# Multivalued Pseudo-Picard Operators and Fixed Point Results

Authors: Gülhan Mınak; Özlem Acar; Ishak Altun
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce the concept of multivalued pseudo-Picard (MPP) operator on a metric space. This concept is weaker than multivalued weakly Picard (MWP) operator, which is given by M. Berinde and V. Berinde (2007). Then, we give both fixed point results and examples for MPP operators. Also, we obtain some ordered fixed point results for multivalued maps as application.


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## Body

## 1. Introduction and Preliminaries

Let(X,d) be a metric space, and let CB(X) denote the class of all nonempty, closed, and bounded subsets of X. It is well known that H:CB(X)×CB(X)→ℝ defined by
(1)H(A,B)=max{supx∈AD(x,B),supy∈BD(y,A)}
is a metric on CB(X), which is called Hausdorff metric, where
(2)D(x,B)=inf{d(x,y):y∈B}.
Let T:X→CB(X) be a map; then, T is called multivalued contraction if for all x,y∈X there exists δ∈[0,1) such that
(3)H(Tx,Ty)≤δd(x,y).In 1969, Nadler [1] proved a fundamental fixed point theorem for multivalued maps: every multivalued contraction on complete metric space has a fixed point.Then, a lot of generalizations of the result of Nadler were given (see, e.g., [2–4]). Two important generalizations of it were given by M. Berinde and V. Berinde [5] and Mizoguchi and Takahashi [6].In [5], M. Berinde and V. Berinde introduced the concept of multivalued weakly Picard operator as follows (for single-valued Picard and weakly Picard operators we refer to [7–9]).Definition 1.
Let(X,d) be a metric space, and let T:X→𝒫(X) (the family of all nonempty subsets of X) be a multivalued operator. T is said to be multivalued weakly Picard (MWP) operator if and only if for each x∈X and any y∈Tx, there exists a sequence {xn} in X such that(i)
x0=x,x1=y,(ii)
xn+1∈Txn,(iii)
the sequence{xn} is convergent and its limit is a fixed point of T.Then M. Berinde and V. Berinde [5] show that every Nadler [1], Reich [10], Rus [11] and Petruşel [12] type multivalued contractions on complete metric space are MWP operators. Mizoguchi and Takahashi [6], proved the following fixed point theorem. This is also an example of MWP operator.Theorem 2.
Let(X,d) be a complete metric space, and let T:X→CB(X) be a multivalued map. Assume that
(4)H(Tx,Ty)≤k(d(x,y))d(x,y)
for all x,y∈X, where k is an ℳ𝒯-function (i.e., it satisfies
(5)limsups→t+k(s)<1
for all t∈[0,∞)). Then T is an MWP operator.In the same paper, M. Berinde and V. Berinde [5] introduced the concepts of multivalued (δ,L)-weak contraction and multivalued (k,L)-weak contraction and proved the following nice fixed point theorems.Theorem 3.
Let(X,d) be a complete metric space, and let T:X→CB(X) be a multivalued (δ,L)-weak contraction; that is, there exist two constants δ∈[0,1) and L≥0 such that
(6)H(Tx,Ty)≤δd(x,y)+LD(y,Tx)
for all x,y∈X. Then T is an MWP operator.Theorem 4.
Let(X,d) be a complete metric space, and let T:X→CB(X) be a multivalued (k,L)-weak contraction; that is, there exist an ℳ𝒯-function k and a constant L≥0 such that
(7)H(Tx,Ty)≤k(d(x,y))d(x,y)+LD(y,Tx)
for all x,y∈X. Then T is an MWP operator.We can find some detailed information about the single-valued case of (δ,L)-weak contraction and the nonlinear case of it in [13–15].Recently, Samet et al. [16] introduced the notion of α-ψ-contractive mappings and gave some fixed point results for such mappings. Their results are closely related to some ordered fixed point results. Then, using their idea, some authors presented fixed point results for single and multivalued mappings (see, e.g., [16–20]). First, we recall these results. Denote by Ψ the family of nondecreasing functions ψ:[0,∞)→[0,∞) such that ∑n=1∞‍ψn(t)<∞ for all t>0.Definition 5 (see [16]).
Let(X,d) be a metric space, T be a self map on X, ψ∈Ψ and α:X×X→[0,∞) be a function. Then T is called α-ψ-contractive whenever
(8)α(x,y)d(Tx,Ty)≤ψ(d(x,y))
for all x,y∈X.Note that every Banach contraction mapping is anα-ψ-contractive mapping with α(x,y)=1 and ψ(t)=δt for some δ∈[0,1).Definition 6 (see [16]).
T is called α-admissible whenever α(x,y)≥1 implies that α(Tx,Ty)≥1.There exist some examples forα-admissible mappings in [16]. For convenience, we mention in here one of them. Let X=[0,∞). Define T:X→X and α:X×X→[0,∞) by Tx=x for all x∈X and α(x,y)=ex-y for x≥y and α(x,y)=0 for x<y. Then T is α-admissible.Definition 7 (see [17]).
α is said to have (B) property whenever {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n∈ℕ and xn→x, then α(xn,x)≥1 for all n∈ℕ.Theorem 8 (see [16]).
Let(X,d) be a complete metric space, and let T:X→X be an α-admissible and α-ψ-contractive mapping. If there exists x0∈X such that α(x0,Tx0)≥1 and T is continuous, then T has a fixed point.Remark 9.
If we assume thatα has (B) property instead of the continuity of T, then again T has a fixed point (Theorem 2.2 of [16]). If for each x,y∈X there exists z∈X such that α(x,z)≥1 and α(y,z)≥1, then X is said to have (H) property. Therefore, if X has the (H) property in Theorems 2.1 and 2.2 in [16], then the fixed point of T is unique (Theorem 2.3 of [16]).Then, some generalizations ofα-ψ-contractive mappings are given as follows.Definition 10 (see [17]).
T is called Ćirić type α-ψ-generalized contractive mapping whenever
(9)α(x,y)d(Tx,Ty)≤ψ(m(x,y))
for all x,y∈X, where
(10)m(x,y)=max{12d(x,y),d(x,Tx),d(y,Ty),kkkkkkk12[d(x,Ty)+d(y,Tx)]}.Note that every Ćirić type generalized contraction mapping is a Ćirić typeα-ψ-generalized contractive mapping with α(x,y)=1 and ψ(t)=λt for some λ∈[0,1).Theorem 11 (see [17]).
Let(X,d) be a complete metric space, and let T:X→X be an α-admissible and Ćirić type α-ψ-generalized contractive mapping. If there exists x0∈X such that α(x0,Tx0)≥1 and T is continuous or α has (B) property, then T has a fixed point. If X has the (H) property, then the fixed point of T is unique.We can find some fixed point results for single-valued mappings in these directions in [18, 20]. Now we recall some multivalued cases.Definition 12 (see [17, 19]).
Let(X,d) be a metric space, and let T:X→CB(X) be a multivalued mapping. Then T is called multivalued α-ψ-contractive whenever
(11)α(x,y)H(Tx,Ty)≤ψ(d(x,y))
for all x,y∈X, and T is called multivalued α*-ψ-contractive whenever
(12)α*(Tx,Ty)H(Tx,Ty)≤ψ(d(x,y)),
where α*(Tx,Ty)=inf{α(a,b):a∈Tx,b∈Ty}. Similarly if we replace d(x,y) with m(x,y) we can obtain Ćirić type multivalued α-ψ-generalized contractive and Ćirić type multivalued α*-ψ-generalized contractive mappings on X.Definition 13 (see [17, 19]).
Let(X,d) be a metric space, and let T:X→CB(X) be a multivalued mapping.(a)
T is said to be α-admissible whenever each x∈X and y∈Tx with α(x,y)≥1 imply that α(y,z)≥1 for all z∈Ty.(b)
T is said to be α*-admissible whenever each x∈X and y∈Tx with α(x,y)≥1 imply that α*(Tx,Ty)≥1.Remark 14.
It is clear thatα*-admissible maps are also α-admissible, but the converse may not be true as shown in the following example.Example 15.
LetX=[-1,1], and α:X×X→[0,∞) is defined by α(x,x)=0 and α(x,y)=1 for x≠y. Define T:X→CB(X) by
(13)Tx={{-x},x∉{-1,0}{0,1},x=-1{1},x=0.
Leting x=-1 and y=0∈Tx={0,1}, then α(x,y)≥1, but α*(Tx,Ty)=α*({0,1},{1})=0. Thus, T is not α*-admissible. Now we show that, T is α-admissible with the following cases.
Case
1. If x=0, then y=1 and α(x,y)≥1. Also, α(y,z)≥1 since z=-1∈Ty={-1}.
Case
2. If x=-1, then y∈{0,1} and α(x,y)≥1. Also, α(y,z)≥1 for all z∈Ty.
Case
3. If x∉{-1,0}, then y=-x and α(x,y)≥1. Also, α(y,z)≥1 since z=x∈Ty={x}.The multivalued version of the results forα-ψ-contractive mappings is given [17, 19] as follows.Theorem 16 (see [19]).
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be a function, let ψ∈Ψ be a strictly increasing map, and let T:X→CB(X), α-admissible and α-ψ-contractive multifunction on X. Suppose that there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1. Assume that T is continuous or α has (B) property; then T has a fixed point.Theorem 17 (see [17]).
Let(X,d) be a complete metric space, let α:X×X→[0,∞) be a function, let ψ∈Ψ be a strictly increasing map, and let T:X→CB(X), α*-admissible and α*-ψ-contractive multifunction on X. Suppose that there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1. Assume that α has (B) property. then T has a fixed point.The purpose of this paper is to introduce the concept of multivalued pseudo-Picard (MPP) operators and present fixed point results and examples.


