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ISSN: 2347
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Available Online: http://ijmaa.in/
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ISSN: 2347-1557
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Volume 4, Issue 3–A (2016), 93–97.
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International Journal of Mathematics And its Applications
International Journal of Mathematics And its Applications
A Fixed Point Theorem for Meir Keeler Type Contraction
Using Result of Wong, Chi Song
Research Article
Sujata Goyal1∗
1 Department of Mathematics, A.S.College, Khanna, India.
Abstract:
In this paper, we establish a new fixed point theorem for Meir Keeler type contraction using result of Wong [2]. The
presented theorem is an extension of the result of Wong [2].
Keywords: Complete metric space, Fixed point, Meir-Keeler type contraction.
c JS Publication.
1.
Introduction
The Banach contraction principle [1] is the most celebrated fixed point theorem. It is very useful, simple and classical tool
in nonlinear analysis. Moreover this principle has many generalizations (see for example [2–10]). Among the most relevant
results in this direction, one can give that of Meir and Keeler [3] who proved the following fixed point result.
Theorem 1.1. Let (X, d) be a complete metric space and T be a self mapping from X into itself satisfying the following
condition:
∀ > 0, ∃ δ() > 0 such that ≤ d(x, y) < + δ() ⇒ d(T x, T y) < Then T has a unique fixed point z ∈ X. Moreover, for all x ∈ X, the sequence {T n x} converges to z.
Wong [2] proved the following fixed point result:
Theorem 1.2. Let T be a self mapping on a complete metric space (X, d). Suppose there exists symmetric functions αi ,
i=1,2,. . . , 5, of X × X into [0, 1) such that
(a). r = sup{
5
P
αi (x, y) : x, y ∈ X} < 1
i=1
(b). for any x, y in X
d(T x, T y) ≤ a1 d(x, y) + a2 d(x, T x) + a3 d(y, T y) + a4 d(x, T y) + a5 d(y, T x), where ai = αi (x, y)
then T has a unique fixed point.
In this paper, a new fixed point theorem of Meir Keeler type that generalizes Theorem 1.2 of Wong, Chi Song in the case
when each ai is assumed to be the constant in [0, 1/5] has been derived.
∗
E-mail: [email protected]
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A Fixed Point Theorem for Meir Keeler Type Contraction Using Result of Wong, Chi Song
2.
Main Result
Theorem 2.1. Let (X, d) be a complete metric space and T be a mapping of X into itself. We assume that the following
hypothesis holds: given > 0, ∃ δ() > 0 such that
5 ≤ d(x, y) + d(x, T x) + d(y, T y) + d(x, T y) + d(y, T x) < 5 + δ() ⇒ d(T x, T y) < .
(1)
Then T has a unique fixed point z ∈ X. Moreover, for all x ∈ X, the sequence {T n x} converges to z.
To prove the Theorem we need the following lemma :
Lemma 2.2. Let (X, d) be a complete metric space and T be a mapping of X into itself. Then the conditions (A) & (B)
are equivalent :
(A). Given > 0, ∃ δ() > 0 such that 5 ≤ d(x, y)+d(x, T x)+d(y, T y)+d(x, T y)+d(y, T x) < 5+δ() ⇒ d(T x, T y) < .
(B). Given > 0, ∃ δ() > 0 such that d(x, y) + d(x, T x) + d(y, T y) + d(x, T y) + d(y, T x) < 5 + δ() ⇒ d(T x, T y) < .
Proof.
Clearly (B) ⇒ (A). Now suppose condition (A) holds. For convenience, let M (x, y) =
1
{d(x,
5
y) + d(x, T x) +
d(y, T y) + d(x, T y) + d(y, T x)}. If M (x, y) < , then by condition (A), we have d(T x, T y) ≤ M (x, y) and thus
d(T x, T y) < . If M (x, y) ≥ then condition (B) immediately holds.
Proof of Main Theorem. First note that trivially (1) implies
d(T x, T y) <
1
{d(x, y) + d(x, T x) + d(y, T y) + d(x, T y) + d(y, T x) for x 6= y
5
(2)
Let x0 ∈ X and consider the sequence {xn } = {T n x0 }, n ∈ N , we will prove that {xn } is a Cauchy sequence in X. If there
exists t ∈ N such that xt = xt+1 , the clearly xt is a fixed point of T. So assume that xn 6= xn+1 for all n.
