Ann. Acad. Rom. Sci.

Ser. Math. Appl.

ISSN 2066-6594

Vol. 18, No. 2/2026

ON BEST PROXIMITY POINTS OF CYCLIC

ORBITAL PROXIMAL CONTRACTIONS^∗

Kanagajothi Dharumaraj^†

Prabavathy Magadevan^§

Selvaraj Chellachi Premila^‡

Saravanan Karpagam^¶

Communicated by G. Moro¸sanu

10.56082/annalsarscimath.2026.2.263

DOI

Abstract

Let A and B be non-empty subsets of a metric space (X, d). Let

T : A ∪ B → A ∪ B be a map such that T(A) ⊆ B and T(B) ⊆ A

satisfying a certain contractive condition called cyclic orbital proximal

contraction. We give the necessary conditions for the existence of a

unique point ξ ∈ A such that d(ξ, Tξ) is equal to the distance between

A and B. Our main result generalizes the main result of [A.A. Eldred

and P. Veeramani, Existence and convergence of best proximity points,

J. Math. Anal. Appl., 323 (2006), 1001-1006].

Keywords: cyclic map, best proximity point, orbital contractions, uni-

formly convex Banach space.

MSC: 47H10, 54H25.

^∗Accepted for publication on March 18, 2026

^†kanagajothi82@gmail.com, Department of Mathematics, Vel Tech Rangarajan Dr.

Sagunthala R and D Institute of Science and Technology, Chennai, Tamilnadu, India

^‡premilasc77@gmail.com, Department of Mathematics, Saveetha Engineering Col-

lege(Autonomous), Thandalam, Chennai, India

^§prabavathy09@gmail.com, m.prabavathy@adjadmc.ac.in, PG and Research Depart-

ment of Mathematics, A.D.M College for Women(Autonomous), Nagapattinam - 611 001,

Tamilnadu, India

^¶karpagam.saravanan@gmail.com, Department of Mathematics, Bhaktavatsalam

Memorial College for Women, Chennai, Tamilnadu, India

263

On best proximity points of cyclic orbital proximal contractions

264

1 Introduction and preliminaries

Many problems of practical interest are formulated as ﬁxed point equations.

The most commonly used ﬁxed point theorem is given in [2]. The classical

Banach contraction theorem given in [2] states that ”Let T be a self map-

ping of a complete metric space (X, d) such that for some k, 0 < k < 1,

d(Tx, Ty) ≤ kd(x, y), for all x, y ∈ X. Then for any x ∈ X, the sequence

{Tⁿx} converges to a unique ﬁxed point of T in X. ” The mapping T is

called the Banach Contraction. Since the contraction condition is strong,

numerous attempts were made to weaken the condition on the mapping.

One such attempt was made in [3], in which an approximation to the ﬁxed

point was given. In [3], a notion of best proximity point was introduced.

If there does not exist a ﬁxed point, the best proximity point gives an ap-

proximate solution to the ﬁxed point problem. If (X, d) is a metric space

and A and B are non-empty subsets of X, a map T : A ∪ B → A ∪ B such

that T(A) ⊆ B and T(B) ⊆ A is introduced. We call such a map the Cyclic

map. A contraction condition is imposed on the cyclic map as follows:

Deﬁnition 1. (see [3]) Let (X, d) be a metric space. Let A and B be non-

empty subsets of X. Let T : A ∪ B → A ∪ B be a cyclic map. If for some k,

0 < k < 1,

d(Tx, Ty) ≤ kd(x, y) + (1 − k)dist(A, B),

x ∈ A, y ∈ B,

(1)

where dist(A, B) = inf{d(x, y) : x ∈ A, y ∈ B}, then T is said to be a

Cyclic Contraction map.

In [3], a best proximity point is deﬁned for a cyclic map as a point x ∈ A

such that d(x, Tx) = dist(A, B). Note that if dist(A, B) = 0, then equation

(1) reduces to a Banach contraction and the obtained best proximity point

becomes a ﬁxed point. In [3 ], a best proximity point for a cyclic contraction

map is obtained in a uniformly convex Banach space settings as follows:

Theorem 1. (see [3 ]) Let X be a uniformly convex Banach space. Let A and

B be non-empty, closed and convex subsets of X. Let T : A∪B → A∪B be a

cyclic contraction map. Then for any x ∈ A, the sequence {T²ⁿx} converges

to a unique best proximity point of T in A.

