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r
n
lim
d
(
g
(
x
),
x
)
=
0
. (4)
0
0
1
−
r
n
→
∞
Therefore, from (2),(3) and (4), we obtain
*
*
d
(
g
(
x
),
x
)
=
0
*
is a fixed point of
g
.
Furthermore, assume that
x
∈
X
which implies that
*
*
*
*
x
and
y
are fixed points of
g
with
x
≥
y
or
*
*
x
≤
y
, that is,
*
*
*
*
g
(
x
)
=
x
,
g
(
y
)
=
y
.
Since
g
is monotone, one has
*
*
*
*
d
(
x
,
y
)
=
d
(
g
(
x
),
g
(
y
))
*
*
≤
kQ
(
x
,
y
)
*
*
*
*
*
*
*
=
k
max{
d
(
x
,
y
),
d
(
x
,
g
(
x
)),
d
(
x
,
g
(
y
)),
d
(
y
,
g
(
y
))}
*
*
=
kd
(
x
,
y
)
,
*
*
*
*
d
(
x
,
y
)
=
0
x
=
y
,
k
∈
(
0
).
which means that
, i.e.,
where
This completes
the proof.
3 Conclusions
In this paper, a fixed point theorem for quasi-contractive maps is established in
partially ordered sets under certain assumptions. Further, we can discuss the common
fixed point problem of a family of quasi-contractive maps, and apply the fixed point
theory to study optimization, computational algorithms, management, building
mathematical model, economics, variational inequalities, and complementary
problems and so on.
Acknowledgments.
The work is supported by the National Science Foundation of
China (No. 70771080) and the Fundamental Research Fund for the Central
Universities.
References
1.
Ran, A.C.M., Reurings, M.C.B.: A Fixed Point Theorem in Partially Ordered Sets and
Some Applications to Matrix Equations. Proc. Amer. Math. Soc. 132, 1435-1443 (2004)
2.
Nieto, J.J., Rodriguez-Lopez, R.: Contractive Mapping Theorems in Partially Ordered Sets
and Applicationsto Ordinary Differential Equations. Order 22, 223-239 (2005)
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