**1. Introduction**

*Interpolation* is a method of constructing new data points within the range of a discrete set of known data points or the process of estimating the value of a function at a point from its values at nearby points. Although a large number of interpolation schemes are available in the mathematical field of numerical analysis, the majority of these conventional interpolation methods produce *interpolants*, i.e., functions used to generate interpolation, that are differentiable a number of times except possibly at a finite set of points. Taking into account that the *smoothness* of a function is a property measured by the number of continuous derivatives it has over some domain, the aforementioned interpolants are considered smooth.

On the other hand, many real-world and experimental signals are intricate and rarely show a sensation of smoothness in their traces. Consequently, to model these signals, we require interpolants that are nondifferentiable in dense sets of points in the domain. To address this issue, interpolation by fractal (graph of) functions is introduced in [1, 2], which is based on the theory of *iterated function system*. A *fractal interpolation function* can be considered as a continuous function whose graph is the *attractor*, a fractal set, of an appropriately chosen iterated function system. If this graph has a *Hausdorff-Besicovitch dimension* between 1 and 2, the resulting attractor is called *fractal interpolation curved line* or *fractal interpolation curve*. If this graph has a *Hausdorff-Besicovitch dimension* between 2 and 3, the resulting attractor is called *fractal interpolation surface*. Various types of fractal interpolation functions have

been constructed, and some significant properties of them, including calculus, dimension, smoothness, stability, perturbation error, etc., have been widely studied [3–5].

An *iterated function system*, or *IFS* for short, is a collection of a complete metric

. The associated map of subsets

*f <sup>n</sup>*ð Þ *E* for all *E* ∈ Hð Þ *X* ,

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

*be an IFS consisting*

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

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

*n* ¼ 1, 2, … , *N*. It is often convenient to write an IFS formally as *X*; *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>N</sup>*

space ð Þ *X*, *ρ* together with a finite set of continuous mappings, *f <sup>n</sup>* : *X* ! *X*,

where Hð Þ *X* is the metric space of all nonempty, compact subsets of *X* with respect to some metric, e.g., the Hausdorff metric. The map *W* is called the *Hutchinson operator* or the *collage map* to alert us to the fact that *W E*ð Þis formed as a union

If *wn* are contractions with corresponding contractivity factors *sn* for *n* ¼ 1, 2, … , *N*, the IFS is termed *hyperbolic* and the map *W* itself is then a contraction with contractivity factor *s* ¼ max f g *s*1, *s*2, … , *sN* ([2], Theorem 7.1, p. 81). In what

*<sup>k</sup>* the *k*-fold composition *f* ∘*f* ∘⋯∘*f*. **Definition 2.1.** *Let X be a set. A* self-map on *X or a* transformation *is a mapping*

i. *A self-map f on a metric space X*ð Þ , *ρ is called a φ-*contraction*, if there exists a function φ* : ð Þ! 0, þ∞ ð Þ 0, þ∞ *with ϕ*ð Þ¼ 0 0 *and ϕ*ð Þ*t* <*t for all t*>0 *such*

ii. *We say that f is a* Rakotch contraction*, if f is a φ-contraction such that for any*

*<sup>t</sup>* <sup>&</sup>lt;<sup>1</sup> *and the function* ð Þ 0, <sup>þ</sup><sup>∞</sup> <sup>∍</sup> *<sup>t</sup>* ! *<sup>φ</sup>*ð Þ*<sup>t</sup>*

iii. *If f is a φ-contraction for some function φ* : ð Þ! 0, þ∞ ð Þ 0, þ∞ *such that for*

*(or nondecreasing, or continuous), then we call such a function a* Geraghty

*<sup>t</sup>* <sup>&</sup>lt;<sup>1</sup> *and the function* ð Þ 0, <sup>þ</sup><sup>∞</sup> <sup>∍</sup> *<sup>t</sup>* ! *<sup>φ</sup>*ð Þ*<sup>t</sup>*

