\varkappa\_m - \varkappa\_{m-1}}{\varkappa\_M - \varkappa\_0} \varkappa\_0, \\\\ c\_n &= \frac{\varkappa\_n - \varkappa\_{n-1}}{\varkappa\_N - \varkappa\_0}, \quad d\_n = \varkappa\_{n-1} - \frac{\varkappa\_n - \varkappa\_{n-1}}{\varkappa\_N - \varkappa\_0} \varkappa\_0. \end{aligned}$$

Furthermore, for *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*, let mappings *Fm*,*<sup>n</sup>* : *K* ! be continuous with respect to each variable. We consider an IFS of the form f g *K*; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N* in which maps *wm*,*<sup>n</sup>* : *D* � ! *Dm*,*<sup>n</sup>* � are transformations of the special structure

$$w\_{m,n}(\boldsymbol{\kappa}, \boldsymbol{y}, \boldsymbol{z}) := (L\_{m,n}(\boldsymbol{\kappa}, \boldsymbol{y}), F\_{m,n}(\boldsymbol{\kappa}, \boldsymbol{y}, \boldsymbol{z})),$$

where the transformations are constrained by the data according to

$$w\_{m,n}\begin{pmatrix}\varkappa\_0\\\varkappa\_0\\\varkappa\_{0,0}\end{pmatrix} = \begin{pmatrix}\varkappa\_{m-1}\\\varkappa\_{n-1}\\\varkappa\_{m-1,n-1}\end{pmatrix},\quad w\_{m,n}\begin{pmatrix}\varkappa\_0\\\varkappa\_N\\\varkappa\_{0,N}\end{pmatrix} = \begin{pmatrix}\varkappa\_{m-1}\\\varkappa\_n\\\varkappa\_{m-1,n}\end{pmatrix},$$

$$w\_{m,n}\begin{pmatrix}\boldsymbol{\omega}\_{\mathcal{M}}\\ \boldsymbol{\mathcal{y}}\_{0}\\ \boldsymbol{z}\_{\mathcal{M},0}\end{pmatrix} = \begin{pmatrix}\boldsymbol{\omega}\_{m}\\ \boldsymbol{\mathcal{y}}\_{n-1}\\ \boldsymbol{z}\_{m,n-1}\end{pmatrix},\quad w\_{m,n}\begin{pmatrix}\boldsymbol{\omega}\_{\mathcal{M}}\\ \boldsymbol{\mathcal{y}}\_{N}\\ \boldsymbol{z}\_{\mathcal{M},N}\end{pmatrix} = \begin{pmatrix}\boldsymbol{\omega}\_{m}\\ \boldsymbol{\mathcal{y}}\_{n}\\ \boldsymbol{z}\_{m,n}\end{pmatrix}.$$

Let *C*<sup>∗</sup>

Let *C*∗ ∗

For *f* ∈*C*<sup>∗</sup>

<sup>0</sup> ð Þ *<sup>D</sup>* <sup>≔</sup> *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup>

*the IFS defined above. Then,*

2.*f xi*, *y <sup>j</sup>*

*xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>*

**123**

<sup>0</sup> ð Þ *<sup>D</sup>* , we define *<sup>T</sup>* : *<sup>C</sup>*<sup>∗</sup>

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

3.*if G* ⊂ *D* � *is the graph of f, then*

**Example 2.** *Let <sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>t</sup>*

*Let for z*∈½ Þ 0, þ∞ ,

*Tf x*ð Þ¼ , *<sup>y</sup> Fm*,*<sup>n</sup> <sup>u</sup>*�<sup>1</sup>

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

<sup>¼</sup> *em*,*nu*�<sup>1</sup>

<sup>þ</sup> *sm*,*<sup>n</sup> f u*�<sup>1</sup>

� � <sup>¼</sup> *zi*,*<sup>j</sup> for all i* <sup>¼</sup> 0, 1, … , *M and j* <sup>¼</sup> 0, 1, … , *N;*

*G* ¼ ⋃ *M m*¼1 ⋃ *N n*¼1

The most simple example is the following; cf. [12].

