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Maximal and minimal point theorems and Caristi's fixed point theorem
Fixed Point Theory and Applications 2011, 2011:103 doi:10.1186/1687-1812-2011-103
Zhilong Li ()
Shujun Jiang ()
ISSN 1687-1812
Article type Research
Submission date 8 August 2011
Acceptance date 21 December 2011
Publication date 21 December 2011
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Maximal and minimal point
theorems and Caristi’s fixed point
theorem
Zhilong Li
∗
and Shujun Jiang
Department of Mathematics, Jiangxi University of Finance
and Economics, Nanchang, Jiangxi 330013, China
∗
Corresponding author:
E-mail address:

SJ: jiangshujun
Abstract
This study is concerned with the existence of fixed points of
Caristi-type mappings motivated by a problem stated by Kirk.
First, several existence theorems of maximal and minimal points
are established. By using them, some generalized Caristi’s fixed
1
point theorems are proved, which improve Caristi’s fixed point
theorem and the results in the studies of Jachymski, Feng and
Liu, Khamsi, and Li.
MSC 2010: 06A06; 47H10.
Keywords: maximal and minimal point; Caristi’s fixed point
theorem; Caristi-type mapping; partial order.
1 Introduction
In the past decades, Caristi’s fixed point theorem has been generalized
and extended in several directions, and the proofs given for Caristi’s
result varied and used different techniques, we refer the readers to [1–15].
Recall that T : X → X is said to be a Caristi-type mapping [14] pro-
vided that there exists a function η : [0, +∞) → [0, +∞) and a function
ϕ : X → (−∞, +∞) such that
η(d(x, Tx)) ≤ ϕ(x) − ϕ(T x), ∀ x ∈ X,
where (X, d) is a complete metric space. Let  be a relationship defined
on X as follows
(1) x  y ⇐⇒ η(d(x, y)) ≤ ϕ(x) − ϕ(y), ∀ x, y ∈ X.
Clearly, x  T x for each x ∈ X provided that T is a Caristi-type
mapping. Therefore, the existence of fixed points of Caristi-type map-
2
pings is equivalent to the existence of maximal point of (X, ). Assume
that η is a continuous, nondecreasing, and subadditive function with
η

−1
({0}) = {0}, then the relationship defined by (1) is a partial order
on X. Feng and Liu [12] proved each Caristi-type mapping has a fixed
point by investigating the existence of maximal point of (X, ) provided
that ϕ is lower semicontinuous and bounded below. The additivity of η
appearing in [12] guarantees that the relationship  defined by (1) is a
partial order on X. However, if η is not subadditive, then the relation-
ship  defined by (1) may not be a partial order on X, and consequently
the method used there becomes invalid. Recently, Khamsi [13] removed
the additivity of η by introducing a partial order on Q as follows
x 
∗
y ⇐⇒ cd(x, y) ≤ ϕ(x) − ϕ(y), ∀ x, y ∈ Q,
where Q = {x ∈ X : ϕ(x) ≤ inf
t∈X
ϕ(t) + ε} for some ε > 0. Assume
that ϕ is lower semicontinuous and bounded below, η is continuous and
nondecreasing, and there exists δ > 0 and c > 0 such that η(t) ≥ ct
for each t ∈ [0, δ]. He showed that (Q, 
∗
) has a maximal point which is
exactly the maximal point of (X, ) and hence each Caristi-type mapping
has a fixed point. Very recently, the results of [9, 12, 13] were improved by
Li [14] in which the continuity, subadditivity and nondecreasing property
of η are removed at the expense that
3
(H) there exists c > 0 and ε > 0 such that η(t) ≥ ct for each
t ∈ {t ≥ 0 : η(t) ≤ ε}.
From [14, Theorem 2 and Remark 2] we know that the assumptions
made on η in [12, 13] force that (H) is satisfied. In other words, (H) is

necessarily assumed in [12–14]. Meanwhile, ϕ is always assumed to be
lower semicontinuous there.
In this study, we shall show how the condition (H) and the lower
semicontinuity of ϕ could be removed. We first proved several existence
theorems of maximal and minimal points. By using them, we obtained
some fixed point theorems of Caristi-type mappings in a partially ordered
complete metric space without the lower semicontinuity of ϕ and the
condition (H).
2 Maximal and minimal point theorems
For the sake of convenience, we in this section make the following
assumptions:
(H
1
) there exists a bounded below function ϕ : X → (−∞, +∞)
and a function η : [0, +∞) → [0, +∞) with η
−1
({0}) = {0}
such that
(2) η(d(x, y)) ≤ ϕ(x) − ϕ(y),
4
for each x, y ∈ X with x  y;
(H
2
) for any increasing sequence {x
n
}
n≥1
⊂ X, if there exists some
x ∈ X such that x
n