In [6], a notion of cyclic orbital contraction was introduced.

Deﬁnition 2. (see [6]) Let A and B be non-empty subsets of a metric space

(X, d) and T : A ∪ B → A ∪ B be a cyclic map such that for some x ∈ A,

K. Dharumaraj, S.C. Premila, P. Magadevan, S. Karpagam

265

(2)

there exists a k ∈ (0, 1) such that

d(T²ⁿx, Ty) ≤ kd(T²ⁿ⁻¹x, y), n ∈ N, y ∈ A.

Then T is called a Cyclic Orbital contraction.

In [6], the following ﬁxed point theorem is obtained.

Theorem 2. (see [6]) Let A and B be non-empty and closed subsets of a

metric space (X, d) and T : A ∪ B → A ∪ B be a cyclic orbital contraction.

Then A ∩ B is non-empty and T has a unique ﬁxed point.

In this paper, we attempt to combine Theorem 1 and Theorem 2 and

obtain a new best proximity point result by introducing a notion of Cyclic

Orbital Proximal Contraction map. For further generalizations of Banach

contraction theorem, one may refer to [1], [4], [5], [7], [10], [11], [13] and

other papers found in the literature.

In [3], the following lemmas are proved which are useful to prove the

main result.

Lemma 1. (see [3]) Let A be a non-empty, closed and convex subset and

B be a non-empty, closed subset of a uniformly convex Banach space. Let

{x_n} and {z_n} be sequences in A and {y_n} be a sequence in B satisfying

(1) k z_n− y_nk−→ dist(A, B),

(2) for every ꢀ > 0, there exists N₀∈ N such that for all m > n ≥ N₀,

k x_m− y_nk≤ dist(A, B) + ꢀ.

Then for every ꢀ > 0, there exists N₁∈ N, such that for all m > n ≥ N₁,

k x_m− z_nk≤ ꢀ.

Lemma 2. (see [3]) Let A be a non-empty, closed and convex subset and

B be a non-empty, closed subset of a uniformly convex Banach space. Let

{x_n} and {z_n} be sequences in A and {y_n} be a sequence in B satisfying

(1)k x_n− y_nk−→ dist(A, B),

(2) k z_n− y_nk−→ dist(A, B).

Then k x_n− z_nk−→ 0.

2 Main results

We now give a notion of cyclic orbital proximal contraction.

On best proximity points of cyclic orbital proximal contractions

266

Deﬁnition 3. Let A and B be non-empty subsets of a metric space (X, d)

and T : A ∪ B → A ∪ B be a cyclic map such that for some x ∈ A, there

exists a k ∈ (0, 1) such that

d(T²ⁿx, Ty) ≤ kd(T²ⁿ⁻¹x, y) + (1 − k)dist(A, B), n ∈ N, y ∈ A.

(3)

Then T is called a Cyclic Orbital Proximal contraction.

We note that if dist(A, B) = 0 then a cyclic orbital proximal contraction

becomes a cyclic orbital contraction. In that case, A ∩ B is non-empty and

we obtain a unique ﬁxed point, which is stated in Theorem 2.

Deﬁnition 4. Let A and B be non-empty subsets of a metric space (X, d)

and T : A ∪ B → A ∪ B be a cyclic map such that for some x ∈ A,

d(T²ⁿx, Ty) ≤ d(T²ⁿ⁻¹x, y), n ∈ N, y ∈ A.

(4)

Then T is called a cyclic orbital nonexpansive map.

Note that if T is a cyclic orbital proximal contraction then it is a cyclic

orbital nonexpansive map. Now we prove a useful convergence result.

Proposition 1. Let A and B be non-empty subsets of a metric space (X, d)

and T : A ∪ B → A ∪ B be a cyclic orbital proximal contraction with an

x ∈ A satisfying equation (3). Then, for all y ∈ A, we have the following:

d(T²ⁿx, T²ⁿ⁺¹y) −→ dist(A, B) as n −→ ∞.