*<sup>s</sup>*1,1ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

*<sup>s</sup>*2,1ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

½ �� 0, 1 ½ �! 0, 1 *that interpolates the given data xi*, *y <sup>j</sup>*

1 þ *z*

1 þ 3*z*

*not Banach contractions on* ½ Þ 0, þ∞ *. So, there exists a continuous function f* :

Let *dm*,*<sup>n</sup>* : *D* ! be a function such that max ð Þ *<sup>x</sup>*,*<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>*∣*dm*,*<sup>n</sup>*ð Þ *x*, *y* ∣ ≤1,

<sup>0</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* be the set of continuous functions *<sup>f</sup>* : *<sup>D</sup>* ! such that

� � <sup>¼</sup> *<sup>z</sup>* <sup>∗</sup> , <sup>∗</sup> ,

� � <sup>¼</sup> *<sup>z</sup>* <sup>∗</sup> , <sup>∗</sup> ,

� � <sup>¼</sup> *<sup>z</sup>* <sup>∗</sup> , <sup>∗</sup> ,

� � <sup>¼</sup> *<sup>z</sup>* <sup>∗</sup> , <sup>∗</sup>

*z* ∗ ∗ ≔ *z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *zM*,*<sup>j</sup>* ¼ *zi*,*N:*

*<sup>n</sup>* ð Þ*<sup>y</sup>* , *f u*�<sup>1</sup>

*<sup>n</sup>* ð Þ*<sup>y</sup>* � � � �

*wm*,*<sup>n</sup>*ð Þ *G :*

, *<sup>s</sup>*1,2ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

, *<sup>s</sup>*2,2ð Þ*<sup>z</sup>* <sup>≔</sup> *<sup>z</sup>*

*Then, s*1,1*, s*1,2*, s*2,1*, s*2,2 *are Rakotch contractions (with the same function φ) that are*

1 þ 2*z* ,

> 1 þ 4*z :*

> > , *zi*,*<sup>j</sup>*

� � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o.

<sup>1</sup>þ*<sup>t</sup> for t* <sup>∈</sup>ð Þ 0, <sup>þ</sup><sup>∞</sup> *. Let a set of data*

� � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o *be given, where* <sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>&</sup>lt;*x*<sup>1</sup> <sup>&</sup>lt;*x*<sup>2</sup> <sup>¼</sup> 1, 0 <sup>¼</sup> *<sup>y</sup>*<sup>0</sup> <sup>&</sup>lt;*y*<sup>1</sup> < *y*<sup>2</sup> ¼ 1 *and zi*,*<sup>j</sup>* ∈½ � 0, 1 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Let for all i* ¼ 0, 1, 2 *and j* ¼ 0, 1, 2, *z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *z*2,*<sup>j</sup>* ¼ *zi*,2 ¼ 0*:*

<sup>0</sup> ð Þ! *D B D*ð Þ by

*<sup>m</sup>* ð Þþ *<sup>x</sup> <sup>f</sup> <sup>m</sup>*,*<sup>n</sup>v*�<sup>1</sup>

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup> *<sup>n</sup>* ð Þ*<sup>y</sup>* � � � � <sup>þ</sup> *hm*,*<sup>n</sup>*

1.*there is a unique continuous function f* : *D* ! *which is a fixed point of T*;

**Corollary 2.1** *(see* [18]*) Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote*

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

� � <sup>¼</sup> *zi*,*<sup>j</sup>*, *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>*; *<sup>j</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* n o⊂*C*∗ ∗ ð Þ *<sup>D</sup>* .

*<sup>n</sup>* ð Þþ *<sup>y</sup> gm*,*<sup>n</sup>u*�<sup>1</sup>

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

*<sup>m</sup>* ð Þ *<sup>x</sup> <sup>v</sup>*�<sup>1</sup> *<sup>n</sup>* ð Þ*y*

*f x*0, 1ð Þ � *λ y*<sup>0</sup> þ *λyN*

*f xM*, 1ð Þ � *λ y*<sup>0</sup> þ *λyN*

*f* ð Þ 1 � *λ x*<sup>0</sup> þ *λxM*, *y*<sup>0</sup>

*f* ð Þ 1 � *λ x*<sup>0</sup> þ *λxM*, *yN*

for all *λ*∈½ � 0, 1 , where for all *i* ¼ 0, 1, … , *M* and *j* ¼ 0, 1, … , *N*,

