A countable fractal interpolation scheme involving Rakotch contractions
aa r X i v : . [ m a t h . D S ] F e b A countable fractal interpolation schemeinvolving Rakotch contractions
Cristina Maria PACURAR
Faculty of Mathematics and Computer Science,Transilvania University of Bra¸sov, Bulevardul Eroilor 29, Bra¸sovemail:[email protected]
Abstract
The main result of this paper states that for a given countablesystem of data ∆, there exists a countable iterated function systemconsisting of Rakotch contractions, such that its attractor is the graphof a fractal interpolation function corresponding to ∆. In this way,on the one hand, we generalize a result due to N. Secelean (see
Thefractal interpolation for countable systems of data , Univ. Beograd.Publ. Elektrotehn. Fak. Ser. Mat., (2003), 11–19) by consideringcountable systems consisting of Rakotch contractions rather than Ba-nach contractions. On the other hand, we generalize a result due toS. Ri (see A new idea to construct the fractal interpolation function ,Indag. Math., (2018), 962-971) by considering countable (ratherthan finite) systems consisting of Rakotch contractions. Some exem-plifications are provided. Key words: fractal interpolation function, countable iterated functionsystem, Rakotch contractions, Matkowski contractions
AMS 2010 Subject Classification : 28A80, 41A05, 58F12
Fractal interpolation is a special method for constructing a continuous func-tion which passes through all of the points of a given system of points. For agiven set of data { ( x i , y i ) ∈ I × R , i = { , . . . , N − }} , where I = [ x , x N ] is a1losed real interval and x i < x i +1 for all i = { , . . . , N } , the fractal interpola-tion function (FIF) is a continuous function f : I → R which interpolates thegiven data such that its graph is the attractor of an iterated function system,a notion due to Hutchinson (see [10]). Fractal interpolation functions wereintroduced by Barnsley (see [1], [2]) and have been intensively studied eversince.The main difference between fractal interpolation and other types of in-terpolation techniques is that the interpolation function obtained is not nec-essarily differentiable at any point, thus, being closer to natural world phe-nomena and providing a more powerful tool in fitting real-world data. Acomprehensive survey on FIFs is that of Navascu´es et al. (see [17]).In the development of the theory of FIFs, there have been many general-izations of Barnsley’s result. Among these directions of research, we mentionthe hidden variable FIFs, introduced by Barnsley for systems of data whichare not self-referential (see [1], [3], [6], [4]) and the extension to higher di-mensional cases of FIFs (see [15], [5], [7], [33], [32], [21], [23], [30]).Another direction of interest regarding FIFs is related to the fixed pointresult which guarantees the existence of the FIF. While most of the extensionsrely on the Banach fixed point theorem (following Barnsley’s results) in orderto prove the existence of a FIF, there have been recent results which usedifferent fixed point results. In this respect, Ri has used Rakotch contractionsto obtain new results (see [20]), Kim et al. resorted to Geraghty contractions(see [14]) and Ri and Drakopoulos extended the results to surfaces (see [22]).A different direction related to FIFs is to extend the finite set of pointswhich are interpolated to a countable set. Thus, countable fractal inter-polation has been introduced by Secelean (see [25]) based on countable it-erated function systems (see [8], [24], [28]). In [25], Secelean proved theexistence of the FIF for a countable iterated function system for a set of data∆ = { ( x n , y n ) ∈ I × R , n ≥ } where ( x n ) n ≥ is a strictly increasing boundedsequence and ( y n ) n ≥ is a convergent sequence. These results were extendedby Gowrisankar and Uthayakumar (see [9]) for systems of data where ( x n ) n ≥ is a monotone bounded sequence and ( y n ) n ≥ is a bounded sequence. Thisdirection has been further developed in several papers (see [27], [26], [31]).By combining these two lines of research initiated by Secelean and Ri,in this paper we present a new fractal interpolation scheme for countablesystems of data and countable iterated function systems composed of Rakotchcontractions. Thus, the present paper extends the results from [25] and[20]. Although the techniques that we used in our proofs are similar to2hose from [25] and [20], the countable iterated function systems for Rakotchcontractions requires highly more effort and subtleties. Let (
X, d ) be a compact metric space.We denote by P cp ( X ) the set of all non-empty compact subsets of X .We consider the Hausdorff metric h : P cp ( X ) × P cp ( X ) → [0 , ∞ ), definedas h ( A, B ) = max { sup x ∈ A inf y ∈ B d ( x, y ) , sup x ∈ B inf y ∈ A d ( x, y ) } for A, B ∈ P cp ( X ).For A ⊂ X , by diam ( A ) we denote the diameter of A . Definition 2.1.
