International Mathematical Forum, 4, 2009, no. 36, 1783 - 1791
Fixed Point Theorems of Contractive Type Mappings in a 2-Metric Space Mantu Saha Department of Mathematics, The University of Burdwan Burdwan-713104, West Bengal, India
[email protected] Debashis Dey Koshigram Union Institution, Koshigram-713150, Burdwan West Bengal, India
[email protected] Abstract Here we have proved Some fixed point theorems for a class of contractive type mappings in a setting of 2-metric space using the notion of (C, 1) summability.
Mathematics Subject Classification: 47H10, 54H25 Keywords: fixed point, 2-metric space, (C, 1) summable
1
Introduction
The introduction of a 2-metric space was initially introduced by G¨ahler in a series of papers ([1]-[2]) in 1963-1965. Then about a decade after, Iseki[3] found some basic fixed point results in a setting of 2-metric space. After that some important fixed pointic results are obtained by Rhoades[6], Saha et al. ([7]-[8]) in this space. In the present paper we deal with a mixed type of mappings[5] as well as larger than the class of Kannan[4] type mappings in a 2-metric space where summable by Cesaro means of the first order (C, 1) plays a key role in proof of fixed point theorems.
2
Preliminary Notes
Definition 2.1 Let X be a non empty set. A real valued function d on X × X × X is said to be a 2-metric on X if
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(i) given distinct elements x,y of X, there exists an element z of X such that d(x, y, z) = 0 (ii) d(x, y, z) = 0 when at least two of x, y, z are equal, (iii) d(x, y, z) = d(x, z, y) = d(y, z, x) for all x, y, z in X and (iv) d(x, y, z) ≤ d(x, y, w) + d(x, w, z) + d(w, y, z) for all x, y, z, w in X. When d is a 2-metric on X, then the ordered pair (X, d) is called a 2-metric space. Definition 2.2 A sequence {xn } in X is said to be a Cauchy sequence if for each a ∈ X, lim d(xn , xm , a) = 0 as n, m → ∞. Definition 2.3 A sequence {xn } in X is convergent to an element x ∈ X if for each a ∈ X, lim d(xn , x, a) = 0 n→∞
Definition 2.4 A complete 2-metric space is one in which every Cauchy sequence in X converges to an element of X. Definition 2.5 Let {ak } be a sequence of real constants and
n
ak = Sn .
k=0
If the sequence {Sn } converges to S, then we associate the number S with the series
∞
n=0
an as its sum. It may happen that {Sn } is not convergent but the
sequence {σn } where σn =
(S0 + S1 + ....... + Sn ) (n + 1)
(1)
converges to L (say), as n → ∞. Then we can consider the number L as a conventional sum of the series
∞
n=0
sequence {Sn }.
an , or as a conventional limit of the
Definition 2.6 Let {Sn } be a sequence of real numbers and let {σn } be a sequence of arithmetic means defined by (1), if {σn } converges to L, then {Sn } is said to be summable by Cesaro means of the first order or briefly summable (C, 1) to L. If the series
∞
n=0
an has partial sums Sn and the sequence {Sn } is
summable (C, 1) to L, then the series
∞ n=0
an is said to be summable (C, 1) and
L is said to be it’s (C, 1) sum. If a sequence {Sn } is convergent to S, then it is summable (C, 1) to S.
