Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 302830, 17 pages
doi:10.1155/2012/302830
Research Article
On Approximate Coincidence Point Properties and
Their Applications to Fixed Point Theory
Wei-Shih Du
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan
Correspondence should be addressed to Wei-Shih Du, wsdu@nknucc.nknu.edu.tw
Received 29 October 2011; Revised 21 January 2012; Accepted 23 January 2012
Academic Editor: Giuseppe Marino
Copyright q 2012 Wei-Shih Du. 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.
We first establish some existence results concerning approximate coincidence point properties
and approximate fixed point properties for various types of nonlinear contractive maps in
the setting of cone metric spaces and general metric spaces. From these results, we present
some new coincidence point and fixed point theorems which generalize Berinde-Berinde’s fixed
point theorem, Mizoguchi-Takahashi’s fixed point theorem, and some well-known results in the
literature.
1. Introduction
Let X, d be a metric space. For each x ∈ X and A ⊆ X, let dx, A infy∈A dx, y. Denote
by NX the class of all nonempty subsets of X, CX the collection of all nonempty closed
subsets of X, and CBX the family of all nonempty closed and bounded subsets of X. A
function H : CBX × CBX → 0, ∞ defined by
HA, B max sup dx, A, sup dx, B
x∈B
1.1
x∈A
is said to be the Hausdorff metric on CBX induced by the metric d on X. Let T : X → NX
be a multivalued map. A point x in X is a fixed point of T if x ∈ T x. The set of fixed points
of T is denoted by FT . Let g : X → X be a single-valued self-map and T : X → NX be
a multivalued map. A point x in X is said to be a coincidence point see, for instance, 1, 2
of g and T if gx ∈ T x. The set of coincidence point of g and T is denoted by COPg, T .
Throughout this paper, we denote by N and R the sets of positive integers and real numbers,
respectively.
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Let f be a real-valued function defined on R. For c ∈ R, we recall that
lim supfx inf sup fx,
ε>0 0<|x−c|<ε
x→c
1.2
lim supfx inf sup fx.
ε>0 0<x−c<ε
x → c
Definition 1.1 see 2–9. A function ϕ : 0, ∞ → 0, 1 is said to be an MT-function or
R-function if
lim sup ϕs < 1,
s → t
∀t ∈ 0, ∞.
1.3
It is obvious that if ϕ : 0, ∞ → 0, 1 is a nondecreasing function or a nonincreasing
function, then ϕ is an MT-function. So the set of MT-functions is a rich class.
Very recently, Du 7 first proved some characterizations of MT-functions.
Theorem D see 7. Let ϕ : 0, ∞ → 0, 1 be a function. Then, the following statements are
equivalent.
a ϕ is an MT-function.
1
∈ 0, 1 and εt
2
∈ 0, 1 and εt
3
∈ 0, 1 and εt
4
∈ 0, 1 and εt
b For each t ∈ 0, ∞, there exist rt
1
s ∈ t, t εt .
c For each t ∈ 0, ∞, there exist rt
2
s ∈ t, t εt .
d For each t ∈ 0, ∞, there exist rt
3
s ∈ t, t εt .
e For each t ∈ 0, ∞, there exist rt
4
s ∈ t, t εt .
1
> 0 such that ϕs ≤ rt
1
2
> 0 such that ϕs ≤ rt
3
> 0 such that ϕs ≤ rt
4
> 0 such that ϕs ≤ rt
2
3
4
for all
for all
for all
for all
f For any nonincreasing sequence {xn }n∈N in 0, ∞, we have 0 ≤ supn∈N ϕxn < 1.
g ϕ is a function of contractive factor [5]; that is, for any strictly decreasing sequence {xn }n∈N
in 0, ∞, we have 0 ≤ supn∈N ϕxn < 1.
It is worth to mention that there exist functions which are not MT-functions. For
example, let ϕ : 0, ∞ → 0, 1 be defined by
⎧
π
sin t
⎪
⎨
, if t ∈ 0,
,
t
2
ϕt :
⎪
⎩
0,
otherwise.
Since lim sups → 0 ϕs 1, ϕ is not an MT-function.
