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How Are Fractal Interpolation Functions Related to Several Contractions?

By SongIl Ri and Vasileios Drakopoulos

Submitted: December 23rd 2019Reviewed: April 27th 2020Published: August 21st 2020

DOI: 10.5772/intechopen.92662

Downloaded: 99

Abstract

This chapter provides an overview of several types of fractal interpolation functions that are often studied by many researchers and includes some of the latest research made by the authors. Furthermore, it focuses on the connections between fractal interpolation functions resulting from Banach contractions as well as those resulting from Rakotch contractions. Our aim is to give theoretical and practical significance for the generation of fractal (graph of) functions in two and three dimensions for interpolation purposes that are not necessarily associated with Banach contractions.

Keywords

  • attractor
  • contraction
  • fixed point
  • iterated function system
  • fractal interpolation

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, 4, 5].

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, 7, 8, 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, 14, 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 Xρand Yσbe metric spaces. A mapping T:XYis called a Hölder mapping of exponent or order a, if

σTxTycρxya

for x,yX, a0and for some constant c. Note that, if a>1, the functions are constants. Obviously, c0. The mapping Tis called a Lipschitz mapping, if amay be taken to be equal to 1. If c=1, Tis said to be nonexpansive. A Lipschitz function is a contraction with contractivity factor c, if c<1. We call Tcontractive, if for all x,yXand xy, we have σTxTy<ρxy. Note that ‘contraction contractive nonexpansive Lipschitz’.

An iterated function system, or IFS for short, is a collection of a complete metric space Xρtogether with a finite set of continuous mappings, fn:XX, n=1,2,,N. It is often convenient to write an IFS formally as Xf1f2fNor, somewhat more briefly, as Xf1N. The associated map of subsets W:HXHXis given by:

WE=n=1NfnEforallEHX,

where HXis the metric space of all nonempty, compact subsets of Xwith respect to some metric, e.g., the Hausdorff metric. The map Wis called the Hutchinson operator or the collage map to alert us to the fact that WEis formed as a union or ‘collage’ of sets.

If wnare contractions with corresponding contractivity factors snfor n=1,2,,N, the IFS is termed hyperbolic and the map Witself is then a contraction with contractivity factor s=maxs1s2sN([2], Theorem 7.1, p. 81). In what follows, we abbreviate by fkthe k-fold composition fff.

Definition 2.1. Let Xbe a set. A self-map on Xor a transformation is a mapping from Xto itself.

  1. A self-map fon a metric space Xρis called a φ-contraction, if there exists a function φ:0+0+with ϕ0=0and ϕt<tfor all t>0such that for all x,yX, ρfxfyφρxy.

  2. We say that fis a Rakotch contraction, if fis a φ-contraction such that for any t>0, αtφtt<1and the function 0+tφttis nonincreasing.

  3. If fis a φ-contraction for some function φ:0+0+such that for any t>0, αtφtt<1and the function 0+tφttis nonincreasing (or nondecreasing, or continuous), then we call such a function a Geraghty contraction.

From [14], we have the following.

Theorem 2.1. Let Xbe a complete metric space and Xf1Nbe an IFSconsisting of Rakotch or Geraghty contractions. Then there is a unique nonempty compact set KHXsuch that

K=n=1NfnK.

2.1 Fractal interpolation in R

Let Nbe a positive integer greater than 1 and I=x0xNR. Let a set of interpolation points xiyiI×R:i=01Nbe given, where x0<x1<<xNand y0,y1,,yNR. Set In=xn1xnIand define, for all n=1,2,,N, contractive homeomorphisms Ln:IInby

Lnxanx+bn,

where the real numbers an,bnare chosen to ensure that LnI=In.

Let φ:0+0+be a nondecreasing continuous function such that for any t>0, αtφtt<1and the function 0+tφttis nonincreasing. Let dn:IRbe a continuously differentiable function such that

maxxIdnx1.

Now, consider an IFS of the form I×Rwnn=12Nin which the maps are nonlinear transformations of the special structure

wnxy=LnxFnxy=anx+bncnx+dnxsny+en,

where the transformations are constrained by the data according to

wnx0y0=xn1yn1,wnxNyN=xnyn

for n=1,2,,N, and snare some Rakotch or Geraghty contractions.

Let us denote by CDthe linear space of all real-valued continuous functions defined on D, i.e., CD=f:DR | fcontinuous. Let CICIdenote the set of continuous functions f:IRsuch that fx0=y0and fxN=yN, that is,

CIfCI:fx0=y0fxN=yN.

