A natural selection of a graphic contraction transformation in fuzzy metric spaces

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1 Available online a J. Nonlinear Sci. Appl., (208), Research Aricle Journal Homepage: A naural selecion of a graphic conracion ransformaion in fuzzy meric spaces Hanan Alolaiyana,, Naeem Saleemb, Mujahid Abbasc,d a Deparmen of Mahemaics, King Saud Universiy, Saudi Arabia. b Deparmen of Mahemaics, Universiy of Managemen and Technology, Lahore, Pakisan. c Deparmen of Mahemaics, Governmen College Universiy, Lahore, Pakisan. d Deparmen of Mahemaics, King Abdulaziz Universiy, P. O. Box 80203, Jeddah 2589, Saudi Arabia. Communicaed by C. Vero Absrac In his paper, we sudy sufficien condiions o find a verex v of a graph such ha T v is a erminal verex of a pah which sars from v, where T is a self graphic conracion ransformaion defined on he se of verices. Some examples are presened o suppor he resuls proved herein. Our resuls widen he scope of various resuls in he exising lieraure. Keywords: Graphic conracion, fuzzy meric space, naural selecion. 200 MSC: 47H0, 47H04, 47H07, 54H25, 54C60. c 208 All righs reserved.. Inroducion and preliminaries Zadeh [20] inroduced he noion of fuzzy ses, a new way o represen vagueness and uncerainies in daily life. Kramosil and Michalek [4] inroduced he noion of a fuzzy meric by using coninuous norms, which generalizes he concep of a probabilisic meric space o fuzzy siuaion. Moreover, George and Veeramani ([5, 6]) modified he concep of fuzzy meric spaces and obained a Hausdorff opology for his kind of fuzzy meric spaces. Romaguera [7] inroduced Hausdorff fuzzy meric on a se of nonempy compac subses of a fuzzy meric space. In he sequel, he leers N, R+, and R will denoe he se of naural numbers, he se of posiive real numbers, and he se of real numbers, respecively. Following definiions and known resuls will be needed in he sequel. Corresponding auhor addresses: holayan@ksu.edu.sa (Hanan Alolaiyan), naeem.saleem2@gmail.com (Naeem Saleem), mujahid.abbas@up.ac.za (Mujahid Abbas) doi: /jnsa Received: Revised: Acceped:

2 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), Definiion. ([9]). A binary operaion : [0, ] 2 [0, ] is called a coninuous -norm if () is associaive and commuaive; (2) : [0, ] 2 [0, ] is coninuous (i is coninuous as a mapping under he usual opology on [0, ] 2 ); (3) a = a for all a [0, ]; (4) a b c d whenever a c and b d. Some basic examples of coninuous -norms are (minimum -norm), ( produc -norm), and L (Lukasiewicz -norm), where, for all a, b [0, ], a b = min{a, b}, a b = ab, a L b = max{a + b, 0}. I is easy o check ha L. In fac for all coninuous -norm. Definiion.2 ([5, 6]). Le X be a nonempy se and a coninuous -norm. A fuzzy se M on X X (0, ) is said o be a fuzzy meric on X if for any x, y, z X and s, > 0, he following condiions hold (i) M(x, y, ) > 0; (ii) x = y if and only if M(x, y, ) = for all > 0; (iii) M(x, y, ) = M(y, x, ); (iv) M(x, z, + s) M(x, y, ) M(y, z, s) for all, s > 0; (v) M(x, y, ) : (0, ) (0, ] is coninuous. The riple (X, M, ) is called a fuzzy meric space. Each fuzzy meric M on X generaes Hausdorff opology τ M on X whose base is he family of open M-balls {B M (x, ε, ) : x X, ε (0, ), > 0}, where B M (x, ε, ) = {y X : M(x, y, ) > ε}. Noe ha a sequence {x n } converges o x X (wih respec o τ M ) if and only if lim n M(x n, x, ) = for all > 0. Since for each x, y X, M(x, y, ) is a nondecreasing funcion on (0, ) (see [7]). Moreover every fuzzy meric space X (in he sense of George and Veeramani [5]) is merizable, ha is, here exiss a meric d on X which induces a opology ha agrees wih τ M ([8]). Conversely, if (X, d) is a meric space and M d : X X (0, ) (0, ] is defined as follows: M d (x, y, ) = + d(x, y) for all > 0, hen (X, M d, ) is a fuzzy meric space, called he sandard fuzzy meric space induced by he meric d (see [5]). The opologies induced by he sandard fuzzy meric and he corresponding meric are he same ([9]). A sequence {x n } in a fuzzy meric space X is said o be a Cauchy sequence if for each ε (0, ), here exiss n 0 N such ha M(x n, x m, ) > ε for all n, m n 0. A fuzzy meric space X is complee ([6]) if every Cauchy sequence converges in X. A subse A of X is closed if for each convergen sequence {x n } in A wih x n x, we have x A. A subse A of X is compac if each sequence in A has a convergen subsequence. Definiion.3 ([7]). A fuzzy meric M is said o be coninuous on X 2 (0, ) if lim M(x n, y n, n ) = M(x, y, ), n whenever {(x n, y n, n )} is a sequence in X 2 (0, ) which converges o a poin (x, y, ) X 2 (0, ). Proposiion.4 ([7]). Le (X, M, ) be a fuzzy meric space. Then M is a coninuous funcion on X X (0, ).

