# Fractals of GeneralizedF-Hutchinson Operator in b-Metric Spaces

Authors: Talat Nazir; Sergei Silvestrov; Xiaomin Qi
Journal: Journal of Operators
 (2016)

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## Abstract

The aim of this paper is to construct a fractal with the help of a finite family of generalizedF-contraction mappings, a class of mappings more general than contraction mappings, defined in the setup of b-metric space. Consequently, we obtain a variety of results for iterated function system satisfying a different set of contractive conditions. Our results unify, generalize, and extend various results in the existing literature.


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## Body

## 1. Introduction and Preliminaries

Iterated function systems are method of constructing fractals and are based on the mathematical foundations laid by Hutchinson [1]. He showed that Hutchinson operator constructed with the help of a finite system of contraction mappings defined on Euclidean space Rn has closed and bounded subset of Rn as its fixed point, called attractor of iterated function system (see also [2]). In this context, fixed point theory plays significant and vital role to help in construction of fractals.Fixed point theory is studied in environment created with appropriate mappings satisfying certain conditions. Recently, many researchers have obtained fixed point results for single and multivalued mappings defined on metrics spaces. Banach contraction principle [3] is of paramount importance in metrical fixed point theory with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. This initiated several researchers to extend and enhance the scope of metric fixed point theory. As a result, Banach contraction principles have been extended either by generalizing the domain of the mapping [4–10] or by extending the contractive condition on the mappings [11–15]. There are certain cases when the range X of a mapping is replaced with a family of sets possessing some topological structure and consequently a single-valued mapping is replaced with a multivalued mapping. Nadler [16] was the first who combined the ideas of multivalued mappings and contractions and hence initiated the study of metric fixed point theory of multivalued operators; see also [17–19]. The fixed point theory of multivalued operators provides important tools and techniques to solve the problems of pure, applied, and computational mathematics which can be restructured as an inclusion equation for an appropriate multivalued operator.The concept of metric has been generalized further in one to many ways. The concept ofb-metric space was introduced by Czerwik in [20]. Since then, several papers have been published on the fixed point theory of various classes of single-valued and multivalued operators in b-metric space [20–30].In this paper, we construct a fractal set of iterated function system, a certain finite collection of mappings defined onb-metric space which induce compact valued mappings defined on a family of compact subsets of b-metric space. We prove that Hutchinson operator defined with the help of a finite family of generalized F-contraction mappings on a complete b-metric space is itself generalized F-contraction mapping on a family of compact subsets of X. We then obtain a final fractal obtained by successive application of a generalized F-Hutchinson operator in b-metric space.Definition 1.
LetX be a nonempty set and let b≥1 be a given real number. A function d:X×X→R+ is said to be a b-metric if, for any x,y,z∈X, the following conditions hold:(b1)
d(x,y)=0 if and only if x=y,(b2)
d(x,y)=d(y,x),(b3)
d(x,y)≤bd(x,z)+d(z,y).The pair (X,d) is called b-metric space with parameter b≥1.
Ifb=1, then b-metric space is metric spaces. But the converse does not hold in general [20, 21, 25].Example 2 (see [31]).
Let(X,d) be a metric space, and ρ(x,y)=(d(x,y))p, where p>1 is a real number. Then ρ is b-metric with b=2p-1.
Obviously conditions (b1) and (b2) of above definition are satisfied. If 1<p<∞, then the convexity of the function f(x)=xp(x>0) implies (1)a+b2p≤12ap+bp,and hence a+bp≤2p-1(ap+bp) holds. Thus, for each x,y,z∈X we obtain (2)ρx,y=dx,yp≤dx,z+dz,yp≤2p-1dx,zp+dz,yp=2p-1ρx,z+ρz,y.So condition (b3) of the above definition is satisfied and ρ is b-metric.
IfX=R (set of real numbers) and d(x,y)=x-y is the usual metric, then ρ(x,y)=(x-y)2 is b-metric on R with b=2 but is not a metric on R.Definition 3 (see [24]).
Let(X,d) be b-metric space. Then a subset C⊆X is called(i)
closed if and only if, for each sequence{xn} in C which converges to an element x, we have x∈C (i.e., C=C¯),(ii)
compact if and only if for every sequence of elements ofC there exists a subsequence that converges to an element of C,(iii)
bounded if and only ifδ(C)≔sup⁡{d(x,y):x,y∈C}<∞.
