A generalization of Istratescu's fixed point theorem for convex contractions
aa r X i v : . [ m a t h . C A ] D ec A generalization of Istr˘at¸escu’s fixed point theoremfor convex contractions
Radu MICULESCU and Alexandru MIHAIL
Abstract . In this paper we prove a generalization of Istr˘at¸escu’s theorem forconvex contractions. More precisely, we introduce the concept of iterated functionsystem consisting of convex contractions and prove the existence and uniqueness ofthe attractor of such a system. In addition we study the properties of the canonicalprojection from the code space into the attractor of an iterated function systemconsisting of convex contractions. : Key words and phrases : convex contractions, fixed points, iterated func-tion systems consisting of convex contractions
1. Introduction
Banach-Caccioppoli-Picard contraction principle, which is an extremelyuseful tool in nonlinear analysis, says that any contraction f : ( X, d ) → ( X, d ), where (
X, d ) is a complete metric space, has a unique fixed point x ∗ and lim n →∞ f [ n ] ( x ) = x ∗ for every x ∈ X . Besides its great features (theuniqueness of the fixed point and the possibility to approximate it by themeans of Picard iteration) there exists a drawback of this result, namelythat the contraction condition is too strong.The natural question if there exist contraction-type conditions that do noimply the contraction condition and for which the existence and uniqueness ofthe fixed point are assured was answered, among others, by V. Istr˘at¸escu whointroduced and studied the convex contraction condition (see [5], [6] and [7]).More precisely, a continuous function f : ( X, d ) → ( X, d ), where (
X, d ) is acomplete metric space, is called convex contraction if there exist a, b ∈ (0 , a + b < d ( f [2] ( x ) , f [2] ( y )) ≤ ad ( f ( x )) , f ( y )) + bd ( x, y ) forevery x, y ∈ X . Istr˘at¸escu proved that any convex contraction has a uniquefixed point x ∗ ∈ X (and lim n →∞ f [ n ] ( x ) = x ∗ for every x ∈ X ) and provided anexample of convex contraction which is not contraction. V. Ghorbanian, S.Rezapour and N. Shahzad [8] generalized Istr˘at¸escu’s results to complete or-dered metric spaces. M. A. Miandaragh, M. Postolache and S. Rezapour [16]introduced the concept of generalized convex contraction and proved some1heorems about approximate fixed points of these contractions. Extend-ing these results, A. Latif, W. Sintunavarat and A. Ninsri [12] introduceda new concept called partial generalized convex contraction and establishedsome approximate fixed point results for such mappings in α -complete metricspaces. For more results along these lines of generalization one can also see[10].Let us recall that an iterated function system on a complete metric space( X, d ), denoted by S = ( X, ( f k ) k ∈{ , ,...,n } ), consists of a finite family of con-tractions ( f k ) k ∈{ , ,...,n } , where f k : X → X . The function F S : K ( X ) →K ( X ) defined by F S ( C ) = n ∪ k =1 f k ( C ), for all C ∈ K ( X ) -the set of non-emptycompact subsets of X -, which is called the set function associated to S , turnsout to be a contraction (with respect to the Hausdorff-Pompeiu distance) andits unique fixed point, denoted by A S , is called the attractor of the system S . As iterated function systems represent one of the main tools to generatefractals, the extending problem of the notion of iterated function system wastreated by several authors. Let us mentions some contributions along theselines of research. Given a complete metric space ( X, d ) and a finite familyof functions f , f , ..., f n : X → X , L. M´at´e [15] proved the existence of aunique A ∈ K ( X ) such that A = n ∪ i =1 f i ( A ) under weaker contractivity condi-tions (for example d ( f i ( x ) , f i ( y )) ≤ ϕ ( d ( x, y )), where ϕ : [0 , ∞ ) → [0 , ∞ ) isan upper continuous non-decreasing function with the property that ϕ ( t ) < t for each t > X, d ) be a metricspace and