On generalized weak partial metric space Mohamed A. Barakat and Ahmed M. Zidan Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt Email:
[email protected] Email:
[email protected]
Abstract In this work, we introduce a new concept of generalized G p -metric space and weak partial metric spaces w called G w p -metric space. The main result of our paper is prove fixed point theorem in a complete G p -metric space. An illustrative example is given to support our results. Keywords Fixed point, G p -metric space, G w p -metric space, weakly increasing map
1
Introduction
In 1996, Matthews, semantics expert, announced in [20] an analog of Banach’s principle in a new space he called a partial metric space. Matthews’s innovative approach was quickly adopted and improved by fixed point theorists (see, e.g. [3],[7],[18],[9],[10], [11],[12],[13],[14],[15],[16], [17],[25]). After that Romaguera [26], [27] proved the Caristi type fixed point theorem on this space and Fixed point theorems for generalized contractions on partial metric spaces. Discoveries of mathematicians following Matthews’s studies revealed new techniques that can be implemented to produce of Banach’s principle. A closer look to the work of these distinguished mathematicians following Matthews’s studies reveals that their discoveries can be categorized in terms of the techniques implemented to produce the analogs of Banach’s principle. The first technique is to introduce new space structures with certain properties which guarantee the existence and uniqueness of fixed points of contractions. In addition to Matthews’s investigations, cone metric spaces, D-metric spaces, and G -metric spaces (see, e.g. [1],[22],[23],[24],[21]) constitute a few of the examples to the first approach. The second technique is to introduce mappings defined on metric spaces satisfying certain new contractive conditions. For example, cyclic contractions and weak ϕ contractions can be listed as a few. As another example to the first approach mentioned above, Zand and Nezhad [28] recently introduced G p -metric spaces which are a combination of the notions of partial metric spaces and G -metric spaces (see, e.g. [5],[6]). Then they proved a number of fixed point theorems on these new spaces for certain type of contractions. In this paper, We introduce the notion of G w p -partial metric space and some results of Altun, et al [4] extended to the class of G w -partial metric space. Also, we present an application to support our results. p Definition 1.1. [22] Let X be a non-empty set. Suppose that G : X × X × X → R + satisfies: (a) G ( x, y, z) = 0 if x = y = z, (b) G ( x, y, z) > 0, ∀ x, y, z ∈ X , x 6= y, (c) G ( x, x, y) ≤ G ( x, y, z), ∀ x, y, z ∈ X , y 6= z, (d) G ( x, y, z) = G ( x, z, y) = G ( y, z, x) = ...,(symmetry in all three variables), (e) G ( x, y, z) ≤ G ( x, a, a) + G (a, y, z), ∀ x, y, z, a ∈ X . Then G is called a G -metric on X and ( X ,G ) is called a G -metric space. Definition 1.2. [28] Let X be a non-empty set. Suppose that G p : X × X × X → R + satisfies: (a) x = y = z if G p ( x, x, x) = G p ( y, y, y) = G p ( z, z, z) = G p ( x, y, z) ∀ x, y, z ∈ X (b) G p ( x, x, x) ≤ G p ( x, x, y) ≤ G p ( x, y, z), ∀ x, y, z ∈ X , y 6= z (c) G p ( x, y, z) = G p ( x, z, y) = G p ( y, z, x) = ...,(symmetry in all three variables) (d) G p ( x, y, z) ≤ G p ( x, a, a) + G p (a, y, z) − G p (a, a, a), ∀ x, y, z, a ∈ X . Then G p is called a G p -Metric on X and ( X ,G p ) is called a G p -partial metric space. Example 1.1. [28] Let X = [0, ∞) and define G p ( x, y, z) = max{ x, y, z} for all x.y.z ∈ X Then ( X ,G p ) is a G p metric space 1
