Fixed Point Result in Probabilistic Space

International Mathematical Forum, 4, 2009, no. 43, 237-242 Fixed Point Result in Probabilistic Space Vanita Ben Dhagat, Satyendra Nath and Nirmal Kumar Jai Narain college of Technology, Bairasia Road, Bhopal M.P., India vanita_dhagat@yahoo.co.in Abstract In this paper we prove minimization theorem in the generating space of quasi probabilistic metric space. Also we prove common fixed point theorem for the space which satisfy the minimization theorem. Mathematics Subject Classification: 54H25, 47H0 Keywords: Generating space of quasi probabilistic metric family, quasi compatible mapping, Minimization theorem, common fixed point Introduction and Preliminaries Minimization theorem in generating space of quasi metric space was given in [4].Let X be a non empty set and {d α : α (0,]} be family of mapping d α from XxX into R +. {X, d α } is called generating space of metric family if it satisfy following axioms: (GM ) For any two distinct points x and y in X such that d α (x,y) 0, α (0,] (GM 2) - d α (x,y) = 0 if x=y and α (0,] (GM 3) - d α (x,y) = d α (y,x) For all x, y in X and α (0,] (GM 4) - For any α (0,] there exists α,α 2 (0,α] such that α +α 2 α and so d α (x,y) d α (x,z) + d α2 (z,y) (GM 5) - d α (x,y) is non increasing and left continuous in α and x,y,z in X. Throughout this paper, we assume that k:(0,] (0, ) is a non decreasing function satisfying the condition K = Sup k(α) α

238 V. Ben Dhagat, S. Nath and N. Kumar Probabilistic metric space was first introduced by Menger []. Later many authors Schweizer and Sklar [3] and Mishara [2] and others. Throughout this paper D is the set of all left continuous distribution functions. A function Δ:[0,]x[0,] D is called a t-norm if the following conditions are satisfied: (T-) Δ(a,b) = Δ(b,a), (T-2) Δ(a,) = a, (T-3) Δ(a,Δ(b,c)) = Δ(Δ(a,b),c), (T-4) Δ(a,b) Δ(c,d) for a c and b d. Definition. A triple (X,F,Δ) is called a Menger probabilistic metric space if X is a nonempty set, Δ is a t-norm and F:XxX D is a mapping satisfying the following conditions: (PM-) Fx,y(t) = for all t>0 if and only if x=y, (PM-2) Fx,y(0) = 0, (PM-3) Fx,y = Fy,x, (PM-4) Fx,y(s+t) Δ( Fx,y(s), Fx,y(t)) for all x,y,z X, s,t 0. Remark: If Δ satisfies the condition supδ(t,t) =, then there exists topology T on X such that (X,T ) is a Hausdorff topological space and the family of sets U(p) = {Up(ε,λ):ε>0,λ (0,]}, p X, is a basis of neighborhood of the point p for T, where Up(ε,λ) = { : Fx,p(ε) >-λ}. Usually, the topology T is called (ε,λ) topology on (X,F,Δ). Proposition : Let (X,F,Δ) be a Menger probabilistic metric space with the t-norm Δ satisfying the condition: supδ(t,t) =. () t> For any α (0,], we define d α :XxX R + as follows: d α (x,y) = inf { t>0: Fx,y(t) >-α} (2) Then (i) (X, d α : α (0,]) is a generating space of quasi metric family and (ii) the topology T(d α ) on (X, d α : α (0,]) coincide with the (ε,λ) topology T on (X,F,Δ). Proof : (i) From the definition of (d α : α (0,]), it is easy to see that d α : α (0,]) satisfies the condition (QM-) and (QM-2) in definition. Besides, it follows clearly that (d α ) is non-increasing in α. Next we prove that d α is left continuous in α. For any given α (0,] and ε>0, from definition d α, there exists a t >0 such that t < d α (x,y) + ε and Fx,y(t ) >-α. Letting δ = Fx,y(t ) - (-α ) >0 and λ (α -δ,α ], we have -α < -λ < -(α -δ) = Fx,y(t ), which implies that t {t>0: Fx,y(t) >-λ}. Hence we have d α (x,y) d λ (x,y) = inf {t>0: Fx,y(t) >-λ} t < d α (x,y)+ε, which shows that d α is left continuous in α.