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

<sup>0</sup> ð Þ *D* : *f xi*, *y <sup>j</sup>*

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

Let *B D*ð Þ denote the set of bounded functions *f* : *D* ! and

$$B^\*\left(D\right) = \{ f \in B(D) : f\left(\mathbf{x}\_0, \mathbf{y}\_0\right) = \mathbf{z}\_{0,0}, \ f\left(\mathbf{x}\_0, \mathbf{y}\_N\right) = \mathbf{z}\_{0,N}, \}$$

$$f\left(\mathbf{x}\_M, \mathbf{y}\_0\right) = \mathbf{z}\_{M,0}, f\left(\mathbf{x}\_M, \mathbf{y}\_N\right) = \mathbf{z}\_{M,N}\}.$$

Let *<sup>B</sup>*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup> *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* be the set of bounded functions that pass through the given interpolation points *xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � � <sup>∈</sup>*<sup>K</sup>* <sup>¼</sup> *<sup>D</sup>* � ½ � *<sup>a</sup>*, *<sup>b</sup>* : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>*; *<sup>j</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* n o, that is,

$$B^{s,\*}\left(D\right) = \left\{ f \in B^\*\left(D\right) : f\left(x\_i, y\_j\right) = z\_{ij}, i = 0, 1, \ldots, M; j = 0, 1, \ldots, N \right\}.$$

Define an operator *<sup>T</sup>* : *<sup>B</sup>*<sup>∗</sup> ð Þ! *<sup>D</sup> B D*ð Þ for all *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* by

$$Tf(\boldsymbol{x}, \boldsymbol{y}) = F\_{m,n}(\boldsymbol{u}\_m^{-1}(\boldsymbol{x}), \boldsymbol{\upsilon}\_n^{-1}(\boldsymbol{y}), f\left(\boldsymbol{u}\_m^{-1}(\boldsymbol{x}), \boldsymbol{\upsilon}\_n^{-1}(\boldsymbol{y})\right))$$

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*. In [18], we see the following.

**Theorem 2.3.** *Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote the IFS defined above. Assume that the maps Fm*,*<sup>n</sup> are Rakotch or Geraghty contractions with respect to the third variable, and uniformly Lipschitz with respect to the first and second variable. Then,*

1.*there is a unique bounded function f* : *D* ! *which is a fixed point of T*;

$$2. f\left(\mathbf{x}\_i, \mathbf{y}\_j\right) = z\_{i,j} \\ for \ i = 0, 1, \ldots, M \text{ and } j = 0, 1, \ldots, N;$$

3.*if G* ⊂ *D* � *is the graph of f, then*

$$G = \bigcup\_{m=1}^{M} \bigcup\_{n=1}^{N} w\_{m,n}(G).$$

Let for all *i* ¼ 0, 1, … , *M* and *j* ¼ 0, 1, … , *N*, *z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *zM*,*<sup>j</sup>* ¼ *zi*,*<sup>N</sup>* and define

$$F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = e\_{m,n}\mathbf{x} + f\_{m,n}\mathbf{y} + \mathbf{g}\_{m,n}\mathbf{x}\mathbf{y} + s\_{m,n}(\mathbf{z}) + h\_{m,n}\mathbf{y}$$

where *sm*,*<sup>n</sup>* are Rakotch or Geraghty contractions. Let

$$\begin{aligned} \mathbf{C}^\*(D) &= \{ f \in C(D) : f\left(\mathbf{x}\_0, \mathbf{y}\_0\right) = \mathbf{z}\_{0,0}, f\left(\mathbf{x}\_0, \mathbf{y}\_N\right) = \mathbf{z}\_{0,N}, \\ f\left(\mathbf{x}\_M, \mathbf{y}\_0\right) &= \mathbf{z}\_{M,0}, f\left(\mathbf{x}\_M, \mathbf{y}\_N\right) = \mathbf{z}\_{M,N}\} \end{aligned}$$

and

$$\mathcal{C}^{s,\*}\left(\mathcal{D}\right) = \left\{ f \in \mathcal{C}^\*\left(\mathcal{D}\right) : f\left(\mathbf{x}\_i, \boldsymbol{\mathcal{y}}\_j\right) = \mathbf{z}\_{i,j}, i = \mathbf{0}, \mathbf{1}, \ldots, \mathbf{M}; j = \mathbf{0}, \mathbf{1}, \ldots, N\right\}.$$

