A new contraction principle in menger spaces

Acta Mathematica Sinica, English Series Aug., 2008, Vol. 24, No. 8, pp. 1379–1386 Published online: July 5, 2008 DOI: 10.1007/s10114-007-6509-x Http://www.ActaMath.com

Acta Mathematica Sinica, English Series The Editorial Office of AMS & Springer-Verlag 2008

A New Contraction Principle in Menger Spaces

Binayak S. CHOUDHURY

Krishnapada DAS

Department of Mathematics, Bengal Engineering and Science University, Shibpur, P.O. - B. Garden, Shibpur, Howrah - 711103, West Bengal, INDIA E-mail: [email protected] [email protected] Abstract In the present work we introduce a new type of contraction mapping by using a specific function and obtain certain fixed point results in Menger spaces. The work is in line with the research for generalizing the Banach’s contraction principle. We extend the notion of altering distance function to Menger Spaces and obtain fixed point results. Keywords

Menger space, ϕ-contraction, fixed point

MR(2000) Subject Classification 54H25, 54E70

1

Introduction

The paper is intended to prove a new contraction mapping principle in certain probabilistic metric spaces, namely Menger spaces. As is well known that Banach Contraction Principle is one of the most important results of functional analysis, generalization of this principle in general metric spaces has been intensively investigated and currently is also an active branch of research. To cite a few examples, in [1] a new contraction principle was addressed by Khan et al., where they used a control function on the metric function; in [2] and [3] generalized Banach contraction conjecture has been established independently and in [4] Kirk investigated asymptotic contraction in metric spaces. Particularly, the work of Khan et al., in [1] initiated the study of a new category of fixed point theorems. Furthermore, contractive types of mappings occupy a very important position in the fixed point theory in metric spaces. Several types of contractions and their interrelations have been discussed in the review paper due to Rhoades [5] and comparison of various definitions of contraction mappings may be seen in [6]. Probabilistic metric spaces have been introduced as a probabilistic generalization of metric spaces. Schweizer and Sklar [7] have investigated several of these structures. Particularly, a lot of work has been done on the existence of fixed points of mappings in such spaces. In the following we review some notions connected with probabilistic metric spaces. Definition 1.1 A mapping F : R → R+ is called a distribution function if it is non-decreasing and left continuous with inf t∈R F (t) = 0 and supt∈R F (t) = 1, where R+ denotes the set of nonnegative real numbers. Received September 25, 2006, Accepted October 15, 2007 The present work is partially supported by Government of India under UGC Major Research Project No. F.812/2003 (SR) dated 30.03.2003

Choudhury B. S., and Das K.

1380

Definition 1.2 (Probabilistic metric space [7–8]) A probabilistic metric space (P M space) is an ordered pair (S, F ), where S is a non-empty set and F is a function defined on S × S to the set of distribution functions which satisfies the following conditions : (i) Fxy (0) = 0, (ii) Fxy (t) = 1 for all t > 0 iff x = y, (iii) Fxy (t) = Fyx (t) for all t ∈ R, (iv) If Fxy (t1 ) = 1 and Fyz (t2 ) = 1, then Fxz (t1 + t2 ) = 1. Definition 1.3 (t-norm [7–8]) the following :

A t-norm is a function T : [0, 1] × [0, 1] → [0, 1] which satisfies

(i) T (1, a) = a, T (0, 0) = 0, (ii) T (a, b) = T (b, a), (iii) T (c, d) ≥ T (a, b) whenever c ≥ a and d ≥ b, (iv) T (T (a, b), c) = T (a, T (b, c)). Definition 1.4 (Menger space [7–8]) A Menger Space is a triplet (S, F, T ), where S is a non-empty set, F is a function defined on S × S to the set of distribution functions such that the following are satisfied : (i) Fxy (0) = 0 for all x, y ∈ S, (ii) Fxy (s) = 1 for all s > 0 iff x = y, (iii) Fxy (s) = Fyx (s) for all x, y ∈ S, (iv) Fxy (u + v) ≥ T (Fxz (u), Fzy (v)) for all u, v ≥ 0 and x, y, z ∈ S where T is a t-norm. Menger spaces are special types of probabilistic metric spaces as can be seen from the definition. It is also well known that these spaces are also probabilistic generalizations of metric spaces. Any metric space can be treated as a Menger space if we put Fxy (t) = H(t − d(x, y)), where H is defined as  1 if s > 0, H(s) = 0 if s ≤ 0 and T is defined as T (a, b) = min{a, b}. If (S, F, T ) is a Menger space with continuous t-norm then the topology induced by the family {Uλ (p) : p ∈ S,  > 0, λ > 0} is called the (-λ)-topology, where Uλ (p) = {q ∈ S : Fpq () > 1 − λ} is called the (-λ)-neighborhood of p. A sequence {xn } ⊂ S converges to some point x ∈ S in the (-λ)-topology if and only if given  > 0, λ > 0 we can find a positive integer N,λ such that, for all n > N,λ , Fxn x () ≥ 1 − λ.