→ x as n → ∞, then x
n
 x for each
n ≥ 1;
(H
3
) for each x ∈ X, the set {y ∈ X : x  y} is closed;
(H
4
) η is nondecreasing;
(H
5
) η is continuous and lim inf
t→+∞
η(t) > 0;
(H
6
) there exists a bounded above function ϕ : X → (−∞, +∞)
and a function η : [0, +∞) → [0, +∞) with η
−1
({0}) = {0}
such that (2) holds for each x, y ∈ X with x  y;
(H
7
) for any decreasing sequence {x
n
}
n≥1
⊂ X, if there exists some
x ∈ X such that x

n
→ x as n → ∞, then x  x
n
for each
n ≥ 1;
(H
8
) for each x ∈ X, the set {y ∈ X : y  x} is closed.
Recall that a point x
∗
∈ X is said to be a maximal (resp. minimal)
point of (X, ) provided that x = x
∗
for each x ∈ X with x
∗
 x (resp.
x  x
∗
).
Theorem 1. Let (X, d, ) be a partially ordered complete metric space.
If (H
1
) and (H
2
) hold, and (H
4
) or (H
5
) is satisfied, then (X, ) has a
maximal point.

Proof. Case 1. (H
4
) is satisfied. Let {x
α
}
α∈Γ
⊂ F be an increasing chain
5
with respect to the partial order . From (2) we find that {ϕ(x
α
)}
α
∈
Γ
is
a decreasing net of reals, where Γ is a directed set. Since ϕ is bounded
below, then inf
α∈Γ
ϕ(x
α
) is meaningful. Let {α
n
} be an increasing sequence
of elements from Γ such that
(3) lim
n→∞
ϕ(x
α
n
) = inf

α∈Γ
ϕ(x
α
).
We claim that {x
α
n
}
n≥1
is a Cauchy sequence. Otherwise, there exists a
subsequence {x
α
n
i
}
i≥1
⊂ {x
α
n
}
n≥1
and δ > 0 such that x
α
n
i
 x
α
n
i+1
for

each i ≥ 1 and
(4) d(x
α
n
i
, x
α
n
i+1
) ≥ δ, ∀ i ≥ 1.
By (4) and (H
4
), we have
(5) η(d(x
α
n
i
, x
α
n
i+1
)) ≥ η(δ), ∀ i ≥ 1.
Therefore from (2) and (5) we have
ϕ(x
α
n
i
) − ϕ(x
α
n

i+1
) ≥ η(δ), ∀ i ≥ 1,
which indicates that
(6) ϕ(x
α
n
i+1
) ≤ ϕ(x
α
n
1
) − iη(δ), ∀ i ≥ 1.
Let i → ∞ in (6), by (3) and η
−1
({0}) = {0} we have
inf
α∈Γ
ϕ(x
α
) = lim
i→∞
ϕ(x
α
n
i
) ≤ −∞.
6
This is a contradiction, and consequently, {x
α
n

}
n
≥
1
is a Cauchy sequence.
Therefore by the completeness of X, there exists x ∈ X such that x
α
n
→
x as n → ∞. Moreover, (H
2
) forces that
(7) x
α
n
 x, ∀ n ≥ 1.
In the following, we show that {x
α
}
α∈Γ
has an upper bound. In fact,
for each α ∈ Γ, if there exists some n ≥ 1 such that x
α
 x
α
n
, by (7) we
get x
α
 x

α
n
 x, i.e., x is an upper bound of {x
α
}
α∈Γ
. Otherwise, there
exists some β ∈ Γ such that x
α
n
 x
β
for each n ≥ 1. From (2) we find
that ϕ(x
β
) ≤ ϕ(x
α
n
) for each n ≥ 1. This together with (3) implies that
ϕ(x
β
) = inf
α∈Γ
ϕ(x
α
) and hence ϕ(x
β
) ≤ ϕ(x
α
) for each α ∈ Γ. Note that

{ϕ(x
α
)}
α∈Γ
is a decreasing chain, then we have β ≥ α for each α ∈ Γ.
Since {x
α
}
α∈Γ
is an increasing chain, then x
α
 x
β
for each α ∈ Γ. This
shows that x
β
is an upper bound of {x
α
}
α∈Γ
.
By Zorn’s lemma we know that (X, ) has a maximal point x
∗
, i.e.,
if there exists x ∈ X such that x
∗
 x, we must have x = x
∗
.
Case 2. (H

