NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 (2006), 109 114 A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE Abdolrahma Razai (Received September 2004) Abstract. I this article, a fixed poit theorem i the Meger probabilistic metric space is proved. I additio, the existece of a periodic poit i this space is show. Fially, two questios would arise. 1. Itroductio The otio of a probabilistic metric space was itroduced by Meger [4] i 1942. Fixed poit theory i this space is studied by may authors, for istace [2], [5], [7] ad so o. It is ecessary to metio that fixed poit theorems are mai tools for mathematicias to study the problem of existece of a solutio for a system of differetial equatios i probabilistic metric space, for istace see [2]. The propose of this paper is to preset a fixed poit theorem i Meger probabilistic metric space. Due to do this ad for the sake of coveiece, first, we recall some defiitios ad otatios i [1], [2], [3], [6] ad [7]. Defiitio 1.1. A mappig F : R R + is called a distributio fuctio if it is odecreasig ad left-cotiuous with if t R F (t) = 0 ad sup t R F (t) = 1. Let D + be the set of all distributio fuctios F such that F (0) = 0 ( F is a odecreasig, left-cotiuous mappig from R ito [0, 1] such that sup x R F (x) = 1). Defiitio 1.2. A probabilistic metric space (briefly, PM-space) is a order pair (S, F ) where S is a oempty set ad F : S S D (F + (p, q) is deoted by F p,q ) for every (p, q) S S satisfies the followig coditios: 1. F u,v (x) = 1 for all x > 0 if ad oly if u = v (u, v S). 2. F u,v (x) = F v,u (x) for all u, v S ad x R. 3. If F u,v (x) = 1 ad F v,w (y) = 1, the F u,w (x + y) = 1 for u, v, w S ad x, y R. If oly 1 ad 2 hold, the ordered pair (S, F ) is said to be a probabilistic semimetric space. Defiitio 1.3. A mappig T : [0, 1] [0, 1] [0, 1] is called a triagular orm (abbreviated, a t-orm) if the followig coditios are satisfied: 1991 Mathematics Subject Classificatio 47H10, 54H25. Key words ad phrases: Cotractive mappig, Fixed poit, Meger space, Triagular orm.
110 ABDOLRAHMAN RAZANI (I) T (a, 1) = a for every a [0, 1], (II) T (a, b) = T (b, a) for every a, b [0, 1], (III) a b, c d T (a, c) T (b, d) (a, b, c, d [0, 1]), (IV) T ( a, T (b, c) ) = T ( T (a, b), c ) (a, b, c [0, 1]). Example 1.4. The followig are the four basic t-orms: (I) The miimum t-orm, T M, is defied by T M (x, y) = mi(x, y), (II) The product t-orm, T P, is defied by T P (x, y) = x.y, (III) The Lukasiewicz t-orm, T L, is defied by T L (x, y) = max(x + y 1, 0), (IV) The weakest t-orm, the drastic product, T D, is defied by { mi(x, y) if max(x,y)=1, T D (x, y) = 0 otherwise. As regards the poitwise orderig, we have the iequalities T D < T L < T P < T M. Defiitio 1.5. A Meger probabilistic metric space (briefly, Meger PM-space) (see [6]) is a triple (S, F, T ), where (S, F ) is a probabilistic metric space, T is a triagular orm (abbreviated t-orm) ad the followig iequality holds: for all u, v, w S ad every x > 0, y > 0. F u,v (x + y) T (F u,w (x), F w,v (y)), (1) Schweizer, Sklar ad Thorp [7] proved that if (S, F, T ) is a Meger PM-space with sup 0<t<1 T (t, t) = 1, the (S, F, T ) is a Hausdorff topological space i the topology τ iduced by the family of (ε, λ) eighborhoods where {U p (ε, λ) : p S, ε > 0, λ > 0}, U p (ε, λ) = {u S : F u,p (ε) > 1 λ}. Defiitio 1.6. Let (S, F, T ) be a Meger PM-space with sup 0<t<1 T (t, t) = 1. (1) A sequece {u } i S is said to be τ coverget to u S (we write u u) if for ay give ε > 0 ad λ > 0, there exists a positive iteger N = N(ε, λ) such that F u,u(ε) > 1 λ wheever N. (2) A sequece {u } i S is called a τ Cauchy sequece if for ay ε > 0 ad λ > 0, there exists a positive iteger N = N(ε, λ) such that F u,u m (ε) > 1 λ, wheever, m N. (3) A Meger PM-space (S, F, T ) is said to be τ-complete if each τ-cauchy sequece i S is τ-coverget to some poit i S.