f , f , ..., f n : X → P cl ( X ) be set-valued mappings on X , where P cl ( X ) designates the family of all nonempty closed subsets of X . The system F = ( f , f , ..., f n ) is called an iterated multifunction system and the operator ˆ F : P cl ( X ) → P cl ( X ) given by ˆ F ( Y ) = n ∪ i =1 f i ( Y ), where f i ( Y ) = ∪ x ∈ Y f i ( x ) foreach i ∈ { , , ..., n } , is called the Barnsley-Hutchinson operator generatedby F . A fixed point of this operator is called a multivalued large fractal. C.Chifu and A. Petru¸sel [3] obtained existence and uniqueness results for mul-tivalued large fractals (see also [20]). G. Gw´o´zd´z- Lukowska and J. Jachymski[9] developed the Hutchinson-Barnsley theory for finite families of mappingson a metric space endowed with a directed graph. E. Llorens-Fuster, A.Petru¸sel and J.-C. Yao [14] gave existence and uniqueness results for self-2imilar sets of a mixed iterated function system. M. Boriceanu, M. Botaand A. Petru¸sel [2] extended the Hutchinson-Barnsley theory to the case ofset-valued mappings on a b -metric space. For other related results see [1],[11], [17], [19], [22], [24], [25] and [26].In this paper we introduce the concept of iterated function system consist-ing of convex contractions and prove the existence and uniqueness of the at-tractor of such a system obtaining in this way a generalization of Istr˘at¸escu’sconvex contractions fixed point theorem (see Theorem 3.2). Moreover westudy the properties of the canonical projection from the code space into theattractor of an iterated function system consisting of convex contractions(see Theorem 3.6).
2. Preliminaries
Given a function f : X → X , by f [ n ] we mean the composition of f byitself n times.Given a set X and a family of functions ( f i ) i ∈ I , where f i : X → X ,by f α α ....α n we mean f α ◦ f α ◦ ... ◦ f α n and by Y α α ....α n we understand f α α ....α n ( Y ), where Y ⊆ X and α , α , ...., α n ∈ I .Given a set X , by P ∗ ( X ) we denote the family of all nonempty subsetsof X . For a metric space ( X, d ), by K ( X ) we denote the set of non-emptycompact subsets of X .Given two sets A and B , by B A we mean the set of functions from A to B . Given a set I , Λ( I ) denotes I N ∗ and Λ n ( I ) denotes I { , ,...,n } . Hence theelements of Λ( I ) can be written as infinite words α = α α α ... and theelements of Λ n ( I ) as finite words α = α α ....α n . By Λ ∗ ( I ) we denote the setof all finite words, namely Λ ∗ ( I ) = ∪ n ∈ N ∗ Λ n ( I ) ∪ { λ } , where λ is the emptyword. Λ( I ) can be seen as a metric space with the distance d Λ defined by d Λ ( α, β ) = n where n is the natural number having the property that α k = β k for k < n and α n = β n if α = α α α ...α n α n +1 ... = β = β β β ...β n β n +1 ... and d Λ ( α, α ) = 0. By αβ we understand the concatenation of the words α ∈ Λ ∗ and β ∈ Λ ∪ Λ ∗ . For α ∈ Λ ∪ Λ n and m ≤ n , [ α ] m def = α α ....α m . For i ∈ I , let us consider the function F i : Λ( I ) → Λ( I ) given by F i ( α ) = iα forall α ∈ Λ( I ). 3 efinition 2.1. For a metric space ( X, d ) , we consider on P ∗ ( X ) thegeneralized Hausdorff-Pompeiu pseudometric h : P ∗ ( X ) × P ∗ ( X ) → [0 , + ∞ ] defined by h ( A, B ) = max( d ( A, B ) , d ( B, A )) = inf { η ∈ [0 , ∞ ] | A ⊆ N η ( B ) and B ⊆ N η ( A ) } where d ( A, B ) = sup x ∈ A d ( x, B ) = sup x ∈ A ( inf y ∈ B d ( x, y )) and N η ( A ) = { x ∈ X | there exists y ∈ X such that d ( x, y ) < η } , for every A, B ∈ P ∗ ( X ). Proposition 2.2 (see [23]) . If H and K are two nonempty subsets ofthe metric space ( X, d ) , then h ( H, K ) = h ( H, K ). Proposition 2.3 (see [23]) . If ( H i ) i ∈ I and ( K i ) i ∈ I are two families ofnonempty subsets of the metric space ( X, d ) , then h ( ∪ i ∈ I H i , ∪ i ∈ I K i ) = h ( ∪ i ∈ I H i , ∪ i ∈ I K i ) ≤ sup i ∈ I h ( H i , K i ). Theorem 2.4 (see [23]) . If the metric space ( X, d ) is complete, then ( K ( X ) , h ) is a complete metric space. Definition 2.5.