Let CICICIbe the set of continuous functions that pass through the given data points xiyiI×R:i=01N, that is,

CIfCI:fxi=yii=01N.

Define a metric dCIon the space CIby

dCIghmaxxx0xNgxhx

for all g,hCI. Define a mapping T:CICIfor all fCIby

TfxFnLn1xfLn1x=cnLn1x+dnLn1xsnfLn1x+en

for xxn1xnand n=1,2,,N. From [17], we have the following.

Theorem 2.2. Let I×Rwnn=12Ndenote the IFS defined above. Let each snbe a bounded Rakotch or Geraghty contraction. Then,

  1. there is a unique continuous function f:IRwhich is a fixed point of T;

  2. fxi=yifor all i=0,1,,N;

  3. if GI×Ris the graph of f, then

G=n=1NwnG.

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

Example 1. Let φtt1+tfor t0+. Let a set of data xiyi:i=01Nbe given, where 0=x0<x1<<xN=1and yi01for all i=0,1,,N. Let for all n=1,2,,N, dnxxn.Let for y0+and n=1,2,,N, snyy1+ny.That is, each snis a Rakotch contraction (with the same function φ) that is not a Banach contraction on 0+. Let for all n=1,2,,N,

wnxyanx+bncnx+dnxsny+en,

where

an=xnxn1,bn=xn1,cn=ynyn1,en=yn1.

Then, there exists a continuous function f:01Rthat interpolates the given points xiyi:i=01N. Moreover, the graph Gof fis invariant with respect to 01×Rw1w2wN, i.e.,

G=n=1NwnG.

2.2 Fractal interpolation in R2

Let M, Nbe two positive integers greater than 1. Let us represent the given set of interpolation points as xiyjzi,jK:i=01Mj=01N, where x0<x1<<xM, y0<y1<<yNand zi,jabfor all i=0,1,,Mand j=0,1,,N. Set I=x0xMRand J=y0yNR. Throughout this section, we will work in the complete metric space K=D×R, where D=I×J, with respect to the Euclidean, or to some other equivalent, metric.

Set Im=xm1xm, Jn=yn1yn, Dm,n=Im×Jnand let um:IIm, vn:JJn, Lm,n:DDm,nbe defined for m=1,2,,Mand n=1,2,,N, by

Lm,nxy=umxvny=amx+bmcny+dn.

Thus, for m=1,2,,Mand n=1,2,,N,

am=xmxm1xMx0,bm=xm1xmxm1xMx0x0,cn=ynyn1yNy0,dn=yn1ynyn1yNy0y0.

Furthermore, for m=1,2,,Mand n=1,2,,N, let mappings Fm,n:KRbe continuous with respect to each variable. We consider an IFS of the form Kwm,nm=12Mn=12Nin which maps wm,n:D×RDm,n×Rare transformations of the special structure

wm,nxyzLm,nxyFm,nxyz,

where the transformations are constrained by the data according to

wm,nx0y0z0,0=xm1yn1zm1,n1,wm,nx0yNz0,N=xm1ynzm1,n,
wm,nxMy0zM,0=xmyn1zm,n1,wm,nxMyNzM,N=xmynzm,n

for m=1,2,,Mand n=1,2,,N.

Let BDdenote the set of bounded functions f:DRand

BD={fBD:fx0y0=z0,0,fx0yN=z0,N,fxMy0=zM,0,fxMyN=zM,N}.

Let BDBDbe the set of bounded functions that pass through the given interpolation points xiyjzi,jK=D×ab:i=01Mj=01N, that is,

BD=fBD:fxiyj=zi,ji=01Mj=01N.

Define an operator T:BDBDfor all fBDby

Tfxy=Fm,num1xvn1yfum1xvn1y

for xyDm,n, m=1,2,,Mand n=1,2,,N. In [18], we see the following.

Theorem 2.3. Let D×Rwm,nm=12Mn=12Ndenote the IFS defined above. Assume that the maps Fm,nare 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:DRwhich is a fixed point of T;

  2. fxiyj=zi,jfor i=0,1,,Mand j=0,1,,N;

  3. if GD×Ris the graph of f, then

G=m=1Mn=1Nwm,nG.

