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Best Proximity Point of Generalized θ − φ−Proximal Non-self Contractions

Mohamed Rossafi1,∗ , Abdelkarim Kari2
1Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Fes, Morocco

rossafimohamed@gmail.com
2Faculty of Sciences Ben M’Sik, Hassan II University, Casablanca, Morocco

abdkrimkariprofes@gmail.com
∗Correspondence: rossafimohamed@gmail.com

Abstract. In this manuscript, motivated and inspired by results of Best proximity point of generalized
F -proximal non-self contractions, we introduce the concept of generalized θ−φ−proximal contractionand prove new best proximity results for these contractions in the setting of a metric space. Our resultsgeneralize and extend many recent results appearing in the literature. An example is being given todemonstrate the usefulness of our results.

1. Introduction
It is well known that the Banach contraction theorem is the first outstanding result in thefield of the fixed point theory that ensure the existence of unique fixed point in complete metricspaces. Due to its importance, various mathematics steadied many interesting extensions andgeneralizations [7,8,12,14]. One of the famous generalizations of the Banach contraction principle [2]for existence of fixed point for self-mapping on metric space is the theorem by Zheng et al. [14] andthe contraction introduced by Jleli and Samet in [6].Best proximity point theorem analyses the condition under which the optimisation problem,namely infx∈A d(x, T x), has a solution. The point x is called the best proximity of T : A → B, if

d(x, T x) = d(A,B), where {d(A,B) = inf d(x, y) : x ∈ A, y ∈ B}. Note that the best proximitypoint reduces to a fixed point if T is a self-mapping. Various best proximity point results wereestablished on such spaces [1, 9, 12].Sankar Raj [10] and Zhang et al. [13] defined the notion of P−property and weak P−propertyrespectively. Beg et al. [4] defined the concept of generalized F -proximal non-self contractions andobtained some best proximity point theorems for self-mappings.In this paper, inspired by the idea of generalized F -proximal non-self contractions, introducedby Beg et al. [4] in metric spaces, we prove a new existence of best proximity point for generalized
Received: 21 Jan 2024.
Key words and phrases. P -property, best proximity point, generalized θ − φ-proximal contraction.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.13
https://orcid.org/0000-0002-5662-6921
https://orcid.org/0000-0003-0088-9404


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 2

θ − φ−proximal contraction defined on a closed subset of a complete metric space. Our theoremsextend, generalize and improve many existing results.
2. Preliminaries

Let (A,B) be a pair of non empty subsets of a metric space (X, d). We adopt the followingnotations:
d(A,B) = {inf d (a, b) : a ∈ A, b ∈ B};
A0 = { a ∈ A there exists b ∈ A such that d (a, b) = d (A,B)};
B0 = { b ∈ B there exists a ∈ A such that d (a, b) = d (A,B)}.

Definition 2.1. [5] Let T : A → B be a mapping. An element x∗ is said to be a best proximitypoint of T if
d (x∗, T x∗) = d (A,B) .

Definition 2.2. [10] Let (A,B) be a pair of non empty subsets of a metric space (X, d) such that
A0 is non empty. Then the pair (A,B) is to have P -property if and only ifd (x1, y1) = d (A,B)

d (x2, y2) = d (A,B)
⇒ d(x1, x2) = d(y1, y2)

where x1, x2 ∈ A0 and y1, y2 ∈ B0.
Definition 2.3. [3] A set B is called approximately compact with respect to A if every sequence
{xn} of B with d(y , xn)→ d(y , B) for some y ∈ A has a convergent subsequence.
Definition 2.4. [6] Let Θ be the family of all functions θ : ]0,+∞[ → ]1,+∞[ such that

(θ1) θ is strictly increasing;
(θ2) For each sequence xn ∈ ]0,+∞[;

lim
n→0

xn = 0, if and only if lim
n→∞

θ (xn) = 1;

(θ3) θ is continuous.
Definition 2.5. [14] Let Φ be the family of all functions φ: [1,+∞[ → [1,+∞[, such that

(φ1) φ is increasing;
(φ2) For each t ∈ ]1,+∞[, l imn→∞φn(t) = 1;
(φ3) φ is continuous.