Given a metric space ( X, d ) , an operator f : X → X iscalled a Picard operator if f has a unique fixed point x ∗ ∈ X and lim n →∞ f [ n ] ( x ) = x ∗ , for every x ∈ X , where by f [ n ] we mean the composition of f with itselfn-times. Definition 2.2 (see Matkowski [16], Rakotch [18], Jachymski [11], Rhoades[19]) . i) Let ϕ : [0 , ∞ ) → [0 , ∞ ) and ( X, d ) a metric space. A map f : X → X is called a ϕ -contraction if d ( f ( x ) , f ( y )) ≤ ϕ ( d ( x, y )) , for all x, y ∈ X .ii) Given a metric space ( X, d ) , a map f : X → X is called Matkowskicontraction if it is a ϕ -contraction, where ϕ : [0 , ∞ ) → [0 , ∞ ) is non-decreasing and lim n →∞ ϕ [ n ] ( t ) = 0 for all t > . ii) Given a metric space ( X, d ) , a map f : X → X is called Rakotchcontraction if it is a ϕ -contraction, where ϕ : [0 , ∞ ) → [0 , ∞ ) is suchthat the function α : (0 , ∞ ) → (0 , ∞ ) , given by α ( t ) = ϕ ( t ) t for every t > is non-increasing and α ( t ) < for every t ∈ (0 , ∞ ) . Remark 2.1 (see Remark 2.2 from [20], [12] and [6]) . Given a metric space ( X, d ) , a map f : X → X is a Rakotch contraction if and only if it is a ϕ -contraction for some non-decreasing ϕ : [0 , ∞ ) → [0 , ∞ ) such that thefunction α : (0 , ∞ ) → (0 , ∞ ) , given by α ( t ) = ϕ ( t ) t for every t ∈ (0 , ∞ ) isnon-increasing and α ( t ) < for every t ∈ (0 , ∞ ) . Remark 2.2. i) Each Banach contraction is a Rakotch contraction (fora function ϕ given by ϕ ( t ) = αt for every t > , where ϕ ∈ [0 , ).ii) Each Rakotch contraction is a Matkowski contraction. Theorem 2.1 (see Matkowski [16]) . Given a complete metric space ( X, d ) ,if f : X → X is a Matkowski contraction, then f has a unique fixed point x ∗ ∈ X and lim n →∞ f [ n ] ( x ) = x ∗ for each x ∈ X . Remark 2.3.
The above theorem says that each Matkowski contraction ona complete metric space is a Picard operator.
Definition 2.3.
Let ( X, d ) be a compact metric space and f n : X → X becontinuous functions for every n ∈ N . The pair S = (( X, d ) , ( f n ) n ≥ ) iscalled a countable iterated function system (for short, CIFS). The fractal operator associated to the CIFS S = (( X, d ) , ( f n ) n ≥ ) is thefunction F S : P cp ( X ) → P cp ( X ), defined as F S ( K ) = [ n ≥ f n ( K )for every K ∈ P cp ( X ).If the fractal operator F S is Picard, then we say that the CIFS S hasattractor and the fixed point of F S is called the attractor of the CIFS S . Theorem 2.2 (see Secelean [29] Theorem 3.7) . If the constitutive functions f n of the CIFS S = (( X, d ) , ( f n ) n ≥ ) are Matkowski contractions, for every n ≥ , then the fractal operator F S is a Matkowski contraction. In particular,the CIFS has attractor. .4 Countable systems of data and interpolation func-tions Given a compact metric space (
Y, d ), let us consider the countable system ofpoints ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } . (1)If the sequence ( x n ) n ≥ is strictly increasing and bounded, and the se-quence ( y n ) n ≥ is convergent, then the system of points defined in relation(1) is called a countable system of data.We set the notations a = x , b = lim n →∞ x n , m = y and M = lim n →∞ y n . Definition 2.4.
