**Connection 13**

1. In Theorem 2.2, we can see that

$$a\_n = \frac{\mathbf{x}\_n - \mathbf{x}\_{n-1}}{\mathbf{x}\_N - \mathbf{x}\_0}, \quad b\_n = \frac{\mathbf{x}\_N \mathbf{x}\_{n-1} - \mathbf{x}\_0 \mathbf{x}\_n}{\mathbf{x}\_N - \mathbf{x}\_0}$$

$$c\_n = \frac{\mathbf{y}\_n - \mathbf{y}\_{n-1}}{\mathbf{x}\_N - \mathbf{x}\_0} - \frac{d\_n(\mathbf{x}\_N)s\_n\left(\mathbf{y}\_N\right) - d\_n(\mathbf{x}\_0)s\_n\left(\mathbf{y}\_0\right)}{\mathbf{x}\_N - \mathbf{x}\_0},$$

$$f\_n = \frac{\mathbf{x}\_N \mathbf{y}\_{n-1} - \mathbf{x}\_0 \mathbf{y}\_n}{\mathbf{x}\_N - \mathbf{x}\_0} - \frac{\mathbf{x}\_N d\_n(\mathbf{x}\_0)s\_n\left(\mathbf{y}\_0\right) - \mathbf{x}\_0 d\_n(\mathbf{x}\_N)s\_n\left(\mathbf{y}\_N\right)}{\mathbf{x}\_N - \mathbf{x}\_0}.$$

**Figure 1.**

*The graph of a fractal interpolation function (a) that is associated with Banach contractions, (b) that is not necessarily associated with Banach contractions.*

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

2. In Corollary 2.1, we can see that

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.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

, *bn* <sup>¼</sup> *xNxn*�<sup>1</sup> � *<sup>x</sup>*0*xn*

� *xNdn*ð Þ *<sup>x</sup>*<sup>0</sup> *sn <sup>y</sup>*<sup>0</sup>

*The graph of a fractal interpolation function (a) that is associated with Banach contractions, (b) that is not*

� *dn*ð Þ *xN sn yN*

*xN* � *x*<sup>0</sup>

*xN* � *x*<sup>0</sup>

� *dn*ð Þ *<sup>x</sup>*<sup>0</sup> *sn <sup>y</sup>*<sup>0</sup>

� *<sup>x</sup>*0*dn*ð Þ *xN sn yN*

*xN* � *x*<sup>0</sup>

,

*:*

(continuous) 'contraction functions' were used.

*Mathematical Theorems - Boundary Value Problems and Approximations*

1. In Theorem 2.2, we can see that

*an* <sup>¼</sup> *xn* � *xn*�<sup>1</sup> *xN* � *x*<sup>0</sup>

*cn* <sup>¼</sup> *yn* � *yn*�<sup>1</sup> *xN* � *x*<sup>0</sup>

*<sup>f</sup> <sup>n</sup>* <sup>¼</sup> *xNyn*�<sup>1</sup> � *<sup>x</sup>*0*yn xN* � *x*<sup>0</sup>

**Connection 13**

**Figure 1.**

**130**

*necessarily associated with Banach contractions.*

*am* <sup>¼</sup> *xm* � *xm*�<sup>1</sup> *xM* � *x*<sup>0</sup> , *bm* <sup>¼</sup> *xMxm*�<sup>1</sup> � *<sup>x</sup>*0*xm xM* � *x*<sup>0</sup> , *cn* <sup>¼</sup> *yn* � *yn*�<sup>1</sup> *yN* � *y*<sup>0</sup> , *dn* <sup>¼</sup> *yNyn*�<sup>1</sup> � *<sup>y</sup>*0*yn yN* � *y*<sup>0</sup> , *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> � *sm*,*<sup>n</sup>*ð Þ *zM*,*<sup>N</sup> :*

#### **Figure 2.**

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

3. In Corollary 2.2, we can see that (compare with above coefficients)

$$\mathcal{G}\_{m,n} = \frac{(z\_{m,n} - z\_{m-1,n}) - (z\_{m,n-1} - z\_{m-1,n-1})}{(\mathbf{x}\_M - \mathbf{x}\_0)(\mathbf{y}\_N - \mathbf{y}\_0)},$$

$$\mathbf{c}\_{m,n} = \frac{\mathbf{y}\_N(z\_{m,n-1} - z\_{m-1,n-1}) - \mathbf{y}\_0(z\_{m,n} - z\_{m-1,n})}{(\mathbf{x}\_M - \mathbf{x}\_0)(\mathbf{y}\_N - \mathbf{y}\_0)},$$

$$f\_{m,n} = \frac{\mathbf{x}\_M(z\_{m-1,n} - z\_{m-1,n-1}) - \mathbf{x}\_0(z\_{m,n} - z\_{m,n-1})}{(\mathbf{x}\_M - \mathbf{x}\_0)(\mathbf{y}\_N - \mathbf{y}\_0)},$$

$$h\_{m,n} = \frac{\mathbf{x}\_0 \mathbf{y}\_0 z\_{m,n} - \mathbf{x}\_0 \mathbf{y}\_N z\_{m,n-1} - \mathbf{x}\_M \mathbf{y}\_0 z\_{m-1,n} + \mathbf{x}\_M \mathbf{y}\_N z\_{m-1,n-1}}{(\mathbf{x}\_M - \mathbf{x}\_0)(\mathbf{y}\_N - \mathbf{y}\_0)}.$$

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**Figures 1(a)** and **2(a)** are associated with Banach contractions, whereas **Figures 1(b)** and **2(b)** are not necessarily associated with Banach contractions.