0n the other hand Heckmann [8] introduced the concept of weak partial metric space as follows. Definition 1.3. [8] A weak partial metric space (for short WPMS) on a nonempty set X is a function p w : X × X → R+ (nonnegative real) such that for all x, y, z ∈ X : ( p 1w ) x = y ⇔ p w ( x, x) = p w ( x, y) = p w ( y, y) (T0 -separation axiom), ( p 2w ) p w ( x, y) = p w ( y, x) (symmetry) ( p 3w ) p w ( x, y) ≤ p w ( x, z) + p w ( z, y) − p w ( z, z) (modified triangular inequality). Then p w is called a weak partial metric on X and ( X , p w ) is called a a weak partial metric space. Remark 1.1. [4] If p partial metric on X , then the functions d p , d w : X × X → R+ given by
d p ( x, y) = 2 p( x, y) − p( x, x) − p( y, y)
(1.1)
and
d w ( x, y)
=
max{ p( x, y) − p( x, x), p( x, y) − p( y, y)}
=
p( x, y) − min{ p( x, x), p( y, y)}
(1.2)
are ordinary metrics on X Remark 1.2. Also Heckmann [8] shows that, if p w is a weak partial metric on X , then for all x, y ∈ X , we have the following weak small self-distance property
p w ( x, y) ≥
p w ( x, x) + p w ( y, y) . 2
weak small self-distance property shows that WPMS are not far from small self-distance axiom. It is clear that every PMS is a WPMS, but the converse is not true. Example 1.2. [8] Let X = R+ and let p w : X × X → R+ define by p w ( x, y) = for all x, y ∈ X , so d p is not a metric on X .
x+ y 2 .
Then it is clear that d p ( x, y) = 0
Proposition 1.1. [28] Let ( X ,G p ) is a G p -partial metric space, then for any x, y, z ∈ X and a ∈ X , the following statement are holds (i) G p ( x, y, z) ≤ G p ( x, x, y) + G p ( x, x, z) − G p ( x, x, x) ; (ii) G p ( x, y, y) ≤ 2G p ( x, x, y) − G p ( x, x, x); (iii)G p ( x, y, z) ≤ G p ( x, a, a) + G p ( y, a, a) + G p ( z, a, a) − 2G p (a, a, a); (iv)G p ( x, y, z) ≤ G p ( x, a, z) + G p (a, y, z) − G p (a, a, a) Proposition 1.2. [28] Every G p -metric space ( X ,G p ), define a metric space ( X , D G p ) Where
D G p ( x, y) = G p ( x, y, y) + G p ( y, x, x) − G p ( x, x, x) − G p ( y, y, y) for all x, y ∈ X Definition 1.4. [28] Let G p be G p -metric space and let { xn } be a sequence of points of X . A point x ∈ X is said to be the limit of the sequence { xn } or xn → x if limn,m→∞ G p ( x, xm , xn ) = G p ( x, x, x). Proposition 1.3. [28] Let ( X ,G p ) is a G p -metric space, Then, for any sequence { xn } in X and a point x ∈ X , the following are equivalent that ( A ) { xn } is G p -convergent to x; (B) G p ( xn , xn , x) → G p ( x, x, x) as n → ∞ ( c) G p ( xn , x, x) → G p ( x, x, x) as n → ∞. Remark 1.3. We note that if ( X ,G p ) be G p -metric space, Then
G p ( x, y, z) = 0 ⇒ x = y = z But the converse is not true. Example 1.3. [28] Let X = [0, 1) and let G p : X × X × X → [0, ∞) be a function, where G p ( x, y, z) = max{ x, y, z} for all x, y, z ∈ X .
2
Definition 1.5. [19] Ψ denoted all function ψ : [0, +∞) → [0, +∞) which satisfy (i) ψ is continuous and non-decreasing, (ii) ψ( t) = 0 if and only if t = 0, (iii) ψ( t + s) ≤ ψ( t) + ψ( s), ∀ t, s ∈ [0, +∞). Again, Let Φ denoted all function ϕ : [0, +∞) → [0, +∞) which satisfy lim ϕ( t) > 0 for all r > 0 and lim ϕ( t) = 0. t→ r
t→0+
Altun, et al [4] proved the following result.
Theorem 1.1. [4] Let ( X , p) be a complete W P MS and let f : X → X be a map such that 1 p w ( f x, f y) ≤ φ(max{ p w ( x, y), p w ( x, f x), p w ( y, f y), [ p w ( x, f y) + p w ( y, f x)]}) 2
(1.3)
for all x, y ∈ X where φ : [0, ∞) → [0, ∞) is comparison function. Then f has a unique fixed point.
