Cmjv04i03p0153

J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

Best Proximity Point Theorems for Rational Expression in Complete Metric Spaces M. R. YADAV1, B. S. THAKUR2 and A. K. SHARMA3 1,2

School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur, Chhattisgarh, INDIA. 3 Seth Phoolchand College, Navapara Rajim, Dist. Raipur, Chhattisgarh, INDIA. (Received on: May 13, 2013) ABSTRACT In this paper, we first introduce the new notion of rational cyclic contraction and then establish some convergence theorems of best proximity points for two non-self mappings in the frame work of complete metric space. Our results generalized and improve some main results in the literature. An example is given to support our main results.

Keywords: Cyclic contraction, best proximity point, rational cyclic contraction, and complete metric space.

2000 Mathematics Subject Classification: 41A65, 46B20, 47H10.

1. INTRODUCTION Fixed point theory plays an important role in furnishing a uniform treatment to solve various equations of the form Tx = x for self-mappings T defined on subsets of metric spaces. Given two nonempty subsets A and B of a metric space, consider a non-self-mapping T from A to B. Because T is not a self-mapping, the equation Tx = x is unlikely to have a solution. Therefore, it is of primary importance to seek an element x that in some

sense is closest to Tx. That is, when the equation Tx = x has no solution, one tries to determine an approximate solution x subject to the condition that the distance between x and Tx is minimal. Best approximation theorems and best proximity point theorems are relevant in this perspective. W. A. Kirk et al.7 established the first result in this interesting area. Then after, many authors obtained many results in this area1,2,3,5,8, 10, 11.

Recently, Karapinar and Erhan6 studied some proximity points by using different types cyclic contraction.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

154

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

We first define the cyclic map and the best proximity point.

Definition 1.1. Let A and B be non-empty subsets of a metric space (X, d) and T : A ∪ B → A ∪ B . Then T is called cyclic map if T(A) ⊆ B and T(B) ⊆ A . A point x ∈ A ∪ B is called a best proximity point if d( x, Tx ) = d(A, B) , where d(A, B) = inf {d( x, y) : x ∈A, y ∈B} .

In 2003, Kirk et al.7 gave the following fixed point theorem for a cyclic map.

Theorem 1.2. Let A and B be nonempty closed subsets of a complete metric space (X, d). Suppose that T : A ∪ B → A ∪ B is a cyclic map such that d(Tx, Ty) ≤ k d(x, y) for all x ∈A, y∈B . If k∈[0, 1) , then T has a unique fixed point in A ∪ B .

In 2011, Karapinar and Erhan6 introduced the following cyclic contractions: Definition 1.3. Let A and B be nonempty subsets of a metric space (X, d). A cyclic map T : A ∪ B → A ∪ B is said to be a Kannan type cyclic contraction if there exists k∈(0, 1 / 2) such that

d(Tx, Ty) ≤ k(d(x, Tx) + d(y, Ty)) for all x ∈ A, y ∈ B .

Definition 1.4. Let A and B be nonempty subsets of a metric space (X, d). A cyclic map T : A ∪ B → A ∪ B is said to be a Reich type cyclic contraction if there exists k ∈ (0, 1/3) such that

d(Tx, Ty) ≤ k(d(x, y) + d(x, Tx) + d(y, Ty))

for all x ∈ A, y ∈ B .

Furthermore, Karapinar and Erhan6 obtained the following results: Theorem 1.5. Let A and B be non-empty closed subsets of a complete metric space (X, d) and let T : A ∪ B → A ∪ B be a Kannan type cyclic contraction. Then T has a unique fixed point in A ∩ B .

Theorem 1.6. Let A and B be non-empty closed subsets of a complete metric space (X, d) and let T : A ∪ B → A ∪ B be a Reich type cyclic contraction. Then T has a unique fixed point in A ∩ B .

In 1977, Jaggi4 proved the following fixed point theorem.

