@ Appl. Gen. Topol. 21, no. 1 (2020), 57-70 doi:10.4995/agt.2020.11992 c© AGT, UPV, 2020 Existence of Picard operator and iterated function system Medha Garg and Sumit Chandok School of Mathematics, Thapar Institute of Engineering & Technology, Patiala-147004, Punjab, India. (sumit.chandok@thapar.edu) Communicated by E. A. Sánchez-Pérez Abstract In this paper, we define weak θm− contraction mappings and give a new class of Picard operators for such class of mappings on a complete metric space. Also, we obtain some new results on the existence and uniqueness of attractor for a weak θm− iterated multifunction system. Moreover, we introduce (α,β,θm)− contractions using cyclic (α,β)− admissible mappings and obtain some results for such class of mappings without the continuity of the operator. We also provide an illustrative example to support the concepts and results proved herein. 2010 MSC: 47H10; 54H25; 46J10; 46J15. Keywords: Picard operator; fixed point; weak θm-contraction; iterated func- tion system. 1. Introduction The iterated function system (IFS) is the main generator of fractals. It is introduced by Hutchinson [7] and generalized by Barnsley [2]. An IFS is a finite family of contractions {fi}Ni=1 on a complete metric space (M,d). For an IFS there is always a non-empty set A ⊂ M such that A = ⋃N i=1 fi(A), such A is known as attractor of the respective IFS. In this paper, we study the concept of weak θ -contraction used by Imdad and Alfaqih [8] which is an extension of θ -contraction (or JS contraction) in- troduced by Jleli and Samet [9]. We consider the family Θ1,2,4 and introduce Received 17 June 2019 – Accepted 11 October 2019 http://dx.doi.org/10.4995/agt.2020.11992 M. Garg and S. Chandok weak θm-contraction and prove that every (continuous) weak θm -contraction is a Picard operator in section 3. In section 4, we study about iterated multi- function system (IMS) and obtain some results on the existence and uniqueness of attractor for a weak θm− IMS. Also, we obtain some results on (α,β,θm)− contractions using cyclic (α,β)− admissible mappings without the continuity of the operator in the last section. 2. Preliminaries In this section, we recall some notations, basic definitions and results to be used in the sequel. Definition 2.1 (see [12, 13]). Let (M,d) be a metric space and f : M → M be a self mapping. A sequence {un} defined by un = fnu0 is called a Picard sequence based at the point u0 ∈ M. A self-mapping f is said to be a Picard operator if it has a unique fixed point z ∈ M and z = lim n→∞ fnu for all u ∈ M. Definition 2.2 (see [12, 13]). Let (M,d) be a metric space, and let K(M) be the class of all non-empty compact sets of M. The function η : K(M) × K(M) → [0,∞) define by η(A,B) = max{D(A,B),D(B,A)} where D(A,B) = supa∈A infb∈B d(a,b) for all A,B ∈ K(M) is a metric known as Hausdorff- Pompeiu metric. It is well known that if (M,d) is complete then (K(M),η) is also complete. Alizadeh et al. [1] introduced the notion of cyclic (α,β)-admissible mapping which is defined as follows: Definition 2.3. Let M be a nonempty set, f be a self-mapping on M and α,β : M → [0,∞) be two mappings. We say that f is a cyclic (α,β)-admissible mapping if x ∈ M with α(x) ≥ 1 implies β(fx) ≥ 1 and β(x) ≥ 1 implies α(fx) ≥ 1. The following results will be needed in the proof of our main results. Lemma 2.4 ([10]). Let (M,d) be a metric space and let {xn} be a sequence in M such that (2.1) lim n→∞ d (xn,xn+1) = 0. If {xn} is not a Cauchy sequence in M, then there exist ε > 0 and two sequences {m (k)} and {n (k)} of positive integers such that n (k) > m (k) > k and the following sequences tend to ε+ when k → +∞: (2.2) d ( xm(k),xn(k) ) , d ( xm(k),xn(k)+1 ) , d ( xm(k)−1,xn(k) ) , d ( xm(k)−1,xn(k)+1 ) , d ( xm(k)+1,xn(k)+1 ) . Remark 2.5. Let {xn}n∈N be a sequence in a metric space (X,d) . If for all n ∈ N holds d (xn+1,xn) < d (xn,xn−1), then n 6= m implies xn 6= xm whenever n,m ∈ N. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 58 Existence of Picard operator and iterated function system Lemma 2.6 ([14]). Let A,B,C ∈ K(M). Then we have the following: (i) A ⊂ B if and only if D(A,B) = 0; (ii) D(A,B) ≤ D(A,C) + D(C,B). Lemma 2.7 ([15]). If {Ei}i∈τ and {Fi}i∈τ are finite collection of elements in K(M), then η( ⋃ i∈τ Ei, ⋃ i∈τ Fi) ≤ sup i∈τ η(Ei,Fi). 3. Weak θm-contraction Now we use the definition of an auxiliary function and utilize the same to introduce weak θm-contraction. Definition 3.1 (see [6, 8, 9]). Let θ : (0,∞) → (1,∞) be a function and consider the following conditions: Θ1 : θ is non-decreasing. Θ2: for each sequence {αn} in (0,∞), lim n→∞ θ(αn) = 1 ⇔ lim n→∞ (αn) = 0. Θ3: there exist r ∈ (0, 1) and l ∈ (0,∞) such that lim α→0+ θ(α)−1 αr = l; Θ4: θ is continuous. The following notations to be used in the sequel. • Θ1,2,3 the family of all θ that satisfy Θ1 − Θ3. • Θ1,2,4 the family of all θ that satisfy Θ1, Θ2 and Θ4. • Θ2,3 the family of all θ that satisfy Θ2 and Θ3. • Θ2,4 the family of all θ that satisfy Θ2 and Θ4. • Θ2 the family of all θ that satisfy Θ2. Example 3.2 ([6]). Define θ : (0,∞) → (1,∞) by θ(α) = e √ α, for all α ∈ (0,∞).Then θ ∈ Θ1,2,3,4. Example 3.3 ([6]). Define θ : (0,∞) → (1,∞) by θ(α) = eα, for all α ∈ (0,∞).Then θ ∈ Θ1,2,3. Example 3.4 ([8]). The following function θ : (0,∞) → (1,∞) are in Θ2,4: (1) θ(α) = e α 2 +sinα; (2) θ(α) = αr + 1,r ∈ (0,∞). Example 3.5. Define θ : (0,∞) → (1,∞) by θ(α) = 2 √ α2 − 1√ α , for all α ∈ (0,∞).Then θ ∈ Θ1,2,4. Now, we define weak θm-contraction mapping. Definition 3.6. Let (M,d) be a metric space and f : M → M is a self- mapping. A mapping f is called a weak θm-contraction if there exist a θ ∈ Θ2,4 (or θ ∈ Θ1,2,4) and h ∈ (0, 1), such that for all u,v ∈ M, we have d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(M(u,v))]h,(3.1) c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 59 M. Garg and S. Chandok where M(u,v) = max{d(u,fu),d(v,fv),d(u,v)}. Remark 3.7. Here, we note that weak θm-contraction mapping has at most one fixed point. Assume that f has another fixed point say v ∈ M, d(u,v) > 0. Using (3.1) we have θ(d(u,v)) = θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h = [θ(d(u,v))]h, which is a contradiction. Lemma 3.8. Let (M,d) be a metric space and f : M → M is a weak θm- contraction. Suppose that there exists a Picard sequence {un}⊆ M defined by un+1 = f nu0 = fun for all n ∈ N ∪{0}. Then d(un,un+1) → 0 as n → ∞, where un 6= un+1 (Here θ ∈ Θ2,4 or Θ1,2,4). Proof. Let u0 ∈ M be an arbitrary point. Define the Picard sequence as {un}⊆ M by un+1 = fnu0 = fun for all n ∈ N∪{0}. Assume that un 6= un+1 for all n ∈ N∪{0}. Applying (3.1) we have, for all n ∈ N∪{0}, θ(d(un,un+1)) = θ(d(fun−1,fun)) ≤ [θ(max{d(un−1,fun−1),d(un,fun),d(un−1,un)})]h = [θ(max{d(un,un+1),d(un,un−1)})]h Case 1: When d(un,un+1) > d(un,un−1), then we have θ(d(fun−1,fun)) = θ(d(un,un+1)) ≤ [θ(d(un,un+1)]h, but α ≥ αh,∀α ∈ R+,h ∈ (0, 1). Thus we