1.4
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Let g : X → X be a single-valued self-map. Recall that a multivalued map T : X →
NX is called
1 a multivalued k-contraction 1, 10, 11, if there exists a number 0 < k < 1 such that
H T x, T y ≤ kd x, y ,
∀x, y ∈ X,
1.5
2 a multivalued θ, L-almost contraction 1, 12, if there exist two constants θ ∈ 0, 1
and L ≥ 0 such that
H T x, T y ≤ θd x, y Ld y, T x ,
∀x, y ∈ X,
1.6
3 a generalized multivalued almost contraction 1, 12, if there exists an MT-function ϕ
and L ≥ 0 such that
H T x, T y ≤ ϕ d x, y d x, y Ld y, T x ,
∀x, y ∈ X,
1.7
4 a multivalued almost g-contraction 1, 12, 13, if there exist two constants θ ∈ 0, 1
and L ≥ 0 such that
H T x, T y ≤ θd gx, gy Ld gy, T x ,
∀x, y ∈ X,
1.8
5 a generalized multivalued almost g-contraction 1, if there exists an MT-function ϕ
and L ≥ 0 such that
H T x, T y ≤ ϕ d gx, gy d gx, gy Ld gy, T x ,
∀x, y ∈ X,
1.9
In 1989, Mizoguchi and Takahashi 14 proved the following fixed point theorem
which is a generalization of Nadler’s fixed point theorem 10 and the celebrated Banach
contraction principle see, e.g., 11. It is worth to mention that Mizoguchi-Takahashi’s fixed
point theorem gave a partial answer of Problem 9 in Reich 15 and it’s primitive proof is
difficult. Recently, Suzuki 16 presented a very simple proof of Mizoguchi-Takahashi’s fixed
point theorem.
Theorem MT Mizoguchi and Takahashi. Let X, d be a complete metric space, T : X →
CBX be a multivalued map, and ϕ : 0, ∞ → 0, 1 be a MT-function. Assume that
H T x, T y ≤ ϕ d x, y d x, y ,
Then, FT /
∅.
∀x, y ∈ X,
1.10
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Since then a number of generalizations in various different directions of MizoguchiTakahashi’s fixed point theorem have been investigated by several authors in the past. In
2007, M. Berinde and V. Berinde 12 proved the following interesting fixed point theorem to
generalize Mizoguchi-Takahashi’s fixed point theorem.
Theorem BB M. Berinde and V. Berinde. Let X, d be a complete metric space, T : X → CBX
be a multivalued map, ϕ : 0, ∞ → 0, 1 be a MT-function, and L ≥ 0. Assume that
H T x, T y ≤ ϕ d x, y d x, y Ld y, T x ,
∀x, y ∈ X,
1.11
that is, T is a generalized multivalued almost contraction. Then, FT / ∅.
In 3, the author established some new fixed point theorems for nonlinear multivalued contractive maps by using τ 0 -metrics see 3, Def. 1.2, τ 0 -metrics see 3, Def.
1.3, and MT-functions. Applying those results, the author gave the generalizations of
Berinde-Berinde‘s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s
fixed point theorem, and Banach contraction principle, Kannan’s fixed point theorems, and
Chatterjea’s fixed point theorems for nonlinear multivalued contractive maps in complete
metric spaces; for more detail, one can refer to 3.
Very recently, Du 17 first introduced the concepts of TVS-cone metric and TVS-cone
metric space to improve and extend the concept of cone metric space in the sense of Huang
and Zhang 18.
Definition 1.2 see 17. Let X be a nonempty set and Y a locally convex Hausdorff t.v.s. with
its zero vector θ, K a proper, closed, convex, and pointed cone in Y , and K a partial ordering
with respect to K defined by
x K y ⇐⇒ y − x ∈ K.
1.12
A vector-valued function p : X × X → Y is said to be a T V S-cone metric, if the following
conditions hold:
C1 θ K px, y for all x, y ∈ X and px, y θ if and only if x y,
C2 px, y py, x for all x, y ∈ X,
C3 px, z K px, y py, z for all x, y, z ∈ X.
The pair X, p is then called a TVS-cone metric space.
In this paper, we first establish some existence results concerning approximate coincidence point property and approximate fixed point property for various types of nonlinear
contractive maps in the setting of cone metric spaces and general metric spaces. From these
results, we present some new coincidence point and fixed point theorems which generalize
Berinde-Berinde’s fixed point theorem and Mizoguchi-Takahashi’s fixed point theorem. Our
results generalize and improve some recent results in 1–6, 10–19 and references therein.
Journal of Applied Mathematics
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2. Preliminaries
Let E be a topological vector space t.v.s. for short with its zero vector θE . A nonempty subset
K of E is called a convex cone if K K ⊆ K and λK ⊆ K for λ ≥ 0. A convex cone K is said to
be pointed if K ∩ −K {θE }. For a given proper, pointed, and convex cone K in E, we can
define a partial ordering K with respect to K by
x K y ⇐⇒ y − x ∈ K.
2.1
x ≺K y will stand for x K y and x /
y, while x K y will stand for y − x ∈ int K, where int K
denotes the interior of K.
In the following, unless otherwise specified, we always assume that Y is a locally
convex Hausdorff t.v.s. with its zero vector θ, K is a proper, closed, convex, and pointed
cone in Y with int K /
∅, K is a partial ordering with respect to K and e ∈ int K is fixed.
