ON A NEW CLASS OF MULTIVALUED WEAKLY PICARD OPERATORS ON COMPLETE METRIC

14 Pages·2015·0.1 MB·English

TAIWANESEJOURNALOF MATHEMATICS
Vol. 19, No. 3,pp. 659-672,June2015
DOI:10.11650/tjm.19.2015.4752
Thispaperisavailableonlineat http://journal.taiwanmathsoc.org.tw
ON A NEW CLASS OF MULTIVALUED WEAKLY PICARD OPERATORS
ON COMPLETE METRIC SPACES
Ishak Altun, Murat Olgun and Gu¨lhan Mınak
Abstract. In the present paper, the concept of nonlinear F-contraction for mul-
tivalued mappings in metric spaces is introduced and considering the new proof
technique,whichwasusedforsinglevaluedmapsbyWardowski[D. Wardowski,
Fixed points of a new type of contractive mappings in complete metric spaces,
FixedPointTheoryAppl. 2012,2012:94,6pp.], wedemonstratethatmultivalued
nonlinear F-contractions of C´iric´ type are weakly Picard operators on complete
metricspaces. Finally,wegiveanontrivialexampletoguaranteethatourresultis
veritable generalization of recent result of C´iric´ [Lj. B. C´iric´, Multi-valuednon-
linear contraction mappings, Nonlinear Analysis, 71 (2009), 2716-2723]. Also,
we show that many fixed point resultsin the literaturecan not be applied to this
example.
1. INTRODUCTIONAND PRELIMINARIES
Let (X,d)be a metric space. P(X) denotes the familyof all nonemptysubsetsof
X, C(X) denotes the family of all nonempty, closed subsets of X, CB(X) denotes
the family of all nonempty, closed and bounded subsets of X and K(X) denotes the
family of all nonempty compact subsets of X. It is clear that K(X) ⊆ CB(X) ⊆
C(X)⊆ P(X). For A,B ∈ C(X), let
(cid:2) (cid:3)
H(A,B)= max supd(x,B),supd(y,A) ,
x∈A y∈B
where d(x,B) = inf{d(x,y): y ∈ B}. Then H is called generalized Pompeiu-
Hausdorff distance on C(X). It is well known that H is a metric on CB(X), which
is called Pompeiu-Hausdorff metric induced by d. We can find detailed information
aboutthePompeiu-Hausdorffmetricin [1, 5, 8]. An elementx∈ X issaid tobe fixed
point of a multivalued mapping T :X → P(X) if x ∈ Tx.
Following the Banach contraction principle, Nadler [11] first initiatedthe studyof
fixed point theorems for multivaluedlinear contraction mappings.
ReceivedApril 28,2014,acceptedAugust8, 2014.
CommunicatedbyChongLi.
2010MathematicsSubjectClassification: Primary54H25;Secondary,47H10.
Key words and phrases: Weakly Picard operator, Fixed point, Multivalued mappings, Nonlinear F-
contractions,Completemetric space.
659
660 IshakAltun, Murat OlgunandGu¨lhanMınak
Theorem 1. (Nadler [11]). Let (X,d) be a complete metric space and T : X →
CB(X). Assume that there there exists L∈ [0,1) such that
H(Tx,Ty)≤ Ld(x,y)
for all x,y ∈ X. Then T has a fixed point.
Then many fixed point theorists studiedon fixed points of multivaluedcontractive
maps as follows:
Theorem 2. (Reich [13]). Let (X,d) be a complete metric space and T : X →
K(X). Assume that there exists a map ϕ:(0,∞)→ [0,1) such that
limsupϕ(t)< 1, ∀s > 0;
t→s+
and
H(Tx,Ty)≤ ϕ(d(x,y))d(x,y).
for all x,y ∈ X with x (cid:8)= y. Then T has a fixed point.
In [14, 15], Reich asked the question as if the above theorem is also true for the
map T : X → CB(X). The partial affirmative answer was given by Mizoguchi and
Takahashi [10]. They proved the following theorem.
Theorem 3. (Mizoguchi-Takahashi [10]). Let (X,d) be a complete metric space
and T : X → CB(X). Assume that there exists a map ϕ :(0,∞)→ [0,1) such that
limsupϕ(t)< 1, ∀s ≥ 0;
t→s+
and
H(Tx,Ty)≤ ϕ(d(x,y))d(x,y).
for all x,y ∈ X with x (cid:8)= y. Then T has a fixed point.