(1.1)

A sequence {xn } is said to be a Cauchy sequence in S if given  > 0, λ > 0 there exists a positive integer N,λ such that Fxn xm () ≥ 1 − λ for all m, n > N,λ .

(1.2)

It is observed that the ‘≥’ in (1.1) and (1.2) can be replaced by ‘>’. Most of the literatures use ‘>’ in the inequalities (1.1) and (1.2). For our convenience of the proof we have used ‘≥’ in these two inequalities. A Menger space (S, F, T) is said to be complete if every Cauchy sequence is convergent.

A New Contraction Principle in Menger Spaces

1381

The idea of contraction in PM space was introduced by Sehgal and Bharucha–Reid [9]. Definition 1.5 Probabilistic q-contraction [8] Let (S, F ) be a probabilistic metric space. A mapping f : S → S is a probabilistic q-contraction (q ∈ (0, 1)) if Ff uf v (x) ≥ Fuv ( xq ) for every u, v ∈ S and every x ∈ R. The following theorem was proved by Sehgal and Bharucha–Reid [9]. Theorem 1.6 [9] Let (S, F, TM ) be a complete Menger space where TM (a, b) = min{a, b} and f : S → S is a probabilistic q-contraction. Then there exist a unique fixed point x of the mapping f and x = limn→∞ f n p for every p ∈ S. The q-contraction was further generalized by Mihet [10], where he introduced (-λ)-contraction. Hadzic in [11] by way of generalization of the definition given by Mihet described a contraction which has been called (q, q1 ) -contraction of (-λ)-type. Definition 1.7 Let (S, F ) be a probabilistic metric space. A mapping f : S → S is said to be a (q, q1 )-contraction of (-λ)-type, where q, q1 ∈ (0, 1), if the following implication holds for every p, p1 ∈ S : (∀ > 0)(∀λ ∈ (0, 1))(Fp1 ,p2 () ≥ 1 − λ ⇒ Ff p1 ,f p2 (q) ≥ 1 − q1 λ). Fixed point result of such mapping was also established in [8, 10]. Another generalization of contraction principle in PM space was obtained in [12] and was applied for establishing certain results of differential equations in PM space. Common fixed point results for a sequence of mutually contractive mappings in metric spaces have been established in [13]. In [14] several fixed point results in PM spaces are described. In [1] a new category of contractive fixed point problems in metric spaces was addressed by Khan et al. They introduced altering distance function, which is a control function that alters the distance between two points in a metric space. The definition is as follows: Definition 1.8 Altering distance function [1] ψ : [0, ∞) → [0, ∞)

An altering distance function is a function

(i) which is monotone increasing and continuous and (ii) ψ(t) = 0 if and only if t = 0. Khan et al proved the following result. Theorem 1.9 [1]

Let (X, d) be a complete metric space, ψ be an altering distance function

and let f : X → X be a self mapping which satisfies the following inequality ψ(d(f x, f y)) ≤ cψ(d(x, y))

(1.3)

for all x, y  X and for some 0 < c < 1. Then f has a unique fixed point. In fact, Khan et al proved a more general theorem (Theorem 2 in [1]) of which the above result is a corollary. The above result (Theorem 1.9) was further generalized in a number of works. Also other fixed point results of functions satisfying various types of contractive conditions involving altering distances have been established. Some of these results are noted in [15–20] and [21]. The altering distance function has also been generalized to functions of two and three variables in

Choudhury B. S., and Das K.

1382

[22] and has been applied to establish new fixed point results. Altering distances have also been applied to set-valued mappings [23]. In the present work we propose the notion of an altering distance function in the context of Menger spaces and obtain fixed point results for self-mapping in a complete Menger space, which satisfy a contractive inequality through this function. Finally we show that the above mentioned result of [1] can be obtained from our resulting theorem in Menger space. This is proved through a construction of the postulated function in the corresponding Menger space starting from ψ in Definition 1.8. Definition 1.10 A function ϕ : R → R+ is said to satisfy the condition ∗ if it satisfies the following conditions : (i) ϕ(t) = 0 if and only if t = 0, (ii) ϕ(t) is increasing and ϕ(t) → ∞ as t → ∞, (iii) ϕ is left continuous in (0, ∞), (iv) ϕ is continuous at 0. Definition 1.11