In the above mentioned framework, an interpolation func-tion corresponding to the countable system of data ∆ is a continuous function f : [ a, b ] → Y , such that f ( x n ) = y n , for each n ≥ . Note that f ( b ) = f ( lim n →∞ x n ) f continuous = lim n →∞ f ( x n )= lim n →∞ y n = M. ( f n ) n associated to ∆ Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable system of data.For each n ≥
1, let l n : [ a, b ] → [ x n − , x n ] be a homeomorphism for whichthere exists L n ∈ [0 ,
1) such thati) | l n ( x ) − l n ( x ′ ) | ≤ L n | x − x ′ | for every x, x ′ ∈ [ a, b ];ii) l n ( a ) = x n − and l n ( b ) = x n ;5ii) sup n ≥ L n < . For each n ≥
1, let W n : [ a, b ] × Y → Y be a continuous function suchthatj) W n ( a, m ) = y n − and W n ( b, M ) = y n ;jj) lim n →∞ diam ( Im W n ) = 0.For n ≥
1, we define f n : [ a, b ] × Y → [ a, b ] × Y as f n ( x, y ) = ( l n ( x ) , W n ( x, y )) , for every x ∈ [ a, b ] and y ∈ Y . Let us consider C ([ a, b ]) = { f : [ a, b ] → Y | f ( a ) = m and f ( b ) = M, f - continuous } endowed with the uniform metric d C ([ a,b ]) . Remark 2.4.
The space ( C ([ a, b ]) , d C ([ a,b ]) ) is a complete metric space. Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable system of data.For f ∈ C ([ a, b ]), we consider the function T f : [ a, b ] → Y given as follows: T f ( x ) = ( W n ( l − n ( x ) , f ( l − n ( x ))) , if x ∈ [ x n − , x n ] M, if x = b. Claim 1. T f is well defined. Indeed, since x n ∈ [ x n − , x n ], by definition of T f , on the one hand wehave T f ( x n ) = W n ( l − n ( x n ) , f ( l − n ( x n )))= W n ( b, f ( b ))= W n ( b, M )= y n x n ∈ [ x n , x n +1 ], we have T f ( x n ) = W n +1 ( l − n +1 ( x n ) , f ( l − n +1 ( x n )))= W n +1 ( a, f ( a ))= W n +1 ( a, m )= y n , for all n ≥ Claim 2. T f ∈ C ([ a, b ]) . Indeed, on the one hand we have T f ( a ) = W ( l − ( a ) , f ( l − ( a )))= W ( a, f ( a ))= W ( a, m )= y = m and by definition T f ( b ) = M. On the other hand, since W n are continuous, it is clear that T f is continu-ous on ( x n − , x n ) for all n ≥
1. We need to prove that T f is right continuousat a , continuous at x n for all n ≥ b .For n ≥
1, we havelim x ց x n T f ( x ) = lim x ց x n W n +1 ( l − n +1 ( x ) , f ( l − n +1 ( x )))= W n +1 ( a, f ( a ))= W n +1 ( a, m )= y n = T f ( x n )and lim x ր x n T f ( x ) = lim x ր x n W n ( l − n ( x ) , f ( l − n ( x )))= W n ( b, f ( b ))= W n ( b, M )= y n = T f ( x n ) , T f is continuous on ( a, b ).Since lim x ց a T f ( x ) = lim x ց a W ( l − ( x ) , f ( l − ( x )))= W ( a, f ( a ))= W ( a, m )= y = T f ( a ) , we infer that T f is right continuous at a .Now we prove that T f is left continuous at b .Let ε > n →∞ y n = M and lim n →∞ diam ( Im W n ) = 0, there exists n ε ∈ N suchthat d ( M − y n ) < ε diam ( W n ) < ε n ≥ n ≥ n ε .For x ∈ ( x n ε , b ), as ( x n ) n is a strictly increasing sequence and lim n →∞ x n = b ,there exists n x ∈ N , n x ≥ n ε such that x ∈ [ x n x , x n x +1 ], so we have d ( T f ( x ) , T f ( b )) ≤ d ( T f ( x ) , T f ( x n x )) + d ( y n x , M )= d ( W n x +1 ( l − n x +1 ( x ) , f ( l − n x +1 ( x ))) , W n x +1 ( l − n x +1 ( x n x ) , f ( l − n x +1 ( x n x ))))+ d ( M, y n x ) ≤ diam ( ImW n x +1 ) + d ( M, y n x ) ( ) & ( ) ≤ ε ε ε. Hence, lim x ր b T f ( x ) = T f ( b ) , i.e. T f is left continuous at b , which concludes the proof that T f is continuouson [ a, b ]. 8hus, from Claim 1 and Claim 2, the operator T : C ([ a, b ]) → C ([ a, b ]),defined as T ( f ) = T f for every f ∈ C ([ a, b ]) is well defined. Theorem 3.1.
Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable systemof data. If the functions W n are Matkowski contractions with respect to thesecond argument, i.e. there exists a non-decreasing function ϕ : [0 , ∞ ) → [0 , ∞ ) such that lim n →∞ ϕ n ( t ) = 0 for all t > and d ( W n (( x, y )) , W n (( x, y ′ ))) ≤ ϕ ( d ( y, y ′ )) (4) for all x ∈ [ a, b ] and y, y ′ ∈ Y , then T is a Matkowski contraction.Proof. Let g, h ∈ C ([ a, b ]).It is obvious that0 = d ( M, M ) = d ( T g ( b ) , T h ( b )) ≤ ϕ ( d C ([ a,b ]) ( g, h )) . (5)Let x ∈ [ a, b ) and n ≥ x ∈ [ x n − , x n ].Then, we have d ( T g ( x ) , T h ( x )) = d ( W n ( l − n ( x ) , g ( l − n ( x ))) , W n ( l − n ( x ) , h ( l − n ( x )))) ( ) ≤ ϕ ( d ( g ( l − n ( x )) , h ( l − n ( x )))) ≤ ϕ ( sup u ∈ [ a,b ] d ( g ( u ) , h ( u )))= ϕ ( d C ([ a,b ]) ( g, h )) . (6)Via (5) and (6), we get d C ([ a,b ]) ( T g, T h ) = sup x ∈ [ a,b ] d ( T g ( x ) , T h ( x )) ≤ ϕ ( d C ([ a,b ]) ( g, h )) , which concludes our proof. 9 heorem 3.2. Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable system ofdata such that W n are Lipschitz with respect to the first variable and Rakotchcontractions in the second variable, i.e. there exists L > , and a non-decreasing function ϕ : [0 , ∞ ) → [0 , ∞ ) , satisfying ϕ ( t ) t < for all t > and t → ϕ ( t ) t is non-increasing, such that d ( W n (( x, y )) , W n (( x ′ , y ′ ))) ≤ L | x − x ′ | + ϕ ( d ( y, y ′ )) for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , n ≥ .Then, f n are Rakotch contractions with respect to the metric d θ describedby d θ (( x, y ) , ( x ′ , y ′ )) := | x − x ′ | + θd ( y, y ′ ) for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , where θ = − sup n ≥ L n L +1) ∈ (0 , .Proof. For all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , ( x, y ) = ( x ′ , y ′ ) and n ≥
1, we have d θ ( f n ( x, y ) , f n ( x ′ , y ′ )) = d θ (( l n ( x ) , W n ( x, y )) , ( l n ( x ′ ) , W n ( x ′ , y ′ )))= | l n ( x ) − l n ( x ′ ) | + θd ( W n ( x, y ) , W n ( x ′ , y ′ )) ≤ L n | x − x ′ | + θ ( L | x − x ′ | + ϕ ( d ( y, y ′ )))= ( L n + θL ) | x − x ′ | + θ ϕ ( d ( y, y ′ )) | x − x ′ | + d ( y, y ′ ) ( | x − x ′ | + d ( y, y ′ )) ϕ non-decreasing ≤ ( L n + θL ) | x − x ′ | + θ ϕ ( | x − x ′ | + d ( y, y ′ )) | x − x ′ | + d ( y, y ′ ) ( | x − x ′ | + d ( y, y ′ ))= (cid:20) L n + θ (cid:18) L + ϕ ( | x − x ′ | + d ( y, y ′ )) | x − x ′ | + d ( y, y ′ ) (cid:19)(cid:21) | x − x ′ | + θ ϕ ( | x − x ′ | + d ( y, y ′ )) | x − x ′ | + d ( y, y ′ ) d ( y, y ′ ) t → ϕ ( t ) t non-increasing ≤ (cid:20) L n + θ (cid:18) L + ϕ ( | x − x ′ | + d ( y, y ′ )) | x − x ′ | + d ( y, y ′ ) (cid:19)(cid:21) | x − x ′ | + θ ϕ ( | x − x ′ | + θd ( y, y ′ )) | x − x ′ | + θd ( y, y ′ ) d ( y, y ′ )10 ( t ) t < , ( ∀ ) t> ≤ [sup n ≥ L n + θ ( L + 1)] | x − x ′ | + θ ϕ ( | x − x ′ | + θd ( y, y ′ )) | x − x ′ | + θd ( y, y ′ ) d ( y, y ′ ) . Thus, we get d θ ( f n ( x, y ) , f n ( x ′ , y ′ )) ≤ max (cid:26) sup n ≥ L n + θ ( L + 1) , ϕ ( d θ (( x, y ) , ( x ′ , y ′ ))) d θ (( x, y ) , ( x ′ , y ′ )) (cid:27) d θ (( x, y ) , ( x ′ , y ′ ))for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , ( x, y ) = ( x ′ , y ′ ).Let us consider the map α : (0 , ∞ ) → (0 ,
1) defined as α ( t ) = max (cid:26) sup n ≥ L n + θ ( L + 1) , ϕ ( t ) t (cid:27) for all t >
0, and n ≥ ϕ ( t ) t < t > n ≥ L n + θ ( L + 1) = sup n ≥ L n + 1 − sup n ≥ L n L + 1) ( L + 1)= sup n ≥ L n + 1 − sup n ≥ L n < , it is clear that α ( t ) ∈ (0 ,
1) for every t > t → ϕ ( t ) t is non-increasing, we infer that α is non-increasing.Thus, considering ψ : [0 , ∞ ) → [0 , ∞ ), given by ψ ( t ) = tα ( t ) for all t ≥ d θ ( f n ( x, y ) , f n ( x ′ , y ′ )) ≤ α ( d θ (( x, y ) , ( x ′ , y ′ ))) d θ (( x, y ) , ( x ′ , y ′ ))= ψ ( d θ (( x, y ) , ( x ′ , y ′ ))) , for every ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y .Since ψ ( t ) t = α ( t ) < t > α is non-increasing, we concludethat f n are Rakotch contractions with respect to d θ .11et ( x n , y n ) ⊆ [ a, b ] × Y . Since Y is compact, there exists a subsequence( y n k ) n of ( y n ) n and y ∈ Y such that lim k →∞ y n k = y . Since [ a, b ] is compact, thereexists a subsequence ( x n kp ) p of ( x n k ) k and x ∈ [ a, b ] such that lim p →∞ x n kp = x .We have lim p →∞ d θ ( x n kp , y n kp ) = ( x, y ) ∈ [ a, b ] × Y . Thus, ([ a, b ] × Y, d θ ) iscompact, and from Theorem 2.2 and Remark 2.2, we get the following: Remark 3.1.
Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable system ofdata. If the functions f n are Rakotch contractions with respect to the metric d θ (in particular, if the conditions stated in Theorem 3.2 are satisfied), thenthe CIFS S = (([ a, b ] × Y, d θ ) , ( f n ) n ≥ ) has attractor, so there exists a unique A S ∈ P cp ([ a, b ] × Y ) such that F S ( A S ) = A S . Theorem 3.3.