In [16] Suzuki gave a simple proof of Mizoguchi Takahashi fixed point theorem
and also an example to show that it is a real generalization of Nadler’s. On the other
hand, Feng and Liu [7] obtained some interesting fixed point results for multivalued
mappings without using the Pompeiu-Hausdorff metric. They proved the following
theorem.
Theorem 4. (Feng-Liu [7]). Let (X,d) be a complete metric space and T :X →
C(X). Assume that the following conditions hold:
(i) the map x → d(x,Tx) is lower semi-continuous;
(ii) there exist b,c ∈ (0,1) with c < b such that for any x ∈ X there is y ∈ Tx
satisfying
Multivalued WeaklyPicardOperators 661
bd(x,y)≤ d(x,Tx)
and
d(y,Ty)≤ cd(x,y).
Then T has a fixed point.
Then Klim and Wardowski [9] generalized Theorem 4 as follows:
Theorem 5. (Klim-Wardowski [9]). Let (X,d) be a complete metric space and
T :X → C(X). Assume that the following conditionshold:
(i) the map x → d(x,Tx) is lower semi-continuous;
(ii) there exists b ∈ (0,1) and a function ϕ:[0,∞)→ [0,b)satisfying
limsupϕ(t)< b, ∀s ≥ 0;
t→s+
(iii) for any x ∈ X, there is y ∈ Tx satisfying
bd(x,y)≤ d(x,Tx)
and
d(y,Ty)≤ ϕ(d(x,y))d(x,y).
Then T has a fixed point.
Consideringthesamedirection,in2009,C´iric´ [6] introducednewmultivaluednon-
linear contractions and establisheda few nice fixed point theoremsfor such mappings,
one of them is as follows:
Theorem 6. Let (X,d) be a complete metric space and T : X → C(X). Assume
that the following conditions hold:
(i) the map x → d(x,Tx) is lower semi-continuous;
(ii) there exists a function ϕ :[0,∞)→ [a,1), 0< a < 1, satisfying
limsupϕ(t)< 1, ∀s ≥ 0;
t→s+
(iii) for any x∈ X, there is y ∈ Tx satisfying
(cid:4)
ϕ(d(x,Tx))d(x,y)≤ d(x,Tx)
and
d(y,Ty)≤ ϕ(d(x,Tx))d(x,y).
Then T has a fixed point.
662 IshakAltun, Murat OlgunandGu¨lhanMınak
On theother hand, Berindeand Berinde[4] introduceda general class of multival-
ued contractions and proved the following fixed point theorems:
Theorem 7. (Berinde-Berinde [4], Theorem 3). Let (X,d) be a complete metric
space and T : X → CB(X) be a multivalued almost contraction, that is, there exist
two constants δ ∈ (0,1) and L ≥ 0 such that
(1.1) H(Tx,Ty)≤ δd(x,y)+Ld(y,Tx)
for all x,y ∈ X. Then T has a fixed point.
Theorem 8. (Berinde-Berinde [4], Theorem 4). Let (X,d) be a complete metric
space and T : X → CB(X). If there exist a constant L ≥ 0 and a function ϕ :
[0,∞)→ [0,1) satisfying
limsupϕ(t)< 1, ∀s ≥ 0,
t→s+
such that
(1.2) H(Tx,Ty)≤ ϕ(d(x,y))d(x,y)+Ld(y,Tx)
for all x,y ∈ X. Then T has a fixed point.
Analyzingtheproofsofabovealltheorems,wecanobservethatthementionedmaps
on complete metric spaces are weakly Picard operators. We know that, a multivalued
map T on a metric space is weakly Picard operator if there exists a sequence {xn} in
X such that xn+1 ∈ Txn for any initial point x0, converges to a fixed point of T.
In the presentpaper, by introducinga new and different class of multivaluedmap-
pingsin metricspaces, we give some multivaluedweaklyPicard operators in complete
metricspaces. Our resultsare extendand generalizemanyfixedpointtheoremsinclud-
ing Theorem 6 and they are based on a new approach of contraction mapping, which
is called F-contraction. The concept of F-contraction for single valued mappings on
complete metric space was introduced by Wardowski [17]. Now, we recall this new
concept and some related results.
Let F : (0,∞)→ R be a function. For the sake of completeness,we willconsider
the following conditions:
(F1) F is strictlyincreasing, i.e., for all α,β∈(0,∞)such thatα<β, F(α)<F(β),
(F2) For each sequence {αn} of positive numbers
lim αn =0 if and only if lim F(αn)= −∞,
n→∞ n→∞
(F3) There exists k ∈ (0,1) such that limα→0+αkF(α) = 0,
(F4) F(infA) = infF(A) for all A ⊂ (0,∞) with infA > 0.
