Best proximity points for proximal contractive type mappings with Cclass functions in Smetric spaces
Ansari and Nantadilok Fixed Point Theory and Applications
Best proximity points for proximal contractive type mappings with C class functions in Smetric spaces
Arslan Hojat Ansari 1 3
Jamnian Nantadilok 0 1
0 Department of Mathematics, Faculty of Science, Lampang Rajabhat University , Lampang, 52100 , Thailand
1 which implies φ(t) = or
2 (t) = . That is, t = , which is a contradiction. Hence, t = . That is
3 Department of Mathematics, Karaj Branch, Islamic Azad University , Karaj , Iran
In this paper, we use the concept of Cclass functions to establish the best proximity point results for a certain class of proximal contractive mappings in Smetric spaces. Our results extend and improve some known results in the literature. We give examples to analyze and support our main results. MSC: Primary 47H10; secondary 54H25 1 Introduction and preliminaries Consider the equation Tx = x. If the equation Tx = x does not possess a solution, then we attempt to resolve the problem of finding an element x such that x is in proximity to Tx. In fact, in the setting of a metric space (X, d), if T : A → X, then a best approximation theorem provides sufficient conditions that confirm the existence of an element x, known as the best approximant, such that d(x, Tx) = d(Tx, A), where d(A, B) := inf{d(x, y) : x ∈ A and y ∈ B} for any nonempty subsets A and B of X. Indeed, a classical best approximation theorem, due to Ky Fan [], states that if K is a nonempty compact convex subset of a Banach space X and T : K → X is a singlevalued continuous mapping, then there exists an element x ∈ K such that d(x, Tx) = inf{d(y, Tx) : y ∈ K }, where d is a metric on X. This result has been generalized by many authors (see[]). In other words, if A and B are two nonempty subsets of a metric space (X, d), then an element x ∈ A is said to be a fixed point of a given map T : A → B if Tx = x. Clearly, T (A) ∩ A = ∅ is a necessary (but not sufficient) condition for the existence of a fixed point of T . If T (A) ∩ A = ∅, then d(x, Tx) > for all x ∈ A, that is, the set of fixed points of T is empty. In such a situation, one often attempts to find an element x which is in some sense closest to Tx. Best proximity point analysis has been developed in this direction. An element x∗ ∈ A is called a best proximity point of T if
fixed points; proximity points; Gmetric space; Smetric space; Bmetric space; proximal contractive mapping; Cclass function

d x∗, Tx∗ = d(A, B).
Indeed, in view of the fact that d(x, Tx) ≥ d(A, B) for all x ∈ A, the global minimum of
the mapping x → d(x, Tx) is attained at a best proximity point. Clearly, if the underlying
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mapping is selfmapping, then a best proximity point reduces to a fixed point. The goal
of best proximity point theory is to furnish sufficient conditions that assure the existence
of such points. For more details on this approach, we refer the reader to [–] and the
references therein.
Mustafa and Sims [] introduced the notion of Gmetric and obtained some
wellknown fixed point results in the setting of Gmetric spaces. Many authors have obtained
fixed point results in the context of Gmetric spaces [–]. In , Sedghi et al. [, ]
introduced a D∗metric space, which is a modification of Dmetric spaces introduced by
Dhage [], and established common fixed point theorems in D∗metric spaces. We note
that every Gmetric is a D∗metric, but in general the converse is not true (see []). In
, Sedghi et al. [] introduced the concept of an Smetric space, a modification of
D∗metric and Gmetric spaces, and gave a generalization of fixed point theorems in Smetric
spaces, but the best proximity point results in Smetric spaces still remain open. Recently,
Ansari [] introduced the concept of Cclass functions which can be used to generalize
many fixed point theorems in the literature (see, for example, []). Later, Nantadilok []
obtained best proximity point results for a certain class of proximal contractive mappings
in complete Smetric spaces. Inspired and motivated by Ansari [] and Nantadilok [],
in this paper, we establish best proximity point results for proximal contractive type
mappings with Cclass functions in the setting of Smetric spaces. We also give examples to
support our results.
Now we collect some necessary definitions and results in this direction. The notion of
Smetric spaces is defined as follows.
Definition . (see []) Let X be a nonempty set. An Smetric on X is a function S :
X → [, ∞) that satisfies the following conditions, for each x, y, z, a ∈ X.
(i) S(x, y, z) ≥ ;
(ii) S(x, y, z) = if and only if x = y = z;
(iii) S(x, y, z) ≤ S(x, x, a) + S(y, y, a) + S(z, z, a).
