In this section, we study the existence and convergence of coupled best proximity points for cyclic contraction pairs. We begin by introducing the notion of property UC* and a coupled best proximity point.

**Definition 3.1**. Let *A* and *B* be nonempty subsets of a metric space (*X*, *d*). The ordered pair (*A*, *B*) satisfies the *property UC** if (*A*, *B*) has property UC and the following condition holds:

If {*x*
_{
n
}} and {*z*
_{
n
}} are sequences in *A* and {*y*
_{
n
}} is a sequence in *B* satisfying:

(1) *d*(*z*
_{
n
}, *y*
_{
n
}) → *d*(*A*, *B*).

(2) For every

*ε* > 0 there exists

*N* ∈ ℕ such that

for all *m* > *n* ≥ *N*,

then, for every

*ε* > 0 there exists

*N*
_{1} ∈ ℕ such that

for all *m* > *n* ≥ *N*
_{1}.

**Example 3.2**. *The following are examples of a pair of nonempty subsets* (*A*, *B*) *satisfying the property UC**.

*(1) Every pair of nonempty subsets A*, *B of a metric space* (*X*, *d*) *such that d*(*A*, *B*) = 0.

*(2) Every pair of nonempty closed subsets A*, *B of a uniformly convex Banach space X such that A is convex* [[38], Lemma 3.7].

**Definition 3.3**. Let

*A* and

*B* be nonempty subsets of a metric space

*X* and

*F* :

*A* ×

*A* →

*B*. A point (

*x*,

*x'*) ∈

*A* ×

*A* is called a

*coupled best proximity point* of

*F* if

It is easy to see that if *A* = *B* in Definition 3.3, then a coupled best proximity point reduces to a coupled fixed point.

Next, we introduce the notion of a cyclic contraction for a pair of two binary mappings.

**Definition 3.4**. Let

*A* and

*B* be nonempty subsets of a metric space

*X*,

*F* :

*A* ×

*A* →

*B* and

*G* :

*B* ×

*B* →

*A*. The ordered pair (

*F*,

*G*) is said to be a

*cyclic contraction* if there exists a non-negative number

*α <* 1 such that

for all (*x*, *x*') ∈ *A* × *A* and (*y*, *y*') ∈ *B* × *B*.

Note that if (*F*, *G*) is a cyclic contraction, then (*G*, *F* ) is also a cyclic contraction.

**Example 3.5**.

*Let X* = ℝ

*with the usual metric d*(

*x*,

*y*) =

*|x - y| and let A* = [2,4]

*and B* = [-4, -2]

*. It easy to see that d*(

*A*,

*B*) = 4

*. Define F* :

*A* ×

*A* →

*B and G* :

*B* ×

*B* →

*A by*
*For arbitrary* (

*x*,

*x*') ∈

*A* ×

*A and* (

*y*,

*y*') ∈

*B* ×

*B and fixed*
,

*we get*
*This implies that* (*F*, *G*) *is a cyclic contraction with*
.

**Example 3.6**.

*Let X* = ℝ

^{2}
*with the metric d*((

*x*,

*y*), (

*x*',

*y*')) = max{

*|x* -

*x*'

*|*,

*|y* -

*y*'

*|*}

*and let A* = {(

*x*, 0): 0 ≤

*x* ≤ 1}

*and B* = {(

*x*, 1): 0 ≤

*x* ≤ 1}

*. It easy to prove that d*(

*A*,

*B*) = 1

*. Define F* :

*A* ×

*A* →

*B and G* :

*B* ×

*B* →

*A by*
*Also for all α* > 0,

*we get*
*This implies that* (*F*, *G*) *is cyclic contraction*.

The following lemma plays an important role in our main results.

**Lemma 3.7**.

*Let A and B be nonempty subsets of a metric space X, F* :

*A* ×

*A* →

*B*,

*G* :

*B* ×

*B* →

*A and* (

*F*,

*G*)

*be a cyclic contraction. If*
*and we define*
*for all n ∈* ℕ ∪ {0}, *then*
,
,
*and*
.

