J. Comp. & Math. Sci. Vol.3 (1), 103-107 (2012)
Kannan’s Theorem in Generalised Quasi Metric Spaces P. SUMATI KUMARI Department of Mathematics, FED – I, K L University. Green Fields, Vaddeswaram, A.P, 522502, India. (Received on: 23rd January, 2012) ABSTRACT Quasi metrics have been used in several places in the literature on domain theory and the formal semantics of programming languages1,5.The notion of a generalized dislocated quasi metric space( gq space) is introduced and an analogue of Kannan’s fixed point theorem4 is obtained from which existence and uniqueness of fixed points for self maps that satisfy the metric analogues of contractive conditions mentioned in4 can be derived. Keywords: Generalised Quasi metric, fixed point,
β -property.
AMS Subject classification: 47H10.
1. INTRODUCTION Rhoades4 collected a large number of variants of Banach’s Contractive conditions on self maps on a metric space and proved various implications or otherwise among them. We pick up a good number of these conditions which ultimately imply Kannan condition4. We prove that these implications hold good for self maps on a gq metric space and prove the gq metric version of Kannan’s result then by deriving the gq analogue’s of fixed point theorems of Hardy and Rogers, Bianchini, Reich, Ciric and others.
We denote the set of non-negative real + numbers by R and set of natural numbers by N. 1.13 Let binary operation ◊ : R + × R + → R + satisfies the following conditions: (I) ◊ is Associative and Commutative, (II) ◊ is continuous w.r.t to the usual metric R + A few typical examples are a ◊ b = max{ a , b }, a ◊ b = a + b , a ◊ b = a b , a ◊ b = a b + a + b and
a ◊ b=
ab for each a , b ∈ R + . max{a, b,1}
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
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P. Sumati Kumari, J. Comp. & Math. Sci. Vol.3 (1), 103-107 (2012)
In what follows we fix a binary operation ◊ that satisfies (l) and (ll)
Definition 1.4: We say that a net { xα / α ∈ ∆ } in X converges to x in(X, gq )
Definition 1.23 A binary operation ◊ on R + is said to satisfy β -property if there exists a positive real number β such that a ◊ b ≤ β max{ a , b } for every a , b ∈
and
+
R . Definition 1.3: Let X be a non empty set. A generalized quasi (simply gq ) metric on X is a function gq : X 2 → R + that satisfies the following conditions: (1) gq ( x , x ) = 0 (2) gq( x , y ) = gq( y , x ) = 0 implies x = y. (3) gq( x , z ) ≤ gq(x, y)◊ gq( y, z) for all x, y, z ∈ X. The pair (X, gq ) is called a generalized quasi metric space.
lim{ xα / α ∈ ∆} = x
write
α
if
lim gq( xα , x) = 0 and lim gq( x , xα ) = 0. Definition 1.5 : A sequence ( x ) in a gq n metric space is a Cauchy sequence if, for all ∈ >0, there corresponds nε ∈ N such that
for all n ≥ m ≥ nε we have gq ( xn , xm )< ∈ .The gq metric space is said to be complete if every Cauchy sequence is convergent. Now we prove the gq metric version of Kannan’s theorem
Theorem 1.6: Let X be a gq complete metric space such that ◊ satisfies β property with β ≤ 1 .If f is continous and gq ( f ( x), f ( y )) < a{gq ( x, f ( x))◊gq( y, f ( y ))} for each
x, y ∈ X , where
Then f has a unique fixed point.
Proof: Consider
gq( f ( x), f 2 ( x)) < a{gq( x, f ( x))◊gq( f ( x), f 2 ( x))} < aβ max{gq ( x, f ( x)), gq( f ( x), f 2 ( x))} < a{gq( x, f ( x)) + gq( f ( x), f 2 ( x))} Thus gq ( f ( x), f 2 ( x)) < γ gq( x, f ( x)) where γ =
a <1 1− a
If m > n then Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
1 0<a< . 2
P. Sumati Kumari, J. Comp. & Math. Sci. Vol.3 (1), 103-107 (2012)
105
gq ( f n ( x ), f m ( x )) ≤ gq ( f n ( x ), f n +1 ( x )) ◊ gq ( f n +1 ( x ), f n + 2 ( x )) ◊...... ◊ gq ( f m −1 ( x ), f m ( x )) ≤ β max{ gq ( f n ( x ), f n +1 ( x )), gq ( f n +1 ( x ), f n + 2 ( x )),......, gq ( f m −1 ( x ), f m ( x ))} ≤ gq ( f n ( x ), f n +1 ( x )) + gq ( f n +1 ( x ), f n + 2 ( x )) + ...... + gq ( f m −1 ( x ), f m ( x )) ≤ γ n (1 + γ + γ 2 + ..... + γ m − n − 1 ) gq ( x , f ( x )) <
γn gq ( x , f ( x )) 1− γ
n Hence { f (x) } is Cauchy’s sequence in (X, gq ) ,hence convergent.