get contradiction. Case 2: When d(un,un−1) > d(un,un+1), we have θ(d(fun−1,fun)) ≤ [θ(d(un,un−1)]h. Hence on the same lines, we have [θ(d(fun−1,fun−2))] h ≤ [θ(max{d(un−1,fun−1),d(un−2,fun−2),d(un−1,un−2)})]h 2 = [θ(max{d(un−1,un),d(un−1,un−2)})]h 2 ≤ [θ(d(un−1,un−2))]h 2 . Proceeding on these lines, we get θ(d(fun,fun−1)) ≤ [θ(d(fun−1,fun−2))]h ≤ [θ(d(fun−2,fun−3))]h 2 ≤ ... ≤ [θ(d(fu0,u0))]h n . Thus, we have θ(d(un,un+1)) ≤ [θ(d(u1,u0))]h n . Now, taking n →∞ we have, lim n→∞ θ(d(un,un+1)) = 1. Using Θ2, we have lim n→∞ d(un,un+1) = 0. � Lemma 3.9. Let (M,d) be a metric space and f : M → M is a weak θm- contraction. Suppose that there exists a Picard sequence {un}⊆ M defined by un+1 = f nu0 = fun for all n ∈ N ∪ {0}. Then Picard sequence {un} is a Cauchy sequence (Here θ ∈ Θ2,4 or Θ1,2,4). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 60 Existence of Picard operator and iterated function system Proof. Let u0 ∈ M be an arbitrary point. Define the Picard sequence as {un}⊆ M by un+1 = fnu0 = fun for all n ∈ N∪{0}. Assume that un 6= un+1 for all n ∈ N ∪{0}. Using Lemma 3.8, we have lim n→∞ d(un,un+1) = 0. Now we have to prove that {un} is a Cauchy sequence. We’ll prove this by contradiction. Assume that {un} is not a Cauchy sequence. Now, since the sequence {un} is not a Cauchy sequence, then by Lemma 2.4, we have d ( um(k),un(k) ) and d ( um(k)+1,un(k)+1 ) tend to ε > 0, as k →∞. Using (3.1), we have θ(d(um(k),un(k))) = θ(d(fum(k)−1,fun(k)−1)) ≤ [θ(max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1), d(um(k)−1,un(k)−1)})]h. Case 1: If max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(um(k)−1,fum(k)−1), then we have θ(d(um(k),un(k))) ≤ [θ(d(um(k)−1,fum(k)−1)]h. Letting k →∞, from Lemma 2.4 and Θ4, we have θ(�) ≤ [θ(0)]h, which is a contradiction. Case 2: If max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(un(k)−1,fun(k)−1), then proceeding the same way as in Case 1 we again get a contradiction. Case 3: If max{d(um(k)−1,fum(k)−1),d(un(k)−1,fun(k)−1),d(um(k)−1,un(k)−1)} = d(um(k)−1,un(k)−1), then we have θ(d(um(k),un(k))) ≤ [θ(d(um(k)−1,un(k)−1)]h. Letting k →∞ and using Lemma 2.4 and Θ4, we obtain θ(�) ≤ [θ(�)]h, which is again a contradiction. Hence Picard sequence {un} is a Cauchy sequence. � Theorem 3.10. Every weak θm-contraction on a complete metric space is a Picard operator. [Here, we consider θ ∈ Θ1,2,4.] Proof. Let u0 ∈ M be an arbitrary point. Define the Picard sequence as {un}⊆ M by un+1 = fnu0 = fun for all n ∈ N∪{0}. If there exist n0 ∈ N∪{0} such that un0 = fun0 , then we are done. Assume that un 6= un+1 for all n ∈ N ∪{0}. Using Lemma 3.9, we have {un} is a Cauchy sequence. Now as (M,d) is a complete metric space so there exist u ∈ M such that {un} converges to u. From (Θ1) and (3.1), it is easy to conclude that θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h ≤ θ(max{d(u,fu),d(v,fv),d(u,v)}) for all u,v ∈ M with d(fu,fv) > 0. Using (Θ1) and above inequality, we have d(fu,fv) ≤ max{d(u,fu),d(v,fv),d(u,v)}. Suppose that u 6= fu. Therefore, we have d(un+1,fu) = d(fun,fu) ≤ max{d(un,fun),d(u,fu),d(un,u)} = max{d(un,un+1),d(u,fu),d(un,u)}. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 61 M. Garg and S. Chandok Taking n → ∞, using Lemma 3.8 we have d(u,fu) ≤ d(u,fu), which is a contradiction. Hence fu = u, thus we get a fixed point. Further, now we prove the uniqueness of the fixed point. Assume that f has another fixed point say v ∈ M, v 6= u. Using (3.1) we have θ(d(fu,fv)) ≤ [θ(max{d(u,fu),d(v,fv),d(u,v)})]h = [θ(d(u,v))]h, which is a contradiction. Hence the result. � Theorem 3.11. Every continuous weak θm-contraction on a complete metric space is a Picard operator. [Here, we consider θ ∈ Θ2,4.] Proof. Let u0 ∈ M be an arbitrary point. Define the Picard sequence as {un}⊆ M by un+1 = fnu0 = fun for all n ∈ N∪{0}. If there exist n0 ∈ N∪{0} such that un0 = fun0 , then we are done. Assume that un 6= un+1 for all n ∈ N ∪{0}. Proceeding as in Theorem 3.10, we have Picard sequence {un} is a Cauchy sequence. Now as (M,d) is a complete metric space so there exist u ∈ M such that {un} converges to u. The continuity of f and uniqueness of limit implies fu = u, thus we get a fixed point. Hence every continuous weak θm-contraction on a complete metric space is a Picard operator. � Example 3.12. Let M = {1, 2, 3}. Define the metric d : M × M → [0,∞) by d(x,y) = |x − y|, for all x,y ∈ M. Define a function f : M → M as f(1) = 2,f(2) = 2,f(3) = 1. Define a function θ : (0,∞) → (1,∞) by θ(t) = e √ t. So θ ∈ Θ1,2,3,4. Case 1. Consider (u,v) = (1, 3). We have θ(d(f1,f3)) = θ(d(2, 1)) = θ(1) = e. Also, [θ(max{d(1,f1),d(3,f3),d(1, 3)})]h = [θ(d(1, 2),d(1, 3))]h = [θ(2)]h = [e √ 2]h. Therefore θ(d(f1,f3)) ≤ [θ(max{d(1,f1),d(3,f3),d(1, 3)})]h, for all h ∈ [ 1√ 2 , 1). Case 2. Consider (u,v) = (2, 3). We have θ(d(f2,f3)) = θ(d(2, 1)) = θ(1) = e. Also, [θ(max{d(2,f2),d(3,f3),d(2, 3)})]h = [θ(d(3, 1),d(2, 3))]h = [θ(2)]h = [e √ 2]h. Therefore θ(d(f1,f3)) ≤ [θ(max{d(2,f2),d(3,f3),d(2, 3)})]h, for all h ∈ [ 1√ 2 , 1). Thus all the conditions of Theorem 3.10 are satisfied and 2 is a unique fixed point of f. Here is to note that when (u,v) = (2, 3) in the above example, then (a) f is not Banach contraction; (b) f is not weak θ-contraction of Imdad et al. [8]; (c) f is weak θm-contraction. (d) f is a Picard operator. Theorem 3.13. Let (M,d) be a complete metric space and let f : M → M be a self mapping. If there exist n ∈ N such that fn is a weak θm -contraction, then f is a Picard operator. Proof. From Theorem 3.10, it is obvious that fn is a Picard operator, thus there exists a unique z ∈ M such that fnz = z and lim m→∞ tm+1 = (f n) m u = z, c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 62 Existence of Picard operator and iterated function system for all u ∈ M. Also, we observe that fn+1z = fnz, that is fn(fz) = fz, thus fz is also a fixed point of fn. Thus fz = z. Further, if z∗ is another fixed point of f, then it must be a fixed point of fn. Hence z = z∗. Therefore f has a unique fixed point. Now, let m be a positive integer greater than n. Then there exist l ≥ 1 and s ∈{0, 1, 2, ...,n−1} such that m = nl + s. Here, we notice that for all u ∈ M, we have lim m→∞ um+1 = lim m→∞ fmu = lim l→∞ fnl(fsu) = lim l→∞ (fn)l(fsu) = z. Hence the result. � Haghi et al. [5], in 2011, proved a lemma by using the axiom of choice as follows: Lemma 3.14. Let M be a nonempty set and f : M → M a function. Then there exist a set E ⊆ M such that f(E) = f(M) and f : E → M is one-to-one. By using above lemma, we prove common fixed point theorems for two self mappings on M as follows: Theorem 3.15. Let (M,d) be a complete metric space and f,g be two self maps on M satisfying d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(max{d(gu,fu),d(gv,fv),d(gu,gv)})]h.