For a metric space ( X, d ) , we consider on P ∗ ( X ) thefunction δ : P ∗ ( X ) × P ∗ ( X ) → [0 , + ∞ ] defined by δ ( A, B ) = sup x ∈ A,y ∈ B d ( x, y ), for all A, B ∈ P ∗ ( X ). Remark 2.6.
For every
A, B ∈ P ∗ ( X ) we have h ( A, B ) ≤ δ ( A, B ).4 roposition 2.7.
Let ( X, d ) be a complete metric space, ( Y n ) n ∈ N ⊆K ( X ) and Y a closed subset of X such that lim n →∞ h ( Y n , Y ) = 0 . Then Y ∈K ( X ) .Proof. It is enough to prove that Y is precompact. To this aim, letus note that for each ε > n ε ∈ N such that h ( Y n ε , Y ) < ε ,so Y ⊆ N ε ( Y n ε ). Since Y n ε ∈ K ( X ) there exist x , ..., x m ∈ X such that Y n ε ⊆ m ∪ i =1 B ( x i , ε ) and therefore Y ⊆ m ∪ i =1 B ( x i , ε ). (cid:3) Proposition 2.8.
Let ( X, d ) be a complete metric space, ( Y n ) n ∈ N ⊆K ( X ) and Y ∈ K ( X ) such that lim n →∞ h ( Y n , Y ) = 0 . Then H def = Y ∪ ( ∞ ∪ n =0 Y n ) ∈K ( X ) .Proof. First of all we prove that H is a closed subset of X .Indeed, if x ∈ H , then there exists a sequence ( x k ) k ∈ N ⊆ H such thatlim k →∞ x k = x .If { k ∈ N | x k ∈ Y } is infinite, then there exists a subsequence ( x k p ) p ∈ N of ( x k ) k ∈ N such that x k p ∈ Y for every p ∈ N . Since Y ∈ K ( X ) thereexists a subsequence ( x k pq ) q ∈ N of ( x k p ) p ∈ N and y ∈ Y such that lim q →∞ x k pq = y .Consequently, as lim q →∞ x k pq = x , we conclude that x = y ∈ Y ⊆ H .If there exists n ∈ N such that { k ∈ N | x k ∈ Y n } is infinite, a similarargument shows that x ∈ H .If none of the above described two cases is valid, then there exist anincreasing sequence ( k p ) p ∈ N ⊆ N , x k p ∈ Y k p and y k p ∈ Y such that d ( x k p , y k p ) < h ( Y k p , Y ) + 1 p .Since Y ∈ K ( X ) there exists ( y k pq ) q ∈ N a subsequence of ( y k p ) p ∈ N and y ∈ Y such that lim q →∞ y k pq = y . As d ( x k pq , y ) < d ( x k pq , y k pq ) + d ( y k pq , y ) ≤ h ( Y k pq , Y ) + 1 p q + d ( y k pq , y )and lim q →∞ h ( Y k pq , Y ) = lim q →∞ p q = lim q →∞ d ( y k pq , y ) = 0,we infer that lim q →∞ x k pq = y . Consequently, as lim q →∞ x k pq = x , we get x = y ∈ Y ⊆ H . 5ow we prove that lim m →∞ h ( m ∪ i =0 Y i , H ) = 0.Indeed h ( m ∪ i =0 Y i , H ) = h (( m ∪ i =0 Y i ) ∪ ( ∞ ∪ i = m +1 Y m ) ∪ Y m , ( m ∪ i =0 Y i ) ∪ ( ∞ ∪ i = m +1 Y i ) ∪ Y )) Proposition 2.3 ≤≤ sup { h ( Y m , Y ) , h ( Y m , Y m +1 ) , h ( Y m , Y m +2 ) , ... } for every m ∈ N . As lim m →∞ h ( Y m , Y ) = 0, we conclude that lim m →∞ h ( m ∪ i =0 Y i , H ) =0. Because m ∪ i =0 Y i ∈ K ( X ) for every m ∈ N and H is closed, using Proposition2.7, we obtain that H is compact. (cid:3)