**122**

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

Let *C*<sup>∗</sup> <sup>0</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* be the set of continuous functions *<sup>f</sup>* : *<sup>D</sup>* ! such that

$$\begin{aligned} f\left(\mathbf{x}\_{0}, (\mathbf{1} - \boldsymbol{\lambda})\mathbf{y}\_{0} + \boldsymbol{\lambda}\mathbf{y}\_{N}\right) &= \mathbf{z}\_{\*, \*}, \\ f\left(\mathbf{x}\_{M}, (\mathbf{1} - \boldsymbol{\lambda})\mathbf{y}\_{0} + \boldsymbol{\lambda}\mathbf{y}\_{N}\right) &= \mathbf{z}\_{\*, \*}, \\ f\left((\mathbf{1} - \boldsymbol{\lambda})\mathbf{x}\_{0} + \boldsymbol{\lambda}\mathbf{x}\_{M}, \mathbf{y}\_{0}\right) &= \mathbf{z}\_{\*, \*}, \\ f\left((\mathbf{1} - \boldsymbol{\lambda})\mathbf{x}\_{0} + \boldsymbol{\lambda}\mathbf{x}\_{M}, \mathbf{y}\_{N}\right) &= \mathbf{z}\_{\*, \*} \end{aligned}$$

for all *λ*∈½ � 0, 1 , where for all *i* ¼ 0, 1, … , *M* and *j* ¼ 0, 1, … , *N*,

$$z\_{\*\;\;\ast} := z\_{0,j} = z\_{i,0} = z\_{M,j} = z\_{i,N}.$$

Let *C*∗ ∗ <sup>0</sup> ð Þ *<sup>D</sup>* <sup>≔</sup> *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> <sup>0</sup> ð Þ *D* : *f xi*, *y <sup>j</sup>* � � <sup>¼</sup> *zi*,*<sup>j</sup>*, *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>*; *<sup>j</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* n o⊂*C*∗ ∗ ð Þ *<sup>D</sup>* . For *f* ∈*C*<sup>∗</sup> <sup>0</sup> ð Þ *<sup>D</sup>* , we define *<sup>T</sup>* : *<sup>C</sup>*<sup>∗</sup> <sup>0</sup> ð Þ! *D B D*ð Þ by

$$Tf(\boldsymbol{x}, \boldsymbol{y}) = F\_{m,n}(\boldsymbol{u}\_m^{-1}(\boldsymbol{x}), \boldsymbol{v}\_n^{-1}(\boldsymbol{y}), f\left(\boldsymbol{u}\_m^{-1}(\boldsymbol{x}), \boldsymbol{v}\_n^{-1}(\boldsymbol{y})\right))$$

$$= \boldsymbol{c}\_{m,n}\boldsymbol{u}\_m^{-1}(\boldsymbol{x}) + f\_{m,n}\boldsymbol{v}\_n^{-1}(\boldsymbol{y}) + \boldsymbol{g}\_{m,n}\boldsymbol{u}\_m^{-1}(\boldsymbol{x})\boldsymbol{v}\_n^{-1}(\boldsymbol{y})$$

$$+ \boldsymbol{s}\_{m,n}\left(\left.f\left(\boldsymbol{u}\_m^{-1}(\boldsymbol{x}), \boldsymbol{v}\_n^{-1}(\boldsymbol{y})\right)\right) + h\_{m,n}$$

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*.

**Corollary 2.1** *(see* [18]*) Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote the IFS defined above. Then,*

1.*there is a unique continuous function f* : *D* ! *which is a fixed point of T*;

$$2. f\left(\mathbf{x}\_i, \mathbf{y}\_j\right) = \mathbf{z}\_{i,j} \\ for \text{ all } i = 0, 1, \dots, M \text{ and } j = 0, 1, \dots, N;$$

3.*if G* ⊂ *D* � *is the graph of f, then*

$$G = \bigcup\_{m=1}^{M} \bigcup\_{n=1}^{N} w\_{m,n}(G).$$

The most simple example is the following; cf. [12].

