Box 55002, Jeddah, Saudi Arabia Correspondence should be addressed to Abdul Latif,latifmath@yahoo.com Received 13 October 2008; Accepted 27 January 2009 Recommended by Hichem Ben-El-Mech
Trang 1Volume 2009, Article ID 487161, 9 pages
doi:10.1155/2009/487161
Research Article
Fixed Points of Generalized Contractive Maps
Abdul Latif1 and Afrah A N Abdou2
1 Department of Mathematics, King Abdulaziz University, P.O Box 80203, Jeddah 21589, Saudi Arabia
2 Girls College of Education, King Abdulaziz University, P.O Box 55002, Jeddah, Saudi Arabia
Correspondence should be addressed to Abdul Latif,latifmath@yahoo.com
Received 13 October 2008; Accepted 27 January 2009
Recommended by Hichem Ben-El-Mechaiekh
We prove some results on the existence of fixed points for multivalued generalized w-contractive
maps not involving the extended Hausdorff metric Consequently, several known fixed point results are either generalized or improved
Copyrightq 2009 A Latif and A A N Abdou This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Throughout this paper, unless otherwise specified, X is a metric space with metric d Let
2X , Cl X, and CBX denote the collection of nonempty subsets of X, nonempty closed subsets of X, and nonempty closed bounded subsets of X, respectively Let H be the
Hausdorff metric on CBX, that is,
H A, B max
sup
x ∈A d x, B, sup
y ∈B d y, A
, A, B ∈ CBX. 1.1
A multivalued map T : X → CBX is called
i contraction 1 if for a fixed constant h ∈ 0, 1 and for each x, y ∈ X,
H Tx, Ty ≤ hdx, y; 1.2
ii generalized contraction 2 if for any x, y ∈ X,
H Tx, Ty ≤ kdx, ydx, y, 1.3
Trang 2where k is a function from 0, ∞ to 0, 1 with lim sup r → tk r < 1, for every t ∈
0, ∞;
iii contractive 3 if there exist constants b, h ∈ 0, 1, h < b such that for any x ∈ X there is y ∈ I x
b satisfying
d y, Ty ≤ hdx, y, 1.4
where I b x {y ∈ Tx : bdx, y ≤ dx, Tx};
iv generalized contractive 4 if there exist b ∈ 0, 1 such that for any x ∈ X there is
y ∈ I x
b satisfying
d y, Ty ≤ kdx, ydx, y, 1.5
where k is a function from 0, ∞ to 0, b with lim sup r → tk r < b, for every t ∈
0, ∞.
An element x ∈ X is called a fixed point of a multivalued map T : X → 2 X if x ∈ Tx.
We denote FixT {x ∈ X : x ∈ Tx}
A sequence{x n } in X is called an orbit of T at x0 ∈ X if x n ∈ Tx n−1 for all n ≥ 1 A map f : X → R is called lower semicontinuous if for any sequence {x n } ⊂ X with x n → x ∈ X imply that f x ≤ lim inf n→ ∞f x n
Using the concept of Hausdorff metric, Nadler Jr 1 established the following fixed point result for multivalued contraction maps which in turn is a generalization of the well-known Banach contraction principle
Then Fix T / ∅.
This result has been generalized in many directions For instance, Mizoguchi and Takahashi2 have obtained the following general form of the Nadler’s theorem
contraction map Then Fix T / ∅.
Another extension of Nadler’s result obtained recently by Feng and Liu3 Without using the concept of the Hausdorff metric, they proved the following result
contractive map Suppose that a real-valued function g on X, g x dx, Tx, is lower
semicontinuous Then Fix T / ∅.
Most recently, Klim and Wardowski4 generalizedTheorem 1.3as follows:
generalized contractive map such that a real-valued function g on X, g x dx, Tx is lower
semicontinuous Then Fix T / ∅.
Trang 3Recently, Kada et al.5 introduced the concept of w-distance on a metric space as
follows
A function ω : X × X → 0, ∞ is called w-distance on X if it satisfies the following for any x, y, z ∈ X:
w1 ωx, z ≤ ωx, y ωy, z;
w2 a map ωx, · : X → 0, ∞ is lower semicontinuous;
w3 for any > 0, there exists δ > 0 such that ωz, x ≤ δ and ωz, y ≤ δ imply
d x, y ≤ .
Using the concept of w-distance, they improved Caristi’s fixed point theorem, Ekland’s
variational principle, and Takahashi’s existence theorem In6, Susuki and Takahashi proved
a fixed point theorem for contractive type multivalued maps with respect to w-distance See
also7 12
Let us give some examples of w-distance5
a The metric d is a w-distance on X.
b Let X be normed space with norm · Then the functions ω1 , ω2: X × X → 0, ∞ defined by ω1x, y x y and ω2x, y y for every x, y ∈ X, are
w-distance
The following lemmas concerning w-distance are crucial for the proofs of our results.