3 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), Lemma.5 ([7]). Le (X, M, ) be a fuzzy meric space. Then for each a X, B K(X) and > 0, here is b 0 B such ha M(a, B, ) = M(a, b 0, ). A sequence { n } of posiive real numbers is said o be s-increasing ([9]) if here exiss n 0 N such ha m+ m + for all m n 0. In a fuzzy meric space (X, M, ), an infinie produc (compare []) is denoed by for all x, y X. M(x, y, ) M(x, y, 2 ) M(x, y, n ) = M(x, y, i ) Definiion.6 ([8]). Le Ω = { η : [0, ] [0, ], η is coninuous, nondecreasing and η() > for (0, ), furher η() = if and only if = or η() = 0 if and only if = 0, overall η(), for all [0, ]}. Le Ψ be a collecion of all coninuous and decreasing funcions ψ : [0, ] [0, ] wih ψ() = 0 if and only if =. A funcion ψ Ψ is said o have propery (α) if for all r, > 0, where is any coninuous -norm. i= r > 0, we have ψ(r ) ψ(r) + ψ(), Now, an example is provided o explain he propery (α). Example.7. Define he mapping ψ : [0, ] [0, ] by ψ() =. Noe ha, i admis propery (α) for differen coninuous -norms. Take =. Suppose ha (s ) = min{s, } = (s ) = s, where s, R {0}. Then ψ(s ) = ψ(s) ψ(s) + ψ(). Similarly, if min{s, } = s, hen we have ψ(s ) = ψ(s) ψ(s) + ψ(). This shows ha propery (α) holds for minimum -norm. If =, hen (s ) = (s ) = s, where s, R {0}. Noe ha ψ(s ) = ψ(s) ψ(s) + ψ(). Thus propery (α) holds for a produc -norm. Suppose ha = L, ha is, s L = max{s +, 0}. If s L = max{s +, 0} = 0, hen ψ(s L ) = ψ(0) ψ(s) + ψ(). If s L = max{s +, 0} = s +, hen we have ψ(s L ) = ψ(s + ) ψ(s) + ψ(), which shows propery (α) holds for Lukasiewicz -norm. On he oher hand, he inerplay beween he preference relaion of absrac objecs of underlying mahemaical srucure and fixed poin heory is very srong and fruiful. This gives rise o an ineresing branch of nonlinear funcional analysis called order oriened fixed poin heory. This heory is sudied in he framework of a parially ordered ses along wih appropriae mappings saisfying cerain order condiions and has many applicaions in economics, compuer science and oher relaed disciplines. Exisence of fixed poins in parially ordered meric spaces was firs invesigaed in 2004 by Ran and Reurings [6], and hen by Nieo and Lopez [5]. Recenly, Azam [4] obained coincidence poins of mappings and relaions saisfying cerain conracive condiions in he seup of a meric space. Jachymski [3] inroduced a new approach in meric fixed poin heory by replacing order srucure wih a graph srucure on a meric space. In his way, he resuls obained in ordered meric spaces are generalized (see also [2] and he reference herein); in fac, Gwodzdz-lukawska and Jachymski [0]