Recall that the nonlinear scalarization function ξe : Y → R is defined by
ξe y inf r ∈ R : y ∈ re − K ,
∀y ∈ Y.
2.2
Theorem 2.1 see 6, 17, 20, 21. For each r ∈ R and y ∈ Y , the following statements are satisfied:
i ξe y ≤ r ⇔ y ∈ re − K,
/ re − K,
ii ξe y > r ⇔ y ∈
iii ξe y ≥ r ⇔ y ∈
/ re − int K,
iv ξe y < r ⇔ y ∈ re − int K,
v ξe · is positively homogeneous and continuous on Y ,
vi if y1 ∈ y2 K (i.e., y2 K y1 ), then ξe y2 ≤ ξe y1 ,
vii ξe y1 y2 ≤ ξe y1 ξe y2 for all y1 , y2 ∈ Y .
Clearly, ξe θ 0. Notice that the reverse statement of (vi) in Theorem 2.1 (i.e., ξe y2 ≤
ξe y1 ⇒ y1 ∈ y2 K) does not hold in general. We illustrate the truth with the following simple
example.
Example A. Let Y R2 , K R2 {x, y ∈ R2 : x, y ≥ 0}, and e 1, 1. Then K is a proper,
∅ and e ∈ int K.
closed, convex, and pointed cone in Y with int K {x, y ∈ R2 : x, y > 0} /
/ re − int K, and y2 0, 0 ∈ re − int K. By
For r 1, it is easy to see that y1 10, −30 ∈
/ y2 K.
applying iii and iv of Theorem 2.1, we have ξe y2 < 1 ≤ ξe y1 while y1 ∈
Definition 2.2 see 17. Let X, p be a TVS-cone metric space, x ∈ X, and {xn }n∈N a sequence
in X.
i {xn } is said to TVS-cone converge to x if, for every c ∈ Y with θ K c, there exists a
natural number N0 such that pxn , x K c for all n ≥ N0 . We denote this by conecone
limn → ∞ xn x or xn → x as n → ∞ and call x the TVS-cone limit of {xn }.
ii {xn } is said to be a TVS-cone Cauchy sequence if, for every c ∈ Y with θ K c, there is
a natural number N0 such that pxn , xm K c for all n, m ≥ N0 .
iii X, p is said to be TVS-cone complete if every TVS-cone Cauchy sequence in X is
TVS-cone convergent in X.
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Journal of Applied Mathematics
In 17, the author proved the following important results.
Theorem 2.3 see 17. Let X, p be a TVS-cone metric spaces. Then, dp : X×X → 0, ∞ defined
by dp : ξe ◦ p is a metric.
Example B. Let X 0, 1, Y R2 , K R2 {x, y ∈ R2 : x, y ≥ 0}, e 1, 1, and θ 0, 0.
Define p : X × X → Y by
p x, y x − y, 8x − y .
2.3
Then, X, p is a TVS-cone complete metric space and
dp x, y ξe p x, y inf r ∈ R : p x, y ∈ re − K 8x − y.
2.4
So dp is a metric on X and X, dp is a complete metric space.
Theorem 2.4 see 17. Let X, p be a TVS-cone metric space, x ∈ X, and {xn }n∈N a sequence in
X. Then, the following statements hold.
cone
a If {xn } TVS-cone converges to x i.e., xn −−−−→ x as n → ∞, then dp xn , x → 0 as
dp
n → ∞ (i.e., xn −−→ x as n → ∞).
b If {xn } is a TVS-cone Cauchy sequence in X, p, then {xn } is a Cauchy sequence (in usual
sense) in X, dp .
Definition 2.5 see 1, 22. Let X, d be a metric space. A multivalued map T : X → NX is
said to have an approximate fixed point property provided
inf dx, T x 0,
x∈X
2.5
or, equivalently, for any > 0, there exists z ∈ X such that
dz, T z ≤ ,
2.6
or, equivalently, for any > 0, there exists x ∈ X such that
T x ∩ Bx , /
∅,
2.7
where Bx, r denotes a closed ball of radius r centered at x.
It is known that every generalized multivalued almost contraction in a metric space
X, d has the approximate fixed point property see 1, Lemma 2.2.
Remark 2.6. It is obvious that FT /
∅ implies that T has the approximate fixed point property.
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Definition 2.7 see 1. Let X, d be a metric space, g : X → X a single-valued self-map,
and T : X → NX a multivalued map. The maps g and T are said to have an approximate
coincidence point property provided
inf d gx, T x 0,
x∈X
2.8
or, equivalently, for any > 0, there exists z ∈ X such that
d gz, T z ≤ .