2
Generalized Weak partial metric space (GWP-metric space)
+ Definition 2.1. Let X be a non-empty set. Suppose that G w p : X × X × X → R satisfies:
w w w (a) x = y = z if G w p ( x, x, x) = G p ( y, y, y) = G p ( z, z, z) = G p ( x, y, z), ∀ x, y, z ∈ X w w w (b) G p ( x, y, z) = G p ( x, z, y) = G p ( y, z, x) = ...,(symmetry in all three variables) w w w (c) G w p ( x, y, z) ≤ G p ( x, a, a) + G p (a, y, z) − G p (a, a, a), ∀ x, y, z, a ∈ X . w w w Then G w p is called a G p -weak partial metric space on X and ( X ,G p ) is called a G p -metric space.
Remark 2.1. If G w : X × X → R+ given by p partial metric on X , then the functions D G w p
D G wp ( x, y)
=
w w w max{G w p ( x, y, y) − G p ( x, x, x),G p ( y, x, x) − G p ( y, y, y)}
=
w w Gw p ( x, y, y) − min{G p ( x, x, x),G p ( y, y, y)}
(2.1)
is ordinary metrics on X w Proposition 2.1. Let ( X ,G w p ) is a G p -partial metric space, then for any x, y, z ∈ X and a ∈ X , it follows that w w (i) G w ( x, y, z ) ≤ G ( x, x, y ) + G ( x, x, z ) − Gw p p p p ( x, x, x) ; w w (ii) G w ( x, y, y ) ≤ 2 G ( x, x, y ) − G ( x, x, x ); p p p w w w w (iii)G w p ( x, y, z) ≤ G p ( x, a, a) + G p ( y, a, a) + G p ( z, a, a) − 2G p (a, a, a).
proof (i)
Gw p ( x, y, z)
=
Gw p ( y, x, z)
≤
w w Gw p ( y, x, x) + G p ( x, x, z) − G p ( x, x, x)
=
w w Gw p ( x, x, y) + G p ( x, x, z) − G p ( x, x, x).
=
Gw p ( y, x, y)
≤
w w Gw p ( y, x, x) + G p ( x, x, y) − G p ( x, x, x)
=
w 2G w p ( x, x, y) − G p ( x, x, x).
proof (ii)
Gw p ( x, y, y)
proof (iii)
Gw p ( x, y, z)
≤
w w Gw p ( x, a, a) + G p (a, y, z) − G p (a, a, a)
≤
w w Gw p ( x, a, a) + G p ( y, a, z) − G p (a, a, a)
≤
w w w Gw p ( x, a, a) + G p ( y, a, a) + G p ( z, a, a) − G p (a, a, a).
w w Definition 2.2. A GW P -metric space ( X ,G w p ) is called a symmetric GW P -metric space if G p ( x, y, y) = G p ( x, x, y) for all x, y ∈ X .
3
w Proposition 2.2. Let ( X ,G w p ) is a G p -partial metric space, then for any a, b ∈ X and a ∈ X , we have
1 w w Gw p ( b, a, a) ≥ (G p (a, a, a) + G p ( b, b, b)) 3 proof w w w w w w w Gw p (a, a, a) ≤ G p (a, b, b) + G p ( b, a, a) − G p ( b, b, b) =⇒ G p (a, a, a) + G p ( b, b, b) ≤ G p ( b, a, a) + 2G p ( b, a, a) 1 w w Therefor, G w p ( b, a, a) ≥ 3 (2G p (a, a, a) + G p ( b, b, b)) w Example 2.1. Let X = [0, ∞) and let G w p : X × X × X → [0, ∞) be a function, where G p ( x, y, z) = w w w x.y.z ∈ X . Then ( X ,G p ) is a G p -metric space. Clearly ( X ,G p ) is not a G p -metric space.
3
x + y+ z 3
for all
Main results
Definition 3.1. Let ( X ,G w p ) be GW P -metric space. Then for x0 ∈ X , r > 0. the GW P -ball of radius r centered at x0 is w BG wp ( x0 , r ) = { y ∈ X : G w p ( x0 , y, y) < r + G p ( x0 , x0 , x0 )}.
