Let (X,d) be a complete metric space,
T
:
X
→
X
T:X \to X
, and
α
:
[
0
,
∞
)
5
→
[
0
,
∞
)
\alpha :[0,\infty )^5 \to [0,\infty )
be nondecreasing with respect to each variable. Suppose that for the function
γ
(
t
)
=
α
(
t
,
t
,
t
,
2
t
,
2
t
)
\gamma (t) = \alpha (t,t,t,2t,2t)
, the sequence of iterates
γ
n
{\gamma ^n}
tends to 0 in
[
0
,
∞
)
[0,\infty )
and
lim
t
→
∞
(
t
−
γ
(
t
)
)
=
∞
{\lim _{t \to \infty }}(t - \gamma (t)) = \infty
. Furthermore, suppose that for each
x
∈
X
x \in X
there exists a positive integer
n
=
n
(
x
)
n = n(x)
such that for all
y
∈
X
y \in X
,
\[
d
(
T
n
x
,
T
n
y
)
⩽
α
(
d
(
x
,
T
n
x
)
,
d
(
x
,
T
n
y
)
,
d
(
x
,
y
)
,
d
(
T
n
x
,
y
)
,
d
(
T
n
y
,
y
)
)
.
d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).
\]
Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.