2.9
It is known that every generalized multivalued almost g-contraction in a metric space
X, d has the approximate coincidence point property provided each T x is g-invariant i.e.,
gT x ⊆ T x for each x ∈ X see 1, Theorem 2.7.
3. Main Results
For any locally convex Hausdorff t.v.s. Y with its zero vector θ, let τ denote the topology of
Y and let Uτ be the base at θ consisting of all absolutely convex neighborhood of θ. Let
L {
: is a Minkowski functional of U for U ∈ Uτ }.
3.1
Then, L is a family of seminorms on Y . For each ∈ L, put
V y ∈ Y : y < 1
3.2
and denote
UL {U : U r1 V 1 ∩ r2 V 2 ∩ · · · ∩ rn V n ,
rk > 0, k ∈ L, 1 ≤ k ≤ n, n ∈ N}.
3.3
Then, UL is a base at θ and the topology ΓL generated by UL is the weakest topology for Y
such that all seminorms in L are continuous and τ ΓL . Moreover, given any neighborhood
Oθ of θ, there exists U ∈ UL such that θ ∈ U ⊂ Oθ ; for more detail, we refer the reader to 23,
Theorem 12.4 in II.12, Page 113.
The following lemmas are very crucial to our main results.
Lemma 3.1. Let Oθ be a neighborhood of θ, ω ∈ K, and {αn } ⊂ 0, ∞ with limn → ∞ αn 0. Suppose
that U ∈ UL satisfies θ ∈ U ⊂ Oθ , then there exists n0 ∈ N, such that ±αn ω ∈ U for all n ≥ n0 .
Proof . Clearly, if ω θ, then ±αn ω θ ∈ U for all n ∈ N, and we are done. Suppose ω ∈
K \ {θ}. Since U ∈ UL ,
U r1 V 1 ∩ r2 V 2 ∩ · · · ∩ rs V s ,
3.4
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Journal of Applied Mathematics
for some ri > 0, i ∈ L and 1 ≤ i ≤ s. Let
δ min{ri : 1 ≤ i ≤ s} > 0,
ρ max{
i ω : 1 ≤ i ≤ s}.
3.5
We need consider two possible cases.
Case 1. If ρ 0, since each i is a seminorm, we have i ω 0 and
i ±αn ω αn i ω 0 < ri
3.6
for all 1 ≤ i ≤ s and all n ∈ N.
Case 2. If ρ > 0, since limn → ∞ αn 0, there exists n0 ∈ N such that αn < δ/ρ for all n ≥ n0 . So,
for each i ∈ {1, 2, . . . , s} and any n ≥ n0 , we obtain
i ±αn ω αn i ω
<
δ
i ω
ρ
3.7
≤δ
≤ ri .
Hence, by Cases 1 and 2, we get that, for any n ≥ n0 , ±αn ω ∈ ri V i for all 1 ≤ i ≤ s.
Therefore, ±αn ω ∈ U for all n ≥ n0 .
Lemma 3.2 see 24, Lemma 2.5. Let E be a t.v.s., K a convex cone with int K / ∅ in E, and
a, b, c ∈ E. Then, the following statements hold.
a int K K ⊆ int K.
b If a K b and b K c, then a K c.
In this section, we first establish an existence theorem related to approximate
coincidence point property for maps in TVS-cone complete metric space which is one of the
main results of this paper. It will have many applications to study metric fixed point theory.
It is worth observing that the following existence theorem does not require the TVScone completeness assumption on TVS-cone metric space X, p.
Theorem 3.3. Let X, p be a TVS-cone metric space, T : X → NX a multivalued map, h : X →
X a self-map, and dp : ξe ◦ p. Suppose that
D1 there exists an MT-function μ : 0, ∞ → 0, 1 such that, for each x ∈ X, if y ∈ T x with
y/
x then there exists z ∈ T y such that
p hy, hz K μ dp hx, hy p hx, hy ,
3.8
D2 T x is h-invariant (i.e., hT x ⊆ T x) for each x ∈ X.
Then, the following statements hold.
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9
a h and T have the dp -approximate coincidence point property on X (i.e., infx∈X dp hx, T x 0).
b There exists a sequence {zn } in X, p such that {zn } is a TVS-cone Cauchy sequence and
limn → ∞ dp zn , zn1 infx∈X dp hx, T x 0.
Proof. By Theorem 2.3, we know that dp is a metric on X. Let x1 ∈ X and x2 ∈ T x1 . If x1 x2 ,
then, by D2, we have hx1 ∈ T x1 . Since
inf dp hx, T x ≤ dp hx1 , T x1 0,
x∈X
3.9
we have infx∈X dp hx, T x 0. Let zn hx1 for all n ∈ N. Then, {zn } ⊂ X is a TVS-cone
Cauchy sequence and limn → ∞ dp zn , zn1 0 infx∈X dp hx, T x. Hence, all conclusions are
θ, then, by D1, there exists x3 ∈ T x2
x1 or px1 , x2 /
proved in the case of x1 x2 . If x2 /
such that
phx2 , hx3 K μ dp hx1 , hx2 phx1 , hx2 .