**Example 2.** *Let <sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>t</sup>* <sup>1</sup>þ*<sup>t</sup> for t* <sup>∈</sup>ð Þ 0, <sup>þ</sup><sup>∞</sup> *. Let a set of data xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o *be given, where* <sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>&</sup>lt;*x*<sup>1</sup> <sup>&</sup>lt;*x*<sup>2</sup> <sup>¼</sup> 1, 0 <sup>¼</sup> *<sup>y</sup>*<sup>0</sup> <sup>&</sup>lt;*y*<sup>1</sup> < *y*<sup>2</sup> ¼ 1 *and zi*,*<sup>j</sup>* ∈½ � 0, 1 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Let for all i* ¼ 0, 1, 2 *and j* ¼ 0, 1, 2,

$$\mathbf{z}\_{0,j} = \mathbf{z}\_{i,0} = \mathbf{z}\_{2,j} = \mathbf{z}\_{i,2} = \mathbf{0}.$$

*Let for z*∈½ Þ 0, þ∞ ,

$$\begin{aligned} s\_{1,1}(z) &:= \frac{z}{1+z}, & s\_{1,2}(z) &:= \frac{z}{1+2z}, \\ s\_{2,1}(z) &:= \frac{z}{1+3z}, & s\_{2,2}(z) &:= \frac{z}{1+4z}. \end{aligned}$$

*Then, s*1,1*, s*1,2*, s*2,1*, s*2,2 *are Rakotch contractions (with the same function φ) that are not Banach contractions on* ½ Þ 0, þ∞ *. So, there exists a continuous function f* : ½ �� 0, 1 ½ �! 0, 1 *that interpolates the given data xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o.

Let *dm*,*<sup>n</sup>* : *D* ! be a function such that max ð Þ *<sup>x</sup>*,*<sup>y</sup>* <sup>∈</sup> *<sup>D</sup>*∣*dm*,*<sup>n</sup>*ð Þ *x*, *y* ∣ ≤1,

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

*xM y*0 *zM*,0 1

CA <sup>¼</sup>

*<sup>B</sup>*<sup>∗</sup> ð Þ¼f *<sup>D</sup> <sup>f</sup>* <sup>∈</sup>*B D*ð Þ : *f x*0, *<sup>y</sup>*<sup>0</sup>

, *zi*,*<sup>j</sup>* � �

� �

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*. In [18], we see the

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

1.*there is a unique bounded function f* : *D* ! *which is a fixed point of T*;

¼ *zi*,*<sup>j</sup> for i* ¼ 0, 1, … , *M and j* ¼ 0, 1, … , *N*;

*G* ¼ ⋃ *M m*¼1 ⋃ *N n*¼1

**Theorem 2.3.** *Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote the IFS defined above. Assume that the maps Fm*,*<sup>n</sup> are Rakotch or Geraghty contractions with respect to the third variable, and uniformly Lipschitz with respect to the first and second*

Define an operator *<sup>T</sup>* : *<sup>B</sup>*<sup>∗</sup> ð Þ! *<sup>D</sup> B D*ð Þ for all *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* by

� � <sup>¼</sup> *zM*,0, *f xM*, *yN*

*Tf x*ð Þ¼ , *<sup>y</sup> Fm*,*<sup>n</sup> <sup>u</sup>*�<sup>1</sup>

*xm yn*�<sup>1</sup> *zm*,*n*�<sup>1</sup> 1

� � <sup>¼</sup> *zM*,*N*g*:*

Let *<sup>B</sup>*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup> *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* be the set of bounded functions that pass through the given

n o

*<sup>n</sup>* ð Þ*<sup>y</sup>* , *f u*�<sup>1</sup>

*wm*,*<sup>n</sup>*ð Þ *G :*

� � <sup>¼</sup> *<sup>z</sup>*0,0, *f x*0, *yN*

n o

� � <sup>¼</sup> *<sup>z</sup>*0,*<sup>N</sup>*,

¼ *zi*,*<sup>j</sup>*, *i* ¼ 0, 1, … , *M*; *j* ¼ 0, 1, … , *N*

*:*

Let for all *i* ¼ 0, 1, … , *M* and *j* ¼ 0, 1, … , *N*, *z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *zM*,*<sup>j</sup>* ¼ *zi*,*<sup>N</sup>* and define