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3
Main Results
These are the main results of the paper. Theorem 3.1 Let X be a complete 2-metric space and 0 ≤ βi,j , γi,j < 1 (i,j=1,2,.......), satisfy (I) for each i, βi,j ≤ α< 1 eventually with respect to j and ∞ βi,i+1 + γi,i+1 (II) is (C, 1) summable. 1 − βi,i+1 i=1 If {Tn } is a sequence of self maps on X satisfying d(Ti (x), Tj (y), a) ≤ βi,j [d(x, Ti (x), a) + d(y, Tj (y), a] + γi,j d(x, y, a)
(2)
for x, y, a ∈ X; i,j=1,2,....... with x = y .Then {Tn } have a unique common fixed point in X. Proof. For any x ∈ X, let xn = Tn (xn−1 ), n = 1, 2, ...... with x = x0 , then d(x1 , x2 , a) = d(T1 (x0 ), T2 (x1 ), a) ≤ β1,2 [d(x0 , T1 (x0 ), a) + d(x1 , T2 (x1 ), a)] + γ1,2 d(x0 , x1 , a) = β1,2 [d(x0 , x1 , a) + d(x1 , x2 , a)] + γ1,2 d(x0 , x1 , a) implies (1 − β1,2 ) d(x1 , x2 , a) ≤ (β1,2 + γ1,2 ) d(x0 , x1 , a) β1,2 + γ1,2 d(x0 , x1 , a) ⇒ d(x1 , x2 , a) ≤ 1 − β1,2 Also d(x2 , x3 , a) = d(T2 (x1 ), T3 (x2 ), a) β2,3 + γ2,3 d(x1 , x2 , a) ≤ 1 − β2,3 β2,3 + γ2,3 β1,2 + γ1,2 d(x0 , x1 , a) ≤ 1 − β2,3 1 − β1,2 In general, d(xn , xn+1 , a) ≤
n i=1
βi,i+1 + γi,i+1 d(x0 , T1 (x0 ), a) 1 − βi,i+1
Again d(xn , xn+2 , a) = d(xn+2 , xn , a) ≤ d(xn+2 , xn , xn+1 ) + d(xn+2 , xn+1 , a) + d(xn+1 , xn , a) = d(xn+2 , xn , xn+1 ) +
1 k=0
d(xn+k , xn+k+1 , a)
(3)
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similarly d(xn , xn+3 , a) ≤ In this fashion, for p > 0 d(xn , xn+p , a) ≤
p−2 k=0
1
d(xn+3 , xn+k , xn+k+1 ) +
k=0
2
d(xn+k , xn+k+1, a)
k=0
d(xn+p , xn+k , xn+k+1 ) +
p−1 k=0
d(xn+k , xn+k+1 , a)
(4)
Now by (3), p−2
d(xn+p , xn+k , xn+k+1 ) = d(xn+p , xn , xn+1 ) + d(xn+p , xn+1 , xn+2 )
k=0
+.............. n n+1 βi,i+1 + γi,i+1 βi,i+1 + γi,i+1 ≤ + 1 − βi,i+1 1 − βi,i+1 i=1 i=1 +.............] d(x0 , T1 (x0 ), xn+p ) (5) Again d(x0 , T1 (x0 ), xn+p ) = d(Tn+p (xn+p−1 ), T1 (x0 ), x0 ) ≤ βn+p,1 [d(xn+p−1 , xn+p , x0 ) + d(x0 , x1 , x0 )] +γn+p,1d(xn+p−1 , x0 , x0 ) = βn+p,1 d(xn+p−1 , xn+p , x0 ) Put n + p − 1 = m then, d(x0 , T1 (x0 ), xm+1 ) ≤ βm+1,1 d(xm , xm+1 , x0 ) m βi,i+1 + γi,i+1 ≤ βm+1,1 d(x0 , T1 (x0 ), x0 ) 1 − βi,i+1 i=1 ⇒ d(x0 , T1 (x0 ), xn+p ) = 0 Then from (5),
p−2 k=0
d(xn+p , xn+k , xn+k+1) = 0
Also from (4) we get, d(xn , xn+p , a) ≤ ≤ ≤
p−1
d(xn+k , xn+k+1, a) k=0 p−1 n+k βi,i+1 + γi,i+1 k=0 i=1 n+p−1 k k=n i=1
d(x0 , T1 (x0 ), a)
1 − βi,i+1
βi,i+1 + γi,i+1 d(x0 , T1 (x0 ), a) 1 − βi,i+1
implies d(xn , xn+p , a) ≤
k n+p−1 k=n
i=1
βi,i+1 + γi,i+1 1 − βi,i+1
1 k
k
d(x0 , T1 (x0 ), a)
(6)
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Fixed point theorems of contractive type mappings ⎛
Put sk =
k i=1
βi,i+1 + γi,i+1 1 − βi,i+1
and ∞
Sk =
k
⎞
⎜ sv ⎟ ⎜ ⎟ ⎜ v=1 ⎟ ⎜ k ⎟ ⎜ ⎟ ⎝ ⎠
∞ βi,i+1 + γi,i+1 Then by (C, 1) summability of we have Sk < ∞ and 1 − βi,i+1 i=1 k=1 hence lim Sk = 0. Without loss of generality, let Sk < 1 for all k.