LetH(X) denote the set of all nonempty compact subsets of X. For A,B∈H(X), let (3)HA,B=max⁡supb∈B⁡db,A,supa∈A⁡da,B,where d(x,B)=inf⁡{d(x,y):y∈B} is the distance of a point x from the set B. The mapping H is said to be the Pompeiu-Hausdorff metric induced by d. If (X,d) is a complete b-metric space, then H(X),H is also a complete b-metric space.For the sake of completeness, we state that the following lemma holds inb-metric space [32].Lemma 4.
Let(X,d) be b-metric space. For all A,B,C,D∈H(X), the following hold:(i)
IfB⊆C, then supa∈A⁡d(a,C)≤supa∈A⁡d(a,B).(ii)
supx∈A∪B⁡d(x,C)=max⁡{supa∈A⁡d(a,C),supb∈B⁡d(b,C)}.(iii)
One hasH(A∪B,C∪D)≤max⁡{H(A,C),H(B,D)}.The following lemmas from [20, 27, 28] will be needed in the sequel to prove the main result of the paper.Lemma 5.
Let(X,d) be b-metric space and CBX denotes the set of all nonempty closed and bounded subsets of X. For x,y∈X and A,B∈CB(X), the following statements hold:(1)
(CB(X),H) is b-metric space.(2)
d(x,B)≤H(A,B) for all x∈A.(3)
One hasd(x,A)≤bd(x,y)+d(y,A).(4)
Forh>1 and a´∈A, there is b´∈B such that da´,b´≤hH(A,B).(5)
For everyh>0 and a´∈A, there is b´∈B such that d(a´,b´)≤H(A,B)+h.(6)
For everyλ>0 and a~∈A, there is b~∈B such that d(a~,b~)≤λ.(7)
For everyλ>0 and a~∈A, there is b~∈B such that d(a~,b~)≤λ implies H(A,B)≤λ.(8)
d(x,A)=0 if and only if x∈A¯=A.(9)
For{xn}⊆X, (4)dx0,xn≤bdx0,x1+⋯+bn-1dxn-2,xn-1+bn-1dxn-1,xn.Definition 6.
Let(X,d) be b-metric space. A sequence {xn} in X is called(i)
Cauchy if and only if, forε>0, there exists n(ε)∈N such that for each n,m≥n(ε) one has d(xn,xm)<ε,(ii)
convergent if and only if there existsx∈X such that for all ε>0 there exists n(ε)∈N such that for all n≥n(ε) one has d(xn,x)<ε. In this case one writes limn→∞⁡xn=x.It is known that a sequence{xn} in b-metric space X is Cauchy if and only if limn→∞⁡d(xn,xn+p)=0 for all p∈N. A sequence {xn} is convergent to x∈X if and only if limn→∞⁡d(xn,x)=0, and b-metric space (X,d) is said to be complete if every Cauchy sequence in X is convergent in X.An et al. [21] studied the topological properties of b-metric spaces and stated the following assertions:(c1)
Inb-metric space (X,d),d is not necessarily continuous in each variable.(c2)
Inb-metric space (X,d), if d is continuous in one variable then d is continuous in other variables.(c3)
An open ball inb-metric space (X,d) is not necessarily an open set. An open ball is open if d is continuous in one variable.Wardowski [33] introduced another generalized contraction called F-contraction and proved a fixed point result as interesting generalization of the Banach contraction principle in complete metric space (see also [34]).LetϜ be the collection of all continuous mappings F:R+→R that satisfy the following conditions:(F1)
F is strictly increasing, that is, for all α,β∈R+ such that α<β implies that F(α)<F(β).(F2)
For every sequence{αn} of positive real numbers, limn→∞⁡αn=0 and limn→∞⁡Fαn=-∞ are equivalent.(F3)
There existsh∈0,1 such that limα→0+⁡αhF(α)=0.Definition 7 (see [33]).
Let(X,d) be a metric space. A self-mapping f on X is called F-contraction if, for any x,y∈X, there exist F∈Ϝ and τ>0 such that(5)τ+Fdfx,fy≤Fdx,y,whenever d(fx,fy)>0.