Lemma 1.5 see 5 Let {x n } and {y n } be sequences in X and let {α n } and {β n } be sequences in
0, ∞ converging to 0 Then, for the w-distance ω on X the following hold for every x, y, z ∈ X:
a if ωx n , y ≤ α n and ω x n , z ≤ β n for any n ∈ N, then y z; in particular, if ωx, y 0
and ω x, z 0, then y z;
b if ωx n , y n ≤ α n and ω x n , z ≤ β n for any n ∈ N, then {y n } converges to z;
c if ωx n , x m ≤ α n for any n, m ∈ N with m > n, then {x n } is a Cauchy sequence;
d if ωy, x n ≤ α n for any n ∈ N, then {x n } is a Cauchy sequence.
Lemma 1.6 see 9 Let K be a closed subset of X and let ω be a w-distance on X Suppose that there
exists u ∈ X such that ωu, u 0 Then ωu, K 0 ⇔ u ∈ K (where ωu, K inf y ∈K ω u, y.)
We say a multivalued map T : X → 2X is generalized w-contractive if there exist a
w-distance ω on X and a constant b ∈ 0, 1 such that for any x ∈ X there is y ∈ J x
b satisfying
ω y, Ty ≤ kωx, yωx, y, 1.6
where J b x {y ∈ Tx : bωx, y ≤ ωx, Tx} and k is a function from 0, ∞ to 0, b with
lim supr → tk r < b, for every t ∈ 0, ∞.
Trang 4Note that if we take ω d, then the definition of generalized w-contractive map
reduces to the definition of generalized contractive map due to Klim and Wardowski 4
In particular, if we take a constant map k h < b, h ∈ 0, 1 then the map T is weakly
contractivein short, w-contractive 8, and further if we take ω d, then we obtain J x
b I x b
and T is contractive3
In this paper, using the concept of w-distance, we first establish key lemma and then obtain fixed point results for multivalued generalized w-contractive maps not involving the
extended Hausdorff metric Our results either generalize or improve a number of fixed point results including the corresponding results of Feng and Liu3, Latif and Albar 8, and Klim and Wardowski4
2 Results
First, we prove key lemma in the setting of metric spaces
Lemma 2.1 Let T : X → ClX be a generalized w-contractive map Then, there exists an orbit
{x n } of T in X such that the sequence of nonnegative real numbers {ωx n , T x n } is decreasing to
zero and the sequence {x n } is Cauchy.
Proof Since for each x ∈ X, Tx is closed, the set J x
b is nonempty for any b ∈ 0, 1 Let x obe
an arbitrary but fixed element of X Since T is generalized w-contractive, there is x1 ∈ J x o
T x o such that
ω
x1, T
x1
≤ kω
x0, x1
ω
x0, x1
, k
ω
x0, x1
< b, 2.1
bω
x0, x1
≤ ωx0, T
x0
Using2.1 and 2.2, we have
ω
x0, T
x0
− ωx1, T
x1
≥ bωx0, x1
− kω
x0, x1
ω
x0, x1
b − kω
x0, x1
ω
x0, x1
> 0. 2.3
Similarly, there is x2 ∈ J x1
b ⊆ Tx1 such that
ω
x2, T
x2
≤ kω
x1, x2
ω
x1, x2
, k
ω
x1, x2
< b, 2.4
bω
x1, x2
≤ ωx1, T
x1
Using2.4 and 2.5, we have
ω
x1, T
x1
− ωx2, T
x2
≥ bωx1, x2
− kω
x1, x2
ω
x1, x2
b − kω
x1, x2
ω
x1, x2
> 0. 2.6
Trang 5From2.5 and 2.1, it follows that
ω
x1, x2
≤ 1
b ω
x1, T x1
≤ 1
b k
ω
x0, x1
ω
x0, x1
≤ ωx0, x1
. 2.7
Continuing this process, we get an orbit{x n } of T in X such that x n1∈ J x n
b ,
bω
x n , x n1
≤ ωx n , T
x n
,
ω
x n1, T
x n1
≤ kω
x n , x n1
ω
x n , x n1
, k
ω
x n , x n1
< b. 2.8
Using2.8, we get
ω
x n , T
x n
− ωx n1, T
x n1
≥ bωx n , x n1
− kω
x n , x n1
ω
x n , x n1
b − kω
x n , x n1
ω
x n , x n1
> 0,
2.9
and thus for all n
ω
x n , T
x n
> ω
x n1, T
x n1
ω
x n , x n1
≤ ωx n−1, x n
Note that the sequences {ωx n , T x n } and {ωx n , x n1} are decreasing, and thus
convergent Now, by the definition of the function k there exists α ∈ 0, b such that
lim sup
n→ ∞ k
ω
x n , x n1
Thus, for any b0∈ α, b, there exists n0∈ N such that