Let (S, F, T ) be a Menger space. A self map f : S → S is said to be

ϕ-contractive if

   t , Ff xf y (ϕ(t)) ≥ Fxy ϕ c

(1.4)

where 0 < c < 1, x, y ∈ S and t > 0 and the functionϕ satisfies the condition *. Fixed point studies in probabilistic metric spaces have a vast literature. A comprehensive survey of this line of research is given in [8]. We use in our result minimum t-norm, which is the strongest t-norm. Fixed point results with minimum t-norm have been proved as for example in [12], [24]. 2

Main Result

Theorem 2.1

Let (S, F, T ) be a complete Menger space with continuous t-norm and f : S → S be ϕ-contractive and for xo ∈ S the sequence {xn } in S be constructed as follows: xn = f xn−1 n = 1, 2, 3, . . . . If the sequence {xn } converges, then it converges to a unique fixed point of f. Proof In view of the conditions (i) and (iv) in Definition 1.10, for s > 0 we can find a positive number r such that s > ϕ(r). Then for s > 0, we have Fxn xn+1 (s) ≥ Ff xn−1 f xn (ϕ(r))    r ≥ Fxn−1 xn ϕ c    r = Ff xn−2 f xn−1 ϕ c    r ≥ Fxn−2 xn−1 ϕ 2 c ···

A New Contraction Principle in Menger Spaces

1383

   r . ≥ Fx0 x1 ϕ n c Therefore,

   r Fxn xn+1 (s) ≥ Fx0 x1 ϕ n . c

(2.1)

Taking n → ∞, we have Fxn xn+1 (s) → 1 as n → ∞. By the hypothesis of the theorem, {xn } is convergent. Let xn → z as n → ∞.

(2.2)

We next prove that z is a fixed point of f. From the property of ϕ, it follows that given  > 0 we can find 1 > 0 such that  > ϕ(1 ) > 0. Then for all n = 0, 1, 2, 3, . . . , Ff zz () ≥ T (Ff zxn (ϕ(1 )), Fxn z ( − ϕ(1 ))) = T (Ff zf xn−1 (ϕ(1 )), Fxn z ( − ϕ(1 )))      1 , Fxn z ( − ϕ(1 )) . ≥ T Fzxn−1 ϕ c Making n → ∞ in the above inequality, by virtue of (2.2) and the fact that T is continuous t-norm, we have for all  > 0, Ff zz () = 1, that is, f z = z. We next show that the fixed point is unique. If possible, let u and v be two fixed points. As in the above corresponding to  > 0, we make a choice of 1 > 0 such that  > ϕ(1 ). Then Fuv () ≥ Ff uf v () ≥ Ff uf v (ϕ(1 ))    1 ≥ Fuv ϕ c    1 = Ff uf v ϕ c    1 . ≥ Fuv ϕ 2 c Proceeding as above we obtain for any  > 0, Fuv () ≥ Fuv (ϕ( cn1 )) → 1 as n → ∞. Therefore u = v. This proves the uniqueness of the fixed point. This completes the proof. Theorem 2.2 Let (S, F, TM ) be a complete Menger space with continuous t-norm TM given by TM (a, b) = min{a, b} and f : S → S be ϕ-contractive. Then f has a unique fixed point. Proof Let x0 ∈ S. Now we construct a sequence {xn } in S as follows: xn = f xn−1

n = 1, 2, 3, . . . .

In view of Theorem 2.1, the proof of the theorem is complete if we can prove that {xn } is a Cauchy sequence. If {xn } is not a Cauchy sequence, then there exist  > 0, λ > 0 such that for every positive integer k there exist positive integers m(k), n(k) ≥ k such that Fxm(k) xn(k) () < 1 − λ.

(2.3)

We can choose m(k) < n(k) and n(k) to be the smallest integer corresponding to m(k) satisfying the above inequality.

Choudhury B. S., and Das K.

1384

From the above statement it follows that there exist  > 0, λ > 0 for which we can construct increasing sequences of integers {n(k)} and {m(k)} satisfying the following: n(k) > m(k), Fxm(k) xn(k)−1 () ≥ 1 − λ

(2.4)

Fxm(k) xn(k) () < 1 − λ.