Fixed point results in probabilistic space 239 Now, we prove that (X, d α : α (0,]) also satisfies the condition (QM-3). By the condition (), for any given α (0,], there exists an μ (0,α] such that Δ(-μ, -μ) > -α Letting d μ (x,z) = σ and d μ (z,y) = β, from (2), for any given ε>0, we have Fx,z (σ+ε) > -μ, Fz,y (β+ε) > -μ and so Fx,y (σ + β + 2ε) Δ( Fx,z (σ+ε),fz,y (β+ε)) Δ(-μ, -μ) > -α. Hence we have d α (x,y) (σ + β + 2ε) = d μ (x,z) + d μ (z,y) +2ε. d α (x,y) d μ (x,z) + d μ (z,y). (ii) to prove the condition, it is enough to prove that for any ε>0 and α (0,], d α (x,y) < ε if and only if Fx,y (ε) > -α. In fact if d α (x,y) < ε, from (2), we have Fx,y (ε-μ) > -α. Conversely, if Fx,y (ε) > -α, since Fx,y is a left continuous distribution function, there exists an μ > 0 such that Fx,y (ε-μ) > -α. Hence d α (x,y) ε-μ < ε. This complete the proof. Remark: Such Menger probabilistic metric space with (2) is called generating space of quasi probabilistic metric space. Definition : Let S and T be mappings from a generating space of quasi metric family (X, d α : α (0,]) into itself. The mappings S and T are said to be quasi compatible if d α (STx n, TSx n ) 0 as n for all α (0,]. Whenever {x n } is a sequence in X such that limsx n = lim Tx n = p, for some p in X. Theorem. Let (X,F,Δ ) and (X 2,F 2,Δ 2 ) be two complete generating space of quasi probabilistic metric space. with t-norm satisfying the condition (). Let f : X X 2 be a mapping, T:X X be a continuous mapping satisfying inf{t>0:f Tx,Ty (t) > -α} inf{t>0:/2(f x,ty (t)+ F Tx,y (t)) > -α} and inf{t>0:f 2 f (Tx), f (Ty) (t) > -α} inf{t>0:/2(f 2 x,ty (t)+ F 2 Tx,y (t))> -α} for all x,y in X and α (0,]. Let ψ: R R be a non decreasing continuous function, bounded from below and φ:f(x ) R a lower semi continuous function, bounded from below. Assume that for any p X with inf ψ(φ(f(x))) < ψ(φ(f(p))), there exists q X with p Tq and Max{/2[inf{t>0:(F q,tp (t)> -α}+ inf{t>0:(f p,tq (t)> -α}], c/2[[inf{t>0:(f 2 f (q),f (Tp) (t)> -α}+ inf{t>0:(f 2 f (p), f (Tq) (t)> -α}]} k(α) [ψ(φ(f(p))) - ψ(φ(f(q)))], where c >0 is a given constant. Then there exists x 0 in X, such that inf ψ(φ(f(x))) = ψ(φ(f(x 0 ))) Proof: Let us suppose inf ψ(φ(f(x))) = ψ(φ(f(y))) for every y X and chose p X

