Abstract
Assume that
A
and
B
are non-void subsets of a metric space, and that
S
:
A
⟶
B
and
T
:
A
⟶
B
are given non-self-mappings. In light of the fact that
S
and
T
are non-self-mappings, it may happen that the equations
S
x
=
x
and
T
x
=
x
have no common solution, named a common fixed point of the mappings
S
and
T
. Subsequently, in the event that there is no common solution of the preceding equations, one speculates about finding an element
x
that is in close proximity to
S
x
and
T
x
in the sense that
d
(
x
,
S
x
)
and
d
(
x
,
T
x
)
are minimum. Indeed, a common best proximity point theorem investigates the existence of such an optimal approximate solution, named a common best proximity point of the mappings
S
and
T
, to the equations
S
x
=
x
and
T
x
=
x
when there is no common solution. Moreover, it is emphasized that the real valued functions
x
⟶
d
(
x
,
S
x
)
and
x
⟶
d
(
x
,
T
x
)
evaluate the degree of the error involved for any common approximate solution of the equations
S
x
=
x
and
T
x
=
x
. Owing to the fact that the distance between
x
and
S
x
, and the distance between
x
and
T
x
are at least the distance between
A
and
B
for all
x
in
A
, a common best proximity point theorem accomplishes the global minimum of both functions
x
⟶
d
(
x
,
S
x
)
and
x
⟶
d
(
x
,
T
x
)
by postulating a common approximate solution of the equations
S
x
=
x
and
T
x
=
x
for meeting the condition that
d
(
x
,
S
x
)
=
d
(
x
,
T
x
)
=
d
(
A
,
B
)
. This work is devoted to an interesting common best proximity point theorem for pairs of non-self-mappings satisfying a contraction-like condition, thereby producing common optimal approximate solutions of certain simultaneous fixed point equations.