Finally we prove a fixed point theorem in a complete metric space by using the concept of generalized w-distance

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 2(2011), Pages 134-139.

A FIXED POINT THEOREM VIA GENERALIZED W-DISTANCE

(COMMUNICATED BY DENNY H. LEUNG)

SUSHANTA KUMAR MOHANTA

Abstract. In this paper we first introduce the concept of generalized w- distance in a metric space and prove a fixed point theorem which generalizes Banach contraction theorem.

1. Introduction

In 1996, W. Takahashi et. al.[5] had introduced the concept of w-distance in a metric space and proved some fixed point theorems in complete metric spaces.

In this paper we first introduce the concept of generalizedw-distance in a metric space. At the beginning of the paper an example is provided to show that the class of generalized w-distance functions is strictly larger than the class of w-distance functions. Finally we prove a fixed point theorem in a complete metric space by using the concept of generalized w-distance. This theorem is a generalization of Banach contraction theorem.

2. Definitions and Examples

Definition 2.1. [5] Let (X, d) be a metric space. Then a function p:X ×X → [0,∞)is called a w- distance onX if the following conditions are satisﬁed : (i)p(x, z)≤p(x, y) +p(y, z)for any x, y, z∈X;

(ii) for anyx∈X, p(x, .) :X →[0,∞) is lower semicontinuous ;

(iii) for any ϵ >0, there exists δ >0 such thatp(z, x)≤δ and p(z, y)≤δ imply d(x, y)≤ϵ.

Clearly every metric is aw-distance but the converse is not true. The following example supports our contention.

2000Mathematics Subject Classification. Primary 54C20; Secondary 47H10.

Key words and phrases. w-distance, generalizedw-distance, fixed point in a metric space.

⃝c2011 Universiteti i Prishtinës, Prishtinë, Kosovë.

Submitted July 30, 2010. Accepted May 11, 2011.

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Example 2.1. [5]Let (X, d) be a metric space. A function p: X×X →[0,∞) deﬁned byp(x, y) =cfor everyx, y∈X is aw-distance onX, wherec is a positive real number. But pis not a metric sincep(x, x) =c̸= 0 for anyx∈X.

Definition 2.2. Let (X, d)be a metric space andj∈N. A functionp:X×X → [0,∞)is called a generalizedw- distance of orderj onX if for all x, z∈X and for all distinct pointsxi∈X, i∈ {1,2,3,· · ·, j}, each of them diﬀerent from xandz, the following conditions are satisﬁed:

(i)p(x, z)≤

∑j

i=0

p(xi, xi+1), where x0=x, xj+1=z;

(ii)f or any x∈X, p(x, .) :X →[0,∞)is lower semicontinuous;

(iii)f or any ϵ >0, there exists δ >0 such that p(z, x)≤δ and p(z, y)≤δ imply d(x, y)≤ϵ.

From Definition 2.2 it follows that every w-distance is a generalizedw-distance of order 1.

Now we consider the following example to show that a generalizedw-distance may not be aw-distance.

Example 2.2. LetX ={1,2,3,4} be a metric space with metricd(x, y) =|x−y| for allx, y∈X. Letp:X×X →[0,∞)be deﬁned by

p(1,2) =p(2,1) = 3, p(1,3) =p(3,1) =p(2,3) =p(3,2) = 1, p(1,4) =p(4,1) =p(2,4) =p(4,2) =p(3,4) =p(4,3) = 2

and p(x, x) = 0.6f or every x∈X.

Then p satisfies condition (i) of Definition 2.2 for j = 2. Also, condition (ii) of Definition2.2 is obvious. To show (iii), for anyϵ >0, putδ= ¹₂. Then

p(z, x)≤δ and p(z, y)≤δ imply d(x, y)≤ϵ.

Thuspis a generalizedw-distance of order2onX but it is not aw-distance onX since it lacks the triangular property:

p(1,2) = 3>1 + 1 =p(1,3) +p(3,2).

3. Main Result

In this section we prove a fixed point theorem in a complete metric space by employing notion of generalizedw-distance. The following Lemma is crucial in the proof of the theorem.

Lemma 3.1. Let (X, d) be a metric space and let p be a generalized w-distance of order j on X. Let {xn} and {yn} be sequences in X, let {αn} and {βn} be sequences in[0,∞) converging to0, and letx, y, z∈X. Then the following hold : (i) Ifp(xn, y)≤αn andp(xn, z)≤βn for any n∈N, theny=z. In particular, if p(x, y) = 0 andp(x, z) = 0, theny=z;

(ii) ifp(x_n, y_n)≤α_n andp(x_n, z)≤β_n for any n∈N, then {y_n}converges to z;

(iii) if p(x_n, x_m) ≤ α_n for any n, m ∈ N with m > n, then {x_n} is a d-Cauchy

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sequence;

(iv) if p(y, xn)≤αn for anyn∈N, then{xn}is a d-Cauchy sequence.

Proof. Proof is similar to that of Lemma 1 [5] and we left it.

Theorem 3.1. Let (X, d) be a complete metric space, let p be a generalized w- distance of orderj onX and let T be a mapping from X into itself. Suppose that there existsr∈[0,1)such that

p(T x, T y)≤r p(x, y) (3.1)

for everyx, y∈X. Then there existsz∈X such thatz=T z. Moreover, ifv=T v, thenp(v, v) = 0.

Proof. Letube an arbitrary element ofX. We consider the sequence{u_n} where u_n = Tⁿu for any n ∈ N. We can suppose that Tⁿu ̸= T^mu for all distinct n, m∈N. In fact, ifTⁿu=T^mufor somem, n∈N,m̸=nthen assumingm > n, we have

T^m⁻ⁿ(Tⁿu) =Tⁿu

i.e., T^ky=y where k=m−n >0and y=Tⁿu.