*f <sup>n</sup>*ð Þ *K :*

or, somewhat more briefly, as *<sup>X</sup>*; *<sup>f</sup>* <sup>1</sup>�*<sup>N</sup>*

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

*W E*ð Þ¼ ⋃

*that for all x*, *y*∈*X, ρ*ð Þ *f x*ð Þ, *f y*ð Þ ≤*φ ρ*ð Þ ð Þ *x*, *y .*

**Theorem 2.1.** *Let X be a complete metric space and X*; *<sup>f</sup>* <sup>1</sup>�*<sup>N</sup>*

*n* ¼ 1, 2, … , *N*, contractive homeomorphisms *Ln* : *I* ! *In* by

where the real numbers *an*, *bn* are chosen to ensure that *Ln*ðÞ¼ *I In*.

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

*of Rakotch or Geraghty contractions. Then there is a unique nonempty compact set*

*K* ¼ ⋃ *N n*¼1

Let *N* be a positive integer greater than 1 and *I* ¼ ½ � *x*0, *xN* ⊂ . Let a set of

*Ln*ð Þ *x* ≔ *anx* þ *bn*,

Let *φ*: ð Þ! 0, þ∞ ð Þ 0, þ∞ be a nondecreasing continuous function such that for

*x*<sup>0</sup> < *x*<sup>1</sup> < ⋯ <*xN* and *y*0, *y*1, … , *yN* ∈ . Set *In* ¼ ½ � *xn*�1, *xn* ⊂*I* and define, for all

*<sup>t</sup>* <sup>&</sup>lt; 1 and the function 0, ð Þ <sup>þ</sup><sup>∞</sup> <sup>∍</sup> *<sup>t</sup>* ! *<sup>φ</sup>*ð Þ*<sup>t</sup>*

<sup>∈</sup> *<sup>I</sup>* � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* be given, where

*N n*¼1

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

*W* : Hð Þ!H *X* ð Þ *X* is given by:

or 'collage' of sets.

*from X to itself.*

follows, we abbreviate by *f*

*<sup>t</sup>*>0*, <sup>α</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>φ</sup>*ð Þ*<sup>t</sup>*

contraction.

**2.1 Fractal interpolation in**

*interpolation points xi*, *yi*

any *<sup>t</sup>*<sup>&</sup>gt; 0, *<sup>α</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>φ</sup>*ð Þ*<sup>t</sup>*

**119**

*K* ∈ Hð Þ *X such that*

*any t*>0*, <sup>α</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>φ</sup>*ð Þ*<sup>t</sup>*

From [14], we have the following.

*Fractal interpolation* is an advanced technique for analysis and synthesis of scientific and engineering data, whereas the approximation of natural curves and surfaces in these areas has emerged as an important research field. *Fractal functions* are currently being given considerable attention due to their applications in areas such as Metallurgy, Earth Sciences, Surface Physics, Chemistry and Medical Sciences. In the development of fractal interpolation theory, many researchers have generalised the notion in different ways [6–9]. Two key issues should be addressed in constructing fractal interpolation functions. They regard to ensuring continuity and the existence of the *contractivity*, or *vertical scaling*, *factors*; see [10, 11]. In [12], nonlinear fractal interpolation surfaces resulting from Rakotch or Geraghty contractions together with some continuity conditions were introduced as well as explicit illustrative examples were given.

The concept of iterated function system was originally introduced as a generalisation of the well-known Banach contraction principle. Since it has become a powerful tool for constructing and analysing fractal interpolation functions, one can use the well-known fixed point results obtained in the fixed point theory in order to construct them in a more general sense. A comparison of various definitions of contractive mappings as well as fixed point theorems that can be used to construct iterated function systems can be found in [13–15]. In [14], the authors proposed some iterated function systems by using various fixed point theorems, but unfortunately, one does not know whether fractal interpolation functions correspond to those may exist or not. As far as we know, the first significant generalisation of Banach's principle was obtained by Rakotch [16] in 1962. Recently, a method to generate nonlinear fractal interpolation functions by using the Rakotch or Geraghty fixed point theorem instead of Banach fixed point theorem was presented in [12, 17, 18].