3.10
If x2 x3 , then the conclusions also hold by following a similar argument as above. If x3 / x2 ,
then there exists x4 ∈ T x3 such that
phx3 , hx4 K μ dp hx2 , hx3 phx2 , hx3 .
3.11
By induction, we can obtain a sequence {xn } in X satisfying the following: for each n ∈ N,
xn1 ∈ T xn ,
phxn1 , hxn2 K μ dp hxn , hxn1 phxn , hxn1 .
3.12
Applying v and vi of Theorem 2.1, the inequality 3.12 implies
dp hxn1 , hxn2 ξe phxn1 , hxn2 ≤ ξe μ dp hxn , hxn1 phxn , hxn1 ≤ μ dp hxn , hxn1 ξe phxn , hxn1 μ dp hxn , hxn1 dp hxn , hxn1 .
3.13
Since μt < 1 for all t ∈ 0, ∞, the sequence {dp hxn , hxn1 }n∈N is strictly decreasing in
0, ∞. Since μ is an MT-function, by g of Theorem D,
0 ≤ sup μ dp hxn , hxn1 < 1.
3.14
n∈N
Let λ : supn∈N μdp hxn , hxn1 . So λ ∈ 0, 1. From 3.12, we get
phxn1 , hxn2 K μ dp hxn , hxn1 phxn , hxn1 K λphxn , hxn1 .
3.15
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Journal of Applied Mathematics
It follows from 3.15 that
phxn1 , hxn2 K λphxn , hxn1 K · · · K λn phx1 , hx2 for each n ∈ N.
3.16
Let zn : hxn , n ∈ N. Applying v and vi of Theorem 2.1, the inequality 3.16 implies
dp zn1 , zn2 ≤ λn dp z1 , z2 for each n ∈ N.
3.17
Since λ ∈ 0, 1, by 3.17, we have limn → ∞ dp zn , zn1 0. By 3.12 and D2, we obtain
zn1 hxn1 ∈ T xn , n ∈ N. It implies that
inf dp hx, T x ≤ dp hxn , T xn ≤ dp zn , zn1 ,
x∈X
∀n ∈ N.
3.18
Since dp zn , zn1 → 0 as n → ∞, it follows from 3.18 that infx∈X dp hx, T x 0 and the
conclusion a is proved.
To see b, it suffices to prove that {zn } is a TVS-cone Cauchy sequence in X, p. For
m, n ∈ N with m > n, it follows from 3.16 that
pzn , zm K
m−1
λn−1
pz1 , z2 .
p zj , zj1 K
1−λ
jn
3.19
Given c ∈ Y with θ K c i.e, c ∈ int K intint K, there exists a neighborhood Nθ of θ such
that c Nθ ⊆ int K. Therefore, there exists Uc ∈ UL with Uc ⊆ Nθ such that
c Uc ⊆ c Nθ ⊆ int K.
3.20
Let ω pz1 , z2 and αn λn−1 /1 − λ, n ∈ N. Since λ ∈ 0, 1, we have {αn } ⊂ 0, ∞ and
limn → ∞ αn 0. Applying Lemma 3.1, there exists n0 ∈ N, such that −αn ω ∈ Uc for all n ≥ n0 .
So, by 3.20, we obtain
c − αn ω ∈ c Uc ⊆ int K
3.21
αn ω K c
3.22
or
for all n ≥ n0 . For m, n ∈ N with m > n ≥ n0 , since pzn , zm K αn ω from 3.19 and αn ω K c,
it follows from Lemma 3.2 that
pzm , zn K c.
3.23
Hence, we prove that {zn } is a TVS-cone Cauchy sequence in X, p. The proof is completed.
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Theorem 3.4. Let X, p be a TVS-cone metric space, T : X → NX a multivalued map, and
dp : ξe ◦ p. Suppose that
D3 there exists an MT-function μ : 0, ∞ → 0, 1 such that, for each x ∈ X, if y ∈ T x with
y/
x, then there exists z ∈ T y such that
p y, z K μ dp x, y p x, y .
3.24
Then, there exists a nonempty proper subset V of X, such that infx∈X dp x, T x infx∈V dp x, T x 0 (i.e., T has the dp -approximate fixed point property on V and X.)