4 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), developed he Huchinson-Barnsley heory for finie families of mappings on a meric space endowed wih a direced graph, furher Abbas e al. obained resuls using graphical conracions (for deails see [ 3]). The following definiions and noaions will be needed in he sequel. Le X be any se and denoes he diagonal of X X. Le G(V, E) be a undireced graph such ha he se V of is verices is a subse of X and E he se of edges of he graph which conains all loops, ha is, E. Also assume ha he graph G has no parallel edges and, hus one can idenify G wih he pair (V, E). Moivaed by he work in [4], we inroduce he following concep of a naural selecion of a ransformaion T : V V. If x, y V, hen (x, y) denoes an edge beween x and y. If verices x and y of a graph are conneced by cerain edges, we say here exiss a pah beween x and y. In his case, we denoe [x, y] a pah which sars from x and erminaes a y (we call verex y a erminal verex and verex x a reference verex). Se E x = The collecion of all erminal verices of edges saring from x and E X = x X E x. A verex w V is called a naural selecion of T : V V if Tw E w, ha is, Tw is a erminal verex of [w, Tw]. Le x and y be wo verices of a graph G. I is a common pracice o assign a cerain weigh o each edge of a graph. The posiive real number obained by calculaing he disance beween x and y can be used as a weigh of an edge joining x and y. In his paper, we assign a fuzzy weigh M(x, y, ) (a number beween 0 and ) o an edge (x, y) a, where is inerpreed as a ime. In his case we have a larger flexibiliy in choosing he weighs specially when one is uncerain or confused a a cerain poin of ime in assigning a weigh o an edge (x, y) a a ime. We esablish an exisence of a verex v of a graph such ha is image under a graphic ransformaion saisfying cerain conracion condiions becomes a erminal verex of a pah saring from v. We give examples o suppor our resuls and o show ha our resuls are poenial generalizaion of comparable resuls in he exising lieraure. M(x, y, ) = for all > 0 if and only if a pah [x, y] defines a loop. Define D (E x, E y, ) = sup{m(u, v, ), u E x, v E y }. 2. Naural selecion of graphic conracions We sar wih he following resul. Theorem 2.. Le T : V V. If here exiss a funcions ψ Ψ having propery (α) such ha ψ(m(tx, Ty, )) kψ(d(e x, E y, )) (2.) holds for all verices x, y V and 0 < k <, hen here exiss w V such ha Tw E w provided ha T(V) E X and E X is complee subspace of fuzzy meric space (X, M, ). Proof. Le x 0 be an arbirarily fixed verex of graph G(V, E) (for simpliciy G). We shall consruc sequences of verices {x n } V, {y n } E X as follows: Le y = Tx 0. Since T(V) E X, we can choose a verex x in V such ha y E x. Le y 2 = Tx. If ψ(d(e x 0, E x, ) = 0, hen we have Tx 0 = Tx which implies ha y 2 E x and hence x becomes he required verex of G. If ψ(d(e x 0, E x, ) 0, hen by inequaliy (2.), we have ψ(m(tx 0, Tx, )) kψ(d(e x 0, E x, )) 0.