3. The main resultsDefinition 3.1.
An iterated function system consisting of convex con-tractions on a complete metric space ( X, d ) is given by a finite family ofcontinuous functions ( f i ) i ∈ I , f i : X → X , such that for every i, j ∈ I thereexist a ij , b ij , c ij ∈ [0 , ∞ ) satisfying the following two properties: α ) a ij + b ij + c ij def = d ij and max i,j ∈ I d ij def = d < β ) d (( f i ◦ f j )( x ) , ( f i ◦ f j )( y )) ≤ a ij d ( x, y ) + b ij d ( f i ( x ) , f i ( y )) + c ij d ( f j ( x ) , f j ( y )) for every i, j ∈ I and every x, y ∈ X .We denote such a system by S = (( X, d ) , ( f i ) i ∈ I ) . One can associate to the system S the function F S : K ( X ) → K ( X ) givenby F S ( B ) = ∪ i ∈ I f i ( B )for all B ∈ K ( X ). Theorem 3.2.
Let S = (( X, d ) , ( f i ) i ∈ I ) be an iterated function systemconsisting of convex contractions. Then: There exists a unique A ∈ K ( X ) such that lim n →∞ h ( F [ n ] S ( B ) , A ) = 0, for every B ∈ K ( X ) . ii) For each ω ∈ Λ( I ) there exists a ω ∈ X such that lim n →∞ h ( f [ ω ] n ( B ) , { a ω } ) = 0, for every B ∈ K ( X ) .Moreover lim n →∞ sup ω ∈ Λ( I ) h ( f [ ω ] n ( B ) , { a ω } ) = 0 for every B ∈ K ( X ) . iii) A = { a ω | ω ∈ Λ( I ) } . iv) For every ( Y n ) n ∈ N ⊆ K ( X ) and Y ∈ K ( X ), the following implicationis valid: lim n →∞ h ( Y n , Y ) = 0 ⇒ lim n →∞ h ( F S ( Y n ) , F S ( Y )) = 0 . v) A is the unique fixed point of F S . Proof .i) For fixed
Y, Z ∈ K ( X ) we define x n ( Y, Z ) = sup ω ∈ Λ n ( I ) δ ( f ω ( Y ) , f ω ( Z ))and y n ( Y, Z ) = max { x n − ( Y, Z ) , x n ( Y, Z ) } for every n ∈ N ∗ . For the sake of simplicity we will denote x n ( Y, Z ) by x n and y n ( Y, Z ) by y n .We claim that the sequence ( y n ) n ∈ N ∗ is decreasing.Indeed, for n ∈ N ∗ and ω ∈ Λ n +1 ( I ) there exist i, j ∈ I and ω ∈ Λ n − ( I )such that ω = ijω . Then, for y ∈ Y and z ∈ Z , we have d ( f ω ( y ) , f ω ( z )) = d ( f ijω ( y ) , f ijω ( z )) ≤≤ a ij d ( f ω ( y ) , f ω ( z )) + b ij d ( f iω ( y ) , f iω ( z )) + c ij d ( f jω ( y ) , f jω ( z )) ≤ a ij x n − + b ij x n + c ij x n ≤ a ij x n − + ( b ij + c ij ) x n ≤≤ d ij max { x n − , x n } = d ij y n ≤ dy n < y n ,so x n +1 = sup ω ∈ Λ n +1 ( I ) δ ( f ω ( Y ) , f ω ( Z )) ≤ dy n < y n . (1)As x n ≤ max { x n − , x n } = y n , (2)we get y n +1 = max { x n , x n +1 } (1) and (2) ≤ y n .Therefore we have y n +2 = max { x n +1 , x n +2 } (1) ≤ max { dy n , dy n +1 } = dy n and consequently y n − ≤ d n − y and y n ≤ d n − y for every n ∈ N ∗ .Thus the series P n ∈ N ∗ y n is convergent, so the series