Lemma 2.6. [14] If φ ∈ Φ Then φ(1)=1, and φ(t) < t .

Definition 2.7. [14]. Let (X, d) be a metric space and T : X → X be a mapping.
T is said to be a θ − φ−contraction if there exist θ ∈ Θ and φ ∈ Φ such that for any x, y ∈ X,

d (Tx, T y) > 0⇒ θ [d (Tx, T y)] ≤ φ [θ (d (x, y))] ,

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 33. Main result
In this section, inspired by the notion of F -proximal contraction of the first kind and second kind,we introduce new generalized θ− φ-proximal first kind and second kind on complete metric space.

Definition 3.1. The mapping T : A→ B is said to be a generalized θ − φ-proximal contraction offirst kind if there exist θ ∈ Θ, φ ∈ Φ and a, b, c, h ≥ 0 with a + b + +2ch, c 6= 1 such thatd (u1, T v1) = d (A,B)

d (u2, T v2) = d (A,B)
⇒ θ(d(u1, u2))

≤ φ [θ [ad (v1, v2) + bd (u1, v1) + cd (u2, v2) + h (d (v1, u2) + d (v2, u1))]]for all u1, u2, v1, v2 ∈ A and u1 6= v1.
Definition 3.2. The mapping T : A→ B is said to be a generalized θ − φ-proximal contraction ofsecond kind if there exist θ ∈ Θ, φ ∈ Φ and a, b, c, h ≥ 0 with a + b + +2ch, c 6= 1 such thatd (u1, T v1) = d (A,B)

d (u2, T v2) = d (A,B)
⇒ θ(d(Tu1, T u2))

≤ φ [θ [ad (Tv1, T v2) + bd (Tu1, T v1) + cd (Tu2, T v2) + h (d (Tv1, T u2) + d (Tv2, T u1))]]for all u1, u2, v1, v2 ∈ A and Tu1 6= Tv1.
Theorem 3.3. Let (X, d) be a complete metric space and (A,B) be a pair of non-void closed
subsets of (X, d). If B is approximately compact with respect to A and T : A → B satisfy the
following conditions :(i) T (A0) ∈ B0 and the pair (A,B) satisfies the weak P -property;(ii) T is a generalized θ − φ-proximal contraction of first kind.

Then there exists a unique u ∈ A such that d(u, Tu) = d(A,B). In addition, for any fixed element
u0 ∈ A0, sequence {un} defined by

d(un+1, T un) = d(A,B),

converges to the proximity point.

Proof. Choose an element u0 ∈ A0. As, T (A0) ∈ B0, therefore there is an element u1 ∈ A0satisfying
d(u1, T u0) = d(A,B).Since T (A0) ∈ B0, there exists u2 ∈ A0 such that
d(u2, T u1) = d(A,B).

Again, since T (A0) ∈ B0, there exists u3 ∈ A0 such that
d(u3, T u2) = d(A,B).

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 4Continuing this process, by induction, we construct a sequence xn ∈ A0 such that
d (un+1, T un) = d(A,B),∀n ∈ N.

Since (A,B) satisfies the P property, we conclude that
d(un, un+1) = d(Tun, T un+1),∀n ∈ N. (3.1)

If un0 = un0+1 for some n0 ∈ N, from (3) one obtains
d (un0 , T un0) = d (un0+1, T un0) = d(A,B) (3.2)

that is, un0 ∈ BPP . Thus, we suppose that d(un, xn+1) > 0 for all n ∈ N.We shall prove that the sequence un is a Cauchy sequence. Let us first prove that
lim
n→∞

d (un, un+1) = 0.

As T is generalized (θ, φ)-proximal contraction of the first kind, we have that
θ (d (un, un+1))

≤ φ [θ [ad (un−1, un) + bd (un−1, un) + cd (un, un+1) + h (d (un−1, un+1) + d (un, un))]]

= φ [θ [ad (un−1, un) + bd (un−1, un) + cd (un, un+1) + h (d (un−1, un+1))]]

≤ φ [θ [ad (un−1, un) + bd (xn−1, xn) + cd (un, un+1) + h (d (un−1, un) + d (un, un+1))]]

= φ [θ [(a + b + h)d (un−1, un) + (c + h)d (un, un+1)]]

Since θ is strictly increasing and by Lemma 2.6, we deduce
d (xn, xn+1) < (a + b + h)d (xn−1, xn) + (c + h)d (xn, xn+1) .

