©2023 Ada Academica https://adac.eeEur. J. Math. Anal. 3 (2023) 16doi: 10.28924/ada/ma.3.16
Best Proximity Points for Generalized Geraghty Quasi-Contraction Type Mappings in Metric

Spaces

J. C. Umudu1,∗, J. O. Olaleru2, H. Olaoluwa2, A. A. Mogbademu2
1Department of Mathematics, Faculty of Natural Sciences, University of Jos, Nigeria

umuduj@unijos.edu.ng
2Department of Mathematics, Faculty of Science, University of Lagos, Nigeria
jolaleru@unilag.edu.ng, holaoluwa@unilag.edu.ng, amogbademu@unilag.edu.ng

∗Correspondence: umuduj@unijos.edu.ng

Abstract. In this paper, we introduce a new concept of α-φ-Geraghty proximal quasi-contractiontype mappings and establish best proximity point theorems for those mappings in proximal T-orbitallycomplete metric spaces. This generalizes and complements the proofs of some known fixed and bestproximity point results.

1. Introduction
Let A and B be two nonempty subsets of a metric space (X,d). A best proximity point of anon-self mapping T : A → B, is the point x ∈ A, satisfying d(x,Tx) = d(A,B). Numerousresults on best proximity point theory were studied by several authors ( [1], [3], [4], [5]) imposingsufficient conditions that would assure the existence and uniqueness of such points. These resultsare generalizations of the contraction principle and other contractive mappings ( [2], [6], [8], [16],[21], [22], [24]) in the case of self-mappings, which reduces to a fixed point if the mapping underconsideration is a self-mapping. The notion of best proximity point was introduced in [14], the classof proximal quasi contraction mappings was introduced in [11] and thereafter, several known resultswere derived ( [10], [12], [13]). Best proximity pair theorems analyse the conditions under which theoptimization problem, namely minx∈Ad(x,Tx) has a solution and is known to have applicationsin game theory. For additional information on best proximity point, see [7], [9], [10], [11], [12], [13],[14], [15], [17], [18], [20], [23].

Definition 1.1 [4]. Let T : X → X be a map on metric space. For each x ∈ X and for any positiveinteger n,
OT (x,n) = {x,Tx,...,Tnx} and OT (x,∞) = {x,Tx,...,Tnx, ...}.

Received: 8 Feb 2023.
Key words and phrases. best proximity; quasi-contraction; metric space.1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.16 2
The set OT (x,∞) is called the orbit of T at x and the metric space X is called T-orbitally completeif every Cauchy sequence in OT (x,∞) is convergent in X.
Quasi contraction mapping is known in literature as one of the most generalized contractive map-pings and is defined as follows.
Definition 1.2 [6]. A mapping T : X → X of a metric space X into itself is said to be a quasi-contraction if and only if there exists a number k, 0 ≤ k < 1, such that

d(Tx,Ty) ≤ k max{d(x,y); d(x,Tx); d(y,Ty); d(x,Ty); d(y,Tx)}

holds for every x,y ∈ X.
Consider the class F of functions β : [0,∞) → [0, 1) satisfying the condition:

lim
n→∞

β(tn) = 1 implies lim
n→∞

tn = 0.

Recently, using these class of functions, Umudu et al. [22] introduced a new class of quasi-contraction type mappings called generalized α-φ-Geraghty quasi-contraction type mappings andproved the existence of its unique fixed point as follows.
Definition 1.3 [22]. Let (X,d) be a metric space and α : X ×X →R+. A mapping T : X → X iscalled a generalized α-Geraghty quasi-contraction type mapping if there exists β ∈ F such thatfor all x,y ∈ X,

α(x,y)(d(Tx,Ty)) ≤ β(MT (x,y))(MT (x,y)), (1)
where MT (x,y) = max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}.
Let Φ denote the class of the functions φ : [0,∞) → [0,∞) which satisfies the following conditions:

(i) φ is nondecreasing;(ii) φ is continuous;(iii) φ(t) = 0 ⇐⇒ t = 0.
Definition 1.4 [22]. Let (X,d) be a metric space and α : X×X →R+. A self mapping T : X → Xis called a generalized α-φ-Geraghty quasi-contraction type mapping if there exists β ∈ F suchthat for all x,y ∈ X,

α(x,y)φ(d(Tx,Ty)) ≤ β(φ(MT (x,y)))φ(MT (x,y)), (2)
where MT (x,y) = max{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}, and φ ∈ Φ.
If φ(t) = t, inequality (2) reduces to inequality (1). The generalized α-φ-Geraghty quasi-contraction type self mapping is a generalization of other quasi-contraction type self mappingsin literature.