5 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), Choose anoher verex x 2 in V such ha y 2 E x 2. If ψ(d(e x, E x 2, ) = 0, hen x 2 is he required verex in V. If ψ(d(e x, E x 2, ) 0, hen by inequaliy (2.), we obain ha ψ(m(tx, Tx 2, )) kψ(d(e x, E x 2, ) 0. Coninuing his way, we can obain wo sequences of verices {x n } V and {y n } E X such ha y n = Tx n, y n E x n and i saisfies: Since y n E x n, y n E x n, we have ψ(m(y n, y n+, )) kψ(d(e x n, E x n, ) 0, n =, 2, 3,.... D(E x n, E x n, ) M(y n, y n, ), which furher implies ha Noe ha ψ(m(y n, y n+, )) kψ(m(y n, y n, )). Tha is, ψ(m(y n, y n+, )) kψ(m(y n, y n, )) k 2 ψ(m(y n 2, y n, )). k n ψ(m(y 0, y, )), n =, 2, 3,.... ψ(m(y n, y n+, )) k n ψ(m(y 0, y, )). On aking limi as n on boh sides of he above inequaliy, we have lim ψ(m(y n, y n+, )) = 0. n Now, we show ha {y n } is a Cauchy sequence. Suppose ha here exis some n 0 N wih m > n > n 0 such ha m ψ(m(y n, y m, )) = ψ(m(y n, y m, a i ), where {a i } is a decreasing sequence of posiive real numbers saisfying m i=n a i =. Thus m M(y n, y m, ) = M(y n, y m, a i ) Furher, we obain ha i=n M(y n, y n+, a n ) M(y n+, y n+2, a n+ ) M(y m, y m, a m ). i=n ψ(m(y n, y m, )) m = ψ(m(y n, y m, a i )) i=n ψ[m(y n, y n+, a n ) M(y n+, y n+2, a n+ ) M(y m, y m, a m )] ψ(m(y n, y n+, a n )) + ψ(m(y n+, y n+2, a n+ )) + + ψ(m(y m, y m, a m )) k n ψ(m(y 0, y, a n )) + k n+ ψ(m(y 0, y, a n+ )) + + k m ψ(m(y 0, y, a m )). (2.2)

6 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), If max{ψ(m(y 0, y, a n )), ψ(m(y 0, y, a n+ )),, ψ(m(y 0, y, a m ))} = ψ(m(y 0, y, b)) for some b {a i : n i m }, hen he inequaliy (2.2) becomes ψ(m(y n, y m, )) k n ψ(m(y 0, y, a n )) + k n+ ψ(m(y 0, y, a n+ )) + + k m ψ(m(y 0, y, a m )) k n ψ(m(y 0, y, b)) + k n+ ψ(m(y 0, y, b)) + + k m ψ(m(y 0, y, b)) (k n + k n+ + + k m )ψ(m(y 0, y, b)) k n ( + k + + k m n )ψ(m(y 0, y, b)) kn k ψ(m(y 0, y, b)), ha is, for all n N, ψ(m(y n, y m, )) kn k ψ(m(y 0, y, b)). On aking limi as n on boh sides of he above inequaliy, we have By coninuiy of ψ, we obain ha 0 lim n,m ψ(m(y n, y m, )) 0. lim M(y n, y m, ) =. n,m Hence {y n } is a Cauchy sequence in E X. Nex we assume ha here exiss a verex z in E X such ha lim n M(y n, z, ) =. Moreover, z E w for some w X. Also, As ψ Ψ, so we have M(z, Tw, ) M(z, y n+, 2 ) M(y n+, Tw, 2 ). ψ(m(z, Tw, )) ψ[m(z, y n+, 2 ) M(y n+, Tw, 2 )] ψ[m(z, y n+, 2 )] + ψ[m(y n+, Tw, 2 )] = ψ[m(z, y n+, 2 )] + ψ[m(tx n, Tw, 2 )] ψ[m(z, y n+, 2 )] + kψ[d(ex n, E w, 2 )] ψ[m(z, y n+, 2 )] + kψ[m(y n, z, 2 )]. On aking limi as n, we have z = Tw, ha is, Tw E w. Example 2.2. Le V = Q Q = R and M : X X (0, ) (0, ] be he fuzzy meric defined by M(x, y, ) =, where d is he usual meric on X. Suppose ha ψ () = for all (0, ). + d(x, y) Define he mapping T : R R by { 2, if x Q, Tx = 0, if x Q. If x, y Q, Tx = Ty = 2 and E x = E y = [0, 4]. Noe ha ψ(m(tx, Ty, )) = ψ() kψ(d(e x, E y, )) = kψ() = 0. If x, y Q, hen Tx = Ty = 0 and E x = E y = [7, 9]. In his case, we have ψ(m(tx, Ty, )) = ψ() kψ(d(e x, E y, )) = kψ() = 0. If x Q and y Q or x Q and y Q, hen E x = [0, 4] and E y = [7, 9] or E x = [7, 9] and E y = [0, 4].