A FIXED POINT THEOREM 111 The rest of the paper is orgaized as follows: I Sectio 2, a fixed poit theorem is give. Fially, the existece of a periodic poit is proved i Sectio 3. 2. Fixed Poit Uder Cotractive Map I this sectio, the defiitio of cotractive map is rewritte ad a iterative theorem is proved. This theorem shows that if we have the existece of a coverget subsequece of a iterate sequece (of a cotractive map) the we ca prove the existece of a fixed poit. I order to do this, we recall the followig defiitio: Defiitio 2.1. Let (S, F, T ) be a Meger PM-space. We say the mappig f : S S is cotractive, if for all u v S, for all every x > 0, but F f(u),(v) F u,v. F f(u),f(v) (x) F u,v (x), (2) Lemma 2.2. Let (S, F, T ) be a Meger PM-space, where T is a cotiuous t orm. Let (u, v, x ) be a sequece i S S R + ad (u, v, x ) (u, v, x 0 ), where u, v S ad x 0 > 0. If F u,v is cotiuous at x 0, the F u,v (x ) F u,v (x 0 ). Proof. Note that lim if F u,v (x ) lim ift ( T ( F u,u(δ), F u,v (x 2δ) ), F v,v (δ) ) ( = T T ( 1, lim if = lim iff u,v (x 2δ) F u,v (x 2δ) ) ), 1 F u,v (x 0 3δ) sice x > x 0 δ evetually, ad so lim iff u,v (x ) F u,v (x 0 ) sice δ > 0 is arbitrary. By the same argumet, lim supf u,v (x ) F u,v (x + 0 ). Sice F u,v is cotiuous at x 0, the F u,v (x ) F u,v (x 0 ). Now we have the mai theorem as follows: Theorem 2.3. Let (S, F, T ) be a Meger PM-space, where T is a cotiuous t orm. If A is a cotractive mappig of S ito itself such that there exists a ( poit u S whose sequece of iterates (A (u)) cotais a coverget subsequece A i (u) ) ; the ξ = lim i A i (u) S is a uique fixed poit. Proof. Suppose A(ξ) ξ ad cosider the sequece ( A i+1 (u) ) which, it ca easily be verified, coverges to A(ξ). Choose x to be such that F A(ξ), A 2 (ξ)(x) > F ξ,a(ξ) (x), with F A(ξ), A2 (ξ) ad F ξ,a(ξ) are both cotiuous at x. This is possible, because there must be some value x 1 > 0 with F A(ξ), A2 (ξ)(x 1 ) > F ξ,a(ξ) (x 1 ),
112 ABDOLRAHMAN RAZANI ad sice F A(ξ),A 2 (ξ) ad F ξ,a(ξ) are both left cotiuous at x 1, we have F A(ξ), A2 (ξ)(x) > F ξ,a(ξ) (x) for all x i some iterval. Each fuctio has at most coutably may discotiuities, so a poit ca be foud i this iterval at which both are cotiuous. Now, costruct eighborhoods B 1, B 2 of ξ, A(ξ), respectively, such that for some ε > 0 ad all u B 1, v B 2 we have For large eough i, we have F A(u), A(v) (x) > F u,v (x) + ε. F A i +1 (u), A i +2 (u)(x) > F A i (u), A i +1 (u)(x) + ε. Applyig A a few more times, gives ad hece F A i +1 (u), A i+1 +1 (u) (x) > F A i (u), A i +1 (u)(x) + ε, F A i (u), A j +1 (u) (x) > F A i (u), A i +1 (u)(x) + (j i)ε for sufficietly large i ad j. Lettig j gives the cotradictio. I order to prove the uiqueess of ξ, suppose there is a η ξ with A(η) = η, the it follows that which is cotradictio. proof of this theorem. F ξ,η (x) = F A(ξ),A(η) (x) < F ξ,η (x), This proves the uiqueess ad, thus, accomplishes the Theorem 2.3 implies some iformatio o the covergece of a sequece of iterates. Remark 2.4. Let all assumptios of Theorem 2.3 hold. If (A (u)), u S, cotais a coverget subsequece (A i (u)), the lim A (u) exists ad coicides with the fixed poit ξ. Proof. We have lim i A i (u) = ξ. Give 1 > δ > 0 there exists, the, a positive itegers N 0 such that i > N 0 implies F ξ,a i (u) (t) > 1 δ. If m = i + l ( i fixed, l variable) is ay positive iteger > i the F ξ,am (u)(x) = F A l (ξ), A i +l (u)(x) > F ξ,a i (u) (x) > 1 δ, which proves the above assertio. 3. Periodic Poits I this sectio, first, we defie a periodic poit or a evetually fixed poit. The we prove the existece of a periodic poit i the Meger PM-space. Fially, two questios would arise. Defiitio 3.1. Let (S, F, T ) be a Meger PM-space, ad f is a self-mappig of S. The ξ is a periodic poit or a evetually fixed poit, if there exists a positive iteger k such that f k (ξ) = ξ.