Thus
d (un, un+1) <

a + b + h

1− c − h (d (un−1, xn)).

If b + b + c + 2h = 1, we have 0 < 1− c − h and so
d (un, un+1) ≤

a + b + h

1− c − h (d (un−1, un)) = d (un−1, un) ,∀n ∈ N;

Consequently,
θ (d (un, un+1)) ≤ φ [θ (d (un−1, un))]

If b + b + c + 2h < 1, we have 0 < 1− c − h and so
d (un, un+1) < d (un−1, un) ,∀n ∈ N;

Consequently,
θ (d (un, un+1)) ≤ φ [θ (d (un−1, un))]

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 5It implies
θ (d (un, un+1)) ≤ φ [θ (d(xn−1, un)]

≤ φ2 [θ (d(un−2, un−1)]

≤ ... ≤ φn [θ (d(u0, u1)] .

Taking the limit as n →∞, we have
1 ≤ θ(d (un, un+1)) ≤ lim

n→∞
φn [θ(d (u0, u1))] = 1.

Since θ ∈ Θ, we obtain
lim
n→∞

d (un, un+1) = 0. (3.3)
Next, we shall prove that {un}n∈N is a Cauchy sequence, i.e, limn→∞ d (un,um) = 0, for all n ∈ N.Suppose to the contrary that exists ε > 0 and sequences n(k) and m(k) of natural numbers suchthat

m(k) > n(k) > k, d
(
xm(k) , xn(k)

)
≥ ε, D

(
um(k)−1 , un(k)

)
< ε. (3.4)

Using the triangular inequality, we find that,
ε ≤ d

(
um(k) , un(k)

)
≤ d

(
um(k) , un(k)−1

)
+ d

(
xn(k)−1, xn(k)

) (3.5)
< ε+ d

(
un(k)−1, un(k)

)
. (3.6)

Then, by 3.4 and 3.22, it follows that
lim
k→∞

d
(
um(k) , un(k)

)
= ε. (3.7)

Using the triangular inequality, we find that,
ε ≤ d

(
um(k) , un(k)

)
≤ d

(
um(k) , un(k)+1

)
+ d

(
xn(k)+1, un(k)

) (3.8)
and

ε ≤ d
(
um(k) , un(k)+1

)
≤ d

(
um(k) , un(k)

)
+ d

(
un(k), un(k)+1

) (3.9)
Then, by (3.25) and (3.9), it follows that

lim
k→∞

d
(
um(k) , un(k)+1

)
= ε. (3.10)

Similarly method, we conclude that
lim
k→∞

d
(
um(k)+1 , un(k)

)
= ε. (3.11)

Using again the triangular inequality,
d
(
um(k)+1 , un(k)+1

)
≤ d

(
xm(k)+1 , xm(k)

)
+ d

(
um(k), un(k)

)
+ d

(
un(k) , un(k)+1

)
. (3.12)

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 6On the other hand, using triangular inequality, we have
d
(
um(k) , un(k)

)
≤ d

(
um(k) , um(k)+1

)
+ d

(
um(k)+1 , un(k)+1

)
+ d

(
un(k)+1 , un(k)

)
. (3.13)

Letting k →∞ in inequality (3.12) and (3.13), we obtain
lim
k→∞

d
(
um(k)+1 , un(k)+1

)
= ε. (3.14)

Substituting u1 = xm(k)+1 , u2 = xn(k)+1 , v1 = um(k) and v1 = un(k) in assumption of the theorem, weget

θ
(
d
(
um(k)+1 , un(k)+1

))
≤ φ


θ



ad
(
um(k) , un(k)

)
+ bd

(
um(k)1, un(k)

)
+ cd

(
un(k)+1, un(k)

)
+ h(d

(
um(k) , un(k)+1

)
+ d

(
un(k) , um(k)+1

)
)




(3.15)

Letting Letting k →∞ in (3.15), and using (θ1), (θ3) , (φ3) and Lemma (2.6) we obtain
θ (ε) ≤ φ [θ (aε+ bε+ cε+ 2hε)] .