Proof. Consider the following:

d(T²ⁿx, T²ⁿ⁺¹y) ≤ kd(T²ⁿ⁻¹x, T²ⁿy) + (1 − k)dist(A, B)

≤ k(kd(T²ⁿ⁻²x, T²ⁿ⁻¹y) + (1 − k)dist(A, B))

+(1 − k)dist(A, B)

= k²d(T²ⁿ⁻²x, T²ⁿ⁻¹y) + (1 − k²)dist(A, B).

Inductively, we have

d(T²ⁿx, T²ⁿ⁺¹y) ≤ k²ⁿd(x, Ty) + (1 − k²ⁿ)dist(A, B).

Since 0 < k < 1, as n −→ ∞, we have the desired result.

We use the following Proposition in proving the main result.

K. Dharumaraj, S.C. Premila, P. Magadevan, S. Karpagam

267

Proposition 2. Let A and B be non-empty and closed subsets of a metric

space (X, d) and T : A ∪ B → A ∪ B be a cyclic orbital proximal contraction

with an x ∈ A satisfying equation (3). If {T²ⁿx} converges to ξ ∈ A, then ξ

is a best proximity point of T in A.

Proof. Given T²ⁿx −→ ξ as n −→ ∞. Since T is cyclic orbital proxi-

mal contraction, by Proposition 1, d(T²ⁿx, T²ⁿ⁺¹y) −→ dist(A, B). Now

dist(A, B) ≤ d(T²ⁿx, Tξ) ≤ d(T²ⁿ⁻¹x, ξ) which tends to dist(A, B) as

n −→ ∞. Therefore, lim_n→∞d(T²ⁿx, Tξ) = dist(A, B). That is, d(ξ, Tξ) =

dist(A, B). Hence ξ is a best proximity point of T in A.

Now we obtain a best proximity point for a cyclic orbital proximal con-

traction, which is the main result.

Theorem 3. Let A and B be non-empty, closed and convex subsets of a

uniformly convex Banach space. Suppose T : A ∪ B → A ∪ B is a cyclic

orbital proximal contraction with an x ∈ A satisfying equation (3). Then

the sequence {T²ⁿx} converges to a best proximity point ξ in A. Moreover,

if y ∈ A, y = x satisﬁes equation (3), then the sequence {T²ⁿy} converges

to the same best proximity point ξ in A.

Proof. If dist(A, B) = 0, then there exists a unique ﬁxed point in A ∩ B as

given in Theorem 2. Therefore, assume dist(A, B) > 0. By Proposition 1,

k T²ⁿx−T²ⁿ⁺¹x k−→ dist(A, B) and k T²ⁿ⁺¹x−T²ⁿ⁺²x k−→ dist(A, B).

By Lemma 2, we have k T²ⁿx−T²ⁿ⁺²x k−→ 0. We now show that for ꢀ > 0

there exists N₀∈ N such that for all m > n ≥ N₀,

k T^2mx − T²ⁿ⁺¹x k≤ dist(A, B) + ꢀ.

Suppose not, then there exists ꢀ > 0 such that for all k ∈ N there exists

m_k> n_k≥ k for which

k T^2mx − T^{2n +1}x k≥ dist(A, B) + ꢀ.

k

This m_kcan be chosen such that it is the least integer greater than n_kto

satisfy the above inequality. Now, we obtain

dist(A, B) + ꢀ ≤ k T^2mx − T^{2n +1}x k

k

≤ k T^2mx − T^{2m −2}x k + k T^{2m −2}x − T^{2n +1}x k .

k

On best proximity points of cyclic orbital proximal contractions

268

Since k T^2mx−T^{2m −2}x k−→ 0 and k T^{2m −2}x−T^{2n +1}x k≤ dist(A, B)+ꢀ

k

as k −→ ∞, we have

lim k T^2mx − T^{2n +1}x k= dist(A, B) + ꢀ.