Proof. Let h ≡ id be the identity map on X. Then, the conditions D1 and D2 as in
Theorem 3.3 hold. Following the same argument as in the proof of Theorem 3.3, we obtain
two sequences {xn } and {zn } in X such that
i xn1 ∈ T xn , zn hxn and zn1 ∈ T xn for each n ∈ N,
ii limn → ∞ dp zn , zn1 0.
Since h ≡ id, {xn } and {zn } are identical, we have limn → ∞ dp xn , xn1 0. Put V {xn }n∈N ⊂ X. Since
inf dp x, T x ≤ inf dp x, T x ≤ dp xn , T xn ≤ dp xn , xn1 ,
x∈X
x∈V
∀n ∈ N,
3.25
we obtain infx∈X dp x, T x infx∈V dp x, T x 0. Hence, T has the dp -approximate fixed point
property on V and X.
New, we establish the following approximate coincidence point property for maps in
general metric spaces by applying Theorem 3.3.
Theorem 3.5. Let X, d be a metric space, T : X → NX a multivalued map, and h : X → X a
self-map. Suppose that
A1 there exists an MT-function μ : 0, ∞ → 0, 1 such that, for each x ∈ X, if y ∈ T x with
y
/ x, then there exists z ∈ T y such that
d hy, hz ≤ μ d hx, hy d hx, hy ,
3.26
A2 T x is h-invariant (i.e., hT x ⊆ T x) for each x ∈ X.
Then, h and T have the approximate coincidence point property on X.
Proof . In Theorem 3.3, let Y R, K 0, ∞ ⊂ R, and e 1. Therefore, the conclusion is
immediate from Theorem 3.3.
The following approximate fixed point property for maps is immediate from
Theorem 3.4.
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Journal of Applied Mathematics
Theorem 3.6. Let X, d be a metric space and T : X → NX a multivalued map. Suppose that
A3 there exists an MT-function μ : 0, ∞ → 0, 1 such that, for each x ∈ X, if y ∈ T x with
y/
x, then there exists z ∈ T y such that
d y, z ≤ μ d x, y d x, y .
3.27
Then, T has the approximate fixed point property on X.
Theorem 3.7. In Theorem 3.6, if the condition (A3) is replaced by (A3)H , where
A3H there exists an MT-function μ : 0, ∞ → 0, 1 such that, for each x ∈ X,
d y, T y ≤ μ d x, y d x, y ,
∀y ∈ T x,
3.28
then, T has the approximate fixed point property on X.
Proof. Define τ : 0, ∞ → 0, 1 by τt 1 μt/2. Then, by 3, Lemma 2.1, τ is also an
MT-function. For each x ∈ X, let y ∈ T x with y /
x. Then, dx, y > 0. By A3H , we have
d y, T y < τ d x, y d x, y .
3.29
It follows that there exists z ∈ T y such that
d y, z < τ d x, y d x, y ,
3.30
which shows that A3 holds. Therefore, the conclusion follows from Theorem 3.6.
Theorem 3.8. Theorems 3.6 and 3.7 are equivalent.
Proof. We have shown that Theorem 3.6 implies Theorem 3.7. So it suffices to prove that
Theorem 3.7 implies Theorem 3.6. If A3 holds, then it is easy to verify that A3H also holds.
Hence, Theorem 3.7 implies Theorem 3.6 and we get the desired result.
Remark 3.9 see 1, Lemma 2.2. Is a special cases of Theorems 3.6 and 3.7.
Let X, p be a TVS-cone metric space. By Theorem 2.3, we know that dp : ξe ◦ p is a
metric on X. So we can obtain the topology Γdp on X induced by dp and hence define dp -open
subsets, dp -closed subsets, and dp -compact subsets of X.
Here, we denote Cdp X by the collection of all nonempty dp -closed subsets of X.
Theorem 3.10. Let X, p be a TVS-cone complete metric space, T : X → Cdp X a multivalued
map, h : X → X a self-map, and dp : ξe ◦ p. Suppose that the conditions (D1) and (D2) as in
Theorem 3.3 hold and further assume one of the following conditions hold:
L1 X, dp is dp -compact and the function f : X → 0, ∞ defined by fx dp hx, T x is
l.s.c.,
L2 G {hx, y ∈ X × X : x ∈ X, y ∈ T x} is a dp -closed subset of X × X.
Then, the following statements hold.
Journal of Applied Mathematics
13
a There exists a nonempty subset M of X, such that M, dp is a complete metric space.
b COPh, T /
∅.
