1The Gandhigram Rural Institute-Deemed University, Dindigul, Tamilnadu, INDIA. Received: 14.06.2013 Revised:29.11.2013 Accepted: 02.12.2013.
Gazi University Journal of Science GU J Sci 27(2):755-759 (2014)
Generalized Well-Posedness and Multivalued Contraction
G. Arockia PRABAKAR1, ♠, Ramasamy UTHAYAKUMAR1
1
The Gandhigram Rural Institute-Deemed University, Dindigul, Tamilnadu, INDIA
Received: 14.06.2013 Revised:29.11.2013 Accepted: 02.12.2013
ABSTRACT In this paper, we establish the fixed point theorems for single and multivalued valued contraction in complete metric space. We investigate an iteration method involving projections which converges to a fixed point using multivalued contraction. Also we prove the generalized well-posedness of the fixed point problem and continuity of single valued contraction in metric space. Key Words: Fixed point; Multivalued mapping; Well-posedness.
1. INTRODUCTION Fixed point theory is an exciting branch of mathematics. In 1922, the Polish mathematician Stefan Banach proved a theorem on the existence and uniqueness of a fixed point in a complete metric space. Various authors have defined contractive mappings on a complete metric space which are generalizations of well-known banach contraction. Rhoades [5] compared and discussed the relation between all the contractions with appropriate examples. We introduce contraction mapping and establish fixed point theorems on complete metric space. The notion of set valued contraction was initiated by Nadler [2] in 1969. He proved that a set valued contraction possesses a fixed point in a complete metric space. Subsequently many authors generalized Nadlers fixed point theorem in different ways. Kunze et al. [3] have introduced an iteration method involving a projection which converges to a fixed point using multivalued nadler contraction.We investigate an ♠
Corresponding author, e-mail: prabakar1985@gmail.com
iteration method involving projections which converges to a fixed point using multivalued contraction. Let of
( X , d ) be a metric space. Let P( X ) be the family all
non-empty
subsets
( )
of
X
and
let
T : X → P X be a multivalued mapping. A point is said to be a fixed point of the multi-valued mapping FT = {x ∈ X : x ∈ T ( x )} and SFT = {x ∈ X :{x} = T ( x )} .
T if x ∈ Tx
or
Definition 1.1 Let
P( X )
( X , d ) be
a metric space and
be the family of all non-empty closed and
bounded subsets of X ,
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GU J Sci, 27(2):755-759 (2014)/ G.Arockia PRABAKAR, Ramasamy UTHAYAKUMAR
Define
d ( x, B ) = inf y∈B d ( x, y ) ,
h( A, B ) = sup x∈A d ( x, B ) for all x ∈ X A, B ∈ P( X ) The Hausdorff metric or Hausdorff distance H d is a function H d : P( X ) × P( X ) → R defined by H d ( A, B ) = max{h( A, B ), h(B, A)} (P( X ), H d ) is called a Hausdorff metric space. Lemma 1.1 [3] Let ( X , d ) be a metric space, x, y ∈ X and A, B, C are subsets of X . Then the following statements hold:
A ⊆ B , then d ( A, C ) ≥ d (B, C ) h( A, C ) ≤ h(B, C ) and h(C , A) ≥ h(C , B ) .
If
T i.e., the {x } sequence by xn+1 = Txn n = 0, 1, 2,.... converges to the fixed point u .
(ii) The picard iteration associated to
and
∞ n n=0 defined
Proof: To prove the existence of the fixed point, we show that for any
x0 ∈ X
the picard iteration
{xn } is a
cauchy sequence.
d ( x1 , x2 ) = d (Tx0 , Tx1 ) ≤ ad ( x0 , x1 ) + b[d ( x0 , Tx0 ) + d (Tx0 , Tx1 )] ≤ ad ( x0 , x1 ) + b[d ( x0 , x1 ) + d ( x1 , x2 )] ≤ (a + b )d ( x0 , x1 ) + bd ( x1 , x2 )
d ( x, A) ≤ d ( x, y ) + d ( y, A)
d ( x1 , x2 ) − bd ( x1 , x2 ) ≤ (a + b )d ( x0 , x1 )
d ( x, A) ≤ d ( x, y ) + d ( y, B ) + h(B, A) .