The function S is called an Smetric on X, and the pair (X, S) is called an Smetric space.
Remark . This notion is a modification of a Gmetric space [] and a D∗metric
space [].
Lemma . (see []) Let (X, S) be an Smetric space. Then S(x, x, y) = S(y, y, x) for all
x, y ∈ X.
Lemma . (see []) Let (X, S) be an Smetric space. Then
for all x, y, z ∈ X.
Definition . (see []) Let (X, S) be an Smetric space.
(i) A sequence {xn} ⊂ X is said to converge to x ∈ X if S(xn, xn, x) → as n → ∞. That
is, for each ε > , there exists n ∈ N such that for all n ≥ n we have S(xn, xn, x) < ε.
We write xn → x for brevity.
(ii) A sequence {xn} ⊂ X is called a Cauchy sequence if S(xn, xn, xm) → as n, m → ∞.
That is, for each ε > , there exists n ∈ N such that for all n, m ≥ n we have
S(xn, xn, xm) < ε.
(iii) The Smetric space (X, S) is said to be complete if every Cauchy sequence is a
convergent sequence.
Some geometric examples for Smetric spaces can be seen in [].
Definition . (see []) Let X be a nonempty set. A Bmetric on X is a function d :
X → [, ∞) if there exists a real umber b ≥ such that the following conditions hold for
all x, y, z ∈ X.
(B) d(x, y) = if and only if x = y.
(B) d(x, y) = d(y, x).
(B) d(x, y) ≤ b[d(x, z) + d(y, z)].
The function d is called a Bmetric on X, and the pair (X, d) is called a Bmetric space.
Theorem . (see []) Let (X, S) be an Smetric space, and let
d(x, y) = S(x, x, y)
for all x, y ∈ X. Then we have
(i) d is a Bmetric on X;
(ii) xn → x in (X, S) if and only if xn → x in (X, d);
(iii) {xn} is a Cauchy sequence in (X, S) if and only if {xn} is a Cauchy sequence in (X, d).
Now we recall the notion of Cclass functions introduced in [] as follows.
Definition . (see []) A mapping f : [, ∞) → R is called a Cclass function if it is
continuous and satisfies the following properties:
() f (s, t) ≤ s;
() f (s, t) = s implies that either s = , or t = for all s, t ∈ [, ∞).
We will denote the family of Cclass functions as C. Note that for some F ∈ C, we have
F(, ) = .
Example . (see []) The following functions F : [, ∞) → R are elements of C, for all
s, t ∈ [, ∞):
() F(s, t) = s – t, F(s, t) = s ⇒ t = ;
() F(s, t) = ms, < m < , F(s, t) = s ⇒ s = ;
() F(s, t) = (+st)r , r ∈ (, ∞), F(s, t) = s ⇒ s = or t = ;
() F(s, t) = log(t + as)/( + t), a > , F(s, t) = s ⇒ s = or t = ;
() F(s, t) = ln( + as)/, a > e, F(s, ) = s ⇒ s = ;
() F(s, t) = (s + l)(/(+t)r) – l, l > , r ∈ (, ∞), F(s, t) = s ⇒ t = ;
() F(s, t) = s logt+a a, a > , F(s, t) = s ⇒ s = or t = ;
() F(s, t) = s – ( ++ss )( +tt ), F(s, t) = s ⇒ t = ;
() F(s, t) = sβ(s), β : [, ∞) → [, ), and is continuous, F(s, t) = s ⇒ s = ;
() F(s, t) = s – k+tt , F(s, t) = s ⇒ t = ;
() F(s, t) = s – ϕ(s), F(s, t) = s ⇒ s = , where ϕ : [, ∞) → [, ∞) is a continuous
function such that ϕ(t) = ⇔ t = ;
() F(s, t) = sh(s, t), F(s, t) = s ⇒ s = , where h : [, ∞) × [, ∞) → [, ∞) is a
continuous function such that h(t, s) < for all t, s > ;
() F(s, t) = s – ( ++tt )t, F(s, t) = s ⇒ t = ;
() F(s, t) = s – ( ++ss )( +tt ), F(s, t) = s ⇒ t = ;
() F(s, t) = √nln( + sn), F(s, t) = s ⇒ s = ;
() F(s, t) = (+ss)r ; r ∈ (, ∞), F(s, t) = s ⇒ s = ;
() F(s, t) = ϑ (s); ϑ : R+ × R+ → R is a generalized MizoguchiTakahashi type
function, F(s, t) = s ⇒ s = .