Proof. Following the same argument as in the proof of Theorem 3.3, one can obtain that there
exist two sequences {xn } and {zn } in X such that for any n ∈ N, zn hxn , zn1 ∈ T xn , {zn }
is a TVS-cone Cauchy sequence in X, p and limn → ∞ dp zn , zn1 infx∈X dp hx, T x 0. By
the TVS-cone completeness of X, p, there exists v ∈ X, such that {zn } TVS-cone converges
to v. On the other hand, applying Theorem 2.4, we obtain that {zn } is a Cauchy sequence in
dp
X, dp and dp zn , v → 0 or zn −−→ v as n → ∞. Let M {zn }n∈N ∪ {v}. Then M, dp is a
complete metric space and the conclusion a holds.
Now, we verify the conclusion b. Suppose that L1 holds. Then the infimum
infx∈X fx is attained. Since infx∈X fx infx∈X dp hx, T x 0, there exists ζ ∈ X such
that
dp hζ, T ζ : fζ inf fx 0.
x∈X
3.31
Since T ζ is a dp -closed subset of X, it implies from 3.31 that hζ ∈ T ζ. Hence, ζ ∈ COPh, T .
Suppose that L2 holds. For any n ∈ N, since zn1 ∈ T xn , we know zn , zn1 ∈ G. Since
dp
G is dp -closed in X × X and zn −−→ v as n → ∞, we have v, v ∈ G. Hence, there exists a ∈ X
such that v ha and v ∈ T a, which say that a ∈ COPh, T . The proof is completed.
Theorem 3.11. Let X, p be a TVS-cone complete metric space, T : X → Cdp X a multivalued
map, and dp : ξe ◦ p. Suppose that the condition (D3) as in Theorem 3.4 holds and further assume
one of the following conditions hold:
H1 T is dp -closed; that is, GrT : {x, y ∈ X × X : y ∈ T x}, the graph of T , is a dp -closed
subset of X × X,
H2 the function f : X → 0, ∞ defined by fx dp x, T x is l.s.c.,
H3 inf{dp x, z dp x, T x : x ∈ X} > 0 for every z ∈
/ FT .
Then, there exists a nonempty subset W of X, such that
a W, dp is a complete metric space,
b FT ∩ W /
∅.
Proof. Following a similar argument as in the proof of Theorem 3.3, we can obtain a sequence
{xn } in X such that, for any n ∈ N,
i xn1 ∈ T xn ,
ii pxn1 , xn2 K γpxn , xn1 , where γ : supn∈N μdp xn , xn1 ∈ 0, 1,
iii pxn , xm K γ n−1 /1 − γ px1 , x2 , for m, n ∈ N with m > n.
iv {xn } is a TVS-cone Cauchy sequence in X, p.
14
Journal of Applied Mathematics
Applying Theorem 2.1, the inequality iii implies that, for m, n ∈ N with m > n, we
have
dp xn , xm ξe pxn , xm γ n−1
≤ ξe
px1 , x2 1−γ
3.32
γ n−1
dp x1 , x2 .
1−γ
Since limn → ∞ γ n−1 /1 − γ 0, it follows from 3.32 that {xn } is a Cauchy sequence in X, dp .
By the TVS-cone completeness of X, p and iv, there exists v ∈ X, such that {xn } TVS-cone
dp
converges to v. Applying Theorem 2.4, we have dp xn , v → 0 or xn −−→ v as n → ∞. Let
W {xn }n∈N ∪ {v}. Then, W, dp is a complete metric space and the conclusion a holds.
Finally, in order to complete the proof, it suffices to show that v ∈ FT . Suppose that
dp
H1 holds. Since xn1 ∈ T xn , we have xn , xn1 ∈ GrT for each n ∈ N. By H1 and xn −−→ v
as n → ∞, we get v ∈ FT .
If H2 holds, by the lower semicontinuity of f and i, we obtain
dp v, T v fv
≤ lim inf fxn n→∞
3.33
≤ lim inf dp xn , T xn n→∞
≤ lim dp xn , xn1 0,
n→∞
which implies dp v, T v 0. Since T v is a dp -closed subset of X, v ∈ FT .
dp
Let H3 holds. Suppose v ∈
/ FT . Since dp is a metric on X and xm −−→ v as m → ∞,
by 3.32, we get
dp xn , v ≤
γ n−1
dp x1 , x2 1−γ
for any n ∈ N.
3.34
From 3.32 and 3.34, we have
0 < inf dp x, v dp x, T x
x∈X
≤ inf dp xn , v dp xn , T xn n∈N
≤ inf dp xn , v dp xn , xn1 n∈N
2γ n−1
dp x1 , x2 n→∞ 1 − γ
≤ lim
0,
which leads a contradiction. Therefore, v ∈ FT . The proof is completed.
3.35
Journal of Applied Mathematics
15
The following result is immediate from Theorem 3.10.