From (F1) and (5), we conclude that (6)dfx,fy<dx,y,∀x,y∈X,fx≠fy;that is, every F-contraction mapping is contractive, and in particular, every F-contraction mapping is continuous.Wardowski [33] proved that, in complete metric space (X,d), every F-contractive self-map has a unique fixed point in X and for every x0 in X a sequence of iterates {x0,fx0,f2x0,…} converges to the fixed point of f.LetΥ be the set of all mapping τ:R+→R+ that is satisfying liminft→0τ(t)>0 for all t≥0.Definition 8.
Let(X,d) be b-metric space. A self-mapping f on X is called a generalized F-contraction if, for any x,y∈X, there exist F∈Ϝ and τ∈Υ such that(7)τdx,y+Fdfx,fy≤Fdx,y,whenever d(fx,fy)>0.Theorem 9.
Let(X,d) be b-metric space and let f:X→X be generalized F-contraction. Then one has the following: (1)
f maps elements in H(X) to elements in H(X).(2)
If, for anyA∈H(X), (8)fA=fx:x∈A,thenf:H(X)→H(X) is a generalized F-contraction mapping on (H(X),H).Proof.
As generalizedF-contractive mapping is continuous and the image of a compact subset under f:X→X is compact, we obtain (9)A∈HXimpliesfA∈HX.To prove (2), let A,B∈H(X) with HfA,fB≠∅. Since f:X→X is a generalized F-contraction, we obtain (10)0<dfx,fy<dx,y∀x,y∈X,x≠y.Thus we have (11)dfx,fB=infy∈Bdfx,fy<infy∈Bdx,y=dx,B.Also (12)dfy,fA=infx∈Adfy,fx<infx∈Ady,x=dy,A.Now (13)HfA,fB=max⁡supx∈Adfx,fB,supy∈Bdfy,fA<max⁡supx∈Adx,B,supy∈Bdy,A=HA,B.Strictly increasing F implies (14)FHfA,fB<FHA,B.Consequently, there exists a function τ:R+→R+ with liminft→0τ(t)>0 for all t≥0 such that (15)τHA,B+FHfA,fB≤FHA,B.Hence f:H(X)→H(X) is a generalized F-contraction.Theorem 10.
Let(X,d) be b-metric space and let {fn:n=1,2,…,N} be a finite family of generalized F-contraction self-mappings on X. Define T:H(X)→H(X) by (16)TA=f1A∪f2A∪⋯∪fNA=⋃n=1NfnA,foreachA∈HX.Then T is a generalized F-contraction on HX.Proof.
We demonstrate the claim forN=2. Let f1,f2:X→X be two F-contractions. Take A,B∈HX with H(T(A),T(B))≠0. From Lemma 4 (iii), it follows that (17)τHA,B+FHTA,TB=τHA,B+FHf1A∪f2A,f1B∪f2B≤τHA,B+Fmax⁡Hf1A,f1B,Hf2A,f2B≤FHA,B.Definition 11.
Let(X,d) be a metric space. A mapping T:HX→HX is said to be a Ciric type generalized F-contraction if, for F∈Ϝ and τ∈Υ such that, for any A, B∈HX with H(T(A),T(B))≠0, the following holds:(18)τMTA,B+FHTA,TB≤FMTA,B,where (19)MTA,B=max⁡HA,B,HA,TA,HB,TB,HA,TB+HB,TA2b,HT2A,TA,HT2A,B,HT2A,TB.Theorem 12.
Let(X,d) be b-metric space and let {fn:n=1,2,…,N} be a finite sequence of generalized F-contraction mappings on X. If T:H(X)→H(X) is defined by (20)TA=f1A∪f2A∪⋯∪fNA=⋃n=1NfnA,foreachA∈HX,then T is a Ciric type generalized F-contraction mapping on HX.Proof.
Using Theorem10 with property (F1), the result follows.An operatorT in above theorem is called Ciric type generalized F-Hutchinson operator.Definition 13.
LetX be a complete b-metric space. If fn:X→X, n=1,2,…,N, are generalized F-contraction mappings, then (X;f1,f2,…,fN) is called generalized F-contractive iterated function system (IFS).