7 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), For, k 3 4, we have ψ(m(tx, Ty, )) = ψ( + 2 ) = + 2 kψ(d(ex, E y, )). Also, T(V) = {0, 2} E x = [0, 4] [7, 9]. Thus all he condiions of Theorem (2.) are saisfied. However, Theorem 3. in [4] does no hold in las case. Theorem 2.3. Le T : V V wih T(V) E X and E X a complee subspace of fuzzy meric space (X, M, ). If here exiss η Ω and k (0, ) such ha for all verices x, y V, we have M(Tx, Ty, k) η(d(e x, E y, )), (2.3) hen here exiss a verex w V such ha Tw E w provided ha for each ε > 0 and an s-increasing sequence { n }, here exiss n 0 in N 0 such ha n n 0 M(x, y, n ) ε for all n n 0. Proof. Le x 0 be an arbirarily fixed verex of graph G(V, E) (for simpliciy G). We shall consruc sequences of verices {x n } V, {y n } E X as follows: Le y = Tx 0. Since T(V) E X, we can choose a verex x in V such ha y E x. Le y 2 = Tx. If η(d(e x 0, E x, ) =, hen we have Tx 0 = Tx which implies ha y 2 E x and hence x becomes he required verex of G. If η(d(e x 0, E x, ), hen by inequaliy (2.3), we have M(Tx 0, Tx, ) η(d(e x 0, E x, )). Choose anoher verex x 2 in V such ha y 2 E x 2. If η(d(e x, E x 2, ) =, hen x 2 is he required verex in V. If η(d(e x, E x 2, ), hen by inequaliy (2.3), we obain ha M(Tx, Tx 2, k) η(d(e x, E x 2, )). Coninuing his way, we can obain wo sequences of verices {x n } V and {y n } E X such ha y n = Tx n, y n E x n and i saisfies: As y n E x n and y n+ E x n+, we have M(y n, y n+, k) η(d(e x n, E x n, )), n =, 2, 3,.... D(E x n, E x n+, ) M(y n, y n+, ). Thus implies ha M(y n, y n+, k) η(m(y n, y n, )) M(y n, y n+, ) η(m(y n, y n, k )) M(y n, y n, k ). Coninuing his way, we have M(y n, y n+, ) M(y n, y n, k ) M(y n 2, y n, Le > 0, ε > 0, m, n N such ha m > n and h i = h n + h n+ + + h m <, we have k 2 ) M(y 0, y, k n ). i(i + ) for i {n, n +,..., m }. As M(y n, y m, ) M(y n, y m, (h n + h n+ + + h m )) M(y n, y n+, h n ) M(y n+, y n+2, h n+ ) M(y m, y m, h m ) M(y 0, y, h n k n ) M(y 0, y, h n+ k n+ ) M(y 0, y, h m k m )

8 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), = M(y 0, y, M(y 0, y, i=n n(n + )k n ) M(y 0, y, i(i + )k i ) = i=n (n + )(n + 2)k n+ ) M(y 0, y, M(y 0, y, i ), ) m(m )km where i = i(i + )k i. Since lim n ( n+ n ) =, herefore { i } is an s-increasing sequence. Consequenly, here exiss n 0 N, such ha for each ε > 0, we have n= M(y 0, y, n ) ε for all n n 0. Hence M(y n, y m, ) ε for all n, m n 0. Thus {y n } is a Cauchy sequence in E X. Nex we assume ha here exiss a verex z in E X such ha lim n M(y n, z, ) =. Moreover, z E w for some w V. Now, implies M(z, Tw, ) M(z, y n+, ( k)) M(y n+, Tw, k) M(z, Tw, ) M(z, y n+, ( k)) M(y n+, Tw, k) M(z, y n+, ( k)) M(Tx n, Tw, k)] M(z, y n+, ( k)) η[d(e x n, E w, k)] M(z, y n+, ( k)) η[m(y n, z, k)]. On aking limi as n, we have z = Tw and hence Tw E w. Example 2.4. Le V = Q Q = R and M : X X (0, ) (0, ] be he fuzzy meric defined by M(x, y, ) =, where d is he usual meric on X. Define he ransformaion T : R R by + d(x, y) {, if x Q, Tx = 0, if x Q. Suppose ha η () = for all (0, ). If x, y Q, hen Tx = Ty = and E x = E y = [0, 4]. Also, M(Tx, Ty, k) = η(d(e x, E y, )), when x, y Q. Then Tx = Ty = 0, and E x = E y = [7, 9]. In his case, we have M(Tx, Ty, k) = η(d(e x, E y, )). If x Q and y Q or x Q and y Q, hen E x = [0, 4] and E y = [7, 9] or E x = [7, 9] and E y = [0, 4]. For k 3 4, we have M(Tx, Ty, ) = η(d(ex, E y, )). Noe ha T(V) = {0, } E x = [0, 4] [7, 9]. Thus all he condiions of Theorem (2.) are saisfied. In he nex heorem, we prove he exisence of a unique coincidence poin of a pair of mappings under a conracive condiion. Theorem 2.5. Le T, S : V V be coninuous mapping wih T(V) S(V) E X and E X a complee subspace of fuzzy meric space (X, M, ). If here exiss η Ω and k (0, ) such ha for all verices x, y V, we have M(Tx, Ty, k) η(m(sx, Sy, )), hen here exiss a verex w V such ha Tw, Sw E w provided ha for each ε > 0 and an s-increasing sequence { n }, here exiss n 0 in N 0 such ha n n 0 M(x, y, n ) ε for all n n 0. Moreover, if eiher T or S is injecive, hen he verex w is unique.