Definition . (see []) A function ψ : [, ∞) → [, ∞) is called an altering distance
function if the following properties are satisfied:
(i) ψ is nondecreasing and continuous,
(ii) ψ (t) = if and only if t = .
denote the class of altering distance functions.
Definition . (see []) An ultra altering distance function is a continuous,
nondecreasing mapping ψ : [, ∞) → [, ∞) such that ψ (t) > , t > and ψ () ≥ .
We let u denote the set of all ultra altering distance functions. We note that every
Smetric on X induces a metric dS on X defined by
dS(x, y) = S(x, x, y) + S(y, y, x),
for all x, y ∈ X.
We show that a metric dS on X defined by () is a Bmetric on X. Conditions (B) and
(B) are easy to check. It follows from the definition of Smetric and Lemma . that
dS(x, y) = S(x, x, y) + S(y, y, x)
≤ S(x, x, z) + S(x, x, z) + S(y, y, z)
+ S(y, y, z) + S(y, y, z) + S(x, x, z)
= dS(x, z) + dS(y, z) + S(x, x, z) + S(y, y, z)
≤ dS(x, z) + dS(y, z) + S(x, x, z) + S(y, y, z)
= dS(x, z) + dS(y, z) .
This shows that dS is a Bmetric.
Definition . (see []) Let (X, S) be an Smetric space, and let A and B be two
nonempty subsets of X. Then B is said to be approximately compact with respect to A
if every sequence {yn} in B, satisfying the condition dS(x, yn) → dS(x, B) for some x in A,
has a convergent subsequence.
Let Φ denote the class of all functions ϕ : [, ∞) → [, ∞) which satisfy
. ϕ continuous and nondecreasing,
. ϕ(t) = if and only if t = ,
. ϕ(t + s) ≤ ϕ(t) + ϕ(s), ∀t, s ∈ [, ∞).
Definition . (see []) Let A and B be two nonempty subsets of an Smetric space
(X, S). Let T : A → B be a nonselfmapping. We say that T is an Sϕψ proximal
contractive mapping, if for all x, y, u, v ∈ A,
dS(u, Tx) = dS(A, B)
dS(v, Ty) = dS(A, B)
holds, where ϕ ∈
Definition . Let A and B be two nonempty subsets of an Smetric space (X, S). An
element x∗ ∈ A is said to be a best proximity point of a nonselfmapping T if dS(x∗, Tx∗) =
dS(A, B).
The main result obtained in [] is the following best proximity point theorem.
Theorem . (see []) Let A, B be two nonempty subsets of an Smetric space (X, S) such
that (A, S) is a complete Smetric space, A is nonempty, and B is approximately compact
with respect to A. Assume that T : A → B is an Sϕψ proximal contractive mapping such
that T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique
element z ∈ A such that dS(z, Tz) = dS(A, B).
2 Main results
Let (X, S) be an Smetric space. Suppose that A and B are nonempty subsets of an Smetric
space (X, S). We will use the following notations:
A = a ∈ A : dS(a, b) = dS(A, B) for some b ∈ B
B = b ∈ B : dS(a, b) = dS(A, B) for some a ∈ A ,
where dS(A, B) = inf{dS(x, y) : x ∈ A, y ∈ B}.
We introduce the following definitions.
Definition . Let A and B be two nonempty subsets of an Smetric space (X, S). Let
T : A → B be a nonselfmapping. We say that T is an S(F, ϕ, ψ )proximal contractive
mapping, if for all x, y, u, v ∈ A,
dS(u, Tx) = dS(A, B)
dS(v, Ty) = dS(A, B)
holds, where F ∈ C, ϕ ∈
Definition . Let A and B be two nonempty subsets of an Smetric space (X, S). Let
T : A → B be a nonselfmapping. We say that T is an S(F, ϕ, ψ )sumproximal contractive
mapping, if for all x, y, u, v ∈ A,
ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ (m(u,u∗,v,x,y))
holds, where F ∈ C, ϕ ∈ , ψ ∈ u and
with a, b, c, d ≥ and a + b + c + d > .
We note that these kind of generalizations make sense, since they extend and cover
those corresponding classes of proximal contractive mappings defined in []. We state
and prove our main results.