Thus generalizedF-contractive iterated function system consists of a complete b-metric space and finite family of generalized F-contraction mappings on X.Definition 14.
A nonempty compact setA⊂X is said to be an attractor of the generalized F-contractive IFS if(a)
T(A)=A,(b)
there is an open setU⊆X such that A⊆U and limn→∞⁡Tn(B)=A for any compact set B⊆U, where the limit is taken with respect to the Hausdorff metric.


## 2. Main Results

We start with the following result.Theorem 15.
Let(X,d) be a complete b-metric space and let {X;fn,n=1,2,…,k} be a generalized F-contractive iterated function system. Then the following hold:(a)
A mappingT:H(X)→H(X) defined by (21)TA=⋃n=1kfnA,∀A∈HX,is Ciric type generalizedF-Hutchinson operator.(b)
OperatorT has a unique fixed point U∈HX; that is, (22)U=TU=⋃n=1kfnU.(c)
For any initial setA0∈HX, the sequence of compact sets {A0,TA0,T2A0,…} converges to a fixed point of T.Proof.
Part (a) follows from Theorem12. For parts (b) and (c), we proceed as follows. Let A0 be an arbitrary element in HX. If A0=TA0, then the proof is finished. So we assume that A0≠TA0. Define (23)A1=TA0,A2=TA1,…,Am+1=TAmfor m∈N.
We may assume thatAm≠Am+1 for all m∈N. If not, then Ak=Ak+1 for some k implies Ak=T(Ak) and this completes the proof. Take Am≠Am+1 for all m∈N. From (18), we have (24)τMTAm,Am+1+FHAm+1,Am+2=τMTAm,Am+1+FHTAm,TAm+1≤FMTAm,Am+1,where (25)MTAm,Am+1=max⁡HAm,Am+1,HAm,TAm,HAm+1,TAm+1,HAm,TAm+1+HAm+1,TAm2b,HT2Am,TAm,HT2Am,Am+1,HT2Am,TAm+1=max⁡HAm,Am+1,HAm,Am+1,HAm+1,Am+2,HAm,Am+2+HAm+1,Am+12b,HAm+2,Am+1,HAm+2,Am+1,HAm+2,Am+2=max⁡HAm,Am+1,HAm+1,Am+2.In case MTAm,Am+1=H(Am+1,Am+2), we have (26)FHAm+1,Am+2≤FHAm+1,Am+2-τHAm+1,Am+2,a contradiction as τ(H(Am+1,Am+2))>0. Therefore MTAm,Am+1=H(Am,Am+1) and we have (27)FHAm+1,Am+2≤FHAm,Am+1-τHAm,Am+1<FHAm,Am+1.Thus {H(Am+1,Am+2)} is decreasing and hence convergent. We now show that limm→∞⁡H(Am+1,Am+2)=0. By property of τ, there exists c>0 with n0∈N such that τ(H(Am,Am+1))>c for all m≥n0. Note that (28)FHAm+1,Am+2≤FHAm,Am+1-τHAm,Am+1≤FHAm-1,Am-τHAm-1,Am-τHAm,Am+1≤⋯≤HA0,A1-τHA0,A1+τHA1,A2+⋯+τHAm,Am+1≤FHA0,A1-n0gives limm→∞⁡FH(Am+1,Am+2)=-∞ which together with (F2) implies that limm→∞⁡H(Am+1,Am+2)=0. By (F3), there exists h∈0,1 such that (29)limn→∞⁡HAm+1,Am+2hFHAm+1,Am+2=0.Thus we have (30)HAm,Am+1hFHAm,Am+1-HAm,Am+1hFHA0,A1≤HAm,Am+1hFHA0,A1-n0-HAm,Am+1hFHA0,A1≤-n0HAm,Am+1h≤0.On taking limit as n→∞ we obtain that limm→∞⁡m[H(Am+1,Am+2)]h=0. Hence limm→∞⁡m1/hH(Am+1,Am+2)=0. There exists n1∈N such that m1/hH(Am+1,Am+2)≤1 for all m≥n1 and hence H(Am+1,Am+2)≤1/m1/h for all m≥n1. For m,n∈N with m>n≥n1, we have (31)HAn,Am≤HAn,An+1+HAn+1,An+2+⋯+HAm-1,Am≤∑i=n∞1i1/h.By the convergence of the series ∑i=1∞‍1/i1/h, we get HAn,Am→0 as n,m→∞. Therefore {An} is a Cauchy sequence in X. Since (H(X),d) is complete, we have An→U as n→∞ for some U∈H(X).