P n ∈ N ∗ x n is convergent(see (2) and use the comparison test) and consequently lim n →∞ x n = 0. Hence,as h ( F [ n ] S ( Y ) , F [ n ] S ( Z )) = h ( ∪ ω ∈ Λ n ( I ) f ω ( Y ) , ∪ ω ∈ Λ n ( I ) f ω ( Z )) Proposition 2.3 ≤≤ sup ω ∈ Λ n ( I ) h ( f ω ( Y ) , f ω ( Z )) Remark 2.6 ≤ sup ω ∈ Λ n ( I ) δ ( f ω ( Y ) , f ω ( Z )) = x n (3)for every n ∈ N ∗ , we get thatlim n →∞ h ( F [ n ] S ( Y ) , F [ n ] S ( Z )) = 0. (4)In particular, for each Y ∈ K ( X ), considering Z = F S ( Y ) ∈ K ( X ) andtaking into account the comparison test and (3), we infer that the series P n ∈ N ∗ h ( F [ n +1] S ( Y ) , F [ n ] S ( Y )) is convergent. Thus we conclude that the sequence8 F [ n +1] S ( Y )) n ∈ N ∗ is Cauchy and, as ( K ( X ) , h ) is complete (see Theorem 2.4),there exists A Y ∈ K ( X ) such thatlim n →∞ h ( F [ n ] S ( Y ) , A Y ) = 0. (5)In the same manner we can prove that if Z ∈ K ( X ), thenlim n →∞ h ( F [ n ] S ( Z ) , A Z ) = 0. (6)From (4), (5) and (6) we obtain that A Y = A Z def = A for every Y, Z ∈K ( X ). Thus lim n →∞ h ( F [ n ] S ( B ) , A ) = 0,for every B ∈ K ( X ) . ii) For ω ∈ Λ( I ) and Y, Z ∈ K ( X ) we have h ( f [ ω ] n ( Y ) , f [ ω ] n ( Z )) Remark 2.6 ≤ δ ( f [ ω ] n ( Y ) , f [ ω ] n ( Z )) ≤ sup ω ∈ Λ n ( I ) δ ( f ω ( Y ) , f ω ( Z )) = x n for every n ∈ N ∗ , so, as lim n →∞ x n = 0, we deduce thatlim n →∞ δ ( f [ ω ] n ( Y ) , f [ ω ] n ( Z )) = lim n →∞ h ( f [ ω ] n ( Y ) , f [ ω ] n ( Z )) = 0. (7)For Y ∈ K ( X ) we have h ( f [ ω ] n ( Y ) , f [ ω ] n +1 ( Y )) Remark 2.6 ≤ δ ( f [ ω ] n ( Y ) , f [ ω ] n +1 ( Y )) ≤≤ δ ( f [ ω ] n ( Y ) , f [ ω ] n ( F S ( Y ))) ≤ x n ( Y, F S ( Y ))for each n ∈ N ∗ , hence, since -as we have seen in the proof of 1)- the series P n ∈ N ∗ x n ( Y, F S ( Y )) is convergent, using the comparison criterion, we infer thatthe series P n ∈ N ∗ h ( f [ ω ] n ( Y ) , f [ ω ] n +1 ( Y )) is convergent. Thus we conclude thatthe sequence ( f [ ω ] n ( Y )) n ∈ N ∗ is Cauchy and as, ( K ( X ) , h ) is complete (seeTheorem 2.4), there exists A ω ( Y ) ∈ K ( X ) such thatlim n →∞ h ( f [ ω ] n ( Y ) , A ω ( Y )) = 0. (8)9n the same manner we can prove that if Z ∈ K ( X ), then there exists A ω ( Z ) ∈ K ( X ) such thatlim n →∞ h ( f [ ω ] n ( Z ) , A ω ( Z )) = 0. (9)From (7), (8) and (9) we obtain that A ω ( Y ) = A ω ( Z ) def = A ω for each Y, Z ∈ K ( X ). Thus lim n →∞ h ( f [ ω ] n ( B ) , A ω ) = 0, (10)for each B ∈ K ( X ) . Since lim n →∞ diam ( f [ ω ] n ( B )) = 0 (11)for each B ∈ K ( X ) (see (7) for Y = Z = B ), we get that diam ( A ω ) = 0. (12)Indeed, using (10) and (11), we infer that for each ε > n ε ∈ N ∗ such that diam ( f [ ω ] nε ( B )) < ε and h ( f [ ω ] nε ( B ) , A ω ) < ε. Therefore there exists η ∈ (0 , ε ) such that A ω ⊆ N η ( f [ ω ] nε ( B )),so diam ( A ω ) ≤ η + diam ( f [ ω ] nε ( B )) < ε .As ε was arbitrarily chosen, we conclude that diam ( A ω ) = 0.From (12) we conclude that there exists a ω ∈ X such that A ω = { a ω } and, from (10), we get lim n →∞ h ( f [ ω ] n ( B ) , { a ω } ) = 0,for each B ∈ K ( X ) . Note that the above limit is uniform with respect to ω ∈ Λ( I ), i.e.lim n →∞ sup ω ∈ Λ( I ) h ( f [ ω ] n ( B ) , { a ω } ) = 0.10ndeed, h ( f [ ω ] n ( B ) , { a ω } ) ≤ m X k = n h ( f [ ω ] k ( B ) , f [ ω ] k +1 ( B )) + h ( f [ ω ] m +1 ( B ) , { a ω } )for every m, n ∈ N , m ≥ n . By passing to limit as m → ∞ , we get h ( f [ ω ] n ( B ) , { a ω } ) ≤ X k ≥ n h ( f [ ω ] k ( B ) , f [ ω ] k +1 ( B )) Remark 2.6 ≤≤ X k ≥ n δ ( f [ ω ] k ( B ) , f [ ω ] k ( F S ( B ))) = X k ≥ n x k ( B, F S ( B ))for every ω ∈ Λ( I ) and every n ∈ N , sosup ω ∈ Λ( I ) h ( f [ ω ] n ( B ) , { a ω } ) ≤ X k ≥ n x k ( B, F S ( B ))for every n ∈ N . As the series P n x n ( B, F S ( B )) is convergent, we concludethat lim n →∞ sup ω ∈ Λ( I ) h ( f [ ω ] n ( B ) , { a ω } ) = 0.iii) As h ( F [ n ] S ( B ) , { a ω | ω ∈ Λ( I ) } ) == h ( ∪ ω ∈ Λ n ( I ) ∪ α ∈ Λ( I ) f [ ωα ] n ( B ) , ∪ ω ∈ Λ n ( I ) ∪ α ∈ Λ( I ) { a ωα | α ∈ Λ( I ) } ) Proposition 2.3 ≤≤ sup ω ∈ Λ n ( I ) sup α ∈ Λ( I ) h ( f ω ( B ) , { a ωα } ),we have h ( A, { a ω | ω ∈ Λ( I ) } ) ≤ h ( A, F [ n ] S ( B )) + h ( F [ n ] S ( B ) , { a ω | ω ∈ Λ( I ) } ) ≤≤ h ( A, F [ n ] S ( B )) + sup ω ∈ Λ n ( I ) sup α ∈ Λ( I ) h ( f ω ( B ) , { a ωα } ) (13)for all n ∈ N ∗ and B ∈ K ( X ) . Since lim n →∞ h ( F [ n ] S ( B ) , A ) = 0(see i)) and lim n →∞ sup ω ∈ Λ n ( I ) sup α ∈ Λ( I ) h ( f ω ( B ) , { a ωα } ) = 011see ii)), by passing to limit in (13), we obtain that h ( A, { a ω | ω ∈ Λ( I ) } ) = 0,i.e. h ( A, { a ω | ω ∈ Λ( I ) } ) = 0(see Proposition 2.2).Hence A = { a ω | ω ∈ Λ( I ) } .iv) Let us consider ( Y n ) n ∈ N ⊆ K ( X ) and Y ∈ K ( X ) such thatlim n →∞ h ( Y n , Y ) = 0.Using Proposition 2.8 we conclude that H def = Y ∪ ( ∞ ∪ n =0 Y n ) ∈ K ( X ).Hence, as the functions f i are continuous, they are uniformly continuous on H , so for each ε > δ ε > d ( f i ( x ) , f i ( y )) < ε i ∈ I and every x, y ∈ H such that d ( x, y ) < δ ε .For each ε > n ε ∈ N such that h ( Y n , Y ) < δ ε n ∈ N , n ≥ n ε .Let us consider i ∈ I and n ∈ N , n ≥ n ε .Since for every x ∈ Y n ⊆ H there exists y ∈ Y ⊆ H such