(3.2) for all u,v ∈ M and θ ∈ Θ2,4 (or θ ∈ Θ1,2,4). If f(M) ⊆ g(M) and g(M) is a complete subset of M then f and g have a unique common fixed point in M. Proof. By using Lemma 3.14, there exist E ⊆ M such that g(E) = g(M) and g : E → M is one-to-one. Define h : g(E) → g(E) by h(gu) = fu. Clearly, h is well defined as g is one-to-one on E. Also, θ(d(h(gu),h(gv))) ≤ [θ(max{d(gu,fu),d(gv,fv),d(gu,gv)})]h, for all gx,gy ∈ g(E). Since g(E) = g(M) is complete, then by using Theorem 3.10, we can easily prove that f and g have a unique common fixed point in M. � 4. Weak θm iterated multifunction system As application of results proved in the last section, we obtain some results on the existence and uniqueness of attractor of iterated multifunction system composed by weak θm-contraction in the setting of complete metric space in this section. In the following section, we consider (M,d) is a complete metric space, N ∈ N and θ ∈ Θ1,2,4. Definition 4.1. Let {fi}Ni=1 be a finite family of self mappings on M. If fi : M → M is a weak θm−contraction (for each i), then the family {fi}Ni=1 is called a weak θm− iterated function system (weak θm−IFS). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 63 M. Garg and S. Chandok The set function G : K(M) → K(M) define by G(B) = ⋃N i=1 fi(Y ) (for all Y ∈ K(M)) is said to be associated Hutchinson operator. A set A ∈ K(M) is called an attractor of the weak θm− IFS if G(A) = A. Let (M,d) be a metric space and F1,F2, ...Fn : M → K(M) be multivalued operator. Then the system F = (F1,F2, ...,Fn) is called an iterated multifunc- tion system (abbreviated as IMS). Definition 4.2. Let {Fi}Ni=1 be a finite family of iterated multifunction system. If Fi : M → K(M) is a weak θm−contraction (for each i), then the family {Fi}Ni=1 is called a weak θm− iterated multifunction system (weak θm−IMS). Define P(M) = {Y ⊂ M : Y is nonempty}. If T : M → P(M) is a multivalued operator then T(Y ) := ⋃ x∈Y T(x),Y ∈ P(M). Let F1,F2, ...Fm : M → K(M) be a finite family of multivalued operators, we define multifractal operator TF generated by the iterated multifunction system F = (F1,F2, ...Fm) by GF : K(M) → K(M), GF (Y ) = ⋃m i=1 Fi(Y ). In this framework, a nonempty compact subset A∗ of M is said to be a multivalued fractal with respect to the iterated multifunctions system F = (F1,F2, ...Fm) if and only if it is a fixed point for the associated multifractal operator. In particular, if the operators Fi = fi are singlevalued, then a fixed point for the fractal operator Gf : K(M) → K(M), Gf (Y ) = ∪mi=1fi(Y ) generated by generated by iterated function system f = (f1,f2, ...fm) is said to be a self similar set or a fractal. Throughout, Fix(f) denotes the set of fixed points of f (see [2, 4, 7]). Definition 4.3. If {Fi}Ni=1 is weak θm− IMS such that Fi : M → K(M) is continuous for i = 1, 2, . . . ,N then the operator GF : K(M) → K(M), GF (Y ) = ⋃N i=1 Fi(Y ) is well defined and is called weak θm− multi-fractal operator. A fixed point of GF is called a multivalued fractal. Now we will use the following lemma to show that a weak θm− multi-fractal operator has a unique multivalued fractal. Lemma 4.4. Let f : M → K(M) is a continuous weak θm- multivalued op- erator. Then the mapping A 7→ f(A) is also a weak θm-multivalued operator from K(M) into itself. Proof. Let A,B ∈ K(M) be such that η(f(A),f(B)) > 0. Assume that η(f(A),f(B)) = D(f(A),f(B)) = sup u∈A inf v∈B D(fu,fv), for all A,B ∈ K(M).(4.1) As f is a continuous weak θm-multivalued operator so there exist h ∈ (0, 1) such that θ(D(fu,fv)) ≤ [θ(max{D(u,fu),D(v,fv),d(u,v)})]h, for all u,v ∈ M. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 64 Existence of Picard operator and iterated function system Now using (4.1), compactness of A, and continuity of f, we can find a ∈ A such that D(f(A),f(B)) = infv∈B D(fa,fv) > 0, so that D(fa,fv) > 0, for all v ∈ B. Hence, for all v ∈ B, we have θ( inf v∈B D(fa,fv)) ≤ θ(D(fa,fv) ≤ [θ(max{D(a,fa),D(v,fv),d(a,v)})]h. Therefore, for all v ∈ B we get (4.2) θ(η(f(A),F(B)) ≤ [θ(max{D(a,fa),D(v,fv),d(a,v)})]h. Case 1: If max{D(a,fa),D(v,fv),d(a,v)} = D(a,fa), then we have: θ(infv∈B D(fa,fv)) ≤ [θ(D(a,fa))]h, Now from (4.2) we have θ(η(f(A),f(B))) ≤ [θ(D(a,fa′))]h ≤ [θ(sup a∈A inf fa∈f(A) d(a,fa))]h = [θ(D(A,A))]h, which is a contradiction. Case 2: If max{D(a,fa),D(v,fv),d(a,v)} = D(v,fv), then proceeding in the same way as in Case 1 we again get a contradiction. Case 3: If max{D(a,fa),D(v,fv),d(a,v)} = d(a,v), then for all v ∈ B we have θ(η(f(A),f(B))) ≤ [θ(d(a,v))]h. Now let v ∈ B be such that d(a,v) = infv∈B d(a,v). From (4.2) we have, θ(η(f(A),f(B))) ≤ [θ(d(a,v))]h, = [θ( inf b∈B d(a,v))]h, ≤ [θ(sup a∈A inf v∈B d(a,v))]h = [θ(D(A,B))]h ≤ [η(A,B)]h. Hence we get the result. � Theorem 4.5. Let (M,d) be a complete metric space and Fi : M → K(M), i = {1, 2, ...,m} be continuous multivalued operator satisfying θ(η(Fiu,Fiv)) ≤ [θ(max{d(u,Fiu),d(v,Fiv),d(u,v)})]h, for all u,v ∈ M and h ∈ (0, 1). Then there exists a unique multivalued fractal with respect to the iterated multifunction system F = (F1,F2, ...Fm), that is, Fix(GF ) = {A∗} and {GnF (A)}n∈N converges to A ∗, for each A ∈ K(M). Proof. First we prove that the operator GF : K(M) → K(M), GF (Y ) = ∪mi=1Fi(Y ) satisfies the conditions of Theorem 3.10. Let B,C ∈ K(M) such that c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 65 M. Garg and S. Chandok 0 < η(GF (B),GF (C)) = η( ⋃m i=1 Fi(B), ⋃m i=1 Fi(C)). Now Lemma 2.7 implies that η(GF (B),GF (C)) = η( m⋃ i=1 Fi(B), m⋃ i=1 Fi(C)) ≤ sup 1≤i≤N η(Fi(B),Fi(C)) = η(Fi0 (B),Fi0 (C)), for some i0 ∈{1, 2, 3, . . . ,N}. Using Θ1 and Lemma 4.4, we have θ(η(GF (B),GF (C)) ≤ θ(η(Fi0 (B),Fi0 (C))) ≤ [θ(η(B,C))]hi0 . Therefore GF is also a continuous weak θm contraction on the complete metric space (K(M),η). Theorem 3.11 ensures the existence and uniqueness of A∗ ∈ K(M) such that GF (A ∗) = A∗ and A∗ = lim n→∞ GnF (B) for all B ∈ K(M). This completes the proof. � In particular, when the operators are single valued, we have the following result. Theorem 4.6. If {fi}Ni=1 is a continuous weak θm -IFS, then it has unique attractor. Moreover, A = lim n→∞ Gn(B) for all B ∈ K(M), the limit being taken with respect to the Hutchinson-Pompeiu metric. Proof. For each i ∈{1, 2, . . .N}, let hi be constant such that hi ∈ (0, 1) and is associated with fi. Let B,C ∈ K(M) such that 0 < η(G(B),G(C)) = η( ⋃N i=1 fi(B), ⋃N i=1 fi(C)). Now Lemma 2.7 implies that η(G(B),G(C)) = η( N⋃ i=1 fi(B), N⋃ i=1 fi(C)) ≤ sup 1≤i≤N η(fi(B),fi(C)) = η(fi0 (B),fi0 (C)), for some i0 ∈{1, 2, 3, . . . ,N}. Using Θ1 and Lemma 4.4, we have θ(η(G(B),G(C)) ≤ θ(η(fi0 (B),fi0 (C))) ≤ [θ(η(B,C))]hi0 . Therefore G is also a continuous weak θm contraction on the complete metric space (K(M),η). Theorem 3.11 ensures the existence and uniqueness of A ∈ K(M) such that G(A) = A and A = lim n→∞ Gn(B) for all B ∈ K(M). This completes the proof. � Example 4.7. Let M=[0, 1] ⊂ R, with the metric given by the usual metric. We define, F : K(M) → K(M) by F(A) = f1(A) ∪f2(A), where f1(x) = 1 3 x, f2(x) = 1 3 x + 2 3 , 0≤x≤1. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 66 Existence of Picard operator and iterated function system First we verify that f1 and f2 are weak θm contraction. Take θ = ex and d(x,y) = |x−y|, thus d(x,f1x) = ∣∣x− x 3 ∣∣ = ∣∣2x 3 ∣∣ for all x ∈ M. Therefore, max{d(x,f1x),d(y,f1y),d(x,y)} = max{2x3 , 2y 3 , |x−y| 3 }. Case 1: If x > y, max{d(x,f1x),d(y,f1y),d(x,y)} = 2x3 . We know that (4.3) x−y 3 ≤ 2xy 3 for all x,y ∈ M. Therefore, e x−y 3 ≤ e 2xy 3 = [e 2x 3 ]y = [e 2x 3 ]h, where h = y ∈ (0, 1). Hence we have θ(d(f1x.f1y)) = e x−y 3 ≤ [θd(x,f1x)]h, h = y ∈ (0, 1). Now choose f2(x) = 1 3 x + 2 3 , d(x,f2x) = ∣∣x−(1 3 x + 2 3 )∣∣ = ∣∣2x 3 − 2 3 ∣∣ for all x ∈ M. As we know that (4.4) x−y 3 − 2 3 ≤ 2xy 3 − 2 3 for all x,y ∈ M. Therefore, e x−y 3 −2 3 ≤ e 2xy 3 −2 3 ≤ e 2xy 3 = [e 2x 3 ]y = [e 2x 3 ]h, we have h = y ∈ (0, 1). Thus we have θ(d(f2x.f2y)) = e x−y 3 −2 3 ≤ [θd(x,f2x)]h,h = y ∈ (0, 1). Case 2: Now take y > x, we have max{d(x,f1x),d(y,f1y),d(x,y)} = d(y,f1y). In this case we also obtain same conclusion as in Case 1. Therefore, d(x,y) 6= max{d(x,f1x),d(y,f1y),d(x,y)}, for any value of x,y ∈ [0, 1]. Hence from both cases we can say that f1 is a weak θm-contraction for θ = ex. In the similar way, we can prove that f2 is also a weak θm-contraction for θ = ex. Thus F = (f1,f2) is iterated multifunction system. The unique fixed point of F must satisfy A = F(A) = f1(A) ∪f2(A). Considering the nature of the two transformations, we get a unique fractal A ⊂ K(M) which is Cantor subset of [0, 1]. 5. Cyclic (α,β)-admissible mappings Definition 5.1. Let (M,d) be a complete metric space, f : M → M be a map- ping and α,β : R → [0,∞) be two functions. Then S is said to be a generalized (α,β,θm)− contraction mapping if f satisfies the following conditions: (1) f is cyclic (α,β)-admissible; (2) there exits a θ ∈ Θ2,4 and h ∈ (0, 1) such that for all u,v ∈ M, we have α(u)β(v) ≥ 1,d(fu,fv) > 0 ⇒ θ(d(fu,fv)) ≤ [θ(M(u,v))]h,(5.1) where M(u,v) = max{d(u,fu),d(v,fv),d(u,v)}. Theorem 5.2. Let (M,d) be a complete metric space, f : M → M be a mapping and α,β : M → [0, 1) be two functions. Suppose that the following conditions hold. (1) f is a generalized (α,β,θm)− contraction mapping; c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 67 M. Garg and S. Chandok (2) There exists an element x0 ∈ M such that α(x0) ≥ 1 and β(x0) ≥ 1; (3) f is continuous; or If sequence {xn} in M converges to x ∈ M with the property α(xn) ≥ 1 (or β(xn) ≥ 1) for all n ∈ N, then α(x) ≥ 1 (or β(x) ≥ 1). Then f is a Picard operator. Proof. Assume that there exist x0 ∈ M such that α(x0) ≥ 1. Define a Picard sequence {xn} by xn+1 = fxn = fnx0, for all n ∈ N ∪ {0}. If there exist n0 ∈ N∪{0} such that un0 = fun0 , then we are done. Assume that un 6= un+1 for all n ∈ N ∪{0}. Assume that there exist x0,x1 ∈ M such that α(x0) ≥ 1 =⇒ β(fx0) = β(x1) ≥ 1 and β(x0) ≥ 1 =⇒ α(fx0) = α(x1) ≥ 1. By continuing above process, we have α(xn) ≥ 1 =⇒ β(fxn) = β(xn+1) ≥ 1 and β(xn) ≥ 1 =⇒ α(fxn) = α(xn+1) ≥ 1. Since α(xm) ≥ 1 =⇒ β(fxm) = β(xm+1) ≥ 1 and β(xm) ≥ 1 =⇒ α(fxm) = α(xm+1) ≥ 1, for all m,n ∈ N with n < m. Moreover, since α(xm) ≥ 1 =⇒ β(xm+2) ≥ 1 and β(xm) ≥ 1 =⇒ α(xm+2) ≥ 1, for all m,n ∈ N with n < m. By continuing this process, we have α(xn) ≥ 1 =⇒ β(xm) ≥ 1 and β(xn) ≥ 1 =⇒ α(xm) ≥ 1, for all m,n ∈ N. Thus α(xn)β(xn+1) ≥ 1, for all n ∈ N∪{0}. Therefore, using (5.1) we have θ(d(fxn,fxn+1)) ≤ [θ(max{d(xn,fxn),d(xn+1,fxn+1),d(xn,xn+1)})]h = [θ(max{d(xn,xn+1),d(xn+1,xn+2),d(xn,xn+1)})]h = [θ(max{d(xn,xn+1),d(xn+1,xn+2)})]h(5.2) Analysis similar to that in the proof of Theorem 3.11 shows that d (xn,xn+1) → 0, as n →∞. Now, we prove that {xn} is a Cauchy sequence. On the contrary, suppose that {xn} is not a Cauchy sequence. By Lemma 2.4, there exist ε > 0 and two sequences {n (k)} and {m(k)} of positive integers such that n(k) > m(k) > k and the sequences {d(xm(k),xn(k))} and {d(xm(k)+1,xn(k)+1)} tend to ε+ > 0 as k →∞. Substituting x = xm(k) and y = xn(k) into the inequality (5.1), we obtain α ( xm(k) ) β ( xn(k) ) ≥ 1 ⇒ θ ( d ( fxm(k),fxn(k) )) ≤ [θ ( M ( xm(k),xn(k) )) ]h,(5.3) where M ( xm(k),xn(k) ) = max { d ( xm(k),xn(k) ) ,d ( xm(k),xm(k)+1 ) ,d ( xn(k),xn(k)+1 )} . Since d ( xm(k),xm(k)+1 ) → 0 and d ( xn(k),xn(k)+1 ) → 0 as k →∞. Then using the fact that α ( xm(k) ) β ( xn(k) ) ≥ 1 holds and that d ( xm(k)+1,xn(k)+1 ) and d ( xm(k),xn(k) ) are both positive numbers, by using the property Θ4, Lemma c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 68 Existence of Picard operator and iterated function system 2.4 and similar arguments as in Theorem 3.11, we obtain α ( xm(k) ) β ( xn(k) ) ≥ 1 ⇒ θ ( d ( fxm(k),fxn(k) )) ≤ [θ ( d ( xm(k),xn(k) )) ]h. For sufficiently large k, k →∞, we get θ(ε) ≤ [θ(ε)]h, which is a contradic- tion. Hence, {xn}n∈N∪{0} is a Cauchy sequence. Now as (M,d) is a complete metric space so there exist x ∈ M such that {xn} converges to x. The continuity of f and uniqueness of limit implies fx = x, thus we get a fixed point. Now, suppose that the sequence {xn} in M converges to x ∈ M with the property α(xn) ≥ 1 (or β(xn) ≥ 1) for all n ∈ N, then α(x) ≥ 1 (or β(x) ≥ 1). Hence α(x)β(x) ≥ 1 Further, we claim that fx = x. Suppose not, that is, fx 6= x. So d(fx,x) > 0 and lim n→∞ d(xn+1,fx) 6= 0. Using (5.1) we have θ(d(xn+1,fx)) = θ(d(fxn,fx)) ≤ [θ(max{d(xn,fxn),d(x,fx),d(xn,x)})]h = [θ(max{d(xn,xn+1),d(x,fx),d(xn,x)})]h.(5.4) Taking n →∞ and using property Θ4, we have θ(d(x,fx)) ≤ [d(x,fx)]h, which is a contradiction. We, thus, obtain that f has a fixed point fx = x. It is easy to prove the uniqueness of fixed point. � Remark 5.3. • Note that, throughout this paper, Lemma 2.4 and the contractive con- ditions imply that the iterative sequence, i.e. Picard sequence is a Cauchy. • For different variants of inequality (3.1), we have many interesting re- sults. For example, when, we replace M(u,v) in (2.1) and (5.1) with M(u,v) = max{d(u,f(u)),d(v,f(v)))} (type of Bianchini [3]), we may extend Theorem 3.11, Theorem 3.13, Theorem 3.15, Theorem 4.6 and Theorem 5.2 to these different variants of inequality. Also when, we replace M(u,v) in (2.1) with M(u,v) = d(u,v), we have the corre- sponding results of Imdad et al. [8]. Acknowledgements. The authors are thankful to the learned referee for valuable suggestions. The second author is also thankful to AISTDF, DST for the research grant vide project No. CRD/2018/000017. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 1 69 M. Garg and S. Chandok References [1] S. Alizadeh, F. Moradlou and P. Salimi, Some fixed point results for (α,β) − (ψ,φ)- contractive mappings, Filomat 28 (2014), 635–647. [2] M. F. Barnsley, Fractals Everywhere, Revised with the Assistance of and with a Foreword by Hawley Rising, III. Academic Press Professional, Boston (1993). [3] R. M. T. Bianchini, Su un problema di S. 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