Let for all i=0,1,,Mand j=0,1,,N, z0,j=zi,0=zM,j=zi,Nand define

Fm,nxyz=em,nx+fm,ny+gm,nxy+sm,nz+hm,n,

where sm,nare Rakotch or Geraghty contractions. Let

CD={fCD:fx0y0=z0,0,fx0yN=z0,N,fxMy0=zM,0,fxMyN=zM,N}

and

CD=fCD:fxiyj=zi,ji=01Mj=01N.

Let C0DCDbe the set of continuous functions f:DRsuch that

fx01λy0+λyN=z,,fxM1λy0+λyN=z,,f1λx0+λxMy0=z,,f1λx0+λxMyN=z,

for all λ01, where for all i=0,1,,Mand j=0,1,,N,

zz0,j=zi,0=zM,j=zi,N.

Let C0DfC0D:fxiyj=zi,ji=01Mj=01NCD. For fC0D, we define T:C0DBDby

Tfxy=Fm,num1xvn1yfum1xvn1y=em,num1x+fm,nvn1y+gm,num1xvn1y+sm,nfum1xvn1y+hm,n

for xyDm,n, m=1,2,,Mand n=1,2,,N.

Corollary 2.1 (see [18]) Let D×Rwm,nm=12Mn=12Ndenote the IFS defined above. Then,

  1. there is a unique continuous function f:DRwhich is a fixed point of T;

  2. fxiyj=zi,jfor all i=0,1,,Mand j=0,1,,N;

  3. if GD×Ris the graph of f, then

G=m=1Mn=1Nwm,nG.

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

Example 2. Let φtt1+tfor t0+. Let a set of data xiyjzi,j:i=0,1,2j=0,1,2be given, where 0=x0<x1<x2=1, 0=y0<y1<y2=1and zi,j01for all i=0,1,2;j=0,1,2. Let for all i=0,1,2and j=0,1,2,

z0,j=zi,0=z2,j=zi,2=0.

Let for z0+,

s1,1zz1+z,s1,2zz1+2z,
s2,1zz1+3z,s2,2zz1+4z.

Then, s1,1, s1,2, s2,1, s2,2are Rakotch contractions (with the same function φ) that are not Banach contractions on 0+. So, there exists a continuous function f:01×01Rthat interpolates the given data xiyjzi,j:i=0,1,2j=0,1,2.

Let dm,n:DRbe a function such that maxxyDdm,nxy1,

dm,nx0y=dm,nxMy=dm,nxy0=dm,nxyN=0

and for some L1,L2>0,

dm,nxydm,nxyL1xx+L2yy.

Let

Fm,nxyz=em,nx+fm,ny+gm,nxy+dm,nxysm,nz+hm,n,

where sm,nis a Rakotch or Geraghty contraction. For fCD, we define T:CDBDby

Tfxy=Fm,num1xvn1yfum1xvn1y=em,num1x+fm,nvn1y+gm,num1xvn1y+dm,num1xvn1ysm,nfum1xvn1y+hm,n

for xyDm,n, m=1,2,,Mand n=1,2,,N.

For the next, see [12] for details.

Corollary 2.2. Let D×Rwm,nm=12Mn=12Ndenote the IFS defined above. If each sm,nbe a bounded function, then.

  1. there is a unique continuous function f:DRwhich is a fixed point of T;

  2. fxiyj=zi,jfor all i=0,1,,Mand j=0,1,,N;

  3. if GD×Ris a graph of f, then

G=m=1Mn=1Nwm,nG.

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

Example 3. Let φtt1+tfor t0+. Let a set of data xiyjzi,j:i=0,1,2j=0,1,2be given, where 0=x0<x1<x2=1, 0=y0<y1<y2=1and zi,j01for all i=0,1,2;j=0,1,2. Here, a set of data points is not necessarily the case that z0,j=zi,0=z2,j=zi,2for all i=0,1,2;j=0,1,2. Let for all i=1,2;j=1,2and xy01×01,

dm,nxy22m+nxm1xmyn1yn.

Let for z0+,

s1,1z11+z,s1,2zz1+z,
s2,1zz1+2z,s2,2zz1+3z.

Then, s1,1, s1,2, s2,1, s2,2are Rakotch contractions (with the same function φ) that are not Banach contractions on 0+. So, there exists a continuous function f:01×01Rthat interpolates the given data xiyjzi,j:i=0,1,2j=0,1,2.

3. Interconnections between FIFs and contractions

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 to the case of Rakotch contractions.