Define sn = d(xn , xn+1 ) ∀ n. By (2), we have
sn = d(xn , xn+1 ) = d(T xn−1 , T xn ) <
1
{d(xn−1 , xn ) + d(xn−1 , xn ) + d(xn , xn+1 ) + d(xn−1 , xn+1 ) + d(xn , xn )}
5
1
{d(xn−1 , xn ) + d(xn−1 , xn ) + d(xn , xn+1 ) + d(xn−1 , xn ) + d(xn , xn+1 )}
5
3
2
= sn−1 + sn
5
5
≤
⇒ sn < sn−1 ∀ n. Thus the sequence {sn } is strictly decreasing. Therefore the sequence {sn } is convergent. Let it converge
to s, where s ≥ 0. If possible, let s > 0. Now 2sn + sn−1 → 5s as n → ∞. According to condition (1), for 5s > 0, there
exists δ(5s) > 0 with the given property. Now for δ(5s) > 0, there exists M such that 5s ≤ 2sM + 3sM −1 < 5s + δ(5s)
⇒ 2d(xM , xM +1 ) + 3d(xM −1 , xM ) < 5s + δ(5s)
⇒ d(xM −1 , xM ) + d(xM −1 , T xM −1 ) + d(xM , T xM ) + d(xM −1 , T xM ) + d(xM , T xM −1 )
= d(xM −1 , xM ) + d(xM −1 , xM ) + d(xM , xM +1 ) + d(xM −1 , xM +1 ) + d(xM , xM )
≤ 2d(xM −1 , xM ) + d(xM , xM +1 ) + d(xM −1 , xM ) + d(xM , xM +1 )
= 2d(xM , xM +1 ) + 3d(xM −1 , xM )
< 5s + δ(5s)
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Sujata Goyal
thus by Lemma 2.2, d(T xM −1 , T xM ) < s ⇒ d(xM , xM +1 ) < s ⇒ sM < s, which is a contradiction. Thus we deduce that
sn → 0 as n → ∞. Now, let δ 1 () = min{δ(), }. By the convergence of the sequence {sn } to 0, there exists L ∈ N such
that
d(xn , xn+1 ) <
Define Ω = {xp : p ≥ L, d(xp , xL ) < + δ
1
()
},
5
δ 1 ()
∀ n≥L
10
(3)
we will show that
T (Ω) ⊂ Ω
let u ∈ Ω, then there exists p ≥ L such that u = xp and d(xp , xL ) <
(4)
5
3
+
δ 1 ()
.
5
If p = L, then clearly T (u) = xL+1 ∈ Ω by
(3). So assume that p > L. Now we have two cases:
Case I:
d(xp , xL ) <
δ 1 ()
5
+
3
5
(5)
We shall show that
≤
δ 1 ()
1
{d(xp , xL ) + d(x, xp+1 ) + d(xL , xL+1 ) + d(xp , xL+1 ) + d(xp+1 , xL )} < +
5
5
(6)
From (3) & (5), we have
≤
3
2
2
3
d(xp , xL ) ≤ d(xp , xL ) + d(xp+1 , xp ) + d(xL+1 , xL )
5
5
5
5
δ 1 ()
4 δ 1 ()
3 5
<
+
+
5 3
5
5
10
=+
⇒≤
δ 1 ()
5
3
d(xp , xL )
5
(7)
and
δ 1 ()
3
2
2
d(xp , xL ) + d(xp+1 , xp ) + d(xL+1 , xL ) < +
5
5
5
5
(8)
From (7), we have,
+
δ 1 ()
1
1
1
1
1
≤ d(xp , xL ) + d(xp , xp+1 ) + d(xp+1 , xL ) + d(xp , xL+1 ) + d(xL+1 , xL )
5
5
5
5
5
5
(9)
From (8), we have,
1
{d(xp , xL ) + d(xp , xp+1 ) + d(xp+1 , xL ) + d(xp , xL+1 ) + d(xL+1 , xL )}
5
1
≤ {d(xp , xL ) + d(xp , xp+1 ) + d(xp+1 , xp ) + d(xp , xL ) + d(xp , xL ) + d(xL , xL+1 ) + d(xL+1 , xL )}
5
3
2
2
= d(xp , xL ) + d(xp+1 , xp ) + d(xL+1 , xL )
5
5
5
δ 1 ()
<+
5
(10)
Combining (9) & (10), (6) is established. From (1), (3) & (6), we have