5
) is satisfied. By lim inf
t→+∞
η(t) > 0, there exists l > δ and c
1
> 0
such that
η(t) ≥ c
1
, ∀ t ≥ l.
Since η is continuous and η
−1
({0}) = {0}, then c
2
= min
t∈[δ,l]
η(t) > 0. Let
7
c = min{c
1
, c
2
}, then by (4) we have
η(d(x
α
n
i
, x
α
n

i+1
)) ≥ c, ∀ i ≥ 1.
In analogy to Case 1, we know that (X, ) has a maximal point. The
proof is complete.
Theorem 2. Let (X, d, ) be a partially ordered complete metric space.
If (H
6
) and (H
7
) hold, and (H
4
) or (H
5
) is satisfied, then (X, ) has a
minimal point.
Proof. Let 
1
be an inverse partial order of , i.e., x  y ⇔ y 
1
x
for each x, y ∈ X. Let φ(x) = −ϕ(x). Then, φ is bounded below since
ϕ is bounded above, and hence from (H
6
) and (H
7
) we find that both
(H
1
) and (H
2

) hold for (X, d, 
1
) and φ. Finally, Theorem 2 forces that
(X, 
1
) has a maximal point which is also the minimal point of (X, ).
The proof is complete.
Theorem 3. Let (X, d, ) be a partially ordered complete metric space.
If (H
1
) and (H
3
) hold, and (H
4
) or (H
5
) is satisfied, then (X, ) has a
maximal point.
Proof. Following the proof of Theorem 1, we only need to show that
(7) holds. In fact, for arbitrarily given n
0
≥ 1, {y ∈ X : x
α
n
0
 y}
is closed by (H
3
). From (2) we know that x
α

n
0
 x
α
n
as n ≥ n
0
and
8
hence x
α
n
∈ {y ∈ X : x
α
n
0
 y} for all n ≥ n
0
. Therefore, we have
x ∈ {y ∈ X : x
α
n
0
 y}, i.e., x
α
n
0
 x. Finally, the arbitrary property of
n
0

implies that (7) holds. The proof is complete.
Similarly, we have the following result.
Theorem 4. Let (X, d, ) be a partially ordered complete metric space.
If (H
6
) and (H
8
) hold, and (H
4
) or (H
5
) is satisfied, then (X, ) has a
minimal point.
3 Caristi’s fixed point theorem
Theorem 5. Let (X, d, ) be a partially ordered complete metric space
and T : X → X. Suppose that (H
1
) holds, and (H
2
) or (H
3
) is satisfied.
If (H
4
) or (H
5
) is satisfied, then T has a fixed point provided that x  Tx
for each x ∈ X.
Proof. From Theorems 1 and 3, we know that (X, ) has a maximal
point. Let x

∗
be a maximal point of (X, ), then x∗  T x
∗
. The
maximality of x
∗
forces x
∗
= T x
∗
, i.e., x
∗
is a fixed point of T . The proof
is complete.
Theorem 6. Let (X, d, ) be a partially ordered complete metric space
and T : X → X. Suppose that (H
6
) holds, and (H
7
) or (H
8
) is satisfied.
9
If (H
4
) or (H
5
) is satisfied, then T has a fixed point provided that T x  x
for each x ∈ X.
Proof. From Theorems 2 and 4, we know that (X, ) has a minimal

point. Let x
∗
be a minimal point of (X, ), then Tx∗  x
∗
. The mini-
mality of x
∗
forces x
∗
= T x
∗
, i.e., x
∗
is a fixed point of T . The proof is
complete.
Remark 1. The lower semicontinuity of ϕ and (H) necessarily assumed
in [9, 12–14] are no longer necessary for Theorems 5 and 6. In what fol-
lows we shall show that Theorem 5 implies Caristi’s fixed point theorem.
The following lemma shows that there does exist some partial order
 on X such that (H
3
) is satisfied.
Lemma 1. Let (X, d) be a metric space and the relationship  defined
by (1) be a partial order on X. If η : [0, +∞) → [0, +∞) is continuous
and ϕ : X → (−∞, +∞) is lower semicontinuous, then (H
3
) holds.
Proof. For arbitrary x ∈ X, let {x
n
}

n≥1
⊂ {y ∈ X : x  y} be a
sequence such that x
n
→ x
∗
as n → ∞ for some x
∗
∈ X. From (1) we
have
(8) η(d(x, x
n
)) ≤ ϕ(x) − ϕ(x
n
).
Let n → ∞ in (8), then
lim sup
n→∞
η(d(x, x
n
)) ≤ lim sup
n→∞
(ϕ(x) − ϕ(x
n
)) ≤ ϕ(x) − lim inf
n→∞
ϕ(x
n
).
10