Theorem 1.7. Let T be a continuous selfmap defined on a complete metric space (X, d). Suppose that T satisfies the following contractive condition:

d(Tx, Ty) ≤ α

d(x, Tx) · d(y, Ty) + β d ( x , y) d(x, y) (1.1)

for all x ∈ A and y∈B , for all x, y ∈X , x ≠ y , and for some α, β ∈ [0, 1) with α + β < 1 , then T has a unique fixed point in X. The purpose of this paper is to first introduce the notion of rational cyclic contraction for best proximity point. We also establish the existence and convergence theorem of best proximity points theorem which is a variant of Theorem 1.7 in the setting of complete metric spaces and gives illustrative examples of our condition.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

155

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

2. PRELIMINARIES

d(A, B) = inf{d(x, y) : x ∈ A, y ∈ B} .

In this section, we first define the cyclic contraction and new notion of rational cyclic contraction. Let (X, d) be a complete metric space and A, B be nonempty subsets of X. A mapping T : X → X is a contraction if and only if for each x, y ∈ X there exists a constant k ∈ (0, 1) such that d(Tx, Ty) ≤ kd(x, y) . Let T : A ∪ B → A ∪ B such that T(A) ⊆ B and T(B) ⊆ A we say that (i) T is cyclic contraction if

Definition 2.1.9 Let (X, d) be a complete metric space and let A and B be nonempty subsets of X. Then a cyclic operator T : A ∪ B → A ∪ B is called weak cyclic Kannan contraction if it satisfies the following condition

d(Tx, Ty) ≤ α d(x, y) + (1 - α) d(A, B),

Definition

for some α ∈ (0, 1) and for all x ∈ A and y∈B where

d(Tx, Ty) ≤ k[d(x, Tx) + d(y, Ty)] + (1 - 2k)d(A, B) ,

(2.1) where k ∈[0, 1/2) ,for all x ∈ A and y∈B .

2.2.

A

pair of mappings T : A ∪ B → A ∪ B is said to form a rational cyclic contraction mapping between A and B if there exists a nonnegative real number 0 ≤ m < 1/2 such that

 d(x, Tx) · d(y, Ty)  d(Tx, Ty) ≤ m d(x, y) +  + (1 - 2m)d(A, B) d(x, y)  

(2.2)

for all x ∈ A and y∈B .

Definition 2.3. A subset K of a metric space is said to be boundedly compact if every bounded sequence in K has a subsequence converging to some element in K. 3. MAIN RESULTS In this section, first, we establish the following convergence theorem for rational cyclic contraction, which is one of the main results in this paper.

Theorem 3.1. Let A and B be two nonempty, closed subsets of a complete

metric space (X, d) . Suppose that the mappings

T:A ∪ B → A ∪ B

form

a

rational cyclic contraction between A and B, for each fixed x 0 ∈ A . Then

d(x n , x n +1 ) → d(A, B) .

Proof. Suppose x 0 ∈ A ∪ B . Define an iterative sequence {x n } by x n = Tx n -1 for all n ∈ N . Since T is a rational cyclic contraction, then from (2.2), we obtain

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

156

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

d(x n +1 , x n + 2 ) = d(Tx n , Tx n +1 )  d(x n , Tx n ) · d(x n +1 , Tx n +1 )  ≤ m  d ( x n , x n +1 ) +  + (1 - 2m)d(A, B) d(x n , x n +1 )    d(x n , x n +1 ) · d(x n +1 , x n + 2 )  ≤ m d(x n , x n +1 ) +  + (1 - 2m)d(A, B) d(x n , x n +1 )   ≤ m [d(x n , x n +1 ) + d(x n +1 , x n + 2 )] + (1 - 2m)d(A, B) which implies that,

m m   d(x n , x n +1 ) + 1  d(A, B) 1− m  1- m  ≤ µd(x n , x n +1 ) + (1 - µ) d(A, B),

d(x n +1 , x n + 2 ) ≤

where

(3.1)

m = µ < 1 . Similarly, we get 1- m

d(x n + 2 , x n + 3 ) = d(Tx n +1 , Tx n + 2 )  d(x n +1 , Tx n +1 ) · d(x n + 2 , Tx n + 2 )  ≤ m d ( x n +1 , x n + 2 ) +  + (1 - 2m)d(A, B) d(x n +1 , x n + 2 )    d(x n +1 , x n + 2 ) · d(x n + 2 , x n + 3 )  ≤ m d(x n +1 , x n + 2 ) +  + (1 - 2m)d(A, B) (3.2) d(x n +1 , x n + 2 )   ≤ m [d(x n +1 , x n + 2 ) + d(x n + 2 , x n + 3 )]+ (1 - 2m)d(A, B) m   m   ≤ d(x n +1 , x n + 2 ) + 1 − d(A, B)  1 − m   1 − m 