Let ∆ = { ( x n , y n ) ∈ R × Y, n ≥ } be a countable system ofdata such that W n satisfy the hypothesis from Theorem 3.2. Then there existsan interpolation function f ∗ corresponding to ∆ such that its graph is the at-tractor of the countable iterated function system S = (([ a, b ] × Y, d θ ) , ( f n ) n ≥ ) .Proof. Since W n are Rakotch contractions with respect to the second argu-ment, from Theorem 3.1 and Remark 2.2, we get that T is a Matkowskicontraction. Thus, it has a unique fixed point f ∗ ∈ C ([ a, b ]). Hence, T f ∗ ( x ) = f ∗ ( x )for all x ∈ [ a, b ].For n ≥ x ∈ [ x n − , x n ], we get T f ∗ ( x ) = W n ( l − n ( x ) , f ∗ ( l − n ( x ))) = f ∗ ( x ) , and T f ( b ) = M = f ∗ ( b ) . As f ∗ ( x n ) = T f ∗ ( x n )= W n ( l − n ( x n ) , f ∗ ( l − n ( x n )))= W n ( b, f ∗ ( b ))= W n ( b, M )= y n n ≥
1, we conclude that f ∗ is an interpolation function correspond-ing to ∆.Let G be the graph of f ∗ . Claim 1. [ n ≥ f n ( G ) ⊆ G. Justification of Claim 1
We have f ∗ ( l n ( x )) = T f ∗ ( l n ( x ))= W n ( l − n ( l n ( x )) , f ∗ ( l − n ( l n ( x ))))= W n ( x, f ∗ ( x )) , (7)for every x ∈ [ a, b ].Thus, we get f n ( x, f ∗ ( x )) = ( l n ( x ) , W n ( x, f ∗ ( x ))) ( ) = ( l n ( x ) , f ∗ ( l n ( x ))) ∈ G (8)for every n ≥ x ∈ [ a, b ], so [ n ≥ f n ( G ) ⊆ G. Since G is closed, we have [ n ≥ f n ( G ) ⊆ G. Claim 2. G ⊆ [ n ≥ f n ( G ) Justification of Claim 2 If x ∈ [ a, b ) then there exists n ≥ x ∈ [ x n − , x n ], so we have( x, f ∗ ( x )) = ( x, f ∗ ( l n ( l − n ( x )))) ( ) = ( l n ( l − n ( x )) , W n ( l − n ( x ) , f ∗ ( l − n ( x )))) ( ) = f n ( l − n ( x ) , f ∗ ( l − n ( x ))) ∈ f n ( G ) ⊆ [ n ≥ f n ( G ) (9)13oreover, we have ( b, f ∗ ( b )) = lim n →∞ ( x n , f ∗ ( x n )) and since ( x n , f ∗ ( x n )) ( ) ∈ f n ( G ), for every n ≥ b, f ∗ ( b )) ∈ [ n ≥ f n ( G ), so G ⊆ [ n ≥ f n ( G ) . Thus, from the above two claims, we get the equality [ n ≥ f n ( G ) = G, i.e. F S ( G ) = G. As d and d θ are equivalent, where d (( x, y ) , ( x ′ , y ′ )) = | x − x ′ | + d (( y, y ′ ))for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , we have G ∈ P cp ([ a, b ] × Y ) and via Remark3.1 we obtain G = A S , i.e. A S = { ( x, f ∗ ( x )) | x ∈ [ a, b ] } . Theorem 3.4.