Moreover, by the continuity of η and the lower semicontinuity of ϕ we
get
η(d(x, x
∗
)) ≤ ϕ(x) − ϕ(x
∗
),
which implies that x  x
∗
, i.e., x
∗
∈ {y ∈ X : x  y}. Therefore,
{y ∈ X : x  y} is closed for each x ∈ X. The proof is complete.
By Theorem 5 and Lemma 1 we have the following result.
Corollary 1. Let (X, d) be a complete metric space and the relation-
ship  defined by (1) be a partial order on X. Let T : X → X be a
Caristi-type mapping and ϕ be a lower semicontinuous and bounded be-
low function. If η is a continuous function with η
−1
({0}) = {0}, and
(H
4
) or lim inf
t→+∞
η(t) > 0 is satisfied, then T has a fixed point.
It is clear that the relationship defined by (1) is a partial order on
X for when η(t) = t . Then, we obtain the famous Caristi’s fixed p oint
theorem by Corollary 1.
Corollary 2 (Caristi’s fixed point theorem). Let (X , d) be a com-
plete metric space and T : X → X be a Caristi-type mapping with

η(t) = t. If ϕ is lower semicontinuous and bounded below, then T has a
fixed point.
Remark 2. From [14, Remarks 1 and 2] we find that [14, Theorem 1]
11
includes the results appearing in [3, 4, 9, 12, 13]. Note that [14, Theorem
1] is proved by Caristi’s fixed point theorem, then the results of [9, 12–14]
are equivalent to Caristi’s fixed point theorem. Therefore, all the results
of [3, 4, 9, 12–14] could be obtained by Theorem 5. Contrarily, Theorem 5
could not be derived from Caristi’s fixed point theorem. Hence, Theorem
5 indeed improve Caristi’s fixed point theorem.
Example 1. Let X = {0} ∪ {
1
n
: n = 2, 3, . . .} with the usual metric
d(x, y) = |x − y| and the partial order  as follows
x  y ⇐⇒ y ≤ x.
Let ϕ(x) = x
2
and
T x =







0, x = 0,
1
n + 1

, x =
1
n
, n = 2, 3, . . . .
Clearly, (X, d) is a complete metric space, (H
2
) is satisfied, and ϕ is
bounded below. For each x ∈ X, we have x ≥ T x and hence x  T x.
Let η(t) = t
2
. Then η
−1
({0}) = {0}, ( H
4
) and (H
5
) are satisfied. Clearly,
(2) holds for each x, y ∈ X with x = y. For each x, y ∈ X with x  y
and x = y, we have two possible cases.
Case 1. When x =
1
n
, n ≥ 2 and y = 0, we have
η(d(x, y)) =
1
n
2
= ϕ(x) − ϕ(y).
12
Case 2. When x =

1
n
, n ≥ 2 and y =
1
m
, m > n, we have
η(d(x, y)) =
(m − n)
2
m
2
n
2
<
m
2
− n
2
m
2
n
2
= ϕ(x) − ϕ(y).
Therefore, (2) holds for each x, y ∈ X with x  y and hence (H
1
) is
satisfied. Finally, the existence of fixed point follows from Theorem 5.
While for each x =
1
n

, n ≥ 2, we have
ϕ(x) − ϕ(T x) =
2n + 1
n
2
(n + 1)
2
<
1
n(n + 1)
= d(x, T x),
which implies that corresponding to the function ϕ(x) = x
2
, T is not a
Caristi-type mapping. Therefore, we can conclude that for some given
function ϕ and some given mapping T , there may exist some function
η such that all the conditions of Theorem 5 are satisfied even though T
may not be a Caristi-type mapping corresponding to the function ϕ.
4 Conclusions
In this article, some new fixed point theorems of Caristi-type mappings
have been proved by establishing several maximal and minimal point
theorems. As one can see through Remark 2, many recent results could
be obtained by Theorem 5, but Theorem 5 could not be derived from
Caristi’s fixed point theorem. Therefore, the fixed point theorems indeed
improve Caristi’s fixed point theorem.
13
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ZL carried out the main part of this article. All authors read and ap-

proved the final manuscript.
Acknowledgments
This study was supported by the National Natural Science Foundation of
China (10701040,11161022,60964005), the Natural Science Foundation of
Jiangxi Province (2009GQS0007), and the Science and Technology Foun-
dation of Jiangxi Educational Department (GJJ11420).
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