9 H. Alolaiyan, N. Saleem, M. Abbas, J. Nonlinear Sci. Appl., (208), Proof. By Theorem (2.3) and y n = Tx n T(V) S(V) E X, hen here exiss y n such ha Tx n = Sy n S(V), we obain lim n y n = w V such ha Tw = Sw, where, Sw = lim n Sy n = lim n Tx n = Tw, x 0 V. For uniqueness, assume ha w, w 2 V, w w 2, Tw = Sw, and Tw 2 = Sw 2. Then M(Tw, Tw 2, k) η(m(sw, Sw 2, )). If S or T is injecive, hen and M (Sw, Sw 2, ), M (Sw, Sw 2, ) > M (Sw, Sw 2, k) = M (Tw, Tw 2, k) η(m (Sw, Sw 2, )) M (Sw, Sw 2, ), which is a conradicion. Theorem 2.6. Le T, S : V V are coninuous mappings wih T(V) S(V) E X and E X is a complee subspace of fuzzy meric space (X, M, ). If here exiss a k (0, ) such ha for all x, y V, we have M(Tx, Ty, k) η(m(sx, Sy, )), where η Ω, hen S and T have a coincidence poin in X. Moreover, if eiher T or S is injecive, hen T and S have a unique coincidence poin in X. Proof. This uniqueness can be proved on he same lines as in proof of Theorem 2.5. Remark 2.7. If in he above heorem we choose X = Y and R = I (he ideniy mapping on X), we obain he Banach conracion heorem. Acknowledgmen The auhors exend heir appreciaion o he Inernaional Scienific parnership program (ISPP) a King Saud Universiy for funding his research work hrough ISPP#0034. References [] M. Abbas, T. Nazir, Common fixed poin of a power graphic conracion pair in parial meric spaces endowed wih a graph, Fixed Poin Theory Appl., 203 (203), 8 pages. [2] M. Abbas, T. Nazir, S. Radenović, Common fixed poins of four maps in parially ordered meric spaces, Appl. Mah. Le., 24 (20), [3] M. Abbas, B. E. Rhoades, Fixed poin heorems for wo new classes of mulivalued mappings, Appl. Mah. Le., 22 (2009), [4] A. Azam, Coincidence poins of mappings and relaions wih applicaions, Fixed Poin Theory Appl., 202 (202), 9 pages., 2.2 [5] A. George, P. Veeramani, On some resuls in fuzzy meric spaces, Fuzzy Ses and Sysems, 64 (994), ,.2, [6] A. George, P. Veeramani, On some resuls of analysis for fuzzy meric spaces, Fuzzy Ses and Sysems, 90 (997), ,.2, [7] M. Grabiec, Fixed poins in fuzzy meric spaces, Fuzzy Ses and Sysems, 27 (988), [8] V. Gregori, S. Romaguera, Some properies of fuzzy meric spaces, Fuzzy Ses and Sysems, 5 (2000), [9] V. Gregori, A. Sapena, On fixed-poin heorems in fuzzy meric spaces, Fuzzy Ses and Sysems, 25 (2002), , [0] G. Gwozdz-Lukawska, J. Jachymski, IFS on a meric space wih a graph srucure and exensions of he Kelisky-Rivlin heorem, J. Mah. Anal. Appl., 356 (2009), [] H. B. Hosseini, R. Saadai, M. Amini, Alexandroff heorem in fuzzy meric spaces, Mah. Sci. Res. J., 8 (2004),

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