Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 175453, 35 pages
doi:10.1155/2010/175453
Research Article
Maximality Principle and General Results of
Ekeland and Caristi Types without Lower
Semicontinuity Assumptions in Cone Uniform
Spaces with Generalized Pseudodistances
Kazimierz Włodarczyk and Robert Plebaniak
Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Ł
´
od
´
z,
Banacha 22, 90-238 Ł
´
od
´
z, Poland
Correspondence should be addressed to Kazimierz Włodarczyk,
Received 31 December 2009; Accepted 8 March 2010
Academic Editor: Tomonari Suzuki
Copyright q 2010 K. Włodarczyk and R. Plebaniak. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Our aim is twofold: first, we want to introduce a partial quasiordering in cone uniform spaces with
generalized pseudodistances for giving the general maximality principle in these spaces. Second,
we want to show how this maximality principle can be used to obtain new and general results of
Ekeland and Caristi types without lower semicontinuity assumptions, which was not done in the
previous publications on this subject.
33 are established. For many applications of these distances, see the papers 30–48 where,
among other things, in metric and uniform spaces with generalized distances 30–34,the
new fixed point theorems of Caristi’ type for dissipative maps with lower semicontinuous
entropies and variational principles of Ekeland type for lower semicontinuous maps are
given.
In this paper, in cone uniform spaces 49, 50, the families of generalized pseudodis-
tances are introduced see Section 2, a partial quasiordering is defined and the general
maximality principle is formulated and proved see Section 3. As applications, in cone
uniform spaces with the families of generalized pseudodistances, the general variational
principle of Ekeland type for not necessarily lower semicontinuous maps and a fixed point
and endpoint theorem of Caristi type for dissipative set-valued dynamic systems with not
necessarily lower semicontinuous entropies are established see Section 4
. Special cases
are discussed and examples and comparisons show a fundamental difference between our
results and the well-known ones in the literature where the standard lower semicontinuity
assumptions are essential see Section 5. Relations between our generalized pseudodistances
and generalized distances are described see Section 6; the aim of this section is to prove
that each generalized distance 30–34 is a generalized pseudodistance and we construct the
examples which show that the converse is not true. The definitions, the results, the ideas
and the methods presented here are new for set-valued and single-valued dynamic systems
in cone uniform, cone locally convex and cone metric spaces and even in uniform, locally
convex, and metric spaces.
2. Generalized Pseudodistances in Cone Uniform Spaces
We define a real normed space to be a pair L, ·, with the understanding that a vector space
L over R carries the topology generated by the metric a, b →a − b, a, b ∈ L.
Let L be a real normed space. A nonempty closed convex set H ⊂ L is called a cone in
L if it satisfies H1∀
s∈0,∞
{sH ⊂ H}, H2H ∩ −H{0},andH3 H
/
cone pseudometrics on XP-family, for short if the following three conditions hold:
P1 ∀
α∈A
∀
x,y∈X
{0
H
p
α
x, y ∧ x y ⇒ p
α
x, y0};
P2 ∀
α∈A
∀
x,y∈X
{p
α
x, yp
α
y, x};
P3 ∀
α∈A
∀
x,y,z∈X
{p
α
x, z
H
p
∈L,0c
α
∃
n
0
n
0
α,c
α
∈N
∀
m∈N;n
0
m
p
α
w
m
,w
c
α
. 2.1
ii We say that a sequence w
m
: m ∈ N in X is a P-Cauchy sequence in X,if
∀
. 2.2
iii If every P-Cauchy sequence in X is P-convergent in X, then X, P is called a P-
sequentially complete cone uniform space.
The following holds.
Theorem 2.3 see 49, Theorem 2.1. Let L be an ordered normed space with normal solid cone H
and let X, P be a Hausdorff cone uniform space with cone H.
a Let w
m
: m ∈ N be a sequence in X and let w ∈ X. The sequence w
m
: m ∈ N is
P-convergent to w if and only if
∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
m∈N;n
0
m
0
n
0
α,ε
α
∈N
∀
m,n∈N;n
0
m<n
p
α
w
m
,w
n
<ε
α
. 2.4
c Each P-convergent sequence is a P-Cauchy sequence.
4 Fixed Point Theory and Applications
Definition 2.4. Let L be an ordered normed space with solid cone H. The cone H
c
2
H
c
1
for some b ∈ L, t here exists
c ∈ L such that lim
m →∞
c
m
− c 0.
Remark 2.5. Every regular cone is normal; see 51.
Definition 2.6. Let L be an ordered normed space with normal solid cone H and let X, P be
a Hausdorff cone uniform space with cone H.