Multivalued WeaklyPicardOperators 663
We denotebyF and F∗ be thesetof all functionsF satisfying(F1)-(F3) and (F1)-
(F4),respectively. ItisclearthatF∗ ⊂ F andsomeexamplesofthefunctio(cid:5)nsbelon(cid:6)ging
F∗ are F1(α) = lnα, F2(α)= α+lnα, F3(α)= −√1α and F4(α)= ln α2 +α . If
we define F5(α)= lnα for α ≤ 1 and F5(α) = 2α for α > 1, then F5 ∈ F\F∗. If F
satisfies (F1), then it satisfies (F4) if and only if it is right continuous.
Let (X,d) be a metric space and T : X → X be a mapping. Then, Wardowski
[17] say that T is an F-contraction if F ∈ F and there exists τ > 0 such that
(1.3) τ +F(d(Tx,Ty))≤ F(d(x,y))
for all x,y ∈ X with d(Tx,Ty) > 0. Also, Wardowski [17] proved that every F-
contractions on complete metric spaces has a unique fixed point. We can find some
detailed information for F-contractions in [17].
By combining the ideas of Wardowski’s and Nadler’s, Altun et al [2] introduced
the concept of multivalued F-contractions and obtained some fixed point results for
mappings of this type on complete metric space.
Definition 1. ([2]). Let (X,d) be a metric space and T : X → CB(X) be a
mapping. Thenwe saythatT is amultivaluedF-contractionif F ∈ F andthereexists
τ > 0 such that
τ +F(H(Tx,Ty))≤ F(d(x,y))
for all x,y ∈ X with H(Tx,Ty)> 0.
By the considering F(α) = lnα, then every multivalued contraction in the sense
of Nadler is also multivalued F-contraction.
Theorem 9. ([2]). Let (X,d)be a complete metric space and T :X → K(X) be
a multivaluedF-contraction, then T has a fixed point in X.
Here, the following question may come to mind: Can we take CB(X) instead
of K(X) in Theorem 9? The answer is negative as shown in Example 1 in [3].
Nevertheless, by adding the condition (F4) on F, we can we take CB(X) instead of
K(X).
Theorem 10. ([2]). Let (X,d) be a complete metric space and T :X → CB(X)
be a multivaluedF-contraction with F ∈ F∗, then T has a fixed point in X.
On the other hand Olgun et al [12] proved the following theorems, one of them is
a generalization of famous Mizoguchi-Takahashi fixed point theorem for multivalued
contractivemappings. Theseresultsarealsononlinearcaseof Theorem9andTheorem
10, respectively.
664 IshakAltun, Murat OlgunandGu¨lhanMınak
Theorem 11. ([12]). Let (X,d) be a complete metric space and T :X → K(X).
If there exist F ∈ F and τ :(0,∞)→ (0,∞) such that
liminfτ(t)> 0, ∀s ≥ 0,
t→s+
satisfying
τ(d(x,y))+F(H(Tx,Ty))≤F(d(x,y))
for all x,y ∈ X with H(Tx,Ty)> 0. Then T has a fixed point in X.
Theorem12. ([12]). Let(X,d)beacompletemetricspaceandT :X → CB(X).
If there exist F ∈ F∗ and τ :(0,∞)→ (0,∞) such that
(1.4) liminfτ(t)> 0, ∀s ≥ 0,
t→s+
satisfying
τ(d(x,y))+F(H(Tx,Ty))≤F(d(x,y))
for all x,y ∈ X with H(Tx,Ty)> 0. Then T has a fixed point in X.
IfweexaminetheproofsofTheorem9,10,11and12,wecanseethatthementioned
maps on complete metric spaces are weaklyPicard operators.
2. MAIN RESULTS
Now, we shall prove a theorem which extends and generalizes Theorem 6.
Theorem 13. Let(X,d)beacompletemetricspace, T :X → C(X)andF ∈ F∗.
Assume that the following conditions hold:
(i) the map x → d(x,Tx) is lower semi-continuous;
(ii) there exists a function τ : (0,∞)→ (0,σ], σ > 0 such that
(2.1) liminfτ(t)> 0, ∀s ≥ 0;
t→s+
(iii) for any x ∈ X with d(x,Tx)> 0, there is y ∈ Tx satisfying
τ(d(x,Tx))
(2.2) F(d(x,y))≤ F(d(x,Tx))+
2
and
(2.3) τ(d(x,Tx))+F(d(y,Ty))≤ F(d(x,y)).