k
ω
x n , x n1
< b0, ∀n > n0 , 2.13
and thus for all n > n0, we have
k
ω
x n , x n1
× · · · × kω
x n1, x n2
< b n −n0
0 . 2.14
Trang 6Also, it follows from2.9 that for all n > n0,
ω
x n , T
x n
− ωx n1, T
x n1
≥ βωx n , x n1
, 2.15
where β b − b0 Note that for all n > n0, we have
ω
x n1, T
x n1
≤ kω
x n , x n1
ω
x n , x n1
≤ 1
b k
ω
x n , x n1
ω
x n , T
x n
≤ 1
b
1
b k
ω
x n , x n1
k
ω
x n−1, x n
ω
x n−1, T
x n−1
≤ 1
b n kω
x n , x n1
× · · · × kω
x1, x2
ωx1, T
x1
k
ω
x n , x n1
× · · · × kω
x n01, x n02
b n −n0
× k
ω
x n0, x n0 1
× · · · × kω
x1, x2
ω
x1, T
x1
2.16
and thus
ω
x n1, T
x n1
<
b0 b
n −n0kωx
n0, x n01
× · · · × kω
x1, x2
ω
x1, T
x1
b n0 . 2.17
Now, since b0 < b, we have lim n→ ∞b0 /bn −n0 0, and hence the decreasing sequence {ωx n , T x n } converges to 0 Now, we show that {x n} is a Cauchy sequence Note that for
all n > n0,
ω
x n , x n1
≤ γ n ω
x o , x1
, n 0, 1, 2, , 2.18
where γ b0 /b < 1 Now, for any n, m ∈ N, m > n > n0 ,
ω
x n , x m
≤m−1
j n
ω
x j , x j1
≤γ n γ n1 · · · γ m−1
ω
x o , x1
≤ γ n
1− γ ω
x o , x1
,
2.19
and thus byLemma 1.5,{x n} is a Cauchy sequence
Trang 7Using Lemma 2.1, we obtain the following fixed point result which is an improved version ofTheorem 1.4and containsTheorem 1.3as a special case
Theorem 2.2 Let X be a complete space and let T : X → ClX be a generalized w-contractive map.
Suppose that a real-valued function g on X defined by g x ωx, Tx is lower semicontinous.
Then there exists v o ∈ X such that gv o 0 Further, if ωv o , v o 0, then v0 ∈ FixT.
Proof Since T : X → ClX is a generalized w-contractive map, it follows fromLemma 2.1
that there exists a Cauchy sequence{x n } in X such that the decreasing sequence {gx n}
{ωx n , T x n } converges to 0 Due to the completeness of X, there exists some v0 ∈ X such
that limn→ ∞x n v o Since g is lower semicontinuous, we have
0≤ gv o
≤ lim inf
n→ ∞ g
x n
and thus, gv o ωv o , T v o 0 Since ωv o , v o 0, and Tv o is closed, it follows from
Lemma 1.6that v0 ∈ Tv0.
As a consequence, we also obtain the following fixed point result
If the real-valued function g on X defined by g x ωx, Tx is lower semicontinous, then there
exists v o ∈ X such that ωv o , T v o 0 Further, if ωv o , v o 0, then v0 ∈ FixT.
ApplyingLemma 2.1, we also obtain a fixed point result for multivalued generalized
w-contractive map satisfying another suitable condition.
Theorem 2.4 Let X be a complete space and let T : X → ClX be a generalized w-contractive
map Assume that
inf{ωx, v ωx, Tx : x ∈ X} > 0, 2.21
for every v ∈ X with v /∈ Tv Then FixT / ∅.
Proof ByLemma 2.1, there exists an orbit{x n } of T, which is a Cauchy sequence in X Due to the completeness of X, there exists v0 ∈ X such that lim n→ ∞x n v o Since ω x n ,· is lower
semicontinuous and x m → v0 ∈ X, it follows from the proof ofLemma 2.1that for all n > n0
ω
x n , v o
≤ lim inf
x n , x m
≤ γ n
1− γ ω
x o , x1
, 2.22
where γ b0 /b < 1 Also, we get
ω
x n , T
x n
≤ ωx n , x n1
≤ γ n ω
x o , x1
Trang 8
Assume that v o / ∈ Tv o Then, we have
0 < inf
ω
x, v o
ωx, T x: x ∈ X
≤ inf ω
x n , v o
ωx n , T
x n
: n > n0
≤ inf
γ n
1− γ ω
x o , x1
γ n ω
x o , x1
: n > n0
2− γ
1− γ ω
x o , x1
inf
γ n : n > n0
0,
2.24
which is impossible and hence v o ∈ FixT.
Assume that
inf{ωx, u ωx, Tx : x ∈ X} > 0, 2.25
for every u ∈ X with u /∈ Tu Then FixT / ∅.
Acknowledgment
The authors thank the referees for their valuable comments and suggestions
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