Connection 1

  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 φt=αt, for some 0α<1. There exist examples of Rakotch contraction maps that are not Banach contraction maps on XRwith respect to the Euclidean metric (see [13]).

  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:0+0+such that additionally φt<tfor t>0and the map tφttis nonincreasing (see [19]).

Connection 2

  1. CIdCI, CIdCIand CIdCIare complete metric spaces, where

    dCIfgmaxxIfxgx

for all f,gCI(see [2]).

  • BDdBD, BDdBDand BDdBDare complete metric spaces, where

    dBDfgsupxyDfxygxy

    for all f,gBD[10].

  • C0D, C0D, CD, CDand CDare closed subspaces of BDwith C0DC0DCDCDBDand C0DCDCDCDBD, and so they are complete metric spaces.

  • Connection 3

    Let dn:IRbe a continuously differentiable function such that

    maxxIdnx1.

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

    dnxdnxLdnxx,

    where x,xI. Hence, dnis Lipschitz continuous function defined on Isatisfying maxxIdnx1, but the converse is not true in general.

    Connection 4

    1. The function dnxsnyis a generalisation of the bivariable function dnxywith vertical scaling factors as (continuous) ‘contraction functions’. In fact, in the case when 0<maxxIdnx<1(see [20], p. 3), obviously,

      dnxy=dnxmaxxIdnxmaxxIdnxy.

    Let sny=maxxIdnxyand dnx=dnxmaxxIdnx. Then dnxy=dnxsny, maxxIdnx=1and snis a Banach (or Rakotch) contraction.

  • The functional condition maxxIdnx1is essential in order to show the difference between Banach contractibility of Fnyand Rakotch contractibility of Fny; compare with [20]. In fact, since φt<tfor any t>0,

    FnxyFnxy=dnxsnysnymaxxIdnxsnysnymaxxIdnxφyymaxxIdnxyy,

    where xy,xyR2. Hence, if maxxIdnx<1, as can be seen, notwithstanding each snis a Rakotch contraction that is not a Banach contraction, each Fnis Banach contraction with respect to the second variable because

    FnxyFnxymaxxIdnxyy.

    On the other hand, if maxxIdnx=1, then we conclude that each Fnis Rakotch contraction with respect to the second variable whenever each snis a Rakotch contraction because

    FnxyFnxymaxxIdnxφyy.

  • In Theorem 2.2, for all xy,xyI×R,

    FnxyFnxy=dnxsnysnysnysnyφyy.

    That is, each wnxyis chosen so that function Fnxyis Rakotch contraction with respect to the second variable.

  • Even though sn:RRare Rakotch contractions, wn:I×Rare not in general Rakotch contractions on the metric space I×Rd0, and thus, the IFSs defined above are not IFSs of [14] (cf. second and third line in p. 215 of [2]).

  • Connection 5

    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’.

    Connection 6

    The boundedness of snis 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<maxxIdnx<1(see [20]). Let MmaxxIcnx+fnand hM1maxxIdnx. Then for all yhh,

    Fnxy=cnx+dnxy+fnM+maxxIdnxyM+maxxIdnxhh.

    So, for all xyI×hh, we can see that Fnxyhh. That is, an IFS of the form I×hhw1Nhas been constructed (cf. [21], p. 1897). Thus Dsn=hhand snymaxxIdnxyis bounded in Dsn. Hence the boundedness of snin Dsnis the essential condition to establish a unique invariant set of an IFS (cf. [21], p. 1897).

    Connection 7

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

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

      φtmaxn=1,2,,Ndnt,

    where dn<1for all i=1,2,,N.

  • In the FIF with vertical scaling factors as (continuous) ‘contraction functions’ (cf. [20], p. 3), for all t>0,

    φtmaxi=1,2,,NmaxxIdnxt,

    where dnxis Lipschitz function defined on Isatisfying supxIdnx<1for all n=1,2,,N.

  • Connection 8

    We refer to fof Theorem 2.2 as a nonlinear FIF. The reason is that the functions Fntake the form

    Fnxy=cnx+dnxsny+en,

    where maxxIdnx1and each snis 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<dnxdn<1or 0<maxxIdnx<1and

    dnxy=dnxmaxxIdnxmaxxIdnxy,

    we can see that

    Fnxy=cnx+dnxy+en=cnx+dnxsny+en,

    where dnxdnxmaxxIdnxand snymaxxIdnxy, and thus, each snis 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 Kwm,n:m=12Mn=12Nunder some conditions, where the maps are transformations of the special structure

      wm,nxyz=umxvnyFm,nxyz=amx+bmcny+dnem,nx+fm,ny+gm,nxy+dm,nxyz+hm,n,

    where dm,nxy<1for all xyDR2. Then for all xyz,xyzK,

    Fm,nxyzFm,nxyz=dm,nxyzdm,nxyzmaxxyDdm,nxyzz.