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The following mappings introduced by Popescu [19] and used by Umudu et al. [22] to establish theexistence of a fixed point will also be needed in this paper.
Definition 1.5 [19]. Let T : X → X be a self-mapping and α : X ×X → R+ be a function. Then
T is said to be α-orbital admissible if α(x,Tx) ≥ 1 implies α(Tx,T2x) ≥ 1.
Definition 1.6 [19]. Let T : X → X be a self-mapping and α : X × X → R+ be a function.Then T is said to be triangular α-orbital admissible if T is α-orbital admissible, α(x,y) ≥ 1 and
α(y,Ty) ≥ 1 imply α(x,Ty) ≥ 1.
The main result obtained in [22] is the following.
Theorem 1.7. Let (X,d) be a T orbitally complete metric space, α : X ×X → R+ be a function,and let T : X → X be a self-mapping. Suppose that the following conditions are satisfied:

(i) T is a generalized α-φ-Geraghty quasi-contraction type mapping;(ii) T is triangular α-orbital admissible mapping;(iii) there exists x1 ∈ X such that α(x1,Tx1) ≥ 1;
Then T has a fixed point x∗ ∈ X and {Tnx1} converges to x∗.
In this paper, we extend the concept of generalized α-φ-Geraghty quasi-contraction typemapping to generalized α-φ-Geraghty proximal quasi-contraction type mapping in the case ofnon-self mappings. More precisely, we study the existence and uniqueness of best proximitypoints for generalized α-φ-Geraghty proximal quasi-contraction for non-self mappings.

2. Preliminaries
We start this section with the following definitions.Let A and B be non-empty subsets of a metric space (X,d). We denote by A0 and B0 the followingsets:

d(A,B) = inf{d(a,b) : a ∈ A, b ∈ B}.

A0 = {x ∈ A : d(x,y) = d(A,B) for some y ∈ B}.
B0 = {y ∈ B : d(x,y) = d(A,B) for some x ∈ A}.

Definition 2.1 [14]. An element x ∈ A is said to be a best proximity point of the non-self-mapping
T : A → B if it satisfies the condition that d(x,Tx) = d(A,B).We denote the set of all best proximity points of T by PT (A), that is,
PT (A) := {x ∈ A : d(x,Tx) = d(A,B)}.

The following were introduced by [11].

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Definition 2.2 [11]. A non-self mapping T : A → B is said to be a proximal quasi-contraction ifand only if there exists a number q, 0 ≤ q < 1, such that{

d(u,Tx) = d(A,B)

d(v,Ty) = d(A,B)
=⇒ d(u,v) ≤ q max{d(x,y); d(x,u); d(y,v); d(x,v); d(y,u)},

where x,y,u,v ∈ A.
If T is a self mapping on A, then Definition 2.2 reduces to Definition 1.2.
Lemma 2.3 [11]. Let T : A → B be a non-self mapping. Suppose that the following conditionshold:

(i) A0 6= ∅;(ii) T (A0) ⊆ B0.Then, for all a ∈ A0, there exists a sequence {xn}⊂ A0 such that{
x0 = a,

d(xn+1,Txn) = d(A,B), ∀n ∈N.Any sequence {xn} ⊂ A0 satisfying the equation in Lemma 2.3 is called a proximal Picardsequence associated to a ∈ A0 and we denote by PP(a) the set of all proximal Picard sequencesassociated to a.
Suppose a ∈ A0 and {xn} ∈ PP(a). For all (i, j) ∈ N2, the following sets are definedby:
OT (xi, j) := {xl : i ≤ l ≤ j + i} and OT (xi,∞) := {xl : l ≥ i}.
Definition 2.4 [11] A0 is said to be proximal T-orbitally complete if and only if every Cauchysequence {xn}∈PP(a) for some a ∈ A0, converges to an element in A0.
If T is a self mapping on A, then the preceding definition reduces to the condition that A is
T-orbitally complete.
The concepts of α-orbital proximal admissible mapping and triangular α-orbital proximaladmissible mapping are hereby introduced as follows.
Definition 2.5 Let T : A → B be a non-self mapping and α : A×A → [0,∞) be a function. Themapping T is said to be α-orbital proximal admissible if


α(x,u) ≥ 1
d(u,Tx) = d(A,B)

d(v,Tu) = d(A,B)

=⇒ α(u,v) ≥ 1,

for all x,u,v ∈ A.