The aim of our article is to provide the connections between several fractal interpolation functions and the contractions used to generate them; it is organised as follows. In Section 2, we recall the results obtained in construction of fractal interpolation curved lines and fractal interpolation surfaces by using Rakotch contractions (or Geraghty contractions) instead of Banach contractions. In Section 3, we only present the connection between fractal interpolation functions by using the Banach contractions and fractal interpolation functions by using the Rakotch contractions because in the case of Geraghty contractions, the existence of fractal interpolation curved lines and fractal interpolation surfaces is similar to the case of Rakotch contractions.

## **2. Preliminaries**

Let ð Þ *X*, *ρ* and ð Þ *Y*, *σ* be metric spaces. A mapping *T*: *X* ! *Y* is called a *Hölder mapping of exponent* or *order a*, if

$$\sigma(T(\mathfrak{x}), T(\mathfrak{y})) \le \mathfrak{c} \left[ \rho(\mathfrak{x}, \mathfrak{y}) \right]^{\mathfrak{a}}$$

for *x*, *y*∈*X*, *a*≥0 and for some constant *c*. Note that, if *a*> 1, the functions are constants. Obviously, *c*≥0. The mapping *T* is called a *Lipschitz mapping*, if *a* may be taken to be equal to 1. If *c* ¼ 1, *T* is said to be *nonexpansive*. A Lipschitz function is a *contraction* with *contractivity factor c*, if *c*< 1. We call *T contractive*, if for all *x*, *y* ∈*X* and *x* 6¼ *y*, we have *σ*ð Þ *T x*ð Þ, *T y*ð Þ <*ρ*ð Þ *x*, *y* . Note that 'contraction ) contractive ) nonexpansive ) Lipschitz'.

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

An *iterated function system*, or *IFS* for short, is a collection of a complete metric space ð Þ *X*, *ρ* together with a finite set of continuous mappings, *f <sup>n</sup>* : *X* ! *X*, *n* ¼ 1, 2, … , *N*. It is often convenient to write an IFS formally as *X*; *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>N</sup>* or, somewhat more briefly, as *<sup>X</sup>*; *<sup>f</sup>* <sup>1</sup>�*<sup>N</sup>* . The associated map of subsets *W* : Hð Þ!H *X* ð Þ *X* is given by:

$$\mathcal{W}(E) = \bigcup\_{n=1}^{N} f\_n(E) \quad \text{for all} \ E \in \mathcal{H}(X),$$

where Hð Þ *X* is the metric space of all nonempty, compact subsets of *X* with respect to some metric, e.g., the Hausdorff metric. The map *W* is called the *Hutchinson operator* or the *collage map* to alert us to the fact that *W E*ð Þis formed as a union or 'collage' of sets.

If *wn* are contractions with corresponding contractivity factors *sn* for *n* ¼ 1, 2, … , *N*, the IFS is termed *hyperbolic* and the map *W* itself is then a contraction with contractivity factor *s* ¼ max f g *s*1, *s*2, … , *sN* ([2], Theorem 7.1, p. 81). In what follows, we abbreviate by *f <sup>k</sup>* the *k*-fold composition *f* ∘*f* ∘⋯∘*f*.

**Definition 2.1.** *Let X be a set. A* self-map on *X or a* transformation *is a mapping from X to itself.*


From [14], we have the following.

**Theorem 2.1.** *Let X be a complete metric space and X*; *<sup>f</sup>* <sup>1</sup>�*<sup>N</sup> be an IFS consisting of Rakotch or Geraghty contractions. Then there is a unique nonempty compact set K* ∈ Hð Þ *X such that*

$$K = \bigcup\_{n=1}^{N} f\_n(K).$$

### **2.1 Fractal interpolation in**

Let *N* be a positive integer greater than 1 and *I* ¼ ½ � *x*0, *xN* ⊂ . Let a set of *interpolation points xi*, *yi* <sup>∈</sup> *<sup>I</sup>* � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* be given, where *x*<sup>0</sup> < *x*<sup>1</sup> < ⋯ <*xN* and *y*0, *y*1, … , *yN* ∈ . Set *In* ¼ ½ � *xn*�1, *xn* ⊂*I* and define, for all *n* ¼ 1, 2, … , *N*, contractive homeomorphisms *Ln* : *I* ! *In* by

$$L\_n(\infty) := a\_n \infty + b\_n \infty$$

where the real numbers *an*, *bn* are chosen to ensure that *Ln*ðÞ¼ *I In*.