*Proof*. For each

*n* ∈ ℕ ∪ {0}, we have

By induction, we see that

For each

*n* ∈ ℕ ∪ {0}, we have

By induction, we see that

By similar argument, we also have
and
for all *n* ∈ ℕ ∪ {0}. □

**Lemma 3.8**.

*Let A and B be nonempty subsets of a metric space X such that* (

*A*,

*B*)

*and* (

*B*,

*A*)

*have a property UC, F* :

*A* ×

*A* →

*B, G* :

*B* ×

*B* →

*A and let the ordered pair* (

*F*,

*G*)

*is a cyclic contraction. If*
and define

*for all n* ∈ ℕ ∪ {0},

*then for ε* > 0,

*there exists a positive integer N*
_{0}
*such that for all m* >

*n* ≥

*N*
_{0},

*Proof*. By Lemma 3.7, we have

*d*(

*x*
_{2n
},

*x*
_{2n+1}) →

*d*(

*A*,

*B*) and

*d*(

*x*
_{2n+1},

*x*
_{2n+2}) →

*d*(

*A*,

*B*). Since (

*A*,

*B*) has a property UC, we get

*d*(

*x*
_{2n
},

*x*
_{2n+2}) → 0. A similar argument shows that

. As (

*B*,

*A*) has a property UC, we also have

*d*(

*x*
_{2n+1},

*x*
_{2n+3}) → 0 and . Suppose that (3.3) does not hold. Then there exists

*ε*' > 0 such that for all

*k* ∈ ℕ, there is

*m*
_{
k
} >

*n*
_{
k
} ≥

*k* satisfying

Letting

*k* →

*∞*, we obtain to see that

By using the triangle inequality we get

which contradicts. Therefore, we can conclude that (3.3) holds. □

**Lemma 3.9**.

*Let A and B be nonempty subsets of a metric space X*, (

*A, B*)

*and* (

*B, A*)

*satisfy the property UC**

*. Let F* :

*A* ×

*A* →

*B, G* :

*B* ×

*B* →

*A and* (

*F, G*)

*be a cyclic contraction. If*
*and define*
*for all n* ∈ ℕ ∪ {0}, *then* {*x*
_{2n
}},
, {*x*
_{2n+1}} *and*
*are Cauchy sequences*.

*Proof*. By Lemma 3.7, we have *d*(*x*
_{2n
}, *x*
_{2n+1}) → *d*(*A*, *B*) and *d*(*x*
_{2n+1}, *x*
_{2n+2}) → *d*(*A*, *B*). Since (*A*, *B*) has a property UC*, we get *d*(*x*
_{2n
}, *x*
_{2n+2}) → 0. As (*B*, *A*) has a property UC*, we also have *d*(*x*
_{2n+1}, *x*
_{2n+3}) → 0.

We now show that for every

*ε* > 0 there exists

*N* such that

for all *m* > *n* ≥ *N*.

Suppose (3.5) not, then there exists

*ε* > 0 such that for all

*k* ∈ ℕ there exists

*m*
_{
k
} >

*n*
_{
k
} ≥

*k* such that

Taking *k* → *∞*, we have
.

By Lemma 3.8, there exists

*N* ∈ ℕ such that

for all

*m* >

*n* ≥

*N*. By using the triangle inequality we get

which contradicts. Therefore, condition (3.5) holds. Since (3.5) holds and *d*(*x*
_{2n
}, *x*
_{2n+1}) → *d*(*A*, *B*), by using property UC* of (*A*, *B*), we have {*x*
_{2n
}} is a Cauchy sequence. In similar way, we can prove that
, {*x*
_{2n+1}} and
are Cauchy sequences. □

Here we state the main results of this article on the existence and convergence of coupled best proximity points for cyclic contraction pairs on nonempty subsets of metric spaces satisfying the property UC*.

**Theorem 3.10**.

*Let A and B be nonempty closed subsets of a complete metric space X such that* (

*A*,

*B*)

*and* (

*B*,

*A*)

*satisfy the property UC**

*. Let F* :

*A* ×

*A* →

*B, G* :

*B* ×

*B* →

*A and* (

*F*,

*G*)

*be a cyclic contraction. Let*
*and define*
*for all n* ∈ ℕ ∪ {0}

*. Then F has a coupled best proximity point* (

*p*,

*q*) ∈

*A* ×

*A and G has a coupled best proximity point* (

*p*',

*q*') ∈

*B* ×

*B such that*
*Moreover, we have x*
_{2n
}→ *p,*
*, x*
_{2n+1 }→ *p*' *and*
.

*Proof*. By Lemma 3.7, we get

*d*(

*x*
_{2n
},

*x*
_{2n+1}) →

*d*(

*A*,

*B*). Using Lemma 3.9, we have {

*x*
_{2n
}} and

are Cauchy sequences. Thus, there exists

*p*,

*q* ∈

*A* such that

*x*
_{2n
}→

*p* and

. We obtain that

Letting

*n* →

*∞* in (3.8), we have

*d*(

*p*,

*x*
_{2n-1}) →

*d*(

*A*,

*B*). By a similar argument we also have

. It follows that

Taking *n* → *∞*, we get *d*(*p*, *F* (*p*, *q*)) = *d*(*A*, *B*). Similarly, we can prove that *d*(*q*, *F* (*q*, *p*)) = *d*(*A*, *B*). Therefore, we have (*p*, *q*) is a coupled best proximity point of *F*.