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Definition 2.6 Let T : A → B be a non-self mapping and α : A × A → [0,∞) be a function.The mapping T is said to be triangular α-orbital proximal admissible if it is α-orbital proximaladmissible and 

α(x,y) ≥ 1
α(y,u) ≥ 1
d(u,Ty) = d(A,B)

=⇒ α(x,u) ≥ 1,

for all x,y,u ∈ A.
Remark 2.7. If T is a self mapping, that is, if A = B, α-orbital proximal admissible mappingreduces to α-orbital admissible mapping while triangular α-orbital proximal admissible mappingreduces to triangular α-orbital admissible mapping defined in [19] .
Example 2.8. Let X be the Euclidean plane R2 and consider the two subsets:

A = {(0, 0), (0, 1), (0, 2), (0, 3)}

B = {(1, 0), (2, 1), (2, 2), (1, 3)}

Define a mapping T : A → B such that T (0, 0) = (1, 0), T (0, 1) = (2, 2), T (0, 2) = (2, 1) and
T (0, 3) = (1, 3).Also define a mapping α : A×A → [0,∞) such that

α(x,y) =


1, if x = y ∈{(0, 0), (0, 3)}
0 elsewhere.

for all x,y ∈ A.
One can see that d(A,B) = 1.
Let u,v,x ∈ A. One can check that

α(x,u) ≥ 1
d(u,Tx) = 1

d(v,Tu) = 1

=⇒ x = u = v ∈{(0, 0), (0, 3)} =⇒ α(u,v) = 1.

Hence, T is α-orbital proximal admissible.
Let u,x,y ∈ A. One can check that


α(x,u) ≥ 1
α(y,u) ≥ 1
d(u,Ty) = 1

=⇒ x = y = u ∈{(0, 0), (0, 3)} =⇒ α(x,u) = 1.

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Thus, T is also triangular α-orbital proximal admissible.
We introduce the following new classes of non-self mappings.
Definition 2.9 Let A and B be two nonempty subsets of a metric space (X,d) and α : A×A →R+be a function. A non-self mapping T : A → B is called a generalized α-φ-Geraghty proximalquasi-contraction type mapping if there exists β ∈ F such that for all x,y,u,v ∈ A,{

d(u,Tx) = d(A,B)

d(v,Ty) = d(A,B)
=⇒ α(x,y)φ(d(u,v)) ≤ β(φ(MT (x,y)))φ(MT (x,y)), (3)

where MT (x,y) = max{d(x,y),d(x,u),d(y,v),d(x,v),d(y,u)}, for all x,y,u,v ∈ A and φ ∈ Φ.
If φ(t) = t, then definition 2.9 reduces to the following.
Definition 2.10 Let A and B be two nonempty subsets of a metric space (X,d) and α : A×A →R+be a function. A non-self mapping T : A → B is called an α-Geraghty proximal quasi-contractiontype mapping if there exists β ∈ F such that for all x,y,u,v ∈ A,{

d(u,Tx) = d(A,B)

d(v,Ty) = d(A,B)
=⇒ α(x,y)d(u,v) ≤ β(MT (x,y))(MT (x,y)), (4)

for all x,y,u,v ∈ A.
where MT (x,y) = max{d(x,y),d(x,u),d(y,v),d(x,v),d(y,u)} for all x,y,u,v ∈ A.

3. Main results
Now we state and prove our main results.

Theorem 3.1. Let A and B be two nonempty subsets of a metric space such that A0 isproximal T-orbitally complete, where T : A → B is a non-self mapping, α : A × A → R+ is afunction and the following conditions are satisfied:
(i) T is a generalized α-φ-Geraghty proximal quasi-contraction type mapping;(ii) T (A0) ⊆ B0 and T is a triangular α-orbital proximal admissible mapping;(iii) there exists x0,x1 ∈ A0 such that d(x1,Tx0) = d(A,B) and α(x0,x1) ≥ 1.Then there exists an element x∗ ∈ A0 such that

d(x∗,Tx∗) = d(A,B).