A FIXED POINT THEOREM 113 Defiitio 3.2. Let (S, F, T ) be a Meger PM-space, a mappig f : S S is called locally cotractive if ad oly if: for all x > 0. u S 0<λ<1 p,q {v S:Fu,v(x)>1 λ} F p,q (x) F f(p),f(q) (x) (3) Defiitio 3.3. Let (S, F, T ) be a Meger PM-space. A mappig f : S S is called λ-uiformly locally cotractive if ad oly if λ does ot deped o u. Theorem 3.4. Let (S, F, T ) be a Meger PM-space, with a cotiuous t orm T defied as T (a, b) = T M (a, b) for a, b [0, 1]. Suppose f is a λ-uiformly locally cotractive self-mappig of S such that there exists a poit u S whose sequece of iterates (f (u)) cotais a coverget subsequece ( f i (u) ), the ξ = lim i f i (u) is a periodic poit of f. Proof. By the coditio (4), there exists a positive iteger N 1 such that for i > N 1 implies F f i (u), ξ (x) > 1 λ for all 0 < λ < 1 ad x > 0. (5) Also f is λ-uiformly locally cotractive, thus the last iequality implies F f i +1 (u),f(ξ)(x) F f i (u),ξ (x) ad so F f i +1 (u),f(v)(x) > 1 λ. After i+1 i iteratios we obtai: Note that F f i+1 (u),f i+1 i (ξ) (x) > 1 λ. F ξ,f i+1 i (ξ) (x) T ( F ξ,f i+1 (u) (x 0 ), F f i+1 (u),f i+1 i (ξ) (x 1 ) ) (4) > T ( (1 λ), (1 λ) ), (6) where x = x 0 +x 1, ad also the last equality is hold because F ξ,f i+1 (u) (x 0 ) > 1 λ by (5) ad F f i+1 (u), f i+1 i (ξ) (x 1 ) > 1 λ by the same argumet as above for x 1 istead of x 0. Now, due to the defiitio of T M, i.e. T M (a, b) = mi{a, b}, we obtai: F ξ,f i+1 i (ξ) (x) > 1 λ. (7) Suppose that η = f i+1 i (ξ) ξ. If we call A = f i+1 i, the by the same proof of Theorem 2.3, we obtai a cotradictio. Hece, puttig k = i+1 i, we have f k (ξ) = ξ as asserted. Corollary 3.5. If, i Theorem 3.4, F ξ,f(ξ) (x) > 1 λ, the k = 1. Ideed hece, f(ξ) ξ cotradicts (4). F f k (ξ), f k+1 (ξ)(x) = F ξ,f(ξ) (x), Questio 1. It is atural to ask whether Theorem 3.4 would remai true if λ- uiformly locally cotractive self-map is substituted by locally cotractive self-map. Questio 2. It is atural to ask whether Theorem 3.4 would remai true if T is defied i geeral case.
114 ABDOLRAHMAN RAZANI Ackowledgmets. I thak the referee, whose helpful commets led to may improvemets i this mauscript. The author would like to thak the Istitute for studies i Theoretical Physics ad Mathematics (IPM), Tehera, Ira, for supportig this research (No. 83340116). Refereces 1. S.S. Chag, Y.J. Cho ad S.M. Kag, Probabilistic Metric Spaces ad Noliear Operator Theory, Sichua Uiv. Press, P. R. Chia, 1994. 2. S.S. Chag, B.S. Lee, Y.J. Cho, Y.Q. Che, S.M. Kag ad J.S. Jug, Geeralized cotractio mappig priciple ad differetial equatios i probabilistic metric spaces, Proc. Amer. Math. Soc. 124 (1996), 2367 2376. 3. O. Had zić, E. Pap ad V. Radu, Geeralized cotractio mappig priciples i probabilistic metric spaces, Acta Math. Hugar, 101 (2003), 131 148. 4. K. Meger, Statistical metrics, Proc. Nat. Acad. of Sci. U.S.A. 28 (1942) 535 537. 5. V. Radu, Some fixed poit theorems probabilistic metric spaces, Lecture Notes i Math. 1233 (1987) 125 133. 6. B. Schweizer ad A. Sklar. Statistical metric spaces, Pacific J. Math. 10 (1960), 313 334. 7. B. Schweizer, A. Sklar ad E. Thorp, The metrizatio of statistical metric spaces, Pacific J. Math. 10 (1960), 673 675. Abdolrahma Razai Departmet of Mathematics, Faculty of Sciece Imam Khomeii Iteratioal Uiversity PO Box: 34194-288 Qazvi IRAN razai@ikiu.ac.ir Istitute for Studies i Theoretical Physics ad Mathematics (IPM) PO Box 19395-5531 Tehra IRAN