that d ( x, y ) < d ( x, Y ) + δ ε d ( x, y ) < h ( Y n , Y ) + δ ε < δ ε δ ε δ ε ,so d ( f i ( x ) , f i ( Y )) ≤ d ( f i ( x ) , f i ( y )) < ε d ( f i ( Y n ) , f i ( Y )) ≤ ε d ( f i ( Y ) , f i ( Y n )) ≤ ε h ( f i ( Y ) , f i ( Y n )) ≤ ε h ( F S ( Y n ) , F S ( Y )) = h ( ∪ i ∈ I f i ( Y n ) , ∪ i ∈ I f i ( Y )) Proposition 2.3 ≤≤ max i ∈ I h ( f i ( Y n ) , f i ( Y )) ≤ ε < ε .Thus for each ε > n ε ∈ N such that h ( F S ( Y n ) , F S ( Y )) < ε for every n ∈ N , n ≥ n ε , i.e.lim n →∞ h ( F S ( Y n ) , F S ( Y )) = 0 . v) Since lim n →∞ h ( F [ n ] S ( A ) , A ) = 0 (see i) for B = A ), using 4) for Y n = F [ n ] S ( A ) ∈ K ( X ) and Y = A ∈ K ( X ), we obtain thatlim n →∞ h ( F [ n +1] S ( A ) , F S ( A )) = 0. (14)Using i), for B = F S ( A ), we infer thatlim n →∞ h ( F [ n +1] S ( A ) , A ) = 0. (15)From (14) and (15) we conclude that F S ( A ) = A .Moreover, if for some A ∈ K ( X ) we have F S ( A ) = A , then F [ n ] S ( A ) = A for each n ∈ N , so lim n →∞ h ( F [ n ] S ( A ) , A ) = 0. Since, according to i), wehave lim n →∞ h ( F [ n ] S ( A ) , A ) = 0, we conclude that A = A . (cid:3) F [ n ] S ( B )) n ∈ N , where B ∈ K ( X ), we have (from the proof of i), the followinginequality: h ( F [ n ] S ( B ) , A ) ≤ d [ n ] − d ( x ( B, F S ( B )) + x ( B, F S ( B ))),for every n ∈ N . Remark 3.3.
By taking in the above Theorem a set I with one element,we get that A has exactly one element which is the fixed point of the convexcontraction that can be approximated by means of Picard iteration. Conse-quently we obtain Istr˘at¸escu’s fixed point theorem for convex contractions. Proposition 3.4 . Let S = (( X, d ) , ( f i ) i ∈ I ) be an iterated function systemconsisting of convex contractions. Then, in the framework of Theorem 3.2,we have lim n →∞ diam ( A [ ω ] n ) = 0 for every ω ∈ Λ( I ). Proof . Take B = A in (11) from the proof of Theorem 3.2. (cid:3) Proposition 3.5.
Let S = (( X, d ) , ( f i ) i ∈ I ) be an iterated function systemconsisting of convex contractions. Then, in the framework of Theorem 3.2,we have ∩ n ∈ N A [ ω ] n = { a ω } for every ω ∈ Λ( I ). Proof . From F S ( A ) = A we infer that A [ ω ] n +1 ⊆ A [ ω ] n for every n ∈ N . Then lim n →∞ h ( A [ ω ] n , ∩ n ∈ N A [ ω ] n ) = 0(see Theorem 1.14 from [23]) and taking into account Theorem 3.2, ii), weconclude that ∩ n ∈ N A [ ω ] n = { a ω } . (cid:3) Using the above two Propositions, the same arguments as the ones usedin the proof of Theorem 4.1 from [18] give us the following:14 heorem 3.6.
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