240 V. Ben Dhagat, S. Nath and N. Kumar with ψ(φ(f(p)))<. Then we define a sequence {p n } X with p = p. Suppose p n is known and consider Wn = {w X : max {/2[inf{t>0:(F w,tpn (t)> -α}+ inf{t>0:(f Tw,pn (t)> -α}], c/2[[inf{t>0:(f 2 f (w),f (T( p n )) (t)> -α}+ inf{t>0:(f 2 f (Tw), f( p n ) (t)> -α}]}..(4) k(α) [ψ(φ(f(p n ))) - ψ(φ(f(w)))] for any α (0,]. Since Wn is nonvoid set therefore there exists w Wn such that p n Tw. We can choose p n+ Wn such that p n Tp n+ and ψ(φ(f(p n+ ))) inf ψ(φ(f(x))) - /2 [[ψ(φ(f(p n ))) inf ψ(φ(f(x)))] (5) we observe that [ψ(φ(f(p n+ )))] is a non increasing lower bounded sequence, hence it is convergent sequence. Now we prove that {p n } and { f(p n )}are cauchy sequences, Max[inf{t>0:(F T( p n),t( p n +) (t)> -α}, c inf{t>0:(f 2 f (T( p n)), f (T( p n +)) (t)> -α}] max { /2[inf{t>0:(F p n,t( p n+) (t)> -α}+ inf{t>0:(f p n+,t( p n) (t)> -α}], c/2[[inf{t>0:(f 2 f ( p n ),f (T( p n+ )) (t)> -α}+ inf{t>0:(f 2 f (T( p n)), f( p n ) (t)> -α}]} k(α) [ψ(φ(f(p n ))) - ψ(φ(f(p n+ )))] Now for all n,m N, n< m there exists μ α, μ = μ(n,m) such that Max[inf{t>0:(F T( p n),t( p m) (t)> -α}, c inf{t>0:(f 2 f (T( p n)), f (T( p m)) (t)> -α}] m- Σ j = n max { /2[inf{t>0:(F p j,t( p j+) (t)> -μ}+ inf{t>0:(f p j+,t( p j) (t)> -μ}], c/2[[inf{t>0:(f 2 f ( p j ),f (T( p j+ )) (t)> -μ}+ inf{t>0:(f 2 f (T( p j)), f( p j+ ) (t)> -μ}]} Hence for all n,m N, n< m: Max[inf{t>0:(F T( p n),t( p m) (t)> -α}, c inf{t>0:(f 2 f (T( p n)), f (T( p m)) (t)> -α}] m- k(μ) Σ [ψ(φ(f(p j ))) - ψ(φ(f(p j+ )))] j = n k(α) [ψ(φ(f(p n ))) - ψ(φ(f(p m )))] for some α with 0< α j+ α k α, j = n,, m- and inf {t>0: F p n,p n+ (t) >-α} inf{t>0:(f p n,t( p n+) (t)> -μ} + inf{t>0:(f T( p n),t( p n+) (t)> -μ} + inf{t>0:(f T( p n), p n+ (t)> -μ { /2[inf{t>0:(F p n,t( p n+) (t)> -μ}+ inf{t>0:(f p n+,t( p n) (t)> -μ}] 3 k(α) [ψ(φ(f(p n ))) - ψ(φ( f(p n+ )))] Then, for n<m inf{t>0:(f T( p n),t( p m) (t)> -α} 3 k(α) [ψ(φ(f(p n ))) - ψ(φ( f(p m )))] Similarly, inf{t>0:(f 2 f (T( p n)), f (T( p n +)) (t)> -α} 3 k(α) [ψ(φ(f(p n ))) - ψ(φ( f(p m )))] Hence {p n } and { f(p n )}are cauchy sequences. Let lim p n = u and lim f(p n ) = v as n. Since f is closed therefore f(u) = v f (X ). By the continuity of ψ and lower semi continuity of φ, we have

Fixed point results in probabilistic space 24 ψ(φ(v)) lim ψ(φ(f (p n ))) = lim ψ(φ(f (p n+ ))) Let inf ψ(φ(f (x))) = λ 0 R. From ψ(φ(f (p n+ ))) λ 0 + /2 {ψ(φ(f (p n ))) - λ 0 }, we have lim ψ(φ(f (p n+ ))) λ 0 /2 + /2 lim ψ(φ(f (p n ))) = λ 0 /2 + /2 lim ψ(φ(f (p n+ ))). Finally, ψ(φ(f (u))) = ψ(φ(v)) lim ψ(φ(f (p n+ ))) λ 0 = inf ψ(φ(f (x))) ψ(φ(f (u))) which is contradiction. Therefore there exists an x 0 X such that inf ψ(φ(f (x))) = ψ(φ(f (x 0 ))). Theorem 2 : Let (X,F,Δ ) and (X 2,F 