i The f amily J {J
α
: X × X → L, α ∈A}is said to be a J-family of cone
pseudodistances on X J-family on X, for short if the following three conditions hold:
J1 ∀
α∈A
∀
x,y∈X
{0
H
J
α
x, y};
J2 ∀
α∈A
0
mn
{
J
α
w
m
,w
n
<ε
α
}
, 2.5
if there exists a sequence v
m
: m ∈ N in X satisfying
∀
α∈A
∀
ε
α
>0
∃
n
0
n
∃
n
0
n
0
α,ε
α
∈N
∀
m∈N; n
0
m
p
α
w
m
,v
m
<ε
α
. 2.7
ii Let the family J {J
J
α
x, yJ
α
y, x}.ByJ1, this gives ∀
α∈A
{J
α
x, x
0}. Thus, we get ∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
m,n∈N; n
0
mn
{J
α
w
<ε
α
} where w
m
x, v
m
y,and
m ∈ N,and,byJ3, ∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
m∈N;n
0
m
{p
α
w
m
,v
m
: m ∈ N in X is J-Cauchy, that is, by
Definition 2.6ii, assume that
∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
m,n∈N; n
0
mn
{
J
α
w
m
,w
n
,w
qm
<ε
α
},andifi
0
∈ N, j
0
∈{0}∪
N, i
0
>j
0
,and
u
m
w
i
0
m
,v
m
w
j
0
m
for m ∈ N, 3.2
then
∀
α∈A
J
α
w
m
,v
m
<ε
α
}
. 3.3
By J3, 3.1 and 3.3,
∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
<ε
α
. 3.4
If M is a normal constant of H, then 3.2 and 3.4 give
∀
α∈A
∀
ε
α
>0
∃
n
0
n
0
α,ε
α
∈N
∀
m∈N; n
0
m
p
α
α
2M
.
3.5
Let α ∈Aand ε
α
> 0 be arbitrary and fixed and let m, n ∈ N satisfy n
0
m<n. We may
suppose that n i
0
n
0
and m j
0
n
0
for some i
0
∈ N and j
0
∈{0}∪N such that i
0
>j
0
. Then,
by P1–P3, ∀
α∈A
{0
0
p
α
w
n
0
,w
i
0
n
0
}.
Hence, using 3.5, ∀
α∈A
{p
α
w
m
,w
n
Mp
α
w
n
0
,w
j
0
n
0
m<n
{p
α
w
m
,w
n
<ε
α
}. Therefore, by
Theorem 2.3b, the sequence w
m
: m ∈ N is P-Cauchy.
Let Λ, ≤
Λ
denote a directed set whose elements will be indicated by the letters λ, η,
and μ. In the sequel, λ<
Λ
η will stand for λ ≤
Λ
η and λ
/
η.
The relation ≤
X
on X which is reflexive i.e., for all x ∈ X the condition x ≤
X
x holds
and transitive i.e., for all x, y, z ∈ X the conditions x ≤
X
i One says that the net w
λ
: λ ∈ Λ in X is J-Cauchy P-
Cauchy in X if ∀
α∈A
∀
c
α
∈L,0c
α
∃
π
0
∈Λ
∀
η,μ∈Λ;π
0
≤
Λ
η≤
Λ
μ
{J
α
w
η
,w
μ
c
α
λ
: λ ∈ Λ in X is J-convergent P-convergent in
X, if there exists w ∈ X such that ∀
α∈A
∀
c
α
∈L,0c
α
∃
π
0
∈Λ
∀
η∈Λ;π
0
≤
Λ
η
{J
α
w
η
,w
c
α
}∀
α∈A
∀
c
∀
α∈A
∀
c
α
∈L,0c
α
∃
π
0
∈Λ
∀
η∈Λ;π
0
≤
Λ
η
{p
α
w
η
,w c
α
}}. The subset E of X is said
to be a closed subset in X if clEE.
v Let X, ≤
X
be a partial quasiordering space. One says that the net w
λ
: λ ∈
: X × X → L, α ∈A}be aJ-family on X and let
X, ≤
X
be a partial quasiordering space.
a Assume that each increasing sequence w
m
: m ∈ N in X is J-Cauchy P-Cauchy.Then
each increasing net w
λ
: λ ∈ Λ in X is J -Cauchy P-Cauchy.
b Assume that each decreasing sequence w
m
: m ∈ N in X is J-Cauchy P-Cauchy.Then
each decreasing net w
λ
: λ ∈ Λ in X is J-Cauchy P-Cauchy.
Proof. a Suppose that there exists an increasing net w
λ
: λ ∈ Λ in X which is not
J-Cauchy, that is, which satisfies ∀
η,μ∈Λ
{η<
Λ
μ ⇒ w
η
≤
X
w
μ
} and
α
0
/
∈ int
H
.
3.6
Assume that π
1
∈ Λ is arbitrary and fixed. By 3.6, there exist η
1
,μ
1
∈ Λ,
π
1
≤
Λ
η
1
≤
Λ
μ
1
, such that J
α
0
2
≤
Λ
η
2
≤
Λ
μ
2
,
such that J
α
0
w
η
2
,w
μ
2
− c
α
0
/
∈ intH and define v
3
w
η
2
and v
4
n
,w
μ
n
− c
α
0
/
∈ intH and define
v
2n−1
w
η
n
and v
2n
w
μ
n
. By induction, this gives ∀
m∈N
{v
m
≤
X
v
m1
} and
∃
α
α
0
/
∈ intH}. Consequently,
there exists an increasing sequence v
n
: n ∈ N in X which is not J-Cauchy.
By Remark 2.7, we get the claim.
b We use a similar argument as in a.
Fixed Point Theory and Applications 7
Let X, ≤
X
be a partial quasiordering space. Set E ⊂ X which is called a chain in X
if any two elements of E are comparable,thatis,x ≤
X
y or y ≤
X
x for all x, y ∈ E. The Zorn
lemma says that every partially ordered set in which every chain has an upper lower bound
contains at least one maximal minimal element.
The main result of this section is the following maximality minimality principle.
Theorem 3.4. Let L be an ordered Banach space with a normal solid cone H and let X, P be a
Hausdorff cone uniform space with cone H.LetJ {J
α
: X × X → L, α ∈A}be aJ-family on X
and let X, ≤
X
be a partial quasiordering space.
A Assume that a
1
λ
: λ ∈ Λ in X is
P-Cauchy.