Ifk= 1, then T y=y andy is a fixed point ofT. Again ifk >1, then

p(y, T y) =p(T^ky, T^k+1y)≤r^kp(y, T y) and beingr <1 one hasp(y, T y) = 0.

Also,

p(y, y) =p(T^ky, T^ky)≤r^kp(y, y) and beingr <1 one hasp(y, y) = 0.

Sincep(y, T y) = 0 andp(y, y) = 0, by using Lemma 3.1(i), we getT y=y i.e.,y is a fixed point ofT.

Thus in the sequel of the proof we can suppose that Tⁿu̸= T^mufor all distinct n, m∈N.

Let us now prove that for alln, m∈N, one has p(Tⁿu, T^n+mu)≤ rⁿ

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j}

. (3.2)

By using (3.1), we have

p(Tⁿu, T^n+mu)≤rⁿp(u, T^mu). (3.3) Ifm≤j, then

p(u, T^mu) ≤ (1 +r+r²+· · ·)p(u, T^mu)

≤ 1

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j} .

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Ifm > j, then there existss∈N such thatm=sj+t, where 0≤t < j.

Ift= 0, then by using (3.1)

p(u, T^mu) ≤ p(u, T u) +p(T u, T²u) +· · ·+p(T^j⁻¹u, T^ju) +p(T^ju, T^mu)

≤ p(u, T u) +rp(u, T u) +· · ·+r^j⁻¹p(u, T u) +r^jp(u, T^m⁻^ju)

=

j−1

∑

q=0

r^qp(u, T u) +r^jp(u, T^m⁻^ju). (3.4) By repeated application of (3.4), we obtain at (s−1)-th step that

p(u, T^mu) ≤

(s−∑1)j−1

q=0

r^qp(u, T u) +r^(s⁻^1)jp(u, T^ju)

≤ (1 +r+r²+· · ·+r^(s⁻^1)j)max{

p(u, Tⁱu) :i= 1,2,· · ·, j}

≤ 1

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j} . Ift̸= 0, then by (3.1)

p(u, T^mu) ≤ p(u, T u) +p(T u, T²u) +· · ·+p(T^j⁻¹u, T^ju) +p(T^ju, T^mu)

≤ p(u, T u) +rp(u, T u) +· · ·+r^j⁻¹p(u, T u) +r^jp(u, T^m⁻^ju)

=

j−1

∑

q=0

r^qp(u, T u) +r^jp(u, T^m⁻^ju). (3.5) By repeated application of (3.5), we obtain ats-th step that

p(u, T^mu) ≤

sj∑−1

q=0

r^qp(u, T u) +r^sjp(u, T^tu)

≤ (1 +r+r²+· · ·+r^sj)max{

p(u, Tⁱu) :i= 1,2,· · ·, j}

≤ 1

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j} . So, ifm > j then it must be the case that

p(u, T^mu)≤ 1

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j} . Now, using (3.3) we have for alln, m∈N,

p(Tⁿu, T^n+mu)≤ rⁿ

1−rmax{

p(u, Tⁱu) :i= 1,2,· · ·, j} .

By Lemma 3.1(iii), {un} is a Cauchy sequence in (X, d) which is a complete metric space. So there exists a pointz∈X such thatz= lim

n un.

Letn∈Nbe fixed. Since{u_m}converges tozandp(u_n, .) is lower semi continuous, one obtains

p(un, z)≤ lim

m→∞inf p(un, um)≤ rⁿ

1−rmax{

p(u, T u), p(u, T²u)} , which implies that,p(un, z)→0 asn→ ∞.

Again, from (3.1)

p(u_n+1, T z) =p(T u_n, T z)≤r p(u_n, z)→ as n→ ∞.

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Thus, by Lemma 3.1(i), p(un+1, T z)→0 andp(un+1, z)→0 imply thatT z =z.

Therefore,zbecomes a fixed point of T.

Ifv=T v, then

p(v, v) =p(T v, T v)≤r p(v, v) and hencep(v, v) = 0.

Corollary 3.1. (Banach Contraction Theorem) Let (X, d) be a complete metric space andT :X →X be a mapping such that

d(T x, T y)≤αd(x, y) (3.6)

for allx, y∈X and0< α <1.ThenT has a unique ﬁxed point inX.

Proof. We see that d is a generalizedw-distance of order 1. So, by Theorem 3.1 there exists z∈X such thatT z=z. Uniqueness follows from condition (3.6).

We now furnish an example which shows that the condition (3.1) in Theorem 3.1 can neither be relaxed.

Example 3.1. Take X = [2,∞)∪ {0,1}, which is a complete metric space with usual metricdof reals. Deﬁne T :X →X where

T x = 0f or x∈(X\ {0})

= 1f or x= 0.

Clearly, T possesses no ﬁxed point inX.

In fact, forx= 0 andy=T x=T0in X, we ﬁnd that d(T x, T y) = 1> r d(x, T x) for any r∈[0,1).

Hence condition(3.1) fails and Theorem3.1 does not hold.

Note: In example above we treat das a generalizedw-distance of order1 inX in reference to Theorem3.1.

Acknowledgement. The author is very grateful to the referee for his helpful sug- gestions to improve the paper.

References

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45(1974) 267-273.

[3] B. Fisher,A fixed point theorem, Math. Mag.48 (1975) 223-225.

[4] R. Kannan,Some results on fixed points-II, Amer. Math. Monthly76 (1969) 405-408.

[5] Osamu Kada, Tomonari Suzuki and Wataru Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japonica44(2)(1996) 381-391.

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Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), West Bengal, Kolkata 700126, India.

E-mail address:[email protected]