(2.5)

and

Equivalently, the construction is finding a point xn(k) in the sequence with n(k) > m(k) which will fall outside the set {z : Fxm(k) z () ≥ 1 − λ} but the point in the sequence preceding the point xn(k) , that is, xn(k)−1 , will fall inside the set. This is guaranteed by the fact the sequence is assumed not to be a Cauchy sequence. Since {x : Fxp (1 ) ≥ 1 − λ} ⊆ {x : Fxp (2 ) ≥ 1 − λ} for all p ∈ S, λ > 0 and 0 < 1 < 2 , it follows that whenever the above construction is possible for  > 0, λ > 0, it is also possible to construct {xm(k) } and {xn(k) } satisfying (2.4) and (2.5) corresponding to  > 0, λ > 0 whenever  < . Again from the properties of ϕ it is immediate that given  > 0 we can find  > 0 such that  > ϕ( ). In view of the above paragraph we make a choice of  in (2.4) and (2.5) such that  = ϕ(1 ) for some 1 > 0 such that ϕ( c1 ) > ϕ(1 ). Such a choice is possible by virtue of conditions (i) and (ii) of Definition 1.10 . We have, from (2.4) and (2.5), Fxm(k) xn(k)−1 (ϕ(1 )) ≥ 1 − λ,

(2.6)

Fxm(k) xn(k) (ϕ(1 )) < 1 − λ.

(2.7)

but

Then 1 − λ > Fxm(k) xn(k) (ϕ(1 )) = Ff xm(k)−1 f xn(k)−1 (ϕ(1 ))    1 . ≥ Fxm(k)−1 xn(k)−1 ϕ c Since ϕ( c1 ) > ϕ(1 ) we make a choice of the positive number η such that η < ϕ( c1 ) − ϕ(1 ),

that is, ϕ( c1 ) − η > ϕ(1 ). In view of (2.1) in Theorem 2.1, we may choose k large enough so that Fxm(k) xm(k)−1 (η) > 1 − λ1 for given 0 < λ1 < λ. With the above choice of η and k, we obtain by virtue of (2.4) and (2.7) and from the above inequality that    1 1 − λ > Fxm(k)−1 xn(k)−1 ϕ c       1 − η , Fxm(k)−1 xm(k) (η) ≥ TM Fxm(k) xn(k)−1 ϕ c ≥ TM (Fxm(k) xn(k)−1 (ϕ(1 )), Fxm(k)−1 xm(k) (η)) ≥ TM (1 − λ, 1 − λ1 ) = 1 − λ (since λ1 < λ implies 1 − λ < 1 − λ1 ), which is a contradiction.

A New Contraction Principle in Menger Spaces

1385

Hence {xn } is a Cauchy sequence. The proof of the theorem is then completed by an application of Theorem 2.1. It is well known that any metric space can be made into a Menger space if we assume Fxy (t) = H(t − d(x, y)), where H is the Heaviside function given by  1 if s > 0, H(s) = 0 if s ≤ 0 and T (a, b) = TM (a, b) = min{a, b}. Furthermore if (X, d) is complete, then the corresponding Menger space is also complete. In the following we show that Theorem 1.9 proved by Khan et al is a corollary to Theorem 2.2. Let ψ be the altering distance function (Def. 1.8) and we define ϕ as ϕ(t) = sup{a : ψ(a) < t}. Let t > 0 and x, y ∈ X. If ϕ(t) − d(x, y) > 0 then we have d(x, y) < ϕ(t) = sup{a : ψ(a) < t}, that is, ψ(d(x, y)) < t (by the continuity of ψ), that is, t − ψ(d(x, y)) > 0. Conversely, if t − ψ(d(x, y)) > 0 then we have ψ(d(x, y)) < t, that is, d(x, y) < sup{a : ψ(a) < t} = ϕ(t) (by the continuity of ψ). Therefore, ϕ(t) − d(x, y) > 0 if and only if t − ψ(d(x, y)) > 0.

(2.8)

Similarly we can prove ϕ(t) − d(x, y) ≤ 0 if and only if t − ψ(d(x, y)) ≤ 0.

(2.9)

From (2.8) and (2.9) we have H(ϕ(t) − d(x, y)) = H(t − ψ(d(x, y))).

(2.10)

Let us assume inequality (1.3) in a complete metric space (X, d). Then we have for all x, y ∈ X, t > 0 and 0 < c < 1, Ff xf y (ϕ(t)) = H(ϕ(t) − d(f x, f y)) = H(t − ψ(d(f x, f y)) (from definition)   t ≥H − ψ(d(x, y)) (by inequality(1.3)) c     t − d(x, y) =H ϕ c    t . ≥ Fxy ϕ c Thus we see that the inequality (1.3) in a complete metric space (X, d) implies the inequality (1.4) in the corresponding complete Menger space. By an application of Theorem 2.2, we can prove Theorem 1.9. Open Question In Theorem (2.2) we have assumed a specific form of the t-norm, that is, T = TM . It remains to be decided whether the theorem is valid for other choices of the t-norm.