Theorem 3.12. Let X, d be a complete metric space, T : X → CX a multivalued map, and
h : X → X a self-map. Suppose that the conditions (A1) and (A2) as in Theorem 3.5 hold and further
assume one of the following conditions hold:
i X, d is compact and the function f : X → 0, ∞ defined by fx dhx, T x is l.s.c.;
ii {hx, y ∈ X × X : x ∈ X, y ∈ T x} is a closed subset of X × X.
Then, COPh, T /
∅.
Theorem 3.13. Let X, d be a complete metric space and T : X → CX a multivalued map.
Suppose that the condition (A3) or (A3)H holds and further assume one of the following conditions
hold:
h1 T is closed,
h2 the function f : X → 0, ∞ defined by fx dx, T x is l.s.c.,
h3 inf{dx, z dx, T x : x ∈ X} > 0 for every z ∈
/ FT ,
h4 for each sequence {xn } in X with xn1 ∈ T xn , n ∈ N and limn → ∞ xn v, we have
limn → ∞ HT xn , T v 0.
Then, FT / ∅.
Proof. The conclusion is immediate from Theorem 3.11 if Y R, K 0, ∞ ⊂ R, e 1, and
one of conditions h1, h2, and h3 holds. Suppose that h4 holds. Following a similar
argument as in the proof of Theorem 3.3, we can construct a Cauchy sequence {xn } in X, d
such that xn1 ∈ T xn , n ∈ N, and {xn } converge to some point v ∈ X. Since the function
d
→ v as n → ∞, by h4, we get
x → dx, T v is continuous, xn1 ∈ T xn and xn −
dv, T v lim dxn1 , T v ≤ lim HT xn , T v 0,
n→∞
n→∞
3.36
which implies v ∈ FT . The proof is completed.
Remark 3.14.
a Let A and B be two topological vector spaces and T : A → 2B a multivalued map.
Recall that T is u.s.c. if and only if for any open set V in B, T V : {x ∈ A : T x ⊆
V } is open in A. It is known that if T is u.s.c. with closed values, then T is closed see
25. Hence, Theorem 3.13 is true if X, · is a Banach space and T : X → CX
is u.s.c.;
b let G be a nonempty subset of a metric space X, d and T : X → CX u.s.c. Then,
the function φ : G → 0, ∞ defined by φx dx, T x is l.s.c.; for detail, see 26,
Lemma 2 or 27, Lemma 3.1;
c it is known that any single-valued map of Kannan’s type or Chatterjea’s type
satisfies h3; for more detail, one can see 28, Corollary 3 or 3, Remark 3.1.
Applying Theorem 3.13, we can prove the following generalization of BerindeBerinde’s fixed point theorem 12.
16
Journal of Applied Mathematics
Theorem 3.15. Let X, d be a complete metric space, T : X → CBX a multivalued map, and
G : X → 0, ∞ a function. Suppose that there exists an MT-function μ : 0, ∞ → 0, 1 such that
H T x, T y ≤ μ d x, y d x, y G y d y, T x ,
∀x, y ∈ X.
3.37
Then, FT / ∅.
Proof. Let x ∈ X. If y ∈ T x, then dy, T x 0. So 3.37 implies the inequality
d y, T y ≤ μ d x, y d x, y ,
∀y ∈ T x.
3.38
Hence, the condition A3H of Theorem 3.13 holds. Let {xn } in X with xn1 ∈ T xn , n ∈ N, and
limn → ∞ xn v. By 3.37, we obtain
lim HT xn , T v ≤ lim ϕdxn , vdxn , v Gvdv, xn1 0,
n→∞
n→∞
3.39
which says that the condition h4 of Theorem 3.13 also holds. Therefore, the conclusion
follows from Theorem 3.13.
Remark 3.16.
a Theorems 3.11, 3.13, and 3.15 all generalize Berinde-Berinde’s fixed point theorem
12.
b In Theorem 3.15, if Gx 0 for all x ∈ X, then we can obtain MizoguchiTakahashi’s fixed point theorem 14.
c In Theorem 3.15, if Gx 0 for all x ∈ X, and μ : 0, ∞ → 0, 1 is defined by
μt γ for some γ ∈ 0, 1, then we can obtain Nadler’s fixed point theorem 10.
d 1, Theorem 2.6 is a special case of Theorem 3.15.
e Notice that, in 1, Theorem 2.6, the authors showed that a generalized multivalued
∅ provided either X, d
almost contraction T in a metric space X, d has FT /
is compact and the function fx dx, T x is l.s.c. or T is closed and compact.
But reviewing Theorem 3.15, we know that the conditions in 1, Theorem 2.6 are
redundant.
Acknowledgments
The author would like to express his sincere thanks to the anonymous referee for their
valuable comments and useful suggestions in improving the paper. This research was
supported partially by Grant no. NSC 100-2115-M-017-001 of the National Science Council
of the Republic of China.
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