Step 2. Let an increasing net w
λ
: λ ∈ Λ in X be arbitrary and fixed. In view of a
1
and
Step 1, w
λ
: λ ∈ Λ is convergent to a w ∈ X and, since X is Hausdorff, w is unique.
Step 3. Let E be a chain in X, ≤
X
.If∃
u∈E
∀
v∈E
{v ≤
X
u}, then E has an upper bound in X.
Step 4. Let E be a chain in X, ≤
X
.If∀
u∈E
∃
v∈E
{u<
X
v}, then denoting ΛE and w
λ
{y ∈ X : w
λ
0
≤
X
y}.By
assumption a
1
, E
0
is complete. Clearly, the net w
λ
: λ ∈ Λ
0
is increasing in X, P-Cauchy,
convergent to w and w ∈ E
0
. This proves that w
λ
0
≤
X
w. Therefore, E has an upper bound in
X.
Step 5. Using Steps 3 and 4 and the Zorn lemma, we conclude that X contains at least one
maximal element.
B We use a similar argument as in A.
4. Variational Principle of Ekeland Type and Fixed Point and
Endpoint Theorem of Caristi Type in Cone Uniform Spaces with
Generalized Pseudodistances
where
X
0
J
{
x ∈ X : ∀
α∈A
{
0 J
α
x, x
}}
,
X
J
{
x ∈ X : ∃
α∈A
{
0 ≺
H
J
α
x, x
,α∈A}is a family of finite positive numbers;
f for each x ∈ X
0
J
, the set Q
J,Ω
x defined by the formula
Q
J,Ω
x
y ∈ X
0
J
: ∀
α∈A
ω
α
y
ε
α
J
α
wε
α
J
α
w
0
,w
H
ω
α
w
0
};
ii ∀
x∈Q
J,Ω
w
0
\{w}
∃
β∈A
{ω
β
w ≺
H
ω
β
xε
β
J
x
if x ∈ X
0
J
,
{
x
}
if x ∈ X
J
4.4
and, by assumption f, for each x ∈ X, Z
J,Ω
x is a closed subset in X.
The proof will be broken into five steps.
Fixed Point Theory and Applications 9
Step 1. The following shrinking property holds:
∀
w
0
∈X
∀
v∈Z
J,Ω
w
0
∈ X
0
J
,wegetv ∈ Z
J,Ω
w
0
Q
J,Ω
w
0
. Hence, by definition
of Q
J,Ω
w
0
, v ∈ X
0
J
. Consequently, u ∈ Z
J,Ω
v implies that u ∈ Q
J,Ω
v and, by
J2,weobtain∀
α∈A
{ω
α
uε
α
H
ω
α
w
0
},thatis,u ∈ Q
J,Ω
w
0
Z
J,Ω
w
0
, which gives 4.5.
Case 2. Assuming that w
0
∈ X
J
,wegetv ∈ Z
J,Ω
w
0
{w
0
}. Hence v u w
0
. This gives
4.5.
Step 2. Let w
u
1
⇐⇒ u
2
u
1
, if u
1
∈ X
J
, or u
2
∈ Q
J,Ω
u
1
, if u
1
∈ X
0
J
. 4.6
Then Z
J,Ω
w
J
and if w
0
∈ X
J
, then u
1
u
2
w
0
∈ X
J
.
Relation ≤
Z
J,Ω
w
0
on Z
J,Ω
w
0
is reflexive. We show that, for each u ∈ Z
J,Ω
w
0
∈ X
J
we
have that u w
0
∈ X
J
and thus we get that u≤
Z
J,Ω
w
0
u.
Relation ≤
Z
J,Ω
w
0
on Z
J,Ω
w
0
is transitive. Indeed, let u ≤
Z
J,Ω
w
0
J
, then u ∈ Q
J,Ω
v and v ∈ Q
J,Ω
z. Hence, using J2,weobtain
∀
α∈A
{ω
α
uε
α
J
α
z, u
H
ω
α
uε
α
J
α
z, vJ
α
v, u
H
ω
α
z} which gives u ≤
0
v and v ≤
Z
J,Ω
w
0
u for
u, v ∈ Z
J,Ω
w
0
. Then, by Remark 4.2, u, v ∈ X
0
J
if w
0
∈ X
0
J
or u, v ∈ X
J
if w
0
∈ X
J
.If
v}∈H and
∀
α∈A
{
ω
α
v
H
ω
α
u
− ε
α
J
α
u, v
}
4.7
are equivalent. Further, u ∈ Q
J,Ω
v means that
∀
α∈A
α
u
ε
α
J
α
v, u
H
ω
α
u
− ε
α
J
α
u, v
}
. 4.9
In conclusion, ∀
α∈A
{−ε
J,Ω
w
0
v}is complete.
Case 1. Assume that w
0
∈ X
0
J
. Therefore Z
J,Ω
w
0
Q
J,Ω
w
0
⊂ X
0
J
. However, v ∈
Z
J,Ω
w
0
⊂ X
0
J
. Hence, by Step 1 and 4.6,weobtainthat{u ∈ Z
0
∈ X
J
. Then Z
J,Ω
w
0
{w
0
} and the set {u ∈ Z
J,Ω
w
0
:
u≤
Z
J,Ω
w
0
v} {w
0
} is complete.
Step 4. Let w
0
∈ X be arbitrary and fixed. Each decreasing with respect to ≤
Z
J,Ω
w
m
∈ Z
J,Ω
w
0
}.