Theorem . Let A, B be two nonempty subsets of an Smetric space (X, S) such that (A, S)
is a complete Smetric space, A is nonempty, and B is approximately compact with respect
to A. Assume that T : A → B is an S(F, ϕ, ψ )proximal contractive mapping such that
T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique element
z ∈ A such that dS(z, Tz) = dS(A, B).
Proof Since the subset A is not empty, we take x in A. Taking Tx ∈ T (A) ⊆ B into
account, we can find x ∈ A such that dS(x, Tx) = dS(A, B). Further, since Tx ∈ T (A) ⊆
B, it follows that there is an element x in A such that dS(x, Tx) = dS(A, B). Recursively,
we obtain a sequence {xn} in A satisfying
dS(xn+, Txn) = dS(A, B),
∀n ∈ N ∪ {}.
dS(u, Tx) = dS(A, B),
dS(v, Ty) = dS(A, B),
ϕ S(xn, xn, xn+) ≤ F ϕ S(xn–, xn–, xn) , ψ S(xn–, xn–, xn)
≤ ϕ S(xn–, xn–, xn) ,
We will show that {xn}n∞= is an SCauchy sequence. Suppose, on the contrary, that there
exist ε > and a subsequence {xnk } of {xn} such that
with nk ≥ mk > k. Further, corresponding to mk , we can choose nk in such a way that it is
the smallest integer with nk > mk and it satisfies (). Hence,
So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent
to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking the limit as
n → ∞ in (), we get
≤ S(xnk , xnk , xnk–) + S(xmk , xmk , xnk–)
Letting k → ∞ in (), we derive that
Again, by Lemmas . and ., we obtain the following inequalities:
S(xmk , xmk , xnk ) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xmk–)
≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xnk–)
+ S(xmk–, xmk–, xnk–)
S(xmk–, xmk–, xnk–) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xmk )
≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xnk )
+ S(xmk , xmk , xnk )
Letting k → ∞ in () and applying (), we find that
From () with u = xmk , x = xmk–, v = xnk and y = xnk–, we have
ϕ S(xmk , xmk , xnk ) ≤ F ϕ S(xmk–, xmk–, xnk–) ,
Taking the limit as k → ∞ in the above inequality, we obtain
That is, {xn}n∞= is a Cauchy sequence. Since (A, S) is a complete Smetric space, there exists
z ∈ A such that xn → z as n → ∞. On the other hand, for all n ∈ N, we can write
dS(z, B) ≤ dS(z, Txn)
≤ dS(z, xn+) + dS(xn+, Txn)
= dS(z, xn+) + dS(A, B).
nl→im∞ dS(z, Txn) = dS(z, B) = dS(A, B).
dS z, y∗ = lim dS(xnk+, Txnk ) = dS(A, B),
k→∞
Taking the limit as n → ∞ in the above inequality, we obtain
Since B is approximately compact with respect to A, the sequence {Txn} has a subsequence
{Txnk } that converges to some y∗ ∈ B. Hence,
and so z ∈ A. Now, since Tz ∈ T (A) ⊆ B, there exists w ∈ A such that dS(w, Tz) =
dS(A, B).
From () with u = xn+, x = xn, v = w and y = z, we have
ϕ S(xn+, xn+, w) ≤ F ϕ S(xn, xn, z) , ψ S(xn, xn, z) .
Taking the limit as n → ∞, we get
ϕ S(p, p, q) ≤ F ϕ S(p, p, q) , ψ S(p, p, q) ,
Example . Let X = [, ∞). Define an Smetric on X by
This implies S(z, z, w) = . That is, w = z. Thus dS(z, Tz) = dS(A, B). Therefore T has a best
proximity point. To prove uniqueness, suppose that p = q such that dS(p, Tp) = dS(A, B)
and dS(q, Tq) = dS(A, B). Now, by () with u = x = p and v = y = q, we get
Let F(s, t) = s – t for all s, t ∈ [, ∞). Also define ϕ, ψ : [, ∞) → [, ∞) by ϕ(t) = t and
ψ (t) = t. Clearly, dS(A, B) = , A = {}, B = {} and T (A) ⊆ B. Let dS(u, Tx) = dS(A, B)
and dS(v, Ty) = dS(A, B), then u = v = , x = , , and y = , . Now since u = v = ,
ϕ(S(u, u, v)) = . Hence,
Therefore, we have
dS(u, Tx) = dS(A, B)
dS(v, Ty) = dS(A, B)
ϕ S(u, u, v) ≤ F ϕ S(x, x, y) , ψ S(x, x, y) .