We derive
ε < ε.

Which is a contradiction. Thus limn,m→∞ d (un, um) = 0, which shows that {xn} is a Cauchysequence. Then there exists z ∈ A such that
lim
n→∞

d (un, u) = 0.

Also,
d (u,B) ≤ d (u, Tun)

≤ d (u, xn+1) + d (un+1, T un)

= d (u, un+1) + d (A,B)

≤ d (u, un+1) + d (u,B) .

Therefore, d (u, Tun)→ d (u,B) . In spite of the fact that B is approximately compact with respectto A , the sequence {Tun} has a subsequence {Tunk} converging to some element v ∈ B. So itturns out that
d(u, v) = lim

n→∞
d
(
unk+1, T unk

)
= d(A,B). (3.16)

Thus u must be an element of A0. Again, since T (A0) ∈ B0, there exists t ∈ A0 such that
d(t, Tu) = d(A,B) (3.17)

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 7for some element t in A. Using the weak p-property and (3.33) we have
d(unk+1, t) = d(Punk , P u),∀nk ∈ N.

If for some n0, d(t, un0+1) = 0, consequently d(Pun0 , T u) = 0. So Pun0 = Tu, hence d(A,B) =

d(u, Tu). Thus the conclusion is immediate. So let for any n ≥ 0, d(t, un+1) > 0. Since T is ageneralized (θ, φ)-proximal contraction of the first kind, it follows from this that
θ(d(t, un+1)) ≤ φ [θ [ad (u, un) + bd (t, u) + cd (un, un+1) + h (d (u, un+1) + d (un, t))]] (3.18)

Since θ and φ are two continuous functions, by letting n →∞ in inequality (3.18), we obtain
θ(d(t, u)) ≤ φ [θ [(b + h) (d (u, t))]]

≤ φ [θ [(d (u, t))]]

< θ(d(t, u)).

It is a contradiction. Therefore, u = t , that
d(u, Tu) = d(t, Tu) = d(A,B).

Uniqueness: Suppose that there is another best proximity point z of the mapping T such that
d(z, T z) = d(A,B).

Since T is a generalized (θ, φ)-proximal contraction of the first kind, it follows from this that
θ(d(z, u)) ≤ φ [θ [ad (z, u) + bd (z, z) + cd (u, u) + h (d (z, u) + d (z, u))]]

= φ [θ [(a + 2h)d (z, u)]] ,

which is a contradiction. Thus, z and u must be identical. Hence, T has a unique best proximitypoint. �

Next, we state and prove the best proximity point theorem for non-self generalized (θ, φ)-proximalcontraction of the second kind.
Theorem 3.4. Let (X, d) be a complete metric space and (A,B) be a pair of non-void closed
subsets of (X, d). If A is approximately compact with respect to B and T : A → B satisfy the
following conditions :(i) T (A0) ∈ B0 and the pair (A,B) satisfies the weak P -property;(ii) T is continuous generalized (θ, φ)-proximal contraction of second kind.

Then there exists a unique u ∈ A such that d(u, Tu) = d(A,B) and un → u, where u0 is any fixed
point in A0 and d(un+1, T un) = d(A,B) for n ≥ 0. Further, if z is another best proximity point of
T , then Tu = Tz .

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 8

Proof. Similar to Theorem 3.3, we can find a sequence {un} in A0 such that
d(un+1, T un) = d(A,B). (3.19)

for all non-negative integral values of n. From the p-property and (3.19) we get
d(un, un+1) = d(Tun−1, T un),∀n ∈ N.

If for some n0, d(un0+1 , un0+2) = 0, consequently d(Tun0 , T un0+1) = 0. So Tun0 = Tun0+1,hence d(A,B) = d(Tun0 , Tn0+1). Thus the conclusion is immediate. So let for any n ≥ 0,
d(Tun, T un+1) > 0. We shall prove that the sequence un is a Cauchy sequence. Let us firstprove that

lim
n→∞

d (un, un+1) = 0.