Case 1. If w
0
∈ X
0
J
, then, for each m ∈ N, w
m
∈ X
0
J
. T herefore, ∀
m∈N
{w
m1
∈
Q
J,Ω
w
m
},thatis,∀
m∈N
∀
α∈A
{ω
H
ω
α
w
m
H
···} and since H is a closed and regular cone,
it follows that
∀
α∈A
∃
u
α
∈H
lim
m →∞
ω
α
w
m
− u
α
0
. 4.10
···
}
. 4.11
Indeed, let m
0
∈ N and α
0
∈Abe arbitrary and fixed. Then, ∀
n∈N
{ω
α
0
w
m
0
−ω
α
0
w
m
0
n
∈ H}.
Consequently, since H is closed, by 4.10, lim
n
{ω
α
0
w
m
,w
n
H
n−1
jm
ε
α
α
J
α
w
j
,w
j1
H
ω
α
w
m
− u
α
−
ω
α
w
n
− u
∈ X
J
, then, for each m ∈ N, w
m
w
0
and w
m
: m ∈ N is P-Cauchy.
Fixed Point Theory and Applications 11
Step 5. Let w
0
∈ D
Ω
∩ X
0
J
be arbitrary and fixed. Then there exists w ∈ Q
J,Ω
w
0
such that {w}
Q
J,Ω
w.
Since w
0
∈ D
Ω
w
0
w} we conclude that V {v ∈
Q
J,Ω
w
0
: v ∈ Q
J,Ω
w} {w}. Therefore, w is an endpoint of Q
J,Ω
in Q
J,Ω
w
0
,that
is, {w} Q
J,Ω
w; we see, by J1,that{w} Q
J,Ω
w gives ∀
α∈A
{J
α
w, w0},that
is, w ∈ X
0
J
. Of course, {w} Q
J
×A, the map ω
α
·ε
α
J
α
x, · : X
0
J
→ H ∪{∞} is X
0
J
,X-lsc
and, for each x ∈ X
0
J
,thesetQ
J,Ω
x is nonempty.
b If J P, then a special case of condition f is a condition f
defined by
f
for each x, α ∈ X ×A, the map ω
α
·ε
α
p
∅;
d the family Ω{ω
α
: X → H ∪{∞},α∈A}satisfies D
Ω
α∈A
domω
α
/
∅;
e {ε
α
,α∈A}is a family of finite positive numbers;
fX, T is a set-valued dynamic system;
g for each x ∈ X
0
J
, the set Q
J,Ω;T
x defined by the formula
Q
J,Ω;T
x
4.12
is a nonempty closed subset in X.
12 Fixed Point Theory and Applications
Then, there exists w ∈ D
Ω
∩ X
0
J
such that
i w ∈ Tw.
Assume, in addition, that
h for each x ∈ X
0
J
, each dynamic process w
m
: m ∈{0}∪N starting at w
0
x and
satisfying ∀
m∈{0}∪N
{w
m1
∈ Tw
m
} satisfies ∀
m∈{0}∪N
{w
m1
≤
X
0
J
u
1
⇐⇒ u
2
∈ Q
J,Ω;T
u
1
. 4.13
Using analogous argumentation as in the proof of Theorem 4.1 we obtain that X
0
J
,≤
X
0
J
is a partial quasiordering space; for each v ∈ X
0
J
,theset{u ∈ X
0
J
J
α
w, w
H
ω
α
w}. Consequently, w ∈ Tw and w ∈ D
Ω
∩ X
0
J
.
Therefore, i holds.
Step 2. Assume that assumptions (a)–(h) hold.
By Step 1, w ∈ Tw where w ∈ X
0
J
is a minimal element of X
0
J
. We prove that {w}
Tw. Otherwise, there exists w
∈ Tw satisfying w
/
w. However, by h, for each dynamic
process w
m
: m ∈{0}∪N starting at w
0
J
×A, the map ω
α
·ε
α
J
α
x, · : Tx ∩ X
0
J
→ H ∪{∞} is
Tx ∩ X
0
J
,X-lsc and, for each x ∈ X
0
J
,thesetQ
J,Ω;T
x is nonempty.
b If J P, then a special case of condition g is a condition g
defined by
g
for each x, α ∈ X×A, the map ω
α
·ε
α
∃
y∈T x
{ε
α
J
α
x, y
H
ω
α
x − ω
α
y}. A dynamic system X, T is called dissipative
if it has an entropy Ω. One says that a family Ω is lsc if, for each α ∈A, ω
α
is lsc.
The notion of a dissipative map in metric space was introduced in 4.
Fixed Point Theory and Applications 13
Remark 4.8. By Definition 4.3, Remarks 4.4 and 4.6, and Definition 4.7, we see that we
established, in particular, the variational principle of Ekeland type for not necessarily lsc
families Ω and endpoint and fixed point theorem of Caristi type for dissipative set-valued
dynamic systems with not necessarily lsc entropies Ω. Consequently, our results are original
in the literature.
5. Examples and Comparisons of Our Results with
the Well-Known Ones
We provide some examples to illustrate the concepts introduced so far.