Choudhury B. S., and Das K.

1386

Acknowledgements

The support is gratefully acknowledged.

References [1] Khan, M. S., Swaleh, M., Sessa, S.: Fixed points theorems by altering distances between the points. Bull. Austral. Math. Soc., 30, 1–9 (1984) [2] Arvanitakis, A. D.: A Proof of generalized Banach contraction conjecture. Proc. Amer. Math. Soc., 131(12), 3647–3656 (2003) [3] Merryfield, J., Rothschild, B., Stein, J. D. :An application of Ramsey’s Theorem to the Banach Contraction Principle. Proc. Amer. Math. Soc., 130, 927–933 (2002) [4] Kirk, W. A.: Fixed points of asymptotic contraction. J. Math. Anal. Appl., 277, 645–650 (2003) [5] Rhoades, B. E.: A comparison of various definitions of contractive mappings., Trans. Amer. Math. Soc., 226–257, 1977 [6] Meszaros, J. : A comparison of various definitions of contractive type mappings. Bull. Cal. Math. Soc., 84, 167–194 (1992) [7] Schweizer, B., Sklar, V.: Probabilistic Metric Space, North-Holland, Amsterdam, 1983 [8] Hadzic, O., Pap, E.: Fixed Point Theory In Probabilistic Metric Spaces, Kluwer Academic Publishers, 2001 [9] Sehgal, V. M., Bharucha-Reid, A. T. : Fixed points of contraction mappings on PM space. Math. Sys. Theory, 6(2), 97–100 (1972) [10] Mihet, D.: Aclass of Sehgal’s Contractions in Probabilistic Metric Spaces, Analele Univ. din Timisoara, Vol. XXXVII, fasc. 1, 1999, Seria Matematica–Informatica, 105–108 [11] Hadzic, O., Pap, E., Radu, V.: Generalized contraction mapping principle in probabilistic metric space. Acta Math. Hungar., 101(1–2), 131–148 (2003) [12] Chang, S. S., Lee, B. S., Cho, Y. S., Chen, Y. Q., Kang, S. M., Jung, J. S: Generalised contraction mapping principle and differential equation in probabilistic metric spaces. Proc. Amer. Math. Soc., 124(8), 2367– 2376 (1996) [13] Choudhury, B. S.: A unique common fixed point theorems for a sequence of self mappings in Menger spaces. Bull. Kor. Math. Soc., 37(3), 569–575 (2000) [14] Chang, S. S., Cho, Y. J., Kang, S. M.: Nonlinear Operator Theory In Probabilistic Metric Spaces, Huntington, NY: Nova Science Publishers. X, 338, 2001 [15] Naidu, S. V. R.: Fixed point theorems by altering distances. Adv. Math. Sci. Appl., 11, 1–16 (2001) [16] Naidu, S. V. R.: Some fixed point theorems in Metric spaces by altering distances. Czechoslovak Mathematical Journal, 53(128), 205–212 (2003) [17] Pathak, H. K., Sharma, R.: A note on fixed point theorems of Khan, Swaleh and Sessa. Math. Edn., 28, 151–157 (1994) [18] Sastry, K. P. R., Babu, G. V. R.: Fixed point theorems in metric space by altering distances. Bull. Cal. Math. Soc., 90, 175–182 (1998) [19] Sastry, K. P. R., Babu, G. V. R.: Some fixed point theorems by altering distances between the points. Ind. J. Pure. Appl. Math., 30(6), 641–647 (1999) [20] Sastry, K. P. R., Babu, G. V. R.: A common fixed point theorem in complete metric spaces by altering distances. Proc. Nat. Acad. Sci. India, 71(A), III 237–242 (2001) [21] Sastry, K. P. R., Naidu, S. V. R., Babu, G. V. R., Naidu, G. A.: Generalisation of fixed point theorems for weekly commuting maps by altering distances. Tamkong Journal of Mathematics, 31(3), 243–250 (2000) [22] Choudhury, B. S., Dutta, P. N.: A unified fixed point result in metric spaces involving a two variable function. FILOMAT, 14, 43–48 (2000) [23] Choudhury, B. S., Upadhyay, A.: An unique common fixed point for a sequence of multivalued mappings on metric spaces, Bulletin of Pure and Applied Science, 19E (2000), 529–533 [24] Singh, B., Jain, S.: A fixed point theorem in Menger space through weak compatibility. J. Math. Anal. Appl., 301, 439–448 (2005)