2,Δ 2 ) be two complete Menger probabilistic metric space with t-norm satisfying the condition (). Let f : X X 2 be a mapping, T:X X be a continuous mapping satisfying inf{t>0:f Tx,Ty (t) > -α} inf{t>0:/2(f x,ty (t)+ F Tx,y (t)) > -α} and inf{t>0:f 2 f (Tx), f (Ty) (t) > -α} inf{t>0:/2(f 2 x,ty (t)+ F 2 Tx,y (t))> -α} for all x,y in X and α (0,]. Let ψ: R R be a non decreasing continuous function, bounded from below and φ:f(x ) R a lower semi continuous function, bounded from below. Let S :X X be continuous mapping. Further, if S and T are quasi compatible and Max{/2[inf{t>0:(F x,tsx (t)> -α}+ inf{t>0:(f Tx,Sx (t)> -α}], c/2[[inf{t>0:(f 2 f (x),f (TSx) (t)> -α}+ inf{t>0:(f 2 f (Tx), f (Sxs) (t)> -α}]} k(α) [ψ(φ(f(x))) - ψ(φ(f(sx)))], for every x and α (0,],where c >0 is a given constant, then there exists unique common fixed point. Proof: By the theorem, we have inf ψ(φ(f (x))) = ψ(φ(f (x 0 ))). Suppose x 0 TSx 0 or Sx 0 Tx 0. Then for some α (0,] 0< /2[inf{t>0:(F x0,tsx0 (t)> -α}+ inf{t>0:(f Tx0,Sx0 (t)> -α}] k(α) [ψ(φ(f(x 0 ))) - ψ(φ(f(sx 0 )))] 0 which is the contradiction. Therefore x 0 = TSx 0 or Sx 0 = Tx 0 and also x 0 = Tx 0. Moreover for every α (0,] there exists μ (0, α] such that inf{t>0:(f x0,tx0 (t)> -α} inf{t>0:(f x0,tsx0 (t)> -μ}+ inf{t>0:(f TSx0,Tx0 (t)> -μ}] = inf{t>0:(f TSx0,Tx0 (t)> -μ}] /2[inf{t>0:(F Sx0,Tx0 (t)> -μ}+ inf{t>0:(f x0,tsx0 (t)> -μ}]] = 0. Thus x 0 = Tx 0 ( = Sx 0 ) Uniqueness can obtain by the given condition. Corollary:: Let (X,F,Δ ) and (X 2,F 2,Δ 2 ) be two complete Menger probabilistic metric space with t-norm satisfying the condition ().Let f : X X 2 be a closed mapping, and φ:f(x ) R a lower semi continuous function, bounded from below. Let S :X X be a mapping such that max{inf{t>0:(f S x,x (t)>-α}, c inf{t>0:(f 2 f (S(x)),f (x) (t)>-α}} k(α) [φ(f(x)) - φ(f(s(x))) for every x and α (0,],where c >0 is a given constant, then there exists unique common fixed point.

242 V. Ben Dhagat, S. Nath and N. Kumar Proof: On taking T = I and ψ = I in theorem 2. References [] K. Menger, Statistical metrics, Procedings of the National Academy of Sciences of the United state of America 28(942), 535-537. [2] S. N. Mishra, Common fixed points of compatible mappings in PM- space, Mathematica Japonica 36 (99) no. 2, 283-289. [3] B. Schweizer and A. Sklar, Statistical metric space, Pacific Journal of Mathematics 0(960), 33-334. [4] W. Takahashi, Fixed Point Theory and Application, Pitman Res. Not.Math. Ser. 252(99), 397-406. Received: March, 2009