Let ξ = lim f
n
n
( x ) . Then
f n+1 (x ) is Cauchy and f ( ξ )= lim f n+1 (x ) n
gq(ξ , f (ξ )) ≤ gq(ξ , f n ( x))◊gq( f n ( x), f n+1 ( x))◊gq( f n +1 ( x), f (ξ )) ≤ β max{gq(ξ , f n ( x)), gq( f n ( x), f n +1 ( x)), gq( f n+1 ( x), f (ξ ))} ≤ gq(ξ , f n ( x)) + gq( f n ( x), f n+1 ( x)) + gq( f n+1 ( x), f (ξ )) < gq( f n ( x), f n+1 ( x)) < γ n gq( x, f ( x)) Since 0< γ < 1 ;
lim γ gq ( x, f ( x) ) =0 n
Similarly gq ( f (ξ ), ξ ) = 0 ⇒ f ( ξ )= ξ
Uniqueness: Suppose f ( ξ )= ξ
Hence gq ( ξ , f ( ξ ) )=0,
and f (η )=η
d (ξ ,η ) = lim d ( f n ( x ),η ) = lim d ( f n+1 ( x ),η ) =0 Similarly, d (η , ξ ) = 0 Hence ξ =η B.E Rhodes4 presented a list of definitions of contractive type conditions for a self map on a metric space ( X , d ) and established implications and nonimplications among them ,there by facilitating to check the implication of any new contractive condition through any one of the condition mentioned in4 so as to derive a fixed point theorem. Among the conditions in4 Kannan’s condition is significant as a good
number of contractive conditions imply Kannan’s condition .These implications also hold good in the present context as well. We state the gq metric version of some of the contractive conditions mentioned in4 and derive various implications and non implications and deduced fixed point theorem for gq metric version of Kannan’s theorem.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
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P. Sumati Kumari, J. Comp. & Math. Sci. Vol.3 (1), 103-107 (2012)
Let (X, gq ) be a gq metric space with a ◊ +
b = a + b for each a , b ∈ R and
f : X → X be a mapping and x , y be any elements of X.
1 such that 2 gq ( f ( x ) , f ( y )) < α [ gq ( x , f ( x )) ◊ gq ( y, f ( y ))] 1 2. (Bianchini): there exists a number h ,0 ≤ h < such that 2 gq ( f ( x ), f ( y )) ≤ h max{ gq ( x , f ( x )) , gq ( y , f ( y ))}
1. (Kannan) :there exists a number α , 0 < α <
3. (Reich) : there exist nonnegative numbers a, b, c satisfying a + b + c < 1 such that gq ( f ( x ), f ( y )) ≤ a gq ( x , f ( x )) ◊ b gq ( y , f ( y )) ◊ c gq ( x , y ) 4. there exist nonnegative functions a, b, c satisfying sup a( x , y )+b ( x , y )+c ( x , y ) x , y∈X
< 1 such that gq ( f ( x ), f ( y )) ≤ a(t) gq ( x , f ( x ))◊b(t) gq ( y , f ( y ))◊ c(t) gq (t) where t=( x , y ) 5. (Hardy and Rogers): gq( f ( x ), f ( y )) ≤ a1 gq ( x, y )◊a 2 gq( x, f ( x))◊a 3 gq ( y , f ( y ))◊a 4 gq ( x, f ( y ))◊a 5 gq( y, f ( x ))
Where
sup { ∑ ai ( x, y ) } < 1 For every x ≠ y
x , y∈ X
i
6. (Ciric): For each x, y ∈ X
gq ( f ( x ), f ( y )) ≤ q ( x, y ) gq ( x , y ) ◊ r ( x, y ) gq ( x , f ( x )) ◊ s ( x, y ) gq ( y , f ( y )) ◊ t ( x, y ) [ gq ( x, f ( y )) ◊ gq ( y, f ( x))] sup {q ( x, y ) + r ( x, y ) + s ( x, y ) + 2t ( x, y )} ≤ λ < 1 x , y∈ X
Theorem 1.7:Let X be a gq complete metric space such that ◊ satisfies β -property with β ≤1. If f is continuous and satisfies any one of the conditions 2 through 6. Then f has a unique fixed point.
Proof: It now follows from Theorem 1.6 that f has a unique fixed point.
ACKNOWLEDGEMENT The author is grateful to Dr. I. Ramabhadra Sarma for his valuable comments and suggestions.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)
P. Sumati Kumari, J. Comp. & Math. Sci. Vol.3 (1), 103-107 (2012)
REFERENCES 1. Pascal Hitzler: Generalised metrices and topology in logic programming semantics, Ph. D Thesis, (2001). 2. John l Kelley: General topology. 3. S. Sedghi: fixed point theorems for four mappings in d*-metric spaces, Thai Journal of Mathematics, Vol 7, November 1:9-19 (2009). 4. B. E. Rhoades: A comparison of various definitions of contractive mappings,
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Trans of the Amer. Math. Society Vol. 226, 257-290 (1977). 5. S. G. Mathews: Metric domains for completeness, Technical report 76, Department of computer science, University of Warwick, U. K., Ph.D Thesis (1985). 6. V. M. Sehgal : On fixed and periodic points for a class of mappings, Journal of the London Mathematical Society (2), 5, 571-576 (1972).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 1, 29 February, 2012 Pages (1-130)