Let *φ*: ð Þ! 0, þ∞ ð Þ 0, þ∞ be a nondecreasing continuous function such that for any *<sup>t</sup>*<sup>&</sup>gt; 0, *<sup>α</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>φ</sup>*ð Þ*<sup>t</sup> <sup>t</sup>* <sup>&</sup>lt; 1 and the function 0, ð Þ <sup>þ</sup><sup>∞</sup> <sup>∍</sup> *<sup>t</sup>* ! *<sup>φ</sup>*ð Þ*<sup>t</sup> <sup>t</sup>* is nonincreasing. Let *dn* : *I* ! be a continuously differentiable function such that

been constructed, and some significant properties of them, including calculus, dimension, smoothness, stability, perturbation error, etc., have been widely

*Mathematical Theorems - Boundary Value Problems and Approximations*

*Fractal interpolation* is an advanced technique for analysis and synthesis of scientific and engineering data, whereas the approximation of natural curves and surfaces in these areas has emerged as an important research field. *Fractal functions* are currently being given considerable attention due to their applications in areas such as Metallurgy, Earth Sciences, Surface Physics, Chemistry and Medical

Sciences. In the development of fractal interpolation theory, many researchers have generalised the notion in different ways [6–9]. Two key issues should be addressed in constructing fractal interpolation functions. They regard to ensuring continuity and the existence of the *contractivity*, or *vertical scaling*, *factors*; see [10, 11]. In [12], nonlinear fractal interpolation surfaces resulting from Rakotch or Geraghty contractions together with some continuity conditions were introduced as well as

The concept of iterated function system was originally introduced as a generalisation of the well-known Banach contraction principle. Since it has become a powerful tool for constructing and analysing fractal interpolation functions, one can use the well-known fixed point results obtained in the fixed point theory in order to construct them in a more general sense. A comparison of various definitions of contractive mappings as well as fixed point theorems that can be used to construct iterated function systems can be found in [13–15]. In [14], the authors proposed some iterated function systems by using various fixed point theorems, but unfortunately, one does not know whether fractal interpolation functions correspond to those may exist or not. As far as we know, the first significant generalisation of Banach's principle was obtained by Rakotch [16] in 1962. Recently, a method to generate nonlinear fractal interpolation functions by using the Rakotch or Geraghty fixed point theorem instead

The aim of our article is to provide the connections between several fractal interpolation functions and the contractions used to generate them; it is organised as follows. In Section 2, we recall the results obtained in construction of fractal interpolation curved lines and fractal interpolation surfaces by using Rakotch contractions (or Geraghty contractions) instead of Banach contractions. In Section 3, we only present the connection between fractal interpolation functions by using the Banach contractions and fractal interpolation functions by using the Rakotch contractions because in the case of Geraghty contractions, the existence of fractal interpolation curved lines and fractal interpolation surfaces is similar to the case of

Let ð Þ *X*, *ρ* and ð Þ *Y*, *σ* be metric spaces. A mapping *T*: *X* ! *Y* is called a *Hölder*

*<sup>σ</sup>*ð Þ *T x*ð Þ, *T y*ð Þ <sup>≤</sup>*c*½ � *<sup>ρ</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup> <sup>a</sup>*

for *x*, *y*∈*X*, *a*≥0 and for some constant *c*. Note that, if *a*> 1, the functions are constants. Obviously, *c*≥0. The mapping *T* is called a *Lipschitz mapping*, if *a* may be taken to be equal to 1. If *c* ¼ 1, *T* is said to be *nonexpansive*. A Lipschitz function is a *contraction* with *contractivity factor c*, if *c*< 1. We call *T contractive*, if for all *x*, *y* ∈*X* and *x* 6¼ *y*, we have *σ*ð Þ *T x*ð Þ, *T y*ð Þ <*ρ*ð Þ *x*, *y* . Note that 'contraction ) contractive )

studied [3–5].

explicit illustrative examples were given.