Proposition 3.1. Let ( X ,G w ( x0 , r ) for each x0 ∈ X , then there exists δ > 0 p ) be GW P -metric space. If y ∈ BG w p such that BG wp ( y, δ) ⊂ BG wp ( x0 , r ). w Proof. Let y ∈ BG wp ( x0 , r ), then G w p ( x0 , y, y) < r + G p ( x0 , x0 , x0 ). w w w Let δ = r + G p ( x0 , x0 , x0 ) − G p ( x0 , y, y), and z ∈ BG p ( y, δ), then w Gw p ( y, z, z) < δ + G p ( y, y, y).
since
Gw p ( x0 , z, z)
≤
w w Gw p ( x0 , y, y) + G p ( y, z, z) − G p ( y, y, y)
0, there is n 0 ∈ N with ² |G w p ( x n , x m , x m ) − a| < 2 for all n, m ≥ n 0 . Hence
D G wp ( xn , xm )
=
w w Gw p ( x n , x m , x m ) − min{G p ( x n , x n , x n ),G p ( x m , x m , x m )}
=
w w Gw p ( x n , x m , x m ) − a + a − min{G p ( x n , x n , x n ),G p ( x m , x m , x m )}
≤
w w |G w p ( x n , x m , x m ) − a| + |a − min{G p ( x n , x n , x n ),G p ( x m , x m , x m )}| ² ² + = ², 2 2
0, then there exists n 0 ∈ N such that D G w p n→∞
5
² 3
for
all n ≥ n 0 . Thus w |G w p ( x n , x n , x n ) − G p ( x, x, x)|
=
w w w max{G w p ( x n , x n , x n ),G p ( x, x, x)} − min{G p ( x n , x n , x n ),G p ( x, x, x)}
=
3{
w w w max{G w p ( x n , x n , x n ),G p ( x, x, x)} + 2 min{G p ( x n , x n , x n ),G p ( x, x, x)}
3 w − min{G w ( x , x , x ) − G ( x, x, x ) }} n n n p p w Gw p ( x n , x n , x n ) + 2G p ( x, x, x)
≤
w − min{G w p ( x n , x n , x n ),G p ( x, x, x)}] 3 w w 3[G w p ( x n , x, x) − min{G p ( x n , x n , x n ),G p ( x, x, x)}]
=
3D G wp ( xn , x, x) < ²
=
3[
whenever n ≥ n 0 . This shows that ( X ,G w p ) is complete. 1 Now we prove that every Cauchy sequence { xn } in ( X , D G wp ) is a Cauchy sequence ( X ,G w p )). Let ² = 2 .
Then there exists n 0 ∈ N such that D G wp ( xn , xm )
0, there exists n ² ∈ N such that w w w max{|G w p ( x n , x, x) − G p ( x n , x n , x n )|, |G p ( x n , x, x) − G p ( x, x, x)|} < ²
whenever n ≥ n ² . As a consequence we have
D G wp ( xn , xm )
=
w w Gw p ( x n , x, x) − min{G p ( x n , x n , x n ),G p ( x, x, x)}
=
w w |G w p ( x n , x, x) − min{G p ( x n , x n , x n ),G p ( x, x, x)}|
n
D G wp ( xn , xm )
≤
D G wp ( xn , xn+1 ) + . . . + D G wp ( xm−1 , xm )
By taking limit at n → ∞ with 3.8 then { xn } is a Cauchy sequence in the metric space ( X , D G wp ). Since ( X ,G w p ) is complete then from Lemma 3.1, the sequence { xn } converges in the metric space ( X , D G wp ), say lim D G wp ( xn , x) = n→∞
0. Again from Lemma 3.1, we have w w Gw p ( x, x, x) = lim G p ( x n , x, x) = lim G p ( x n , x m , x m ). n→∞
(3.9)
n→∞
Since { xn } is a Cauchy sequence in the metric space ( X , D G wp ), we have lim D G wp ( xn , xm ) = 0. n,m→∞