First, we give the example of J-family. Let L be an ordered normed space with cone
H ⊂ L, let the family P {p
α
: X×X → L, α ∈A}be a P-family, and let X, P be a Hausdorff
x,y,z∈X
{J
α
x, y
H
J
α
x, z
J
α
z, y}, therefore condition J2 holds. For proving that J3 holds we assume that the
sequences {x
m
} and {y
m
} in X satisfy 2.5 and 2.6. Then, in particular, 2.6 yields
∀
α∈A
∀
0<ε
α
<c
α
∃
m
0
m
0
α,ε
∀
m m
x
m
y
m
a
. 5.2
From 5.1 and 5.2,weget∀
α∈A
∀
0<ε
α
<c
α
∃
m
∈N
∀
m m
{p
α
x
m
, if x y ∈ W,
2, 2
, if x
/
y ∨ x y
/
∈ W,
x, y ∈ X. 5.3
By Example 5.1, the family J {J} is a J-family. We see that X
0
J
1/2, 1
/
∅.
14 Fixed Point Theory and Applications
Let ε ∈ 0, ∞ be arbitrary and fixed. Defining ω : X → L as follows:
ω
x
⎧
⎪
⎪
⎪
⎪
⎪
2
3
, 1
,
ε ·
3, 3
for x ∈
1
3
,
2
3
,
5.4
we observe that ∀
x∈X
{0
H
ωx} and D
Ω
domω
/
∅.
Let T : X → 2
X
⎪
⎪
⎩
0,
1
2
∪
1
2
, 1
if x 0,
{
1
}
if x ∈
0,
1
2
∪
1
2
, 1
ε
0, 0
ε
2, 2
ε
2, 2
H
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
ε
2, 2
ω
x is nonempty
andclosedinX.
Case 2. If x 1, then Q
J,Ω;T
1{1}. Indeed, Tx{3/4, 1} and ω1εJ1, 1
ε0, 0ε0, 0ε0, 0
H
ε0, 0ω1. Hence, 1 ∈ Q
J,Ω;T
1.Weseethat3/4
/
∈ Q
J,Ω;T
1.
Indeed, if 3/4 ∈ Q
J,Ω;T
1, then ω3/4εJ1, 3/4
H
ω1. On the other hand, by C ase 1,it
follows that ω1εJ3/4, 1
H
ω3/4. Consequently, ω3/4εJ1, 3/4
H
ω1 ≺
H
ω1
εJ3/4, 1
H
ω3/4 which is impossible. Hence, Q
J,Ω;T
m
: m ∈{0}∪N starting at w
0
1 such that w
1
3/4andw
m
1for
each m 2, satisfies ∀
m∈{0}∪N
{w
m1
∈ Q
J,Ω;T
w
m
}. Hence,
ω
3
4
εJ
1,
3
4
ω
m∈{0}∪N
{w
m1
∈ Q
J,Ω;T
w
m
}. Hence,
ω
1
εJ
3
4
, 1
ω
w
1
εJ
w
0
,w
1
holds.
Example 5.3. Let L, H, X,P, X, p : X × X → L, W 1/2, 1, J {J} and Ω{ω} be such
as in Example 5.2.LetT : X → 2
X
be defined by
T
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0,
1
2
.
5.9
By considerations analogous to those for Example 5.2, we prove that assumptions a–
g are satisfied.
We show that assumption h also holds. Indeed, let x ∈ X
0
J
1/2, 1 be arbitrary
and fixed. Then each dynamic process w
m
: m ∈{0}∪N starting at w
0
x and satisfying
∀
m∈{0}∪N
{w
m1
∈ Tw
m
} is of the following form.
Case 1. w
0
x ∈ 1/2, 1 and, for each m 1, w
m
1;
Case 2. For each m ∈{0}∪N, w
m
1.
2, 2
H
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
ε
2, 2
ω
x
ω
w
0
if x ∈
2
{w
m1
∈ Q
J,Ω;T
w
m
}. We proved that, there exists w 1 ∈
X
0
J
1/2, 1 such that w 1 ∈ T1{1},thatis,w 1 is the endpoint of T in X.
Remark 5.4. There exist examples of cone uniform spaces X,P and the maps T that
Theorem 4.5 holds simultaneously for some J
/
P see, Example 5.2; then X 0, 1 and
X
0
J
1/2, 1 and for J P see Example 5.5, then X X
0
J
0, 1. However, in general,
this does not hold see, e.g., Examples 5.6 and 5.7.
Example 5.5. Let L, H, X,P, X, p : X × X → L and T : X → 2
X
be as in Example 5.2.Let
ε ∈ 0, ∞ be arbitrary and fixed and let Ω{ω}, where ω : X → L is of the form
ω
x
⎪
⎪
⎩
ε ·
0, 0
for x 1,
ε ·
4, 4
for x ∈
0, 1
\
1
4
,
1
2
,
3
4
,
ε ·
we calculate that ω1/4εp0, 1/4ε2, 2ε1/4, 1/2ε9/4, 5/2
H
ε4, 4
ω0; b for y 3/4 ∈ T0, we calculate ω3/4εp0, 3/4ε1, 1ε3/4, 3/2
ε7/4, 5/2
H
ε 4, 4ω0; c for y 1 ∈ T0, we calculate ω1εp0, 1ε0, 0
ε1, 2ε1, 2
H
ε4, 4ω0; d for each y ∈ T0 \{1/4, 3/4, 1}, we calculate
ωyεp0,yε4, 4εy, 2y
H
ε 4, 4ω0. Consequently, Q
J,Ω;T
0{1/4, 3/4, 1}
is nonempty and closed in X.
Case 2. If x 1/2, then T1/2{0, 1} and we have the following. a for y 0 ∈ T1/2,we
get ω0εp1/2, 0ε4, 4ε1/2, 1ε9/2, 5
H
ε 6, 6ω1/2; b for y 1 ∈ T1/2,
we obtain ω1εp1/2, 1ε0, 0ε1/2, 1ε1/2, 1
H
ε6, 6ω1/2. Consequently,
Q
J,Ω;T
1/2{0, 1} is nonempty and closed in X.