    That is, each wm,nxyzis chosen so that function Fm,nxyzis 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 xyDR2, dm,nxysm,nand 0sm,n<1, we can see that each wm,nxyzis chosen so that function Fm,nxyzis Banach contraction with respect to the third variable. Also in [21], since

    dm,nxy=λm,nxx0xMxyy0yNy

    and

    λm,n<16xMx02yNy02,

    we can see that maxxyDdm,nxy<1, and so each wm,nxyzis chosen so that function Fm,nxyzis a Banach contraction with respect to the third variable.

  • In Theorem 2.3, for all xyz,xyzKR3,

    Fm,nxyzFm,nxyz=dm,nxysm,nzsm,nzsm,nzsm,nzφzz.

  • That is, each Fm,nxyzis Rakotch contraction with respect to the third variable. So, each wm,nxyzis chosen so that the function Fm,nxyzis a Rakotch contraction with respect to the third variable.

    Connection 10

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

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

      φtmaxm=1,2,,Mmaxn=1,2,,Ndm,nt,

    where dm,n<1for all m=1,2,,Mand n=1,2,,N.

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

    φtmaxm=1,2,,Mmaxn=1,2,,NmaxxIdm,nxt,

    where dm,nxis Lipschitz function defined on Isatisfying supxIdm,nx<1for all m=1,2,,Mand n=1,2,,N.

  • Connection 11

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

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

    2. There are bivariable discontinuous functions that interpolate the given data; (see for instance [23], p. 630, 631).

    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 and Corollary 2.2.

    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. 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.

    Connection 13

    1. In Theorem 2.2, we can see that

      an=xnxn1xNx0,bn=xNxn1x0xnxNx0cn=ynyn1xNx0dnxNsnyNdnx0sny0xNx0,fn=xNyn1x0ynxNx0xNdnx0sny0x0dnxNsnyNxNx0.

  • In Corollary 2.1, we can see that

    am=xmxm1xMx0,bm=xMxm1x0xmxMx0,cn=ynyn1yNy0,dn=yNyn1y0ynyNy0,gm,n=zm,nzm1,nzm,n1zm1,n1xMx0yNy0,em,n=yNzm,n1zm1,n1y0zm,nzm1,nxMx0yNy0,fm,n=xMzm1,nzm1,n1x0zm,nzm,n1xMx0yNy0,hm,n=x0y0zm,nx0yNzm,n1xMy0zm1,n+xMyNzm1,n1xMx0yNy0sm,nzM,N.

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

    gm,n=zm,nzm1,nzm,n1zm1,n1xMx0yNy0,em,n=yNzm,n1zm1,n1y0zm,nzm1,nxMx0yNy0,fm,n=xMzm1,nzm1,n1x0zm,nzm,n1xMx0yNy0,hm,n=x0y0zm,nx0yNzm,n1xMy0zm1,n+xMyNzm1,n1xMx0yNy0.

  • 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.

    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.

    Figure 2.

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

    4. Conclusions and further work

    We reviewed nonlinear fractal interpolation functions by using the Geraghty fixed point theorem instead of the Banach fixed point theorem (or the Rakotch fixed point theorem) since Banach contraction (or Rakotch contraction) is a special case of Geraghty contraction. Theorems 2.1, 2.2 and 2.3 ensure that attractors of constructed nonlinear iterated function systems are graphs of some continuous functions which interpolate the given data. In particular, Examples 1, 2 and 3 show that our results remain still true under essentially weaker conditions on the maps of iterated function systems. The methods presented here can be directly extended to piecewise fractal interpolation functions that are based on recurrent IFS. A premise for future work is to extend these methods to hidden-variable fractal interpolation surfaces as well as to identify the parameters of such surfaces.

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    SongIl Ri and Vasileios Drakopoulos (August 21st 2020). How Are Fractal Interpolation Functions Related to Several Contractions? [Online First], IntechOpen, DOI: 10.5772/intechopen.92662. Available from:

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