Alqahtani et al. Journal of Inequalities and Applications (2018) 2018:158 Page 3 of 15
Some fundamental concepts, like convergence, Cauchy sequence, and completeness in
a extended-bMS, are defined as follows [13].
Definition 1.3 ([13]) Let (X,d
θ
) be an extended-bMS.
(i) A sequence x
n
in X is said to converge to x ∈X if, for every >0,thereexists
N = N() ∈N such that d
θ
(x
n
,x)< for all n ≥N.Inthiscase,wewrite
lim
n→∞
x
n
= x.
(ii) A sequence x
n
in X is said to be Cauchy if, for every >0,thereexistsN = N() ∈N
such that d
θ
(x
m
,x
n
)< for all m,n ≥N.
Definition 1.4 An extended-b-metric space (X,d
θ
)iscompleteifeveryCauchysequence
in X is convergent.
Lemma 1.1 Let (X,d
θ
) be a complete extended-bMS. If d
θ
is continuous, then every con-
vergent sequence has a unique limit.
Theorem 1.1 ([13]) Let (X,d
θ
) be an extended-bMS such that d
θ
is a continuous func-
tional. Let T : X → Xsatisfy
d
θ
(Tx,Ty)≤kd
θ
(x,y)(1)
for all x,y ∈ X, where k ∈ [0,1) is such that, for each x
0
∈ X, lim
n,m→∞
θ(x
n
,x
m
)<
1
k
, here
x
n
= T
n
x
0
, n =1,2,....Then T has precisely one fixed point u. Moreover, for each y ∈ X,
T
n
y →u.
For our purposes, we need to recall the following definition which is proposed by
Popescu [17].
Definition 1.5 Let T : X → X and α : X × X → [0,∞). We say that T is an α-orbital
admissible if, for all x,y ∈X,wehave
α(x,Tx) ≥1 ⇒ α
Tx,T
2
x
≥1. (2)
Definition1.6 AsetX is regularwith respect to mapping α : X×X →[0,∞) if,whenever
{x
n
}is a sequenceinX such that α(x
n
,x
n+1
) ≥1andα(x
n+1
,x
n
) ≥1forall n andx
n
→x ∈X
as n →∞, then there exists a subsequence {x
n(k)
} of {x
n
} such that α(x
n(k)
,x) ≥ 1and
α(x,x
n(k)
) ≥1 for all n.
2 (S,T) orbital cyclic
Definition 2.1 Suppose that T,S are two self-mappings on a complete extended-bMS
(X,d
θ
). Suppose also that there are two functions α,β : X ×X → [0,∞)suchthat,forany
x ∈X,
α(x,Tx) ≥1 ⇒ β(Tx,STx) ≥1and
β(x,Sx) ≥1 ⇒ α(Sx,TSx) ≥1.
(3)
Then we say that the pair S,T is an (α,β)-orbital-cyclic admissible pair.
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