Case 3. If x ∈ 0, 1/2 ∪ 1/2, 1, then Tx{1} and we have
ω
1
⎩
ε
3
4
,
3
2
H
ε
2, 2
ω
1
4
if x
1
4
,
ε
1 − x, 2
1 − x
,
1
2
H
ε
1, 1
ω
3
4
if x
3
4
.
5.12
Fixed Point Theory and Applications 17
Consequently, for each x ∈ 0, 1/2 ∪ 1/2, 1 the set Q
J,Ω;T
x{1} is nonempty and closed
in X.
Case 4. If x 1, then the set Q
J,Ω;T
1{1} is nonempty and closed in X.
We proved that assumption g holds. Assumption h does not hold see Exam-
ple 5.2.
0
J
1/2, 1
/
∅.
Let ε ∈ 0, ∞ be arbitrary and fixed. Defining ω : X → L by the formula
ω
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ε ·
3
,
5.14
we see that ∀
x∈X
{0
H
ωx} and D
Ω
domω
/
∅.
Let T : X → 2
X
be of the form
T
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
if x 0,
1
2
, 1
if x ∈
0,
1
2
,
{
0
}
if x
1
2
,
0,
1
2
∪
{
1
0, 0
ε
2, 2
ε
2, 2
H
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
ε
2, 2
ω
x
if x ∈
Case 2. If x 1, then T1{0, 1} and we see that Tx ∩ X
0
J
{1} and ω1εJ1, 1
ε0, 0ε0, 0ε0, 0
H
ε0, 0ω1. Consequently, Q
J,Ω;T
1{1} is nonempty and
closed in X.
Therefore, assumptions a–g of Theorem 4.5 are satisfied and there exists w 1 ∈
X
0
J
1/2, 1 such that 1 ∈ T1{0, 1}. Thus Theorem 4.5 holds for J
/
P.Itiseasytoshow
that h does not hold.
Example 5.7. Let L, H, X, p, P {p}, X,P and T : X → 2
X
be such as in Example 5.6.
However, let J P by Remark 2.7,itisJ -family on X. Of course, X
0
J
X
/
∅.Thus
assumptions a–c, e and f of Theorem 4.5 are satisfied.
Suppose that there exists Ω{ω} satisfying d and g. Then, for x 0, the
set Q
⎨
⎩
0, 0
if x y ∈ W,
2,
2
if x
/
y ∨ x y
/
∈ W,
x, y ∈ X. 5.17
By Example 5.1, J {J} is a J-family. Moreover, we see that X
0
J
W
/
∅.Letε>0be
arbitrary and fixed. Define ω : X → L as follows:
Fixed Point Theory and Applications 19
ω
x
⎧
⎪
⎪
⎪
⎩
ε ·
4n, 4n
for x ∈
−
2n 1
, −2n
,n 3,
ε ·
3n, 6n
for x ∈
−2n, −
2n − 1
,n∈ N,
ε ·
ε ·
0, 0
if x
1
2
,
ε ·
2, 2
if x ∈
−
1
2
∪ Z,
ε ·
4n, 8n
if x ∈
n, n 1
,n 3.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
,n∈ N,
−2n 2ifx −2n, n ∈ N,
−4n 5
2
if x ∈
−2n, −
2n − 1
,n∈ N,
1
2
if x ∈
−1, 2
,
2n 1ifx 2n, n ∈ N,
4n − 3
2
if x ∈
2n, 2n 1
Case 2. For x ∈ −5, −4, we have Tx−4 · 2 3/2 −5/2 ∈ −3, −2 ⊂ X
0
J
and ω−5/2
εJx, −5/2ε4, 4ε2, 2ε6, 6
H
ε 6, 6ωx, which gives Q
J,Ω;T
x{−5/2}.
Case 3. For x ∈ −3, −2, we have Tx−4·13/2 −1/2 ∈ X
0
J
and ω−1/2εJx, −1/2
ε2, 2ε2, 2ε4, 4
H
ε4, 4ωx, which gives Q
J,Ω;T
x{−1/2}.
Case 4. For x ∈ −2n, −2n − 1, n 2, we have Tx−4n 5/2 ∈ −2n − 1, −2n −
1 − 1 ⊂ X
0
J
and ω−4n 5/2εJx, −4n 5/2ε3n − 1, 6n − 1 ε2, 2
ε3n − 1, 6n − 4
H
ε 3n, 6nωx, which gives Q
J,Ω;T
x{−4n 5/2}.
20 Fixed Point Theory and Applications
Case 5. For x −2, −1, we have Tx1/2 ∈ X
⎩
ε
0, 0
ε
2, 2
H
ε
6, 6
ω
x
if x
/
−
1
2
,
ε
0, 0
ε
ε16, 32ωx, which gives Q
J,Ω;T
x{5/2}.
Case 9. For x ∈ 2n, 2n 1, n 3, we have Tx4n − 3/2 ∈ 2n − 1, 2n − 11 ⊂ X
0
J
and ω4n − 3/2εJx, 4n − 3/2ε42n − 1, 82n − 1 ε2, 2ε8n − 6, 16n −
14
H
ε42n, 82n ωx, which gives Q
J,Ω;T
x{4n − 3/2}.