Rakotch contractions.

*mapping of exponent* or *order a*, if

nonexpansive ) Lipschitz'.

**118**

**2. Preliminaries**

of Banach fixed point theorem was presented in [12, 17, 18].

*Mathematical Theorems - Boundary Value Problems and Approximations*

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

*<sup>n</sup>* <sup>¼</sup> 1, 2, … , *N, sn*ð Þ*<sup>y</sup>* <sup>≔</sup> *<sup>y</sup>*

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

where

*points xi*, *yi*

f g ½ �� 0, 1 ; *w*1, *w*2, … , *wN , i.e.,*

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

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

0, 1, … , *N*. Set *I* ¼ ½ � *x*0, *xM* ⊂ and *J* ¼ *y*0, *yN*

Set *Im* <sup>¼</sup> ½ � *xm*�1, *xm* , *Jn* <sup>¼</sup> *yn*�<sup>1</sup>, *yn*

the Euclidean, or to some other equivalent, metric.

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

*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>

are transformations of the special structure

*x*0 *y*0 *z*0,0 1

CA <sup>¼</sup>

0

B@

0

B@

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

**121**

<sup>1</sup>þ*ny :That is, each sn is a Rakotch contraction (with the same*

*function φ) that is not a Banach contraction on* ½ Þ 0, þ∞ *. Let for all n* ¼ 1, 2, … , *N,*

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

*wn*ð Þ *x*, *y* ≔ ð Þ *anx* þ *bn*,*cnx* þ *dn*ð Þ *x sn*ð Þþ *y en* ,

*an* ¼ *xn* � *xn*�1, *bn* ¼ *xn*�1, *cn* <sup>¼</sup> *yn* � *yn*�1, *en* <sup>¼</sup> *yn*�1*:*

*Then, there exists a continuous function f* : ½ �! 0, 1 *that interpolates the given*

*G* ¼ ⋃ *N n*¼1

*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

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

*Lm*,*<sup>n</sup>* : *D* ! *Dm*,*<sup>n</sup>* be defined for *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*, by

� � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* � �*. Moreover, the graph G of f is invariant with respect to*

*wn*ð Þ *G :*

Let *M*, *N* be two positive integers greater than 1. Let us represent the given set of

will work in the complete metric space *K* ¼ *D* � , where *D* ¼ *I* � *J*, with respect to

*Lm*,*<sup>n</sup>*ð Þ¼ *x*, *y* ð Þ¼ *um*ð Þ *x* , *vn*ð Þ*y* ð Þ *amx* þ *bm*,*cny* þ *dn :*

, *bm* <sup>¼</sup> *xm*�<sup>1</sup> � *xm* � *xm*�<sup>1</sup>

, *dn* <sup>¼</sup> *yn*�<sup>1</sup> � *yn* � *yn*�<sup>1</sup>

Furthermore, for *m* ¼ 1, 2, … , *M* and *n* ¼ 1, 2, … , *N*, let mappings *Fm*,*<sup>n</sup>* : *K* !