Moreover, if α(x,y) ≥ 1 for all x,y ∈ PT (A), then x∗ is the unique best proximity point of T .
Proof.Let x0,x1 ∈ A0 be such that d(x1,Tx0) = d(A,B) and α(x0,x1) ≥ 1.

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T (A0) ⊆ B0 and there exists x2 ∈ A0 such that d(x2,Tx1) = d(A,B). Now, we have

α(x0,x1) ≥ 1
d(x1,Tx0) = d(A,B),

d(x2,Tx1) = d(A,B).

Since T is α-orbital proximal admissible, α(x1,x2) ≥ 1. Thus, we have
d(x2,Tx1) = d(A,B) and α(x1,x2) ≥ 1.
By induction, we can construct a sequence {xi}⊆ A0 such that

d(xi+1,Txi) = d(A,B) and α(xi,xi+1) ≥ 1, f or all i ∈N. (5)
For all i ≥ 0 

α(xi,xi+1) ≥ 1
α(xi+1,xi+2) ≥ 1
d(xi+2,Txi−1) = d(A,B),

=⇒ α(xi,xi+2) ≥ 1,

Since T is triangular α-orbital proximal admissible. Thus by induction, α(xi,xj) ≥ 1 for all i, jsuch that 0 ≤ i < j.Therefore for any i ∈N, we have
α(xi−1,xj−1) ≥ 1
d(xi,Txi−1) = d(A,B),

d(xj,Txj−1) = d(A,B)

for all i, j such that 1 ≤ i < j.Clearly, if xi+1 = xi for some i ∈ N from inequality (5), xi will be a best proximity point, sohenceforth, in this proof, we assume d(xi,xi+1) > 0, ∀ i ∈N.
From inequality (3), we have

φ(d(xi,xj)) ≤ α(xi−1,xj−1)φ(d(xi,xj))

≤ β(φ(MT (xi−1,xj−1)))φ(MT (xi−1,xj−1)) (6)
1 ≤ i < j where

φ(MT (xi−1,xj−1)) ≤ φ(max{d(xi−1,xj−1),d(xi−1,xi),d(xj−1,xj),

d(xi−1,xj),d(xj−1,xi)})

≤ φ(δ[OT (xi−1,n)]), f or i ≤ j ≤ n + i.

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Note that the case φ(MT (xi−1,xj−1)) = φ(d(xi,xj)) is impossible. Indeed, by inequality (6),

φ(d(xi,xj)) ≤ β(φ(MT (xi−1,xj−1)))φ(MT (xi−1,xj−1))

≤ β(φ(d(xi,xj)))φ(d(xi,xj))

< φ(d(xi,xj)),

is a contradiction. Thus, we conclude that φ(d(xi,xj)) < φ(d(xi−1,xj−1)) for all 0 < i < j and sothe sequence {φ(d(xi,xj))} is positive and decreasing. Consequently, there exists r ≥ 0 such that
lim
i,j→∞

φ(d(xi,xj)) = r.

We claim that r = 0. Suppose, on the contrary, that r > 0. Then we have
φ(d(xi,xj))

φ(d(xi−1,xj−1))
≤ β(φ(MT (xi−1,xj−1))) ≤ 1 f or each i, j ∈N such that i < j.

Then, since β ∈ F ,
lim
i,j→∞

β(φ(MT (xi−1,xj−1))) = 1,

implying that
lim
i,j→∞

φ(MT (xi−1,xj−1)) = 0, (7)
and so by inequality (6)

lim
i,j→∞

φ(d(xi,xj)) = 0,

which is a contradiction.
Now, by the continuity property of φ,

φ

(
lim
i,j→∞

(d(xi,xj))

)
= φ(0). (8)

But φ(t) = 0 if and only if t = 0 and so (8) gives
lim
i,j→∞

(d(xi,xj)) = 0.

Therefore, {xn} is a Cauchy sequence in A0 and since A0 is proximal T-orbitally complete, thereexists x∗ ∈ A0 such that lim
i→∞

xi = x
∗. Also, since T (A0) ⊆ B0, then there exists y ∈ A0 such that

d(y,Tx∗) = d(xi,Txi−1) = d(A,B) ∀n ∈N, ∀i ≥ 0.