By this way if we continue, we concluded that from (3.1) and (3.2), we have

d(x n + 2 , x n + 3 ) ≤ µ 2 d(x n , x n +1 ) + (1 - µ 2 )d(A, B) .

Hence inductively, we have

d(x n , x n +1 ) ≤ µd(x n , x n -1 ) + (1 - µ)d(A, B) ≤ µ 2 d(x n -1 , x n - 2 ) + (1 - µ 2 )d(A, B) ≤ . . . .... ... ≤ µ n d(x 0 , x1 ) + (1 - µ n )d(A, B) Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

Since µ ∈ [0, 1) , we have So

the

last

lim µ n = 0 .

n →∞

inequality

implies lim d( x n , x n +1 ) → d(A, B) .

that

157

Proof. Suppose {x 2n } be a subsequence k

of {x 2n } converges to x in A. Now, we deduce that

n→∞

This completes the proof.

Theorem 3.2. Let A and B be two nonempty, closed subsets of a complete metric space (X, d) . Suppose that the T : A ∪ B → A ∪ B form a mappings rational cyclic contraction between A and B, x 0 ∈ A and defined iteration let

d(A, B) ≤ d(x, x 2n k −1 ) ≤ d(x, x 2n k ) + d(x 2n k , x 2n k −1 ) ≤ d(x, x 2n k ) + d(A, B).

x n = Tx n -1 . Let the sequence {x 2n } has a

Thus d(x, x 2n k −1 ) converges to d(A,B) and by Theorem 3.1 d(x 2n k , x 2n k −1)

convergent subsequence in A. Then there exists x ∈ A such that d(x, Tx) = d(A, B) .

also converges to d(A,B). Since T is a rational cyclic contraction, then, we obtain

d(A, B) ≤ d(x 2n k , Tx) = d(Tx 2n k − 1 , Tx)  d(x 2n k − 1 , Tx 2n k − 1 ) · d(x, Tx)  ≤ m  d(x 2n k − 1 , x) +  + (1 − 2 m ) d ( A , B ) d(x 2n k − 1 , x)    d(x 2n k − 1 , x 2n k ) · d(x, Tx)  ≤ m  d(x 2n k − 1 , x) +  + (1 − 2 m ) d ( A , B ) d(x 2n k − 1 , x)    d(A, B) · d(x, Tx)  ≤ m  d(A, B) +  + (1 − 2 m ) d ( A , B ) d(A, B)   ≤ d (A , B)

Taking n → ∞ above inequality, we have

d(x, Tx) = d(A, B) , that is x is a best proximity point of T. This completes the proof.

Theorem 3.3. Let A and B be two nonempty, closed subsets of a complete

metric space (X, d) . Suppose that the mappings T : A ∪ B → A ∪ B form a rational cyclic contraction between A and B, for each fixed x 0 ∈ A , let defined iteration

x n = Tx n -1 . Then the sequence {x n } is bounded.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

158

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

Proof. It follows from Theorem 3.1, that {d(x 2n -1 , x 2n )} is convergent and hence it is

bounded.

Suppose

{x 2n k }

be

a

subsequence of {x 2n } converges to x in A. Now, d(A, B) ≤ d(x, x 2n k − 1 ) ≤ d(x, x 2n k ) + d(x 2n k , x 2n k − 1 ) ≤ d(x, x 2n k ) + d(A, B).

Thus d(x, x 2n −1 ) converges to d(A,B) k and by Theorem 3.1 d(x 2n , x 2n −1 ) k k also converges to d(A,B).