(1 − b )d (x1 , x2 ) ≤ (a + b )d (x0 , x1 )
Definition 1.2 Let
( X , d ) be a metric space. A map
T : X → X is called banach contraction if there exists 0 ≤ k < 1 such that d (Tx, Ty ) ≤ kd ( x, y ) , for all x, y ∈ X . Definition 1.3 [2] Let
( X , d ) be a metric space. A map
T : X → P( X ) is
called multivalued contraction if
0 ≤ k < 1 such that H d (Tx, Ty ) ≤ kd (x, y ) , for all x, y ∈ X .
there
exists
A, B ∈ P( X ) and a ∈ A then for each k > 0 , there exists b ∈ B such that Lemma 1.2 [2] If
d (a, b ) ≤ H d ( A, B ) + k . 2. MAIN RESULTS Theorem 2.1 Let
( X , d ) be a complete metric space.
T : X → X be a single valued map satisfying d (Tx, Ty ) ≤ ad ( x, y ) + b[d ( x, Tx) + d (Ty, Tx )]
Let
there exists then
a, b ∈ R + with a + 2b < 1 , a + b < 1
a+b d ( x0 , x1 ) 1− b d ( x1 , x2 ) ≤ kd ( x0 , x1 ) a+b k= , a + 2b < 1 0 < k < 1 . 1− b
d ( x1 , x2 ) ≤
and by induction
d ( xn , xn +1 ) ≤ k n d ( x0 , x1 ) d (x n , x m ) ≤ d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + .......... + d ( x m −1 , x m )
(
)
≤ k n + k n +1 + ........ + k m −1 d ( x0 , x1 )
kn d ( x0 , x1 ) 1− k n Since 0 < k < 1 , it results that k → 0 as n → ∞ shows that {xn } is a cauchy sequence. Since ≤
( X , d ) be
a complete metric space, therefore
{xn }
converges to u . lim
u = n → ∞ xn Hence u is a fixed point of T . d (u, Tu ) ≤ d (u, xn +1 ) + d ( xn+1 , Tu )
= d (u, xn +1 ) + d (Txn , Tu ) ≤ d (u, xn+1 ) + ad(xn , u ) + b[d (xn , Txn ) + d (Tu, Txn )] (i)
T has a unique fixed point i.e., FT = u ,
(
) (
= d (u, xn+1 ) + ad(xn , u ) + bd x n , x n + bd Tu, x n
)
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GU J Sci, 27(2):755-759 (2014)/ G.Arockia PRABAKAR, Ramasamy UTHAYAKUMAR
≤ d (u, xn+1 ) + ad(xn , u )
(
+ bk d (x0 , x1 ) + d Tu, x n+1 n
≤ ad ( x0 , x1 ) + b[d ( x0 , Tx0 ) + d (Tx1 , Tx0 )] + k
(1)
)
≤ ad ( x0 , x1 ) + bd ( x0 , x1 ) + bd ( x2 , x1 ) + k
n → ∞ in (1) we get
d (u, Tu ) ≤ bd (u, Tu )
≤ (a + b )d ( x0 , x1 ) + bd ( x2 , x1 ) + k
(1 − b )d (u, Tu ) ≤ 0 b < 1 , d (u, Tu ) = 0
d ( x1 , x2 ) − bd ( x2 , x1 ) ≤ (a + b )d ( x0 , x1 ) + k
⇒ u = Tu .
u and v be u = Tu and v = Tv .
Uniqueness: On the contrary let points of
T
then
two fixed
d (u , v ) = d (Tu, Tv ) ≤ ad (u, v ) + b[d (u, Tu ) + d (Tv, Tu )] ≤ ad (u, v ) + bd (u, u ) + bd (v, u ) ≤ (a + b )d (u, v ) 1 − (a + b )d (u , v ) ≤ 0 a + b < 1, d (u, v ) = 0 ⇒ u = v. Example 2.1
X = [0,1] endowed
d ( x, y ) = x − y
metric
d ( x1 , x2 ) ≤
condition
Where
with Euclidean
k=
a
map
H d (Tx1 , Tx2 ) = 0 then Tx1 = Tx2 i.e. x2 ∈ Tx2 . Let H d (Tx1 , Tx2 ) ≠ 0 . Again by lemma 1.2 there exists x3 ∈ Tx2 .
is
d ( x 2 , x3 ) ≤ kd ( x1 , x 2 ) + k 2 and by induction
d ( xn , xn +1 ) ≤ kd ( xn −1 , xn ) + k satisfied
for
::::
( X , d ) be a complete metric space. Let T X → P ( X ) be a multivalued map satisfying H d (Tx, Ty ) ≤ ad ( x, y ) + b[d ( x, Tx) + d (Ty, Tx )]
Theorem 2.2 Let
a, b ∈ R + with a + 2b < 1 , a + b < 1
≤ k (kd ( xn − 2 , xn −1 ) + k n −1 ) + k n = k 2 d ( xn − 2 , xn −1 ) + kk n −1 + k n ≤ .......
≤ k n d ( x0 , x1 ) + nk n . Since
k