Case 10. For x 3, 4, we have Tx3/2 ∈ 1, 2 ⊂ X
0
J
and ω3/2εJx, 3/2ε6, 6
ε2, 2ε8, 8
H
ε12, 24ωx, which gives Q
J,Ω;T
x{3/2}.
Case 11. For x ∈ 2n1, 2n2, n 2, we have Tx4n−1/2 ∈ 2n−11, 2n−12 ⊂ X
0
J
and ω4n −1/2εJx, 4n−1/2ε42n−11, 82n− 11 ε2, 2ε8n−2, 16n−
6
H
ε8n, 16n
H
ε42n 1, 82n 1 ωx, which gives Q
J
. It is worth noticing that in Example 5.8,
TE{1/2}⊂X
0
J
for E {−1, 0, 1}⊂X \ X
0
J
.
Recall that a map f : X → −∞, ∞ is proper if its effective domain, domf{x :
fx < ∞}, is nonempty. A map f : X → −∞, ∞ is lower semicontinuous on Xwritten:
lsc if the set {x ∈ X : fx r} is a closed subset in X for each r ∈ R.
In the literature, the several variants of the variational principle of Ekeland type for
lsc maps and fixed point and endpoint theorem of Caristi type for dissipative single-valued
and set-valued dynamic systems with lsc entropies in metric and uniform spaces and in
metric and uniform spaces with generalized distances are given and various techniques and
methods of investigations notably based on maximality principle are presented. However,
in all these papers assumptions about lower semicontinuity are essential.
Now, we present comparisons between our results and the well-known ones.
We may read, respectively, the results of Mizoguchi 5 and Aubin and Siegel 4,
concerning the existence of endpoints of dissipative set-valued dynamic systems with lsc
entropy in uniform and metric spaces, respectively, as follows.
Fixed Point Theory and Applications 21
Theorem 5.10 Mizoguchi 5, Theorems 1 and 2. Let X be a Hausdorff complete uniform space
with a family {d
α
: α ∈A}of pseudometrics inducing the topology of X, ω : X → −∞, ∞
be a map which is proper lsc and bounded from below and {ε
α
,α ∈A}be a family of finite positive
d
α
w, w
0
}.
Theorem 5.11 Aubin and Siegel 4. Let X, d be a complete metric space and let X, T be a
set-valued dynamic system. Let ω : X → −∞, ∞ be a map which is proper lsc and bounded from
below. Assume that ∀
x∈X
∀
y∈T x
{dx, y ωx − ωy}.ThenT has an endpoint in X.
The results of Feng and Liu 6 concerning the existence of fixed points and endpoints
of dissipative set-valued maps with lsc entropy in metric spaces, of Caristi type, may be read,
respectively, as follows.
Theorem 5.12 Feng and Liu 6, Theorem 4.2 and Corollary 4.3. Let X, d a complete metric
space, X, T a set-valued dynamic system, ω : X → R a bounded from below and lsc map, and
η : 0, ∞ → 0, ∞ a nondecreasing, continuous, and subadditive map and such that η
−1
{0}
{0}.If∀
x∈X
∃
y∈T x
{ηdx, y ωx − ωy}, then there exists w ∈ X such that w ∈ Tw.If
∀
x∈X
∀
y∈T x
{ηdx, y ωx − ωy}, then there exists w ∈ X such that Tw{w}.
Let η : H → H be such as in Theorem 5.12 and suppose that there exists a proper lsc on X
and bounded from below map ω
2
: X → L satisfying
∀
x∈X
∀
y∈T x
η
d
x, y
ω
2
x
− ω
2
y
. 5.22
Observe that 0 < 1/2 d1/2, 0 ω
1
1/2−ω
1
This gives T0 ⊂{y ∈ X : ω
i
y ω
i
0}, i 1, 2. Also, we see that 0 ∈{y ∈ X : ω
i
y
22 Fixed Point Theory and Applications
ω
i
0}, i 1, 2. Moreover, 5.23 implies that 1/2
/
∈{y ∈ X : ω
i
y ω
i
0}, i 1, 2. Since
X 0, 1T0 ∪{0, 1/2}, we conclude that {y ∈ X : ω
i
y ω
i
0} T0 ∪{0}, i 1, 2,
which gives that the set {y ∈ X : ω
i
y ω
i
0} 0, 1/2 ∪ 1/2, 1, i 1, 2, is open in X.
Therefore, for each i 1, 2, the lsc map ω
i
on X has the property that {y ∈ X : ω
ωx − ωy. For details, see 34, page 130.
The results of V
´
alyi 34,Caristi2,Ekeland3, and Jachymski 7, concerning
dissipative single-valued dynamic systems with lsc entropies, may be read, respectively, as
follows.
Theorem 5.15 V
´
alyi 34, Theorems 5 and 6. Let X, U be a Hausdorff uniform space; let Y, V
be a weakly sequentially complete topological vector space ordered by the closed normal cone K;let
d : X × X → Y satisfy i and ii;letω : X → Y ∪{∞} be bounded from below. Assume that a
the map ω is continuous (lsc); b for each x ∈ X,themapy →
dx, y is continuous (lsc); and c
for each U ∈UthereisaV ∈Vsuch that dx, y ∈ V implies t hat x, y ∈ U.
Fixed Point Theorem of Caristi Type. Assume that a map T : X → X has the property
∀
x∈X
{dx, Tx ≤
K
ωx − ωTx}.ThenT has a fixed point in X.