T being a generalized α-φ-Geraghty proximal quasi-contraction type mapping gives
φ(d(y,xi)) ≤ α(x∗,xi−1)φ(d(y,xi))

≤ β(φ(MT (x∗,xi−1)φ(MT (x∗,xi−1))

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Eur. J. Math. Anal. 10.28924/ada/ma.3.16 9
provided that α(x∗,xi−1) ≥ 1 where

φ(MT (x
∗,xi−1)) = φ(max{d(x∗,xi−1),d(x∗,xi),d(xi−1,xi),d(x∗,y),d(xi−1,y)}).

But taking the limit,
φ(d(y,x∗)) ≤ lim

i→∞
β(φ(MT (x

∗,xi−1)))φ(d(x
∗,y)),

which gives, 1 ≤ lim
i→∞

β(φ(MT (x
∗,xi−1))) = β(φ(d(y,x

∗))) = 1 implying φ(d(y,x∗)) = 0 and
d(y,x∗) = 0 i.e y = x∗. We have d(x∗,Tx∗) = d(y,Tx∗) = d(A,B) and x∗ ∈ A0 is a bestproximity point of T .For uniqueness, suppose the best proximity point of T is not unique. Let x∗, y∗ be two bestproximity points of T with x∗ 6= y∗. Then,

α(x∗,y∗) ≥ 1
d(x∗,Tx∗) = d(A,B)

d(y∗,Ty∗) = d(A,B)


Since T is a generalized α-φ-Geraghty proximal quasi-contraction type mapping,

φ(d(x∗,y∗)) ≤ α(x∗,y∗)φ(d(x∗,y∗))

≤ β(MT (x∗,y∗))φ(MT (x∗,y∗))

< φ(MT (x
∗,y∗))

where
MT (x

∗,y∗) = max{d(x∗,y∗),d(x∗,x∗),d(y∗,y∗),d(x∗,y∗),d(y∗,x∗)}

= d(x∗,y∗).

This gives d(x∗,y∗) < d(x∗,y∗), which is a contradiction. Therefore x∗ = y∗, and the bestproximity point of T is unique.
Corollary 3.2. Let A and B be two nonempty subsets of a metric space such that A0 isproximal T-orbitally complete, where T : A → B is a non-self mapping, α : A × A → R+ is afunction and the following conditions are satisfied:

(i) T is a generalized α-Geraghty proximal quasi-contraction type mapping;(ii) T (A0) ⊆ B0 and T is a triangular α-orbital proximal admissible mapping;(iii) there exists x0,x1 ∈ A0 such that d(x1,Tx0) = d(A,B) and α(x0,x1) ≥ 1.Then there exists an element x∗ ∈ A0 such that
d(x∗,Tx∗) = d(A,B).

Moreover, if α(x,y) ≥ 1 for all x,y ∈ PT (A), then x∗ is the unique best proximity point of T .

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Eur. J. Math. Anal. 10.28924/ada/ma.3.16 10
4. Conclusion

In this paper, we introduced the notion of generalized α-φ-Geraghty proximal quasi-contractiontype mappings which, for a self mapping, reduces to that in Umudu et al. [22]. Equipped withan example, we also introduced α-orbital proximal admissible mappings and triangular α-orbitalproximal admissible mappings which include the admissible mappings defined by Popescu [19].The existence of best proximity point was investigated for the class of mappings in a proximal
T-orbitally complete metric space.

Competing interests:
The authors declare that they have no competing interests.

Authors’ contributions:
All authors contributed equally in the preparation of the paper. The authors read and approvedthe final manuscript.

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s13663-020-00683-z.[23] J.C. Umudu, J.O. Olaleru, A.A. Mogbademu, Best proximity point results for Geraghty p-proximal cyclic quasi-contraction in uniform spaces, Divulgaciones Mat. 21 (2020), 21-31.[24] J. Umudu, A. Mogbademu, J. Olaleru, Fixed point results for Geraghty contractive type operators in uniform spaces,Caspian J. Math. Sci. 11 (2022), 191-202. https://doi.org/10.22080/cjms.2021.3052.

https://doi.org/10.28924/ada/ma.3.16
https://doi.org/10.1186/1687-1812-2013-180
https://doi.org/10.1155/2017/6173468
https://doi.org/10.1186/1687-1812-2014-190
https://doi.org/10.1016/j.na.2011.04.052
https://doi.org/10.1186/s13663-020-00683-z
https://doi.org/10.1186/s13663-020-00683-z
https://doi.org/10.22080/cjms.2021.3052

	1. Introduction
	2. Preliminaries
	3. Main results
	4. Conclusion
	Competing interests:
	Authors' contributions:
	References