*Fm*,*<sup>n</sup>*ð Þ¼ *x*, *y*, *z em*,*nx* þ *f <sup>m</sup>*,*<sup>n</sup>y* þ *gm*,*<sup>n</sup>xy* þ *sm*,*<sup>n</sup>*ð Þþ *z hm*,*<sup>n</sup>*,

� � <sup>¼</sup> *zM*,*<sup>N</sup>*<sup>g</sup>

� �

*<sup>n</sup>* ð Þ*<sup>y</sup>* � � � �

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

*xM yN zM*,*<sup>N</sup>* 1

∈*K* ¼ *D* � ½ � *a*, *b* : *i* ¼ 0, 1, … , *M*; *j* ¼ 0, 1, … , *N*

¼ *zi*,*<sup>j</sup>*, *i* ¼ 0, 1, … , *M*; *j* ¼ 0, 1, … , *N*

*<sup>m</sup>* ð Þ *<sup>x</sup>* , *<sup>v</sup>*�<sup>1</sup>

CA <sup>¼</sup>

*xm yn zm*,*<sup>n</sup>* 1

CA

,

*:*

0

B@

� � <sup>¼</sup> *<sup>z</sup>*0,*N*,

0

B@

� � <sup>¼</sup> *<sup>z</sup>*0,0, *f x*0, *yN*

n o

0

*Mathematical Theorems - Boundary Value Problems and Approximations*

B@

Let *B D*ð Þ denote the set of bounded functions *f* : *D* ! and

0

B@

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

*f xM*, *y*<sup>0</sup>

*<sup>B</sup>*∗ ∗ ð Þ¼ *<sup>D</sup> <sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* : *f xi*, *<sup>y</sup> <sup>j</sup>*

3.*if G* ⊂ *D* � *is the graph of f, then*

*f xM*, *y*<sup>0</sup>

*<sup>C</sup>*∗ ∗ ð Þ¼ *<sup>D</sup> <sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* : *f xi*, *<sup>y</sup> <sup>j</sup>*

where *sm*,*<sup>n</sup>* are Rakotch or Geraghty contractions. Let

*<sup>C</sup>*<sup>∗</sup> ð Þ¼f *<sup>D</sup> <sup>f</sup>* <sup>∈</sup>*C D*ð Þ : *f x*0, *<sup>y</sup>*<sup>0</sup>

� � <sup>¼</sup> *zM*,0, *f xM*, *yN*

interpolation points *xi*, *y <sup>j</sup>*

that is,

following.

*variable. Then,*

2.*f xi*, *y <sup>j</sup>* � �

and

**122**

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$d\_{m,n}(\mathfrak{x}\_0, \mathfrak{y}) = d\_{m,n}(\mathfrak{x}\_{\mathcal{M}}, \mathfrak{y}) = d\_{m,n}(\mathfrak{x}, \mathfrak{y}\_0) = d\_{m,n}(\mathfrak{x}, \mathfrak{y}\_{\mathcal{N}}) = \mathbf{0}$$

*Then, s*1,1*, s*1,2*, s*2,1*, s*2,2 *are Rakotch contractions (with the same function φ) that are*

In this section, we only present the interconnections between FIFs resulting from Banach contractions and FIFs resulting from Rakotch contractions because in the case of Geraghty contractions, the existence of FICs and FISs is derived similarly

1.Each Banach contraction is a Rakotch contraction, since a self-map is a Banach contraction if and only if it is a *φ*-contraction for a function *φ*ðÞ¼ *t αt*, for some 0≤ *α*<1. There exist examples of Rakotch contraction maps that are not Banach contraction maps on *X* ⊂ with respect to the Euclidean metric