ϕ S(u, u, v) ≤ F ϕ S(x, x, y) , ψ S(x, x, y) .
Thus T is an S(F, ϕ, ψ )proximal contractive mapping. All the conditions of Theorem .
hold true, and T has a unique best proximity point. Here, z = is the unique best proximity
point of T .
Remark . If we take F(s, t) = s – t in Theorem ., then our result reduces to
Theorem . in [].
Theorem . Let A, B be two nonempty subsets of an Smetric space (X, S) such that (A, S)
is a complete Smetric space, A is nonempty, and B is approximately compact with respect
to A. Assume that T : A → B is an S(F, ϕ, ψ )sumproximal contractive mapping such that
T (A) ⊆ B. Then T has a unique best proximity point; that is, there exists a unique element
z ∈ A such that dS(z, Tz) = dS(A, B).
Proof Since the subset A is not empty, we take x in A. Taking Tx ∈ T (A) ⊆ B into
account, we can find x ∈ A such that dS(x, Tx) = dS(A, B). Further, since Tx ∈ T (A) ⊆
B, it follows that there is an element x in A such that dS(x, Tx) = dS(A, B). Recursively,
we obtain a sequence {xn} in A satisfying
dS(xn+, Txn) = dS(A, B),
∀n ∈ N ∪ {}.
ϕ S(xn+, xn+, xn+)
≤ F ϕ
+ cS(xn+, xn+, xn+) + dS(xn+, xn+, xn+) ,
+ cS(xn+, xn+, xn+) + dS(xn+, xn+, xn+)
≤ ϕ
+ (c + d)S(xn+, xn+, xn+) ,
S(xn, xn, xn+) ≤ S(xn–, xn–, xn).
So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent
to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking the limit as
n → ∞ in (), we get
which implies ϕ(t) = or ψ (t) = . That is, t = , which is a contradiction. Hence, t = .
That is,
We will show that {xn}n∞= is an SCauchy sequence. Suppose, on the contrary, that there
exist ε > and a subsequence {xnk } of {xn} such that
with nk ≥ mk > k. Further, corresponding to mk , we can choose nk in such a way that it is
the smallest integer with nk > mk satisfying (). Hence,
≤ S(xnk , xnk , xnk–) + S(xmk , xmk , xnk–)
Letting k → ∞, we derive
Again, by using Lemmas . and ., we obtain the following inequalities:
S(xmk , xmk , xnk ) ≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xmk–)
≤ S(xmk , xmk , xmk–) + S(xnk , xnk , xnk–)
+ S(xmk–, xmk–, xnk–)
S(xmk–, xmk–, xnk–) ≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xmk )
≤ S(xmk–, xmk–, xmk ) + S(xnk–, xnk–, xnk )
+ S(xmk , xmk , xnk )
Letting k → ∞ in () and () and applying (), we find that
+ cS(xnk–, xnk–, xmk–) + dS(xnk–, xnk–, xnk ) ,
+ cS(xnk–, xnk–, xmk–) + dS(xnk–, xnk–, xnk )
Taking the limit as k → ∞ in the above inequality, we obtain
This proves that {xn} is a Cauchy sequence in an Smetric space (X, S). Since (A, S) is a
complete metric space, there exists z ∈ A such that {xn} converges to z. As in the proof of
Theorem ., we have dS(w, Tz) = dS(A, B) for some w ∈ A. From () with x = xn–, u = xn,
u∗ = xn+, y = z and v = w, we have
≤ F ϕ
+ cS(z, z, xn+) + dS(z, z, w) ,
+ cS(z, z, xn+) + dS(z, z, w)
Taking the limit as n → ∞ in the above inequality, we get
≤ F ϕ
≤ ϕ
≤ F ϕ
≤ ϕ
which implies ϕ( a+b+c+d S(p, p, q)) = or ψ( a+bb++cc+d S(p, p, q)) = , so S(p, p, q) = . Hence
b+c
p = q, that is, T has the unique best proximity point.
Let F(s, t) = s– t+tk for all s, t ∈ [, ∞). This is a Cclass function. Also define ϕ, ψ : [, ∞) →
[, ∞) by ϕ(t) = t and ψ(t) = t. Clearly, dS(A, B) = , A = {}, B = {} and T(A) ⊆ B.