d(T xp , xL ) ≤ d(T xp , T xL ) + d(T xL , xL ) < +
δ 1 ()
δ 1 ()
5
<
+
⇒ T xp ∈ Ω ⇒ T u ∈ Ω
10
3
5
95
A Fixed Point Theorem for Meir Keeler Type Contraction Using Result of Wong, Chi Song
Case II: d(xp , xL ) <
5
3
d(T xp , xL ) ≤ d(T xp , T xL ) + d(T xL , xL )
1
d(xp , xL ) +
5
3
= d(xp , xL ) +
5
<
Thus d(T xp , xL ) ≤
3 5
( )
5 3
1
d(xp+1 , xp ) +
5
2
d(xp+1 , xp ) +
5
1
1
1
1
1
d(xL , xL+1 ) + d(xL+1 , xp ) + d(xL , xp+1 ) + d(xL+1 , xL )
5
5
5
5
2
d(xL+1 , xL )
5
1
+ 52 ( δ 10() ) + 53 ( δ 10() ) = +
δ 1 ()
10
⇒ T xp ∈ Ω ⇒ T u ∈ Ω. Thus (4) is proved. Now from (4), we
have,
d(xm , xL ) <
δ 1 ()
5
+
∀m≥L
3
5
⇒ d(xm , xn ) ≤ d(xm , xL ) + d(xL , xn ) <
⇒ d(xm , xn ) <
10
2
+ δ 1 () ∀ m, n ≥ L
6
5
10
2
2
+ δ 1 () < 2 + < 3 ∀ m, n ≥ L
6
5
5
Thus the sequence {xn } is a Cauchy sequence. Since X is complete, the sequence xn → z (say). Now
1
{d(z, xn ) + d(z, T z) + d(xn , xn+1 ) + d(z, T xn ) + d(xn , T z)} + d(xn+1 , z)
5
1
1
1
6
4
⇒ d(T z, z) < d(z, xn ) + d(xn , xn+1 ) + d(xn , T z) + d(xn+1 , z)
5
5
5
5
5
d(T z, z) ≤ d(T z, T xn ) + d(T xn , z) <
Letting n → ∞, we get:
4
1
d(T z, z) ≤ d(T z, z) ⇒ d(T z, z) ≤ 0 ⇒ T z = z.
5
5
Uniqueness : Let w be another fixed point of T, then T z = z & T w = w, z 6= w. Now
1
{d(z, w) + d(z, T z) + d(w, T w) + d(z, T w) + d(w, T z)}
5
3
3
⇒ d(z, w) < d(z, w) ⇒ 1 < ,
5
5
d(z, w) = d(T z, T w) <
a contradiction. Thus the uniqueness of fixed point is proved, and it completes the proof of the theorem.
From the main theorem, we immediately get the following corollary which extends the known result of Wong [2] for the case
when each ai is assumed to be the constant in [0, 1/5]
Corollary 2.3 ([2]). Let T be a self mapping on a complete metric space (X, d). Suppose there exist a constant k such that
for any x, y in X,
d(T x, T y) ≤ k{d(x, y) + d(x, T x) + d(y, T y) + d(x, T y) + d(y, T x)}
where k ∈ [0, 1/5] then T has a unique fixed point z ∈ X. Also the sequence {T n x} converges to z for every x ∈ X.
References
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18(3)(1974), 265-276.
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Sujata Goyal
[3] A.Meir and E.Keeler, A theorem on contraction mappings, Journal of Mathematical Analysis and Applications,
28(2)(1969), 326-329.
[4] M.U.Ali, T.Kamran and E.Karapınar, Fixed point of α-ψ-contractive type mappings in uniform spaces, Fixed Point
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