k

k→∞

Consequently,

k T^2mx − T^{2n +1}x k ≤ k T^2mx − T^{2m +2}x k + k T^{2m +2}x − T^{2n +3}x k

k

+ k T^{2n +3}x − T^{2n +1}x k

k

≤ k T^2mx − T^{2m +2}x k +K²k T^2mx − T^{2n +1}x k

k

+ (1 − k²)dist(A, B)+ k T^{2n +3}x − T^{2n +1}x k,

k

dist(A, B) + ꢀ ≤ k²dist(A, B) + ꢀ + (1 − k²)dist(A, B)

= dist(A, B) + k²ꢀ,

which is a contradiction. Therefore, k T^2mx − T²ⁿ⁺¹x k≤ dist(A, B) + ꢀ.

Already we have k T^2mx − T²ⁿ⁺¹x k−→ dist(A, B) as n −→ ∞. Therefore,

by Lemma 1 we have for given ꢀ > 0 there exists n₁∈ N such that

k T^2mx − T²ⁿx k≤ ꢀ for m > n > n₁.

Hence {T²ⁿx} is a Cauchy sequence and converges to a ξ ∈ A. By Proposi-

tion 2, ξ is a best proximity point of T in A. Now, let y ∈ A, y = x, satisfy

the inequality (3). By what we have proved now, the sequence {T²ⁿy}

converges to an η ∈ A, such that η is a best proximity point. That is

k η − Tη k= dist(A, B). Let us show that η = ξ. Suppose that η = ξ. Now,

we get

k T²ξ − Tξ k= lim k T²ⁿ⁺²x − T²ⁿ⁺¹x k= dist(A, B),

n

k ξ − Tξ k= dist(A, B) implies that T²ξ = ξ. Similarly, we have T²η = η.

Also,

k Tξ − η k= lim k T²ⁿ⁺¹x − T²ⁿ⁺²y k≤ lim k T²ⁿx − T²ⁿ⁺¹y k=k ξ − Tη k

n

k Tη − ξ k= lim k T²ⁿ⁺¹y − T²ⁿ⁺²x k≤ lim k T²ⁿy − T²ⁿ⁺¹x k=k η − Tξ k,

n

which implies k Tξ − η k=k Tη − ξ k. Now, η = ξ ⇒k Tη − ξ k> dist(A, B).

Now,

k Tη − ξ k=k Tη − T²ξ k= lim k T²ⁿ⁺¹y − T²ⁿ⁺²x k= dist(A, B),

n

which is a contradiction. Hence, η = ξ. Hence, the theorem.

K. Dharumaraj, S.C. Premila, P. Magadevan, S. Karpagam

269

The following example illustrates the main result.

Example 1. Consider the metric space (R), |.|). Let A = [0, 1] and B =

[2, 3] be the closed intervals. Deﬁne T : A∪B → A∪B as follows: T(0) = 3,

T(3) = 0.

T(x) = (5 − x)/2, ∀x ∈ (0, 1] and T(y) = (4 − y)/2, ∀y ∈ [2, 3).

Clearly T(A) ⊆ B and T(B) ⊆ A and dist(A, B) = 1. It is an easy exercise

to see that for all x ∈ (0, 1], the following inequality is satisﬁed:

|T²ⁿx − Ty| ≤ (1/2)|T²ⁿ⁻¹x − y| + (1/2)dist(A, B).

Thus, T is a cyclic orbital proximal contraction with k = 1/2. We ﬁnd that

for all x ∈ (0, 1], the sequence {T²ⁿx} converges to 1, such that |1 − T(1)| =

dist(A, B),which is the unique best proximity point of T in A.

3 Conclusion

In this paper we have introduced a new map called cyclic orbital proximal

contraction (see Deﬁnition 3) and obtained a unique best proximity point

of the map, where the best proximity point is the limit of Picard type itera-

tive sequence of a point satisfying equation (3). This contractive condition

weakens the contractive condition given in deﬁnition 1 (see equation (1 )).

Moreover, the new map also generalizes the notion of cyclic orbital contrac-

tion. Hence, the main result of this paper generalizes the main result of [3]

and also generalizes Theorem 2 of [6]. We have also illustrated our main

result by giving an example.

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