As T is generalized (θ, φ)-proximal contraction of the second kind, we have that
θ (d (Tun, T un+1))

≤ φ [θ [ad (Tun−1, T un) + bd (Tun−1, T un) + cd (Tun, T un+1) + h (d (Tun−1, T un+1) + d (Tun, T un))]]

= φ [θ [ad (Tun−1, T un) + bd (Tun−1, T un) + cd (Tun, T un+1) + h (d (Tun−1, T un+1))]]

≤ φ [θ [ad (Tun−1, T un) + bd (Tun−1, T un) + cd (Tun, T un+1) + h (d (Tun−1, T un) + d (Tun, T un+1))]]

= φ [θ [(a + b + h)d (Tun−1, T un) + (c + h)d (Tun, T un+1)]]

Since θ is strictly increasing and by Lemma 2.6, we deduce
d (Tun, T un+1) < (a + b + h)d (Tun−1, T un) + (c + h)d (Tun, T un+1) .

Thus
d (Tun, T un+1) <

a + b + h

1− c − h (d (Tun−1, T un)).

If b + b + c + 2h = 1, we have 0 < 1− c − h and so
d (Tun, T un+1) ≤

a + b + h

1− c − h (d (Tun−1, T un)) = d (Tun−1, T un) ,∀n ∈ N;

Consequently,
θ (d (Tun, T un+1)) ≤ φ [θ (d (Tun−1, T un))]

If b + b + c + 2h < 1, we have 0 < 1− c − h and so
d (Tun, T un+1) < d (Tun−1, T un) ,∀n ∈ N;

Consequently,
θ (d (Tun, T un+1)) ≤ φ [θ (d (Tun−1, T un))]

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 9It implies
θ (d (Tun, T un+1)) ≤ φ [θ (d(Tun−1, T un)]

≤ φ2 [θ (d(Tun−2, T un−1)]

≤ ... ≤ φn [θ (d(Tu0, T u1)] .

Taking the limit as n →∞, we have
1 ≤ θ(d (Tun, T un+1)) ≤ lim

n→∞
φn [θ(d (Tu0, T u1))] = 1.

Since θ ∈ Θ, we obtain
lim
n→∞

d (Tun, T un+1) = 0. (3.20)
Next, we shall prove that {Tun}n∈N is a Cauchy sequence, i.e, limn→∞ d (Tun,Tum) = 0, for all
n ∈ N. Suppose to the contrary that exists ε > 0 and sequences Tn(k) and Tm(k) of naturalnumbers such that

Tm(k) > Tn(k) > k, d
(
Tum(k) , T un(k)

)
≥ ε, d

(
Tum(k)−1 , T un(k)

)
< ε. (3.21)

Using the triangular inequality, we find that,
ε ≤ d

(
Tum(k) , T un(k)

)
≤ d

(
Tum(k) , T xn(k)−1

)
+ d

(
Tun(k)−1, T un(k)

) (3.22)
< ε+ d

(
Tun(k)−1, T un(k)

)
. (3.23)

Then, by 3.4 and 3.22, it follows that
lim
k→∞

d
(
Tum(k) , T un(k)

)
= ε. (3.24)

Using the triangular inequality, we find that,
ε ≤ d

(
Tum(k) , T un(k)

)
≤ d

(
Tum(k) , T un(k)+1

)
+ d

(
Tun(k)+1, T un(k)

) (3.25)
and

ε ≤ d
(
Tum(k) , T un(k)+1

)
≤ d

(
Tum(k) , T un(k)

)
+ d

(
Tun(k), T un(k)+1

) (3.26)
Then, by (3.25) and (3.9), it follows that

lim
k→∞

d
(
Tum(k) , T un(k)+1

)
= ε. (3.27)

Similarly method, we conclude that
lim
k→∞

d
(
Tum(k)+1 , T un(k)

)
= ε. (3.28)