Under the framework of Theorem 3.3, we have lim n →∞ T [ n ] ( f ) = f ∗ for every f ∈ C ([ a, b ]) .Proof. Claim 1. G T ( f ) = F S ( G f ) , for every f ∈ C ([ a, b ]) .Justification of Claim For the very beginning, let us note that [ n ≥ f n ( G f ) ⊆ G T ( f ) , (10)14or every f ∈ C ([ a, b ]).Indeed, if y ∈ S n ≥ f n ( G f ), then there exists n y ≥
1, such that y ∈ f n y ( G f ), so there exists x ∈ [ a, b ] having the property that y = f n y ( x, f ( x )).Hence y = ( l n y ( x ) , W n y ( x, f ( x )))= ( l n y ( x ) , W n y ( l − n y ( l n y ( x )) , f ( l − n y ( l n y ( x ))))) l ny ( x ) ∈ [ x ny − ,x ny ] = ( l n y ( x ) , T ( f )( l n y ( x ))) ∈ G T ( f ) , and the proof of (10) is completed.We have F S ( G f ) ⊆ G T ( f ) (11)for every f ∈ C ([ a, b ]).Indeed, if y ∈ F S ( G f ) = S n ≥ f n ( G f ), then there exists ( z k ) k ≥ ⊆ S n ≥ f n ( G f )such that y = lim k →∞ z k .Hence, according to (10), there exists u k ∈ [ a, b ] with the property that y = lim k →∞ ( u k , T ( f )( u k )) ∈ G T ( f ) T ( f ) continuous = G T ( f ) .We have G T f ⊆ F S ( G f ) (12)for every f ∈ C ([ a, b ]).Indeed, let us consider y ∈ G T ( f ) . Then, there exists x ∈ [ a, b ], such that y = ( x, T ( f )( x )). First, let us note that if x ∈ [ a, b ), there exists n x ≥ x ∈ [ x n x − , x n x ], so one can find u x ∈ [ a, b ] suchthat x = l n x ( u x ).Thus, y = ( l n x ( u x ) , T ( f )( l n x ( u x ))) l nx ( x ) ∈ [ x nx − ,x nx ] = ( l n x ( u x ) , W n x ( l − n x ( l n x ( u x )) , f ( l − n x ( l n x ( u x )))))= ( l n x ( u x ) , W n x ( u x , f ( u x )))= f n x ( u x , f ( u x )) ∈ f n x ( G f ) ⊆ [ n ≥ f n ( G f ) ⊆ [ n ≥ f n ( G f ) = F S ( G f ) . Consequently, ( x, T ( f )( x )) ∈ F S ( G f ) (13)15or every x ∈ [ a, b ) and every f ∈ C ([ a, b ]).In addition,( b, T ( f )( b )) = lim x ր b ( x, T ( f )( x )) ∈ F S ( G f ) G f compact = F S ( G f ) (14)for every f ∈ C ([ a, b ]).Relations (13) and (14) ensure (12).Taking into account (11) and (12), the justification of the Claim is com-pleted.Finally, the Claim implies - via the mathematical induction method - that F [ n ] S ( G f ) = G T [ n ] ( f ) (15)for every n ≥ f ∈ C ([ a, b ]).As G f ∈ P cp ([ a, b ] × Y ), we havelim n →∞ F [ n ] S ( G f ) Remark 3.1 = A S so, via (15) and Theorem 3.3, we getlim n →∞ G T [ n ] ( f ) = G f ∗ . We can choose l n ( x ) = x n − x n − b − a x + bx n − − ax n b − a (16)for every x ∈ [ a, b ].It is immediate that | l n ( x ) − l n ( x ′ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) x n − x n − b − a x + bx n − − ax n b − a − x n − x n − b − a x ′ − bx n − − ax n b − a (cid:12)(cid:12)(cid:12)(cid:12) ≤ x n − x n − b − a | x − x ′ | x n − x n − b − a ∈ [0 ,
1) and sup n ≥ x n − x n − b − a <
1. Also, we have l n ( a ) = x n − x n − b − a a + bx n − − ax n b − a = x n − l n ( b ) = x n − x n − b − a b + bx n − − ax n b − a = x n for every n ≥ Y is a compact real interval, we can choose W n in the followingtwo ways: A. W n ( x, y ) = c n x + d n y + g n , (17)where c n = y n − y n − b − a − d n M − mb − a ,g n = by n − − ay n b − a − d n bm − aMb − a and d n ∈ [0 ,
1) such that lim n →∞ d n = 0.Indeed, on the one hand we have W n ( a, m ) = (cid:20) y n − y n − b − a − d n M − mb − a (cid:21) a + d n m + by n − − ay n b − a − d n bm − aMb − a = ay n − ay n − + by n − − ay n b − a − d n aM − am + bm − aM − m ( b − a ) b − a = y n − and W n ( b, M ) = (cid:20) y n − y n − b − a − d n M − mb − a (cid:21) b + d n M + by n − − ay n b − a − d n bm − aMb − a = by n − by n − + by n − − ay n b − a − d n bM − bm + bm − aM − M ( b − a ) b − a = y n for every n ≥