*wm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* ≔ ð Þ *Lm*,*<sup>n</sup>*ð Þ *x*, *y* , *Fm*,*<sup>n</sup>*ð Þ *x*, *y*, *z* ,

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

1

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>* �

where the transformations are constrained by the data according to

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

� �⊂ . Throughout this section, we

� �, *Dm*,*<sup>n</sup>* <sup>¼</sup> *Im* � *Jn* and let *um* : *<sup>I</sup>* ! *Im*, *vn* : *<sup>J</sup>* ! *Jn*,

*xM* � *x*<sup>0</sup>

*y*0*:*

*yN* � *y*<sup>0</sup>

*x*0 *yN z*0,*<sup>N</sup>* 1

CA <sup>¼</sup>

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

CA,

0

B@

0

B@

*x*0,

Now, consider an IFS of the form f g *I* � ; *wn*, *n* ¼ 1, 2, … , *N* in which the maps are nonlinear transformations of the special structure

$$w\_n \begin{pmatrix} \varkappa \\ \jmath \end{pmatrix} = \begin{pmatrix} L\_n(\varkappa) \\ F\_n(\varkappa, \jmath) \end{pmatrix} = \begin{pmatrix} a\_n \varkappa + b\_n \\ c\_n \varkappa + d\_n(\varkappa) s\_n(\jmath) + e\_n \end{pmatrix},$$

where the transformations are constrained by the data according to

$$w\_n \binom{\mathbf{x}\_0}{\mathcal{Y}\_0} = \binom{\mathbf{x}\_{n-1}}{\mathcal{Y}\_{n-1}}, \quad w\_n \binom{\mathbf{x}\_N}{\mathcal{Y}\_N} = \binom{\mathbf{x}\_n}{\mathcal{Y}\_n}$$

for *n* ¼ 1, 2, … , *N*, and *sn* are some Rakotch or Geraghty contractions.

Let us denote by *C D*ð Þ the linear space of all real-valued continuous functions defined on *<sup>D</sup>*, i.e., *C D*ð Þ¼ f g *<sup>f</sup>* : *<sup>D</sup>* ! <sup>j</sup> *<sup>f</sup>* continuous . Let *<sup>C</sup>*<sup>∗</sup> ð Þ*<sup>I</sup>* <sup>⊂</sup>*C I*ð Þ denote the set of continuous functions *f* : *I* ! such that *f x*ð Þ¼ <sup>0</sup> *y*<sup>0</sup> and *f x*ð Þ¼ *<sup>N</sup> yN*, that is,

$$\mathcal{C}^\*(I) \coloneqq \left\{ f \in \mathcal{C}(I) : f(\mathfrak{x}\_0) = \mathfrak{y}\_0, f(\mathfrak{x}\_N) = \mathfrak{y}\_N \right\}.$$

Let *<sup>C</sup>*∗ ∗ ð Þ*<sup>I</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ*<sup>I</sup>* <sup>⊂</sup>*C I*ð Þ be the set of continuous functions that pass through the given data points *xi*, *yi* ∈*<sup>I</sup>* � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* , that is,

$$\mathcal{C}^{\*,\*}\left(I\right) \coloneqq \left\{ f \in \mathcal{C}^\*\left(I\right) : f\left(\mathbf{x}\_i\right) = \mathbf{y}\_i, i = \mathbf{0}, \mathbf{1}, \dots, N \right\}.$$

Define a metric *dC I*ð Þ on the space *C I*ð Þ by

$$d\_{C(I)}(\mathbf{g}, h) := \max\_{\mathbf{x} \in [\mathbf{x}\_0, \mathbf{x}\_N]} |\mathbf{g}(\mathbf{x}) - h(\mathbf{x})|^2$$

for all *<sup>g</sup>*, *<sup>h</sup>*∈*C I*ð Þ. Define a mapping *<sup>T</sup>*: *<sup>C</sup>*<sup>∗</sup> ðÞ!*<sup>I</sup> C I*ð Þ for all *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ*<sup>I</sup>* by

$$\begin{aligned} T\!f^\circ(\varkappa) &:= F\_n\left(L\_n^{-1}(\varkappa), f\left(L\_n^{-1}(\varkappa)\right)\right) \\ &= c\_n L\_n^{-1}(\varkappa) + d\_n\left(L\_n^{-1}(\varkappa)\right) s\_n\left(f\left(L\_n^{-1}(\varkappa)\right)\right) + e\_n \end{aligned}$$

for *x*∈ ½ � *xn*�1, *xn* and *n* ¼ 1, 2, … , *N*. From [17], we have the following.