Variational Priciple of Ekeland Type. For each x ∈ domω there exists w ∈ domω such
that x≤
d,ω
w and w is maximal in {y ∈ X : x ≤
d,ω
y} (i.e., ∀
y∈X\{w}
{ωy
K
/
0
∈ domω, there exists u ∈ X such that: (i)
ωuεdx
0
,u ωx
0
; and (ii) ∀
x∈X\{u}
{ωu <ωxεdx, u}.
Fixed Point Theory and Applications 23
Theorem 5.17 Jachymski 7, Theorem 6. Let X, d be a complete metric space, T : X → X,
ω : X → R a nonnegative lsc map on X, and η : 0, ∞ → 0, ∞ a nondecreasing and subadditive
map, continuous at 0 and such that η
−1
{0}{0}.If∀
x∈X
{ηdx, Tx ωx − ωTx},then
there exists w ∈ X such that Tww.
Example 5.18 shows that Theorem 4.5 is different from Theorems 5.15, 5.16,and5.17.
Example 5.18. Let L R, H 0, ∞, X R, P {d}, dx, y|x − y|,andx, y ∈ X.LetT :
X → X be such as in Example 5.8. It is worth noticing that, by Remark 2.7, J P is J-family.
Suppose that there exists a proper lsc and bounded from below map
ω
1
: X → −∞, ∞
satisfying ∀
x∈X
{dx, Tx ω
1
x − ω
0
, 2n
0
1d2n
0
,T2n
0
ω
1
2n
0
− ω
1
2n
0
1 and, by 5.25,0 <ηd2n
0
, 2n
0
1 ηdx, Tx ω
2
2n
0
−
ω
2
2n
0
1, which gives ω
2
2n
0
1 − ω
2
2n
0
, which gives ω
i
2n
0
<ω
i
2n
0
1, i 1, 2. This is impossible.
The Banach fixed point theorem may be read as follows.
Theorem 5.19 Banach 1. Let X, d be a complete metric space and let T : X → X be a single-
valued map satisfying the condition
∃
0λ<1
∀
x,y∈X
d
T
x
0, 1/4, W
2
3/4, 1,
J
i
x, y
⎧
⎨
⎩
0, 0
if x y ∈ W
i
,
2, 2
if x
/
y ∨ x y
/
∈ W
i
,
x, y ∈ X, i 1, 2. 5.27
24 Fixed Point Theory and Applications
⎪
⎪
⎪
⎪
⎪
⎩
ε ·
0, 0
for x 0,
ε ·
2, 2
for x ∈
0,
1
3
∪
2
3
, 1
,
ε ·
⎪
⎪
⎪
⎩
ε ·
0, 0
for x 1,
ε ·
2, 2
for x ∈
0,
1
3
∪
2
3
, 1
,
ε ·
3, 3
: X → 2
X
be defined as follows:
T
1
x
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
{
0, 1
}
if x 0,
{
0
}
if x ∈
0,
1
4
,
3
4
if x
1
4
,
0,
1
4
}
if x 1.
5.29
Step 1. First, we observe that for J
1
and Ω
1
assertions a–f of Theorem 4.5 hold. We prove
that g holds. We see that X
0
J
1
W
1
0, 1/4 and consider t wo cases
Case 1. For x 0, we have T
1
x{0, 1} and we see that T
1
x ∩ X
0
J
1
{0} and ω
1
0
εJ
1
0, 0ε0, 0ε0, 0
H
;T
1
x{0}.
Consequently, for each x ∈ X
0
J
1
, Q
J
1
,Ω
1
;T
1
x is a nonempty and closed subset of X and
there exists w 0 ∈ X
0
J
1
such that w ∈ T
1
0{0, 1},thatis,w 0 is a fixed point of T
1
in X.
Fixed Point Theory and Applications 25
Step 2. We see that for J
2
and Ω
2
assertions a–f of Theorem 4.5 hold. We prove that g
Case 2. For x 1, we have T
1
x{0, 1} and we see that T
1
x ∩ X
0
J
2
{1} and ω
2
1
εJ
2
1, 1ε0, 0ε 0, 0
H
0, 0ω
2
1, which gives Q
J
2
,Ω
2
;T
1
1{1}.
Consequently, for each x ∈ X
0
J
2
, Q
1
w
m
}
such that w
1
1 ∈{0, 1} T
1
w
0
and w
m
1form 2, does not satisfy ∀
m∈{0}∪N
{w
m1
∈
Q
J
1
,Ω
1
;T
1
w
m
} since w
1
1
/
m
0form 2, does not satisfy ∀
m∈{0}∪N
{w
m1
∈
Q
J
2
,Ω
2
;T
1
w
m
} since w
1
0
/
∈ Q
J
2
,Ω
2
;T
1
w
0
{1}.
B Let T
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
{
0
}
if x ∈
0,
1
3
4
,
{
1
}
if x ∈
3
4
, 1
.
5.30
Step 1. First, we observe that for J
1
and Ω
1
assertions a–f of Theorem 4.5 hold. We prove
that g and h hold. We see that X
0
J
1
W
1
0, 1/4 and consider t wo cases.
Case 1. For x 0, we have T
2
x{0}⊂X
0
ε2, 2
H
ε 2, 2ω
1
x, which gives Q
J
1
,Ω
1
;T
2
x{0}.
Consequently, for each x ∈ X
0
J
1
, Q
J
1
,Ω
1
;T
2
x is a nonempty and closed subset of X,that
is, g holds.
For each x ∈ X
0
J
1
, each dynamic process w
w
m
}.
This gives h.
Copyright: Tài liệu đại học ©