k k
Thus skk ≤ skk ≤ Sk . Now passing on limit n → ∞ in (6), we get d(xn , xn+p , a) → 0. Thus {xn } is a Cauchy sequence and by completeness of X, xn converges to u (say) in X that is lim xn = u ∈ X. So for any positive n integer m, d(u, Tm(u), a) ≤ d(u, Tm(u), xn ) + d(u, xn , a) + d(xn , Tm (u), a)
(7)
Now by (2), d(xn , Tm (u), a) = d(Tn (xn−1 ), Tm (u), a) ≤ βn,m [d(xn−1 , xn , a) + d(u, Tm (u), a)] +γn,m d(xn−1 , u, a) (8) and d(u, Tm (u), xn ) = d(Tn (xn−1 ), Tm (u), u) ≤ βn,m [d(xn−1 , xn , u) + d(u, Tm(u), u)] + γn,m d(xn−1 , u, u) implies d(u, Tm(u), xn ) ≤ βn,m d(xn−1 , xn , u)
(9)
So from (7), (8) and (9), d(u, Tm (u), a) ≤ d(u, xn , a) + βn,m [d(xn−1 , xn , a) + d(u, Tm (u), a)] +γn,m d(xn−1 , u, a) + βn,m d(xn−1 , xn , u) Taking limit as n → ∞, we get d(u, Tm (u), a) ≤ βn,m d(u, Tm (u), a) ≤ αd(u, Tm (u), a) by(I) of Theorem (3.1) As α < 1, it follows that d(u, Tm (u), a) = 0 giving u as a common fixed point of {Tm }. Let v be another common fixed point, then d(u, v, a) = d(Tn (u), Tm (v), a) ≤ βn,m [d(u, Tn (u), a) + d(v, Tm (v), a)] + γn,m d(u, v, a) implies that u = v. So also uniqueness proved.
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Corollary 3.2 . Let X be a complete 2-metric space and 0 ≤ βi,j < 1, (i, j = 1, 2, .........) satisfy (I) for each i, βi,j ≤α < 1 eventually with respect to j and ∞ βi,i+1 (III) is (C, 1) summable. i=1 1 − βi,i+1 If {Tn } is a sequence of self maps on X such that d(Ti (x), Tj (y), a) ≤ βi,j [d(x, Ti (x), a)+d(y, Tj (y), a] for x, y, a ∈ X; i,j=1,2,...., with x = y. Then {Tn } have a unique common fixed point in X. Theorem 3.3 Let X be a complete 2-metric space and 0 ≤ βi,j and γi,j < 1 (i=1,2,.......) satisfy (I) for each i, βi,j ≤ α< 1 eventually with respect to j and ∞ βi,i+1 + γi,i+1 is (C, 1) summable. (II) 1 − βi,i+1 i=1 If {Tn } is a sequence of self maps on X such that for a fixed positive integer p, satisfying (p)
(p)
(p)
(p)
d(Ti (x), Tj (y), a) ≤ βi,j [d(x, Ti (x), a) + d(y, Tj (y), a] + γi,j d(x, y, a)(10) for x, y, a ∈ X; i,j=1,2,....... with x = y .Then {Tn } have a unique common fixed point in X. Proof. For any x0 ∈ X, let xn = Tn(p) (xn−1 ), n=1,2,.........., then (p)
(p)
d(x1 , x2 , a) = d(T1 (x0 ), T2 (x1 ), a) (p)
(p)
≤ β1,2 [d(x0 , T1 (x0 ), a) + d(x1 , T2 (x1 ), a)] + γ1,2 d(x0 , x1 , a) = β1,2 [d(x0 , x1 , a) + d(x1 , x2 , a)] + γ1,2 d(x0 , x1 , a)
+γ1,2 d(x0 , x1 , a) implies d(x1 , x2 , a) ≤ β1,2 1−β1,2 Proceeding as in theorem (3.1) we have
d(xn , xn+1 , a) ≤
n i=1
βi,i+1 + γi,i+1 d(x0 , x1 , a) 1 − βi,i+1
(11)
Let r > 0 be any integer and like theorem (3.1) d(xn , xn+r , a) ≤ (p)
r−2 k=0
d(xn+r , xn+k , xn+k+1) +
(p)
r−1 k=0
d(xn+k , xn+k+1 , a) (p)
Now d(x0 , T1 (x0 ), xn+r ) = d Tn+r (xn+r−1 ), T1 (x0 ), x0
(12)
≤ βn+r,1 [d(xn+r−1 , xn+r , x0 ) + d(x0 , x1 , x0 )] +γn+r,1 d(xn+r−1 , x0 , x0 ) = βn+r,1 d(xn+r−1 , xn+r , x0 )
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Fixed point theorems of contractive type mappings
Put n + r − 1 = m, then (p)
(p)
d(x0 , T1 (x0 ), xn+r ) = d(x0 , T1 (x0 ), xm+1 ) ≤ βm+1,1 d(xm , xm+1 , x0 ) m βi,i+1 + γi,i+1 (p) d(x0 , T1 (x0 ), x0 ) ≤ βm+1,1 1 − βi,i+1 i=1 = 0 Therefore r−2
d(xn+r , xn+k , xn+k+1) = d(xn+r , xn , xn+1 ) + d(xn+r , xn+1 , xn+2 ) + ..........