Then T is a weakly Picard operator.
Multivalued WeaklyPicardOperators 665
Proof. First, we show that T has fixed point in X. Assume the contrary, then
d(x,Tx)> 0 for all x∈ X. Therefore, since τ(t)> 0 for all t > 0 and F ∈ F∗, then
for any x ∈ X there exists y ∈ Tx such that (2.2) holds. Let x0 ∈ X be an initial
point. Then by assumptions (2.2) and (2.3) we can choose x1 ∈ Tx0 such that
τ(d(x ,Tx ))
(2.4) F(d(x0,x1))≤ F(d(x0,Tx0))+ 02 0
and
(2.5) τ(d(x0,Tx0))+F(d(x1,Tx1))≤ F(d(x0,x1)).
From (2.4) and (2.5), we get
τ(d(x ,Tx ))
(2.6) 02 0 +F(d(x1,Tx1))≤ F(d(x0,Tx0)).
Now we choose x2 ∈ Tx1 such that
τ(d(x ,Tx ))
F(d(x ,x ))≤ F(d(x ,Tx ))+ 1 1
1 2 1 1 2
and
τ(d(x ,Tx ))+F(d(x ,Tx ))≤ F(d(x ,x )).
1 1 2 2 1 2
Hence we get
τ(d(x ,Tx ))
1 1 +F(d(x ,Tx ))≤ F(d(x ,Tx )).
2 2 2 1 1
Continuing this process we can choose a sequence {xn} such that xn+1 ∈ Txn satis-
fying
τ(d(x ,Tx ))
(2.7) F(d(xn,xn+1))≤ F(d(xn,Txn))+ n n
2
and
τ(d(x ,Tx ))
(2.8) n2 n +F(d(xn+1,Txn+1))≤ F(d(xn,Txn))
for all n ≥ 0.
Now, we shall show that d(xn,Txn) → 0 as n → ∞. From (2.8), we conclude
that{d(xn,Txn)}isastrictlydecreasingsequenceofpositiverealnumbers. Therefore,
there exists δ ≥ 0 such that
lim d(x ,Tx )= δ.
n n
n→∞
Suppose δ > 0. Then, since F is right continuous, taking the limit infimum on both
sides of (2.8) and having in mind the assumption (2.1), we have
τ(d(x ,Tx ))
liminf n n +F(δ) ≤ F(δ),
2
d(xn,Txn)→δ+
666 IshakAltun, Murat OlgunandGu¨lhanMınak
which is a contradiction. Thus δ = 0, that is,
(2.9) lim d(xn,Txn)= 0.
n→∞
Now we shall show that {xn} is a Cauchy sequence in X. Let
τ(d(x ,Tx ))
α = liminf n n > 0.
2
d(xn,Txn)→δ+
and 0 < q < α. Then there exists n0 ∈ N such that τ(d(xn2,Txn)) > q for all n ≥ n0.
Thus from (2.8),
q+F(d(x ,Tx ))≤ F(d(x ,Tx ))
n+1 n+1 n n
for each n ≥ n0. Hence, by induction, for all n ≥ n0
F(d(x ,Tx )) ≤ F(d(x ,Tx ))−q
n+1 n+1 n n
.
.
.
(2.10) ≤ F(d(xn ,Txn ))−(n+1−n0)q.
0 0
Since 0 < τ(t)≤ σ for all t > 0. From (2.7), we get
F(d(x ,x ))≤ F(d(x ,Tx ))+σ.
n n+1 n n
Thus, by (2.10), for all n ≥ n0
F(d(x ,x )) ≤ F(d(x ,Tx ))+σ
n n+1 n n
(2.11) ≤ F(d(xn ,Txn ))−(n−n0)q+σ.
0 0
From(2.11),wegetlimn→∞F(d(xn,xn+1))= −∞.Thusfrom(F2)wehavelimn→∞
d(xn,xn+1)= 0. Therefore, from (F3) there exists k ∈ (0,1) such that
lim [d(x ,x )]kF(d(x ,x ))= 0.
n n+1 n n+1
n→∞
By (2.11), for all n ≥ n0
[d(x ,x )]kF(d(x ,x ))−[d(x ,x )]kF(d(x ,Tx ))
n n+1 n n+1 n n+1 n n
0 0
(2.12)
≤ −[d(x ,x )]k[(n−n )q+σ]≤ 0.
n n+1 0
Lettingn → ∞ in (2.12), we obtain that
(2.13) lim [d(xn,xn+1)]k[(n−n0)q+σ]= 0.
n→∞
Multivalued WeaklyPicardOperators 667
From (2.13), there exits n1 ∈ N such that [d(xn,xn+1)]k[(n−n0)q+σ] ≤ 1 for all
n ≥ n1. We can take n1 > n0. So, we have, for all n ≥ n1
1
(2.14) d(xn,xn+1)≤ 1 .