In order to show thatU is the fixed point of T, we on the contrary assume that Pompeiu-Hausdorff weight assigned to U and TU is not zero. Now(32)τMTAn,U+FHAn+1,TU=τ+FHTAn,TU≤FMTAn,U,where (33)MTAn,U=maxHAn,U,HAn,TAn,HU,TU,HAn,TU+HU,TAn2b,HT2An,TAn,HT2An,U,HT2An,TU=maxHAn,U,HAn,An+1,HU,TU,HAn,TU+HU,An+12b,HAn+2,An+1,HAn+2,U,HAn+2,TU.Now we consider the following cases:(1)
IfMTAn,U=H(An,U), then, on taking lower limit as n→∞ in (32), we have (34)liminfn→∞⁡τHAn,U+FHTU,U≤FHU,U,a contradiction asliminft→0τ(t)>0 for all t≥0.(2)
WhenMTAn,U=H(An,An+1), then, by taking lower limit as n→∞, we obtain (35)liminfn→∞⁡τHAn,An+1+FHTU,U≤FHU,U,which gives a contradiction.(3)
In caseMTAn,U=H(U,TU), we get(36)τHU,TU+FHTU,U≤FHU,TU,a contradiction asτ(H(U,TU))>0.(4)
IfMTAn,U=H(An,TU)+H(U,An+1)/2b, then, on taking lower limit as n→∞, we have(37)liminfn→∞⁡τHAn,TU+HU,An+12b+FHTU,U≤FHU,TU+HU,U2b=FHU,TU2b,a contradiction asF is strictly increasing map.(5)
WhenMTAn,U=H(An+2,An+1), then(38)liminfn→∞⁡τHAn+2,An+1+FHTU,U≤FHU,U,which gives a contradiction.(6)
In caseMTAn,U=H(An+2,U), then, on taking lower limit as n→∞ in (32), we get(39)liminfn→∞⁡τHAn+2,U+FHTU,U≤FHU,U,a contradiction.(7)
Finally ifMTAn,U=H(An+2,TU), then, on taking lower limit as n→∞, we have(40)liminfn→∞⁡τHAn+2,TU+FHTU,U≤FHU,TU,a contradiction.
Thus,U is the fixed point of T.
To show the uniqueness of fixed point ofT, assume that U and V are two fixed points of T with HU,V being not zero. Since T is F-contraction map, we obtain that (41)τMTU,V+FHU,V=τMTU,V+FHTU,TV≤FMTU,V,where (42)MTU,V=maxHU,V,HU,TU,HV,TV,HU,TV+HV,TU2b,HT2U,U,HT2U,V,HT2U,TV=maxHU,V,HU,U,HV,V,HU,V+HV,U2b,HU,U,HU,V,HU,V=HU,V;that is, (43)τHU,V+FHU,V≤FHU,V,a contradiction as τHU,V>0. Thus T has a unique fixed point U∈H(X).Remark 16.
In Theorem15, if we take S(X) the collection of all singleton subsets of X, then clearly S(X)⊆H(X). Moreover, consider fn=f for each n, where f=fi for any i∈{1,2,3,…,k}; then the mapping T becomes (44)Tx=fx.With this setting we obtain the following fixed point result.Corollary 17.
Let(X,d) be a complete b-metric space and let {X:fn,n=1,2,…,k} be a generalized iterated function system. Let f:X→X be a mapping defined as in Remark 16. If there exist some F∈Ϝ and τ∈Υ such that, for any x,y∈HX with d(f(x),f(y))≠0, the following holds: (45)τMfx,y+Fdfx,fy≤FMfx,y,where (46)MTx,y=max⁡dx,y,dx,fx,dy,fy,dx,fy+dy,fx2b,df2x,y,df2x,fx,df2x,fy,then f has a unique fixed point in X. Moreover, for any initial set x0∈X, the sequence of compact sets {x0,fx0,f2x0,…} converges to a fixed point of f.Corollary 18.