Using again the triangular inequality,
d
(
Tum(k)+1 , T un(k)+1

)
≤ d

(
um(k)+1 , T um(k)

)
+ d

(
Tum(k), T un(k)

)
+ d

(
Tun(k) , T un(k)+1

)
. (3.29)

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 10On the other hand, using triangular inequality, we have
d
(
Tum(k) , T un(k)

)
≤ d

(
Tum(k) , T um(k)+1

)
+ d

(
Tum(k)+1 , T un(k)+1

)
+ d

(
Tun(k)+1 , T un(k)

)
. (3.30)

Letting k →∞ in inequality (3.29) and (3.30), we obtain
lim
k→∞

d
(
Tum(k)+1 , T un(k)+1

)
= ε. (3.31)

Substituting u1 = Tum(k)+1 , u2 = Tun(k)+1 , v1 = Tum(k) and v1 = Tun(k) in assumption of thetheorem, we get

θ
(
d
(
Tum(k)+1 , T un(k)+1

))
≤ φ


θ



ad
(
Tum(k) , T un(k)

)
+ bd

(
Tum(k)1, T un(k)

)
+ cd

(
Tun(k)+1, T un(k)

)
+ h(d

(
Tum(k) , T un(k)+1

)
+ d

(
Tun(k) , T um(k)+1

)
)



(3.32)Letting Letting k →∞ in (3.32), and using (θ1), (θ3) , (φ3) and Lemma (2.6) we obtain
θ (ε) ≤ φ [θ (aε+ bε+ cε+ 2hε)] .

We derive
ε < ε.

Which is a contradiction. Thus limn,m→∞ d (Tun, T um) = 0, which shows that {Tun} is a Cauchysequence. Then there exists v ∈ B such that
lim
n→∞

d (Tun, v) = 0.

Also,
d (v , A) ≤ d (v , Tun)

≤ d (v , un+1) + d (un+1, T un)

= d (v , un+1) + d (A,B)

≤ d (v , un+1) + d (v , A) .

Therefore, d (v , Tun) → d (v , A) . Since A is approximately compact with respect to B , thesequence {un} has a subsequence {unk} converging to some element u ∈ A. So it turns out that
d(u, v) = lim

n→∞
d
(
unk+1, T unk

)
= d(A,B). (3.33)

Because T is a continuous mapping,
d(u, Tu) = lim

n→∞
d(un+1, T un) = d(A,B).

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 11Uniqueness: Suppose that there is another best proximity point z of the mapping T such that
d(z, T z) = d(A,B).

Since T is a generalized (θ, φ)-proximal contraction of the first second, it follows from this that
θ(d(Tz, Tu)) ≤ φ [θ [ad (Tz, Tu) + bd (Tz, T z) + cd (Tu, Tu) + h (d (Tz, Tu) + d (Tz, Tu))]]

= φ [θ [(a + 2h)d (Tz, Tu)]] ,

which is a contradiction. Thus, z and u must be identical. Hence, T has a unique best proximitypoint. �

Theorem 3.5. Let (X, d) be a complete metric space and (A,B) be a pair of non-void closed
subsets of (X, d). Let T : A→ B satisfy the following conditions :(i) T (A0) ∈ B0 and the pair (A,B) satisfies the weak P -property;(ii) T is a generalized (θ, φ)-proximal contraction of the first kind as well as a generalized

(θ, φ)-proximal contraction of the second kind.

Then there exists a unique u ∈ A such that d(u, Tu) = d(A,B) and un → u, where u0 is any fixed
point in A0 and d(un+1, T un) = d(A,B) for n ≥ 0.

Proof. Similar to Theorem 3.3, we find a sequence {un} in A0 such that
d(un+1, T un) = d(A,B)

for all non-negative integral values of n. Similar to Theorem 3.3, we can show that sequence {un}is a Cauchy sequence. Thus converges to some element u in A. As in Theorem 3.4, it can be shownthat the sequence {Tun} is a Cauchy sequence and converges to some element v in B. Therefore,
d(u, v) = lim

n→∞
d(un+1, T un) = d(A,B). (3.34)

Eventually, u becomes an element of A0. In light of the fact that T (A0) ∈ B0,
d(t, Tu) = d(A,B)

for some element t in A. From the p-property framework and (3.34,) we get
d(un+1, t) = d(Tun, T u),∀n ∈ N.