k=0
≤
n βi,i+1
+ γi,i+1 1 − βi,i+1
i=1
+
n+1
βi,i+1 + γi,i+1 1 − βi,i+1
i=1 (p) +..........] d(x0 , T1 (x0 ), xn+r )
= 0 Also from (12) we get d(xn , xn+r , a) ≤ ≤ ≤
r−1
d(xn+k , xn+k+1 , a) k=0 r−1 n+k βi,i+1 + γi,i+1 k=0 i=1 k n+r−1 k=n i=1
1 − βi,i+1
d(x0 , x1 , a)
βi,i+1 + γi,i+1 d(x0 , x1 , a) 1 − βi,i+1
implies d(xn , xn+r , a) ≤
n+r−1 k=n
k βi,i+1
+ γi,i+1 1 − βi,i+1
i=1
1 k
⎛
Put
sk =
k i=1
βi,i+1 + γi,i+1 1 − βi,i+1
and ∞
Sk =
k
k
d(x0 , T1 (x0 ), a)
(13)
⎞
⎜ sv ⎟ ⎜ ⎟ ⎜ v=1 ⎟ ⎜ k ⎟ ⎜ ⎟ ⎝ ⎠
∞ βi,i+1 + γi,i+1 Then by (C, 1) summability of we have Sk < ∞ and 1 − βi,i+1 i=1 k=1 hence lim Sk = 0. Without loss of generality, let Sk < 1 for all k.
Thus
k k sk k
≤
sk k
≤ Sk .
Now taking limit as n → ∞ on (13), we get
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d(xn , xn+r , a) → 0. Thus {xn } is a Cauchy sequence and by completeness of X, xn converges to some u in X that is lim xn = u ∈ X. n Now for any positive integer m, we have by (10) d(xn , Tm(p) (u), a) = d(Tn(p) (xn−1 ), Tm(p) (u), a) ≤ βn,m [d(xn−1 , xn , a) + d(u, Tm(p)(u), a)] + γn,m d(xn−1 , u, a) Taking limit as n → ∞ we get, d(u, Tm(p) (u), a) ≤ βn,m d(u, Tm(p)(u), a) ≤ αd(u, Tm(p)(u), a) As α < 1, it follows that d(u, Tm(p)(u), a) = 0 gives u = Tm(p) (u) Now
(p)
(p)
d(u, Tj (u), a) = d Tj (u), Tj (Tj (u)), a
(p)
(p+1)
(p)
(p)
= d Tj (u), Tj
(u), a
= d Tj (u), Tj (Tj (u)), a
(p)
(p)
≤ βi,j d u, Tj (u), a + d Tj (u), Tj (Tj (u)), a +γj,j d(u, Tj (u), a)
(p)
(p)
= βj,j d u, Tj (u), a + d Tj (u), Tj Tj (u) , a +γj,j d(u, Tj (u), a) implies that d(u, Tj (u), a) = 0 For uniqueness: Let v be another common fixed point of {Tm }.
d(u, v, a) = d Tm(p) (u), Tn(p)(v), a
≤ βm,n d u, Tm(p) (u), a + d v, Tn(p)(v), a
+ γm,n d(u, v, a)
gives d(u, v, a) ≤ γm,n d(u, v, a) < d(u, v, a) leads to a contradiction, showing that u = v.
References [1] G¨ahler, S., (1963), 2-metric Raume and ihre topologische strucktur, Math.Nachr.,26 , 115 - 148. [2] G¨ahler, S., 1965, Uber die unifromisieberkeit 2-metrischer Raume, Math.Nachr.,28, 235 - 244. [3] Iseki, K., (1975), Fixed point theorems in 2-metric space, Math.Seminar.Notes, Kobe Univ.,3, 133 - 136.
Fixed point theorems of contractive type mappings
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[4] Kannan, R., 1968, Some results on fixed points, Bull. Calcutta Math. Soc.,60, 71 - 76. [5] Reich, S., 1971, Some results concerning contraction mapping, Canad.Math.Bull.,14(1), 121-124. [6] Rhoades, B.E., 1979, Contractive type mappings on a 2-metric space, Math.Nachr.,91, 151 - 155. [7] Saha, M. and Dey, D., 2009, On the theory of fixed points of contractive type mappings in a 2-metric space, Int. Journal of Math. Analysis,Vol. 3, no. 6, 283 - 293. [8] Saha, M. and Baisnab, A.P., 1993, Fixed point of mappings with contractive iterate, Proc.Nat.Acad.Sci.India,63A, IV, 645-650. Received: January, 2009