In similar way, we can prove that there exists *p*', *q*' ∈ *B* such that *x*
_{2n+1 }→ *p*' and
. Moreover, we also have *d*(*p*', *G*(*p*', *q*')) = *d*(*A*, *B*) and *d*(*q*', *G*(*q*', *p*')) = *d*(*A*, *B*) and so (*p*', *q*') is a coupled best proximity point of *G*.

Finally, we show that

*d*(

*p*,

*p*') +

*d*(

*q*,

*q*') = 2

*d*(

*A*,

*B*). For

*n* ∈ ℕ ∪ {0}, we have

It follows from (3.9) and (3.10) that

Since

*d*(

*A*,

*B*) ≤

*d*(

*p*,

*p*') and

*d*(

*A*,

*B*) ≤

*d*(

*q*,

*q*'), we have

From (3.11) and (3.12), we get

This complete the proof. □

Note that every pair of nonempty closed subsets *A*, *B* of a uniformly convex Banach space *X* such that *A* is convex satisfies the property UC*. Therefore, we obtain the following corollary.

**Corollary 3.11**.

*Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X, F* :

*A* ×

*A* →

*B, G* :

*B* ×

*B* →

*A and* (

*F*,

*G*)

*be a cyclic contraction. Let*
and define

*for all n* ∈ ℕ ∪ {0}

*. Then F has a coupled best proximity point* (

*p*,

*q*) ∈

*A* ×

*A and G has a coupled best proximity point* (

*p*',

*q*') ∈

*B* ×

*B such that*
*Moreover, we have x*
_{2n
}→ *p,*
*, x*
_{2n+1 }→ *p*' *and*
.

Next, we give some illustrative example of Corollary 3.11.

**Example 3.12**.

*Consider uniformly convex Banach space X* = ℝ

*with the usual norm. Let A* = [1,2]

*and B* = [-2, -1].

*Thus d*(

*A*,

*B*) = 2

*. Define F* :

*A* ×

*A* →

*B and G* :

*B* ×

*B* →

*A by*
*For arbitrary* (

*x*,

*x*') ∈

*A* ×

*A and* (

*y*,

*y*') ∈

*B* ×

*B and fixed*
*, we get*
*This implies that* (

*F*,

*G*)

*is a cyclic contraction with*
.

*Since A and B are convex, we have* (

*A*,

*B*)

*and* (

*B*,

*A*)

*satisfy the property UC**

*. Therefore, all hypothesis of Corollary 3.11 hold. So F has a coupled best proximity point and G has a coupled best proximity point. We note that a point* (1, 1) ∈

*A* ×

*A is a unique coupled best proximity point of F and a point* (

*-*1,

*-*1) ∈

*B* ×

*B is a unique coupled best proximity point of G. Furthermore, we get*
**Theorem 3.13**.

*Let A and B be nonempty compact subsets of a metric space X*,

*F* :

*A*×

*A* →

*B, G* :

*B* ×

*B* →

*A and* (

*F*,

*G*)

*be a cyclic contraction pair. If*
*and define*
*for all n* ∈ ℕ ∪ {0},

*then F has a coupled best proximity point* (

*p*,

*q*) ∈

*A* ×

*A and G has a coupled best proximity point* (

*p*',

*q*') ∈

*B* ×

*B such that*
*Proof*. Since

*x*
_{0},

and

for all

*n* ∈ ℕ ∪ {0}, we have

*x*
_{2n
},

and

*x*
_{2n+1},

for all

*n* ∈ ℕ ∪ {0}. As

*A* is compact, the sequence {

*x*
_{2n
}} and

have convergent subsequences

and

, respectively, such that

By Lemma 3.7, we have

. Taking

*k* →

*∞* in (3.13), we get

. By a similar argument we observe that

. Note that

Taking *k* → *∞*, we get *d*(*p*, *F* (*p*, *q*)) = *d*(*A*, *B*). Similarly, we can prove that *d*(*q*, *F*(*q*, *p*)) = *d*(*A*, *B*). Thus *F* has a coupled best proximity (*p*, *q*) ∈ *A* × *A*. In similar way, since *B* is compact, we can also prove that *G* has a coupled best proximity point in (*p*', *q*') ∈ *B* × *B*. For *d*(*p*, *p*') + *d*(*q*, *q*') = 2*d*(*A*, *B*) similar to the final step of the proof of Theorem 3.10. This complete the proof. □