**Theorem 2.2.** *Let I* f g � ; *wn*, *n* ¼ 1, 2, … , *N denote the IFS defined above. Let each sn be a bounded Rakotch or Geraghty contraction. Then,*

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

$$\text{iii.} f(\mathbf{x}\_i) = \mathbf{y}\_i \text{ for all } i = \mathbf{0}, \mathbf{1}, \dots, N;$$

iii. *if G* ⊂*I* � *is the graph of f, then*

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

An extremely explicit simple example is the following; cf. [12].

**Example 1.** *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*, *yi* : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup> 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>&</sup>lt;*xN* <sup>¼</sup> <sup>1</sup> *and yi* <sup>∈</sup>½ � 0, 1 *for all i* <sup>¼</sup> 0, 1, … , *N. Let for all n* <sup>¼</sup> 1, 2, … , *N, dn*ð Þ *<sup>x</sup>* <sup>≔</sup> *<sup>x</sup><sup>n</sup>: Let for y*∈½ Þ 0, <sup>þ</sup><sup>∞</sup> *and*

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

*<sup>n</sup>* <sup>¼</sup> 1, 2, … , *N, sn*ð Þ*<sup>y</sup>* <sup>≔</sup> *<sup>y</sup>* <sup>1</sup>þ*ny :That is, each sn is a Rakotch contraction (with the same function φ) that is not a Banach contraction on* ½ Þ 0, þ∞ *. Let for all n* ¼ 1, 2, … , *N,*

$$w\_n(\mathfrak{x}, \mathfrak{y}) := (a\_n \mathfrak{x} + b\_n, c\_n \mathfrak{x} + d\_n(\mathfrak{x})\mathfrak{s}\_n(\mathfrak{y}) + e\_n),$$

where

max

are nonlinear transformations of the special structure

*x*0 *y*0 

<sup>¼</sup> *Ln*ð Þ *<sup>x</sup> Fn*ð Þ *x*, *y* 

*Mathematical Theorems - Boundary Value Problems and Approximations*

where the transformations are constrained by the data according to

<sup>¼</sup> *xn*�<sup>1</sup> *yn*�<sup>1</sup> 

for *n* ¼ 1, 2, … , *N*, and *sn* are some Rakotch or Geraghty contractions.

*<sup>C</sup>*∗ ∗ ð Þ*<sup>I</sup>* <sup>≔</sup> *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ*<sup>I</sup>* : *f x*ð Þ¼*<sup>i</sup> yi*

*dC I*ð Þð Þ *<sup>g</sup>*, *<sup>h</sup>* <sup>≔</sup> max *<sup>x</sup>*∈½ � *<sup>x</sup>*0, *xN*

for all *<sup>g</sup>*, *<sup>h</sup>*∈*C I*ð Þ. Define a mapping *<sup>T</sup>*: *<sup>C</sup>*<sup>∗</sup> ðÞ!*<sup>I</sup> C I*ð Þ for all *<sup>f</sup>* <sup>∈</sup>*C*<sup>∗</sup> ð Þ*<sup>I</sup>* by

*<sup>n</sup>* ð Þ *<sup>x</sup>* , *f L*�<sup>1</sup> *<sup>n</sup>* ð Þ *<sup>x</sup>*

*<sup>n</sup>* ð Þþ *<sup>x</sup> dn <sup>L</sup>*�<sup>1</sup>

*G* ¼ ⋃ *N n*¼1

An extremely explicit simple example is the following; cf. [12].

**Theorem 2.2.** *Let I* f g � ; *wn*, *n* ¼ 1, 2, … , *N denote the IFS defined above. Let*

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

*wn*ð Þ *G :*

<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, … , *<sup>N</sup> 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>&</sup>lt;*xN* <sup>¼</sup> <sup>1</sup> *and yi* <sup>∈</sup>½ � 0, 1 *for all i* <sup>¼</sup> 0, 1, … , *N. Let for all n* <sup>¼</sup> 1, 2, … , *N, dn*ð Þ *<sup>x</sup>* <sup>≔</sup> *<sup>x</sup><sup>n</sup>: Let for y*∈½ Þ 0, <sup>þ</sup><sup>∞</sup> *and*

for *x*∈ ½ � *xn*�1, *xn* and *n* ¼ 1, 2, … , *N*. From [17], we have the following.