[(n−n )q+σ]
0 k
In order to show that {xn} is a Cauchy sequence consider m,n ∈ N such that m >
n ≥ n1. Using the triangular inequalityfor the metric and from (2.14), we have
d(x ,x ) ≤ d(x ,x )+d(x ,x )+···+d(x ,x )
n m n n+1 n+1 n+2 m−1 m
m(cid:7)−1 (cid:7)∞ (cid:7)∞ 1
= d(x ,x )≤ d(x ,x )≤
i i+1 i i+1
[(i−n )q+σ]1/k
i=n i=n i=n 0
(cid:8)
By the convergence of the series 1 , passing to limit n,m → ∞,
[(i−n )q+σ]1/k
i>n −σ 0
0 q
we get d(xn,xm) → 0. This yields that {xn} is a Cauchy sequence in (X,d). Since
(X,d) is a complete metric space, the sequence{xn} converges to some point z ∈ X,
that is, limn→∞xn = z. Since x → d(x,Tx)is lower semi-continuous,from (2.9) we
have
0 ≤ d(z,Tz)≤ liminfd(x ,Tx )= 0.
n n
n→∞
Hence d(z,Tz)= 0, which is a contradiction. Therefore, T has a fixed point in X.
It can be seenthat, we can construct a sequence{xn} in X such that xn+1 ∈ Txn
for any initial point x0, converges to a fixed point of T. That is, T is a weaklyPicard
operator.
Remark 1. Let A be a compact subset of a metric space (X,d) and x ∈ X, then
there exists a ∈ A such that d(x,a)= d(x,A).
Remark 2. If we take K(X) instead of CB(X) in Theorem 13, we can remove
the condition (F4) on F. Therefore, by taking into account Remark 1 the proof of the
following theorem is obvious.
Theorem 14. Let (X,d)bea completemetricspace, T : X → K(X)andF ∈ F.
Assume that the following conditions hold:
(i) the map x → d(x,Tx) is lower semi-continuous;
(ii) there exists a function τ :(0,∞)→ (0,σ], σ > 0 such that
liminfτ(t)> 0, ∀s ≥ 0;
t→s+
668 IshakAltun, Murat OlgunandGu¨lhanMınak
(iii) for any x ∈ X with d(x,Tx)> 0, there is y ∈ Tx satisfying
τ(d(x,Tx))
F(d(x,y))≤ F(d(x,Tx))+
2
and
τ(d(x,Tx))+F(d(y,Ty))≤ F(d(x,y)).
Then T is a weakly Picard operator.
Taking into account our results, T is a weakly Picard operator in the following
nontrivial example. We also show that Theorems 1, 3, 4, 5, 6, 8, 10, 12 can not be
applied to this example.
Example 1. Let X = { 1 : n ∈ N}∪{0} and d(x,y) = |x−y|, then (X,d) is
n2
complete metric space. Let T :X → CB(X) be defined by
⎧ (cid:13) (cid:14)
⎪ 1 1
⎨ 0, , x =
Tx= (n+1)2 n2 .
⎪
⎩
{x}, x ∈ {0,1}
It is easy to see that
⎧
⎪ 0, x∈ {0,1}
⎪
⎨
d(x,Tx)=
2n+1 1
⎪
⎪⎩ , x= , n ≥ 2
n2(n+1)2 n2
and it is lower semi-continuous.
Let τ(t) = ln2 and σ = 4, then the condition (ii) of Theorem 13 is satisfied.
Now we show that the condition (iii) of Theorem 13 is satisfied with
⎧
⎪ lnα
⎨ √ , 0< α < e2
F(α)= α .
⎪
⎩
α−e2+ 2, α ≥ e2
e
We can see that F ∈ F∗. Note that supx,y∈Xd(x,y)= 1(cid:15)< e2. If d((cid:16)x,Tx)> 0, then
x = 1 , n ≥ 2. Then we choose y = 1 ∈ Tx = 0, 1 . Therefore, we
n2 (n+1)2 (n+1)2
have
2n+1
d(x,y)= d(x,Tx)=
n2(n+1)2
and
2n+3
d(y,Ty)= .
(n+1)2(n+2)2