Since T : A ∪ B → A ∪ B form a rational cyclic mapping between A and B, we have

d(x 2n k , Tx 0 ) = d(Tx 2n k −1 , Tx 0 )  d(x 2n k −1 , Tx 2n k −1 ) · d(x 0 , Tx 0 )  ≤ m d(x 2n k −1 , x 0 ) +  + (1 − 2 m )d ( A , B) d(x 2n k −1 , x 0 )    d(x 2n k −1 , x 2n k ) · d(x 0 , Tx 0 )  ≤ m d(x 2n k −1 , x 0 ) +  + (1 − 2 m )d ( A , B) d(x 2n k −1 , x 0 )   d(A, B) · d(x 0 , Tx 0 )   ≤ m d(A, B) +  + (1 − 2 m )d ( A , B) d(A, B)   ≤ m [d(x 0 , Tx 0 ) - d(A, B) ]+ d ( A , B)

Since m∈[0, 1) , therefore the sequence

{x n } is bounded. This completes the proof.

Example 3.4. Consider the usual metric space d(x, y) = |x − y|, for all x, y ∈ X . Let X = A = B = [0, 1/2] . Then d(A, B) = 0. It is clear that T(A) ⊆ B and T(B) ⊆ A .

Define a mapping T : A ∪ B → A ∪ B as follows:

if x = 0 0  T(x) =  1 1 if x = , n = 2, 3, ...  n  2 n +1

We now show that T satisfies the condition (2.2) with 0 ≤ m < 1/2 . Putting x =

1 n

and y = 0, n = 2, 3, 4, ... in (2.2), we obtain

1 1   d ( , T ( )). d ( 0 , T ( 0 ))   1 1 n n d(T( ), T(0)) ≤ m  d ( , 0 ) +  + (1 − 2 m ) d ( A , B ) 1 n  n  d( , 0)  n  1 1 ≤ m , 2n + 1 n Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)

M. R. Yadav, et al., J. Comp. & Math. Sci. Vol.4 (3), 153-159 (2013)

159

which implies that n + 1 n ( 2 n + 1)

≤ m

1 n

1 n 2n + 1 1 +

⇒

Taking lim as n → ∞ in the last inequality, we get 0 ≤ m . Thus the inequality (2.2) holds and 0 is a best proximity point of T. REFERENCES 1. M.A. Al-Thafai and N. Shahzas ”Convergence and existence for best proximity points” Nonlinear Analysis 70, 3665-3671(2009).

2. A.A. Eldered and P. Veeramani ”Proximal point-wise contraction” Topology and its Applications 156, 2942-2948 (2009).

3. A.A. Eldered and P. Veeramani ”Convergence and existence for best proximity points” J. Math. Anal. Appl.

323, 1001-1006 (2006).

4. D.S. Jaggi Some unique fixed point theorems, Indian Journal of Pure and Applied Mathematics, Vol. 8, No. 2, 223230 (1977).

5. S. Karpagam and S. Agrawal ”Best proximity points for cyclic Meir-Keeler contraction maps” Nonlinear Analysis, 74, 1040-1046 (2011).

≤ m .

6. E. Karapinar and I.M. Erhan ”Best proximity point on different type contractions” Applied Mathematics and Information Sciences, 5, 342-353 (2011).

7. W.A. Kirk, P.S. Srinavasan and P. Veeramani ”Fixed points for mapping satisfying cyclical contractive conditions” Fixed Point Theory 4, 79-89 (2003).

8. G. Petrushel ”Cyclic representations and periodic points” Studia Uni. BabesBolyai Math. 50, 107-112 (2005).

9. M.A. Petric ”Bets proximity point theorems for weak cyclic Kann contractions” Faculty of Sciences and Mathematixs, University of Nis, Serbia, 25:1, 145-154 (2011).

10. S. Sadiq Basha and P. Veeramani ”Best proximity pair theorems for multifunctions with open fibres” J. Approx.

Theory 103, 119-129 (2000).

11. S. Sadiq Basha and N. Shahzad ”Best proximity point for generalized proximal contractions” Fixed Point theory and Applications,2012:42 (2012).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 3, 30 June, 2013 Pages (135-201)