*wn x y* 

the given data points *xi*, *yi*

Define a metric *dC I*ð Þ on the space *C I*ð Þ by

*Tf x*ð Þ <sup>≔</sup> *Fn <sup>L</sup>*�<sup>1</sup>

ii. *f x*ð Þ¼*<sup>i</sup> yi for all i* ¼ 0, 1, … , *N;*

iii. *if G* ⊂*I* � *is the graph of f, then*

**Example 1.** *Let <sup>φ</sup>*ð Þ*<sup>t</sup>* <sup>≔</sup> *<sup>t</sup>*

*xi*, *yi*

**120**

<sup>¼</sup> *cnL*�<sup>1</sup>

*each sn be a bounded Rakotch or Geraghty contraction. Then,*

*wn*

*<sup>x</sup>*∈*<sup>I</sup>* <sup>∣</sup>*dn*ð Þ *<sup>x</sup>* ∣ ≤ <sup>1</sup>*:*

Now, consider an IFS of the form f g *I* � ; *wn*, *n* ¼ 1, 2, … , *N* in which the maps

, *wn*

Let us denote by *C D*ð Þ the linear space of all real-valued continuous functions defined on *<sup>D</sup>*, i.e., *C D*ð Þ¼ f g *<sup>f</sup>* : *<sup>D</sup>* ! <sup>j</sup> *<sup>f</sup>* continuous . Let *<sup>C</sup>*<sup>∗</sup> ð Þ*<sup>I</sup>* <sup>⊂</sup>*C I*ð Þ denote the set of continuous functions *f* : *I* ! such that *f x*ð Þ¼ <sup>0</sup> *y*<sup>0</sup> and *f x*ð Þ¼ *<sup>N</sup> yN*, that is,

*<sup>C</sup>*<sup>∗</sup> ð Þ*<sup>I</sup>* <sup>≔</sup> *<sup>f</sup>* <sup>∈</sup>*C I*ð Þ : *f x*ð Þ¼ <sup>0</sup> *<sup>y</sup>*0, *f x*ð Þ¼ *<sup>N</sup> yN*

∈*<sup>I</sup>* � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* , that is,

Let *<sup>C</sup>*∗ ∗ ð Þ*<sup>I</sup>* <sup>⊂</sup>*C*<sup>∗</sup> ð Þ*<sup>I</sup>* <sup>⊂</sup>*C I*ð Þ be the set of continuous functions that pass through

*:*

, *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup> :*

∣*g x*ð Þ� *h x*ð Þ∣

*<sup>n</sup>* ð Þ *<sup>x</sup> sn f L*�<sup>1</sup>

*<sup>n</sup>* ð Þ *<sup>x</sup>* <sup>þ</sup> *en*

<sup>¼</sup> *anx* <sup>þ</sup> *bn*

*xN yN* 

*cnx* þ *dn*ð Þ *x sn*ð Þþ *y en* 

> <sup>¼</sup> *xn yn*

,

$$\mathcal{A}\_n = \mathcal{x}\_n - \mathcal{x}\_{n-1}, \quad b\_n = \mathcal{x}\_{n-1},$$

$$\mathcal{c}\_n = \mathcal{y}\_n - \mathcal{y}\_{n-1}, \quad \mathcal{c}\_n = \mathcal{y}\_{n-1}.$$

*Then, there exists a continuous function f* : ½ �! 0, 1 *that interpolates the given points xi*, *yi* � � : *<sup>i</sup>* <sup>¼</sup> 0, 1, … , *<sup>N</sup>* � �*. Moreover, the graph G of f is invariant with respect to* f g ½ �� 0, 1 ; *w*1, *w*2, … , *wN , i.e.,*

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