1. 17n the other hand, we have0 ≤ diam ( Im W n ) = sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y | W n ( x, y ) − W n ( x ′ , y ′ ) | = sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y | ( c n x + d n y + g n ) − ( c n x ′ + d n y ′ + g n ) |≤ sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y | c n || x − x ′ | + | d n || y − y ′ |≤ ( b − a ) c n + diam ( Y ) d n for all n ≥ n →∞ c n = lim n →∞ d n = 0, we get lim n →∞ diam ( Im W n ) = 0.Note that | W n ( x, y ) − W n ( x ′ , y ′ ) | = | c n x + d n y + g n − ( c n x ′ + d n y ′ + g n ) |≤ | c n | | x − x ′ | + d n | y − y ′ |≤ L | x − x ′ | + c | y − y ′ | for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , where L = sup n ≥ | c n | ∈ R and c = sup n ≥ | d n | ∈ [0 , W n are Banach contractions on the second variable, so they areRakotch contractions on the second variable for the comparison function ϕ given by ϕ ( t ) = c · t for every t ≥ B. W n ( x, y ) = c n x + y ny + g n , (18)where c n = y n − y n − b − a − b − a (cid:18) M nM − m nm (cid:19) and g n = y n − − a y n − y n − b − a + ab − a M nM − bb − a m nm . Indeed, on the one hand we have W n ( a, m ) = (cid:20) y n − y n − b − a − b − a (cid:18) M nM − m nm (cid:19)(cid:21) a + m nm + y n − − a y n − y n − b − a + ab − a M nM − bb − a m nm = y n − W n ( b, M ) = (cid:20) y n − y n − b − a − b − a (cid:18) M nM − m nm (cid:19)(cid:21) b + M nM + y n − − a y n − y n − b − a + ab − a M nM − bb − a m nm = y n , for every n ≥ ≤ diam ( Im W n ) = sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y | W n ( x, y ) − W n ( x ′ , y ′ ) | = sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) c n x + y ny + g n (cid:19) − (cid:18) c n x ′ + y ′ ny ′ + g n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup ( x,y ) , ( x ′ ,y ′ ) ∈ [ a,b ] × Y (cid:18) | c n || x − x ′ | + | y − y ′ | ny )(1 + ny ′ ) (cid:19) ≤ ( b − a ) c n + diam ( Y ) 1(1 + n inf Y ) for every n ≥ n →∞ c n = lim n →∞ n inf y ) = 0, we get lim n →∞ diam ( Im W n ) = 0.Note that if Y ⊆ [0 , ∞ ), we have | W n ( x, y ) − W n ( x ′ , y ′ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) c n x + y ny + g n (cid:19) − (cid:18) c n x ′ + y ′ ny ′ + g n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | c n | · | x − x ′ | + | y − y ′ | (1 + ny )(1 + ny ′ ) ≤ L | x − x ′ | + | y − y ′ | | y − y ′ | = L | x − x ′ | + ϕ ( | y − y ′ | )for all ( x, y ) , ( x ′ , y ′ ) ∈ [ a, b ] × Y , where L = sup n ≥ | c n | ∈ R and the comparisonfunction ϕ is given by ϕ ( t ) = t t .Hence, W n are Rakotch contractions on the second variable, but they arenot Banach contractions on the second variable.19n the particular case where Y is a compact real interval we can choose f n in the following two ways: A. f n ( x, y ) = (cid:18) x n − x n − b − a x + bx n − − ax n b − a , (cid:18) y n − y n − b − a − d n M − mb − a (cid:19) x + d n y + by n − − ay n b − a − d n bm − aMb − a (cid:19) . B. f n ( x, y ) = (cid:18) x n − x n − b − a x + bx n − − ax n b − a , (cid:18) y n − y n − b − a − b − a (cid:18) M nM − m nm (cid:19)(cid:19) x + y ny + y n − − a y n − y n − b − a + ab − a M nM − bb − a m nm (cid:19) . The above considerations show that our result is a genuine generalizationof Secelean’s and Ri’s results.
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