2.The Rakotch's functional condition for convergence of a contractive iteration in a complete metric space can be replaced by an equivalent (or another) functional condition; for instance, a map is a Rakotch contraction if and only if it is a *φ*-contraction for some nondecreasing function *φ* : ð Þ! 0, þ∞ ð Þ 0, þ∞

1. *C I*ð Þ, *dC I*ð Þ � �, *<sup>C</sup>*<sup>∗</sup> ð Þ*<sup>I</sup>* , *dC I*ð Þ � � and *<sup>C</sup>*∗ ∗ ð Þ*<sup>I</sup>* , *dC I*ð Þ � � are complete metric spaces,

2. *B D*ð Þ, *dB D*ð Þ � �, *<sup>B</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* , *dB D*ð Þ � � and *<sup>B</sup>*∗ ∗ ð Þ *<sup>D</sup>* , *dB D*ð Þ � � are complete metric spaces,

ð Þ *x*, *y* ∈ *D*

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup> *f x*ð Þ� *g x*ð Þ<sup>∣</sup>

∣ *f x*ð Þ� , *y g x*ð Þ , *y* ∣

<sup>0</sup> ð Þ *<sup>D</sup>* , *<sup>C</sup>*∗ ∗ ð Þ *<sup>D</sup>* , *<sup>C</sup>*<sup>∗</sup> ð Þ *<sup>D</sup>* and *C D*ð Þ are closed subspaces of *B D*ð Þ with

<sup>0</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*∗ ∗ ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>*

such that additionally *<sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>&</sup>lt;*<sup>t</sup>* for *<sup>t</sup>* <sup>&</sup>gt;0 and the map *<sup>t</sup>* ! *<sup>φ</sup>*ð Þ*<sup>t</sup>*

*dC I*ð Þð Þ *f*, *g* ≔ max

*dB D*ð Þð Þ *f*, *g* ≔ sup

<sup>0</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C D*ð Þ⊂*B D*ð Þ and *<sup>C</sup>*∗ ∗

⊂*C D*ð Þ⊂ *B D*ð Þ, and so they are complete metric spaces.

, *zi*,*<sup>j</sup>*

� � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o*.*

*<sup>t</sup>* is nonincreasing

*not Banach contractions on* ½ Þ 0, þ∞ *. So, there exists a continuous function f* :

½ �� 0, 1 ½ �! 0, 1 *that interpolates the given data xi*, *y <sup>j</sup>*

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

**3. Interconnections between FIFs and contractions**

to the case of Rakotch contractions.

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

**Connection 1**

(see [13]).

(see [19]).

**Connection 2**

where

where

3.*C*∗ ∗

**125**

*C*∗ ∗

for all *f*, *g* ∈*C I*ð Þ (see [2]).

for all *f*, *g* ∈ *B D*ð Þ [10].

<sup>0</sup> ð Þ *<sup>D</sup>* , *<sup>C</sup>*<sup>∗</sup>

<sup>0</sup> ð Þ *<sup>D</sup>* <sup>⊂</sup>*C*<sup>∗</sup>

and for some *L*1, *L*<sup>2</sup> >0,

$$|d\_{m,n}(\mathfrak{x}, \mathfrak{y}) - d\_{m,n}(\mathfrak{x}', \mathfrak{y}')| \le L\_1|\mathfrak{x} - \mathfrak{x}'| + L\_2|\mathfrak{y} - \mathfrak{y}'|.$$

Let

$$F\_{m,n}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = e\_{m,n}\mathbf{x} + f\_{m,n}\mathbf{y} + \mathbf{g}\_{m,n}\mathbf{x}\mathbf{y} + d\_{m,n}(\mathbf{x}, \mathbf{y})\mathbf{s}\_{m,n}(\mathbf{z}) + h\_{m,n}\mathbf{z}$$

where *sm*,*<sup>n</sup>* is a Rakotch or Geraghty contraction. For *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ *<sup>D</sup>* , we define *<sup>T</sup>* : *<sup>C</sup>*<sup>∗</sup> ð Þ! *<sup>D</sup> B D*ð Þ by