Let(X,d) be a complete b-metric space and let (X;fn,n=1,2,…,k) be iterated function system where each fi for i=1,2,…,k is a contraction self-mapping on X. Then T:H(X)→H(X) defined in Theorem 15 has a unique fixed point in HX. Furthermore, for any set A0∈HX, the sequence of compact sets {A0,TA0,T2A0,…} converges to a fixed point of T.Proof.
It follows from Theorem10 that if each fi for i=1,2,…,k is a contraction mapping on X, then the mapping T:H(X)→H(X) defined by (47)TA=⋃n=1kfnA,∀A∈HX,is contraction on HX. Using Theorem 15, the result follows.Corollary 19.
Let(X,d) be a complete b-metric space and let (X;fn,n=1,2,…,k) be an iterated function system. Suppose that each fi for i=1,2,…,k is a mapping on X satisfying (48)dfix,fiyedfix,fiy-dx,y≤e-τdx,ydx,y,for all x,y∈X, fix≠fiy, where τ∈Υ. Then the mapping T:H(X)→H(X) defined in Theorem 15 has a unique fixed point in HX. Furthermore, for any set A0∈HX, the sequence of compact sets {A0,TA0,T2A0,…} converges to a fixed point of T.Proof.
TakeFλ=ln⁡λ+λ, λ>0 in Theorem 10; then each mapping fi for i=1,2,…,k on X satisfies (49)dfix,fiyedfix,fiy-dx,y≤e-τdx,ydx,y,for all x,y∈X, fix≠fiy, where τ∈Υ. Again from Theorem 10, the mapping T:H(X)→H(X) defined by (50)TA=⋃n=1kfnA,∀A∈HX,satisfies (51)HTA,TBeHTA,TB-HA,B≤e-τHA,B,for all A,B∈H(X) and HTA,TB≠0. Using Theorem 15, the result follows.Corollary 20.
Let(X,d) be a complete b-metric space and let (X;fn, n=1,2,…,k) be iterated function system. Suppose that each fi for i=1,2,…,k is a mapping on X satisfying (52)dfix,fiydfix,fiy+1≤e-τdx,ydx,ydx,y+1,for all x,y∈X, fix≠fiy, where τ∈Υ. Then the mapping T:H(X)→H(X) defined in Theorem 15 has a unique fixed point in HX. Furthermore, for any set A0∈HX, the sequence of compact sets {A0,TA0,T2A0,…} converges to a fixed point of T.Proof.
By takingFλ=ln⁡λ2+λ+λ, λ>0, in Theorem 10, we obtain that each mapping fi for i=1,2,…,k on X satisfies (53)dfix,fiydfix,fiy+1≤e-τdx,ydx,ydx,y+1,for all x,y∈X, fix≠fiy, where τ∈Υ. Again it follows from Theorem 10 that the mapping T:H(X)→H(X) defined by (54)TA=⋃n=1kfnA,∀A∈HX,satisfies (55)HTA,TBHTA,TB+1≤e-τHA,BHA,BHA,B+1,for all A,B∈H(X), HTA,TB≠0. Using Theorem 15, the result follows.Corollary 21.
Let(X,d) be a complete b-metric space and let (X;fn,n=1,2,…,k) be iterated function system. Suppose that each fi for i=1,2,…,k is a mapping on X satisfying (56)dfix,fiy≤11+τdx,ydx,y2dx,y,for all x,y∈X, fix≠fiy, where τ∈Υ. Then the mapping T:H(X)→H(X) defined in Theorem 15 has a unique fixed point HX. Furthermore, for any set A0∈HX, the sequence of compact sets {A0,TA0,T2A0,…} converges to a fixed point of T.Proof.
TakeFλ=-1/λ, λ>0, in Theorem 10, and then each mapping fi for i=1,2,…,k on X satisfies (57)dfix,fiy≤11+τdx,ydx,y2dx,y,∀x,y∈X,fix≠fiy,where τ∈Υ. Again it follows from Theorem 10 that the mapping T:H(X)→H(X) defined by (58)TA=⋃n=1kfnA,∀A∈HX,satisfies (59)HTA,TB≤11+τHA,BHA,B2HA,B,for all A,B∈H(X), HTA,TB≠0. Using Theorem 15, the result follows.



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