If for some n0, d(t, un0+1) = 0, consequently d(Tun0 , T u) = 0 . So Tun0 = Tu, hence d(A,B) =

d(u, Tu). Thus the conclusion is immediate. So let for any n ≥ 0, d(t, un+1) > 0. Since T is ageneralized (θ, φ)-proximal contraction of the first kind, it can be seen that
θ(d(t, un+1)) ≤ φ [θ (ad(u, un) + bd(t, u) + cd(un, un+1) + h[d(u, un+1) + d(un, t))] . (3.35)

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 12Since θ and φ are two continuous functions, by letting n → ∞ in inequality (3.35), we obtain,
d(u, Tu) = d(t, Tu) = d(A,B). Also, as in the theorem 3.3, the uniqueness of the best proximitypoint of mapping T follows. �

Example 3.6. Let X = {λn : n ∈ N} with the metric d(x, y) = |x − y | for all x, y ∈ X , where thesequence Gn, defined by
λ1 = 1

λ2 = 1 + 2

λ3 = 1 + 2 + 3

...

λn = 1 + 2 + 3 + ...+ n.

We know, (X, d) is a complete metric space. Let A = G3n : n ∈ N and B = G3n−1 : n ∈ N.It is easy to see that d(A,B) = 3, A0 = A and B0 = B. Define a mappings T : A → B, by
T (λ3n) = λ3n−1 for all n ≥ 1. It is clear that A is approximately compact with respect to B,
(A,B) satisfies the p-property, T is continuous and T (A0) ⊆ B0 . We will show that T is an
(θ, φ)-proximal contraction with θ ∈ Θ and φ ∈ Φ that is θ(t) = et and φ(t) = t

1
2 . Observe that,With out of generality, we may assume that n < m, and since

λ3n−1 = 1 + 2 + 3 + ...+ 3n − 1,

λ3m−1 = 1 + 2 + 3 + ...+ 3m − 1,

λ3n = 1 + 2 + 3 + ...+ 3n − 1 + 3n,

λ3m = 1 + 2 + 3 + ...+ 3m − 1 + 3m.

It follow that,
d(T (λ3n), T (λ3m)) = |λ3n−1 − λ3m−1|

= 3n + (3n + 1) + ...+ (3m − 1),

d(λ3n, λ2m) = |λ2n − λ2m|

= 3n + (3n + 1) + ...+ (3m),

and
d(T (λ2n), T (λ3m))− d(λ3n, λ3m) = |λ3n−1 − λ3m−1| − |λ3n − λ3m|

= 3n − 3m.

https://doi.org/10.28924/ada/ma.4.13


Eur. J. Math. Anal. 10.28924/ada/ma.4.13 13So that,
ed(T (λ3n),T (λ3m))−d(λ3n,λ3m)) =

ed(T (λ3n),T (λ3m))

ed(λ3n,λ3m)

= e3n−3m)

= e−3(m−n))

≤ e−3 =
1

e3
.

So that,
ed(T (λ3n),T (λ3m)) + 1 = θ(d(T (λ3n), T (λ3m)))

≤ ed(λ3n,λ3m)
1

e3
+ 1

≤
ed(λ3n,λ3m) + 2

2

= φ [θ(d(λ3n, λ3m))] .

Consequently, T is an generalized (θ, φ)-proximal contraction of the second kind with a = 1,
b = c = h = 0. Thus, all the conditions of Theorem 3.4 are satisfied. Hence, T has a unique bestproximity point and there exist λ3 ∈ A such that

d(λ3, Tλ3) = d(λ3, λ2) = 3 = d(A,B)

Conflict of interestThe authors declare that they have no competing interests.
Authors’ contributionsThe authors equally conceived of the study, participated in its design and coordination, drafted themanuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.13 14

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https://doi.org/10.28924/ada/ma.4.13

	1. Introduction
	2. Preliminaries
	3. Main result
	References