$$\begin{aligned} Tf(\mathbf{x}, \boldsymbol{\uprho}) &= F\_{m,n} \left( u\_m^{-1}(\mathbf{x}), v\_n^{-1}(\boldsymbol{\uprho}), f\left( u\_m^{-1}(\mathbf{x}), v\_n^{-1}(\boldsymbol{\uprho}) \right) \right) \\ &= e\_{m,n} u\_m^{-1}(\boldsymbol{\upkappa}) + f\_{m,n} v\_n^{-1}(\boldsymbol{\uprho}) + g\_{m,n} u\_m^{-1}(\boldsymbol{\upkappa}) v\_n^{-1}(\boldsymbol{\uprho}) \\ &+ d\_{m,n} \left( u\_m^{-1}(\boldsymbol{\upkappa}), v\_n^{-1}(\boldsymbol{\uprho}) \right) s\_{m,n} \left( f\left( u\_m^{-1}(\boldsymbol{\upkappa}), v\_n^{-1}(\boldsymbol{\uprho}) \right) \right) + h\_{m,n} \end{aligned}$$

for ð Þ *x*, *y* ∈ *Dm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*. For the next, see [12] for details.

**Corollary 2.2.** *Let D*f g � ; *wm*,*<sup>n</sup>*, *m* ¼ 1, 2, … , *M*; *n* ¼ 1, 2, … , *N denote the IFS defined above. If each sm*,*<sup>n</sup> be a bounded function, then.*

1.*there is a unique continuous function f* : *D* ! *which is a fixed point of T*;

$$2. f\left(\mathbf{x}\_i, \mathbf{y}\_j\right) = \mathbf{z}\_{i,j} \\ for \text{ all } i = 0, 1, \dots, M \text{ and } j = 0, 1, \dots, N;$$

3.*if G* ⊂ *D* � *is a graph of f, then*

$$G = \bigcup\_{m=1}^{M} \bigcup\_{n=1}^{N} w\_{m,n}(G).$$

An especially simple example is the following; see [12].

**Example 3.** *Let <sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>t</sup>* <sup>1</sup>þ*<sup>t</sup> for t*∈ð Þ 0, <sup>þ</sup><sup>∞</sup> *. Let a set of data xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o *be given, where* <sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>0</sup> <sup>&</sup>lt; *<sup>x</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>x</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*,* <sup>0</sup> <sup>¼</sup> *y*<sup>0</sup> <*y*<sup>1</sup> <*y*<sup>2</sup> ¼ 1 *and zi*,*<sup>j</sup>* ∈½ � 0, 1 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Here, a set of data points is not necessarily the case that z*0,*<sup>j</sup>* ¼ *zi*,0 ¼ *z*2,*<sup>j</sup>* ¼ *zi*,2 *for all i* ¼ 0, 1, 2; *j* ¼ 0, 1, 2*. Let for all i* ¼ 1, 2; *j* ¼ 1, 2 *and x*ð Þ , *y* ∈½ �� 0, 1 ½ � 0, 1 ,

$$d\_{m,n}(\varkappa,\boldsymbol{\wp}) := 2^{2(m+n)} \varkappa^m (1-\varkappa)^m \boldsymbol{\wp}^n (1-\boldsymbol{\wp})^n.$$

*Let for z*∈½ Þ 0, þ∞ ,

$$s\_{1,1}(z) \coloneqq \frac{1}{1+z}, s\_{1,2}(z) \coloneqq \frac{z}{1+z},$$

$$s\_{2,1}(z) \coloneqq \frac{z}{1+2z}, s\_{2,2}(z) \coloneqq \frac{z}{1+3z}.$$

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

*Then, s*1,1*, s*1,2*, s*2,1*, s*2,2 *are Rakotch contractions (with the same function φ) that are not Banach contractions on* ½ Þ 0, þ∞ *. So, there exists a continuous function f* : ½ �� 0, 1 ½ �! 0, 1 *that interpolates the given data xi*, *y <sup>j</sup>* , *zi*,*<sup>j</sup>* � � : *<sup>i</sup>* <sup>¼</sup> 0, 1, 2; *<sup>j</sup>* <sup>¼</sup> 0, 1, 2 n o*.*