Also, since 1 w w Gw p ( x n , x n+1 , x n+1 ) ≥ (G p ( x n , x n , x n ) + 2G p ( x n+1 , x n+1 , x n+1 )) 3 By taking the limit and we obtain by 3.8 we have w limn→∞ G w p ( x n , x n , x n ) = 0 and lim n→∞ G p ( x n+1 , x n+1 , x n+1 ) = 0. w Therefore from the definition D G p we have w Gw ( xn , xm ) + min{G w p ( xn , xm , xm ) = D G w p ( x n , x n , x n ) + G p ( x m , x m , x m )} p
and so lim G w p ( x n , x m , x m ) = 0. Thus from 3.9 we have n,m→∞
w w Gw p ( x, x, x) = lim G p ( x n , x, x) = lim G p ( x n , x m , x m ) = 0. n→∞
(3.10)
n→∞
Now we show that G w p ( x, f x, f x) = 0. Since
Gw p ( x, f x, f x)
≤
w w Gw p ( x, x n+1 , x n+1 ) + G p ( x n+1 , f x, f x) − G p ( x n+1 , x n+1 , x n+1 )
≤
w Gw p ( x, x n+1 , x n+1 ) + G p ( f x n , f x, f x)
Therefor with (3.3) ψ(G w p ( x, f x, f x))
≤
w ψ(G w p ( x, x n+1 , x n+1 ) + G p ( f x n , f x, f x))
≤
w ψ(G w p ( x, x n+1 , x n+1 )) + ψ(G p ( f x n , f x, f x))
≤
w w ψ(G w p ( x, x n+1 , x n+1 )) + ψ(G p ( x n , x, x)) − ϕ(G p ( x n , x, x))
Taking the limit as n → ∞, we have w w ψ(G w p ( x, f x, f x)) = 0 then G p ( x, f x, f x) = 0 =⇒ x = f x. Moreover G p ( x, x, x) = 0 For the uniqueness, suppose y is another fixed point of f . Then we have from 3.3 ψ(G w p ( y, y, y))
=
ψ(G w p ( f y, f y, f y))
≤
ψ( M ( y, y, y)) − ϕ( M ( y, y, y))
≤
w ψ(G w p ( y, y, y)) − ϕ(G p ( y, y, y))
This contradiction. Thus G w p ( y, y, y) = 0, then w Gw ( x, x, x ) = G ( y, y, y ) = 0 =⇒ x = y, therefor, the fixed point is unique. p p Example 3.1. Let X = [ 21 , 1] be a set endowed with order x ¹ y ⇔ y ≤ x. let G w p ( x, y, z) = metric space on X Define by f : X → X by, f ( x) = 2 x − 1, ∀ x ∈ X . Put y = 2x , also ψ( t) = 2 t and ϕ( t) = t2 , t ∈ R + , then we have from Theorem 3.1
x+ y+ z 3
be a G w p -partial
ψ(G w p ( f x, f y, f y)) ≤ ψ( M ( x, y, y)) − ϕ( M ( x, y, y)).
Then we have, ψ(G w p ( f x, f y, f y)) = ψ( since y = 2x . Also
2 x+2 y+2 y−3 ) = 2( 2 x+32 x−3 ) = 83 x − 2 3
½ ¾ 1 w w w w M ( x, y, y) = max G w ( x, y, y ) ,G ( x, f x, f x ) ,G ( y, f y, f y ) , [ G ( x, f y, f y ) + G ( y, f x, f x )] , p p p p 2 p
8
then ½
M ( x, y, y) = max
¾ 5x − 2 x + 2 y 5x − 2 5 y − 2 1 x + 4 y − 2 y + 4x − 2 , , , { + = 3 3 3 2 3 3 3
Therefor ψ( M ( x, y, y)) − ϕ( M ( x, y, y))
= =
5x − 2 5x − 2 ) − ϕ( ) 3 3 5x − 2 5x − 2 2 2( )−( ) , 3 3 ψ(
therefor 8 5 x−2 5 x−2 2 ψ(G w p ( f x, g y, g y)) = 3 x − 2 ≤ ψ( M ( x, y, y)) − ϕ( M ( x, y, y)) = 2( 3 ) − ( 3 ) . Hence all the conditions of Theorem 3.1 are satisfied.
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