Let dS(u, Tx) = dS(A, B), dS(u∗, Tu) = dS(A, B) and dS(v, Ty) = dS(A, B), then we get u = u∗ =
v = , x = , , and y = , . Now since u = u∗ = v = , ϕ(S(u, u, v)) = . Hence,
= ϕ(m(u,u∗,v,x,y)) –
with a, b, c, d ≥ and a + b + c + d > .
Therefore, we have
ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ(m(u,u∗,v,x,y)) .
Thus T is an S(F, ϕ, ψ)sumproximal contractive mapping. All the conditions of
Theorem . hold true and T has a unique best proximity point. Here, z = is the unique best
proximity point of T .
Remark . If we take F(s, t) = (a + b + c + d)s, < a + b + c + d < and ϕ(t) = t in
Theorem ., then our result reduces to Theorem . in [].
ϕ S(u, u, v) ≤ F ϕ(m(u,u∗,v,x,y)), ψ(m(u,u∗,v,x,y)) .
ϕ S(u, u, v) ≤ F ϕ(m(u,v,x,y)), ψ (m(u,,v,x,y)) ,
dS(u, Tx) = dS(A, B)
dS(v, Ty) = dS(A, B)
× aS(x, x, u) + b
S(x,+x,Sy()uS,(ux,, xv),u) + cS(x, x, y) + dS(x, x, u)
Proof Following the same lines as those in the proof of Theorem ., we can construct a
sequence {xn} in A satisfying
dS(xn+, Txn) = dS(A, B),
∀n ∈ N ∪ {}.
From () with x = xn–, u = xn, y = xn and v = xn+, we have
Theorem . Let A, B be two nonempty subsets of an Smetric space (X, S) such that (A, S)
is a complete Smetric space, A is nonempty, and B is approximately compact with respect
to A. Assume that T : A → B is a nonselfmapping such that T (A) ⊆ B and, for x, y, u, v ∈
A,
ϕ S(xn, xn, xn+)
≤ F ϕ
aS(xn–, xn–, xn)
S(xn–, xn–, xn)S(xn–, xn–, xn)
+ S(xn, xn, xn+)
+ cS(xn–, xn–, xn) + dS(xn–, xn–, xn) ,
aS(xn–, xn–, xn)
S(xn–, xn–, xn)S(xn–, xn–, xn)
+ S(xn, xn, xn+)
+ cS(xn–, xn–, xn) + dS(xn–, xn–, xn)
≤ ϕ S(xn–, xn–, xn)
for all n ∈ N ∪ {}. This implies
S(xn, xn, xn+) ≤ S(xn–, xn–, xn).
So, the sequence {S(xn, xn, xn+)} is a decreasing sequence in R+ and thus it is convergent
to t ∈ R+. We claim that t = . Suppose, on the contrary, that t > . Taking limit as n → ∞
in (), we get
which implies ϕ(t) = or ψ (t) = . That is, t = which is a contradiction. Hence, t = .
That is,
Similarly, one can see that {xn} is a Cauchy sequence in an Smetric space (X, S). Due to the
completeness of (A, S), there exists z ∈ A such that {xn} converges to z. As in the proof of
Theorem ., we have dS(w, Tz) = dS(A, B) for some w ∈ A. Now, from () with x = xn–,
u = xn, y = z and v = w, we deduce
≤ F ϕ
aS(xn–, xn–, xn)
S(xn–, xn–, z)S(xn–, xn–, xn)
+ S(xn, xn, w)
+ cS(xn–, xn–, z) + dS(xn–, xn–, xn) ,
aS(xn–, xn–, xn)
S(xn–, xn–, z)S(xn–, xn–, xn)
+ S(xn, xn, w)
+ cS(xn–, xn–, z) + dS(xn–, xn–, xn)
S(p, p, q) ≤ aS(p, p, p) + b
By taking the limit as n → ∞ in the above inequality, we get S(z, z, w) = ; that is, z = w.
Hence, dS(z, Tz) = dS(w, Tz) = dS(A, B); that is, T has a best proximity point. To prove
uniqueness, assume that p = q, such that dS(p, Tp) = dS(A, B) and dS(q, Tq) = dS(A, B). Now,
by () with x = u = p and y = v = q, we have
Remark . By taking F(s, t) = (a + b + c + d)s, < a + b + c + d < and ϕ(t) = t in
Theorem ., our result reduces to Theorem . in [].
Acknowledgements
The second author is grateful to Lampang Rajabhat University for financial support during the preparation of this
manuscript and to the referees for useful suggestions.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors contributed equally in the writing of this paper. All authors read and approved the manuscript.
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