## 2. Main Results

Definition 18.
Let(X,d) be a metric space, and let T:X→𝒫(X) be a multivalued operator. T is said to be multivalued pseudo-Picard (MPP) operator if and only if there exist x0∈X, x1∈Tx0 and a sequence {xn} in X such that(i)
xn+1∈Txn,(ii)
the sequence{xn} is convergent and its limit is a fixed point of T.Remark 19.
It is clear that the operators mentioned in Theorems16 and 17 are MPP operators. Also, note that every MWP operator is an MPP operator, but the converse may not be true as shown in the following examples.Example 20.
LetX=[-1,1] and d(x,y)=|x-y|. Define T:X→CB(X) by
(14)Tx={{0,x2},x∈(-1,1){-x},x∈{-1,1}.
Then T is not an MWP operator. Indeed, letting x=1, then Tx={-1} and so y=-1. Therefore, xn=(-1)n for n≥0, which is not convergent. But T is MPP operator. To see this, let x0=1/2 and x1=(1/4)∈Tx0={0,1/4}. Continuing this way, we can construct a sequence {xn} in X with xn+1∈Txn such that it is convergent to 0, which is a fixed point of T.Example 21.
LetX=[0,∞) and d(x,y)=|x-y|. Define T:X→CB(X) by
(15)Tx={[0,x2],0≤x≤1[x+1,x+2],x>1.
Then T is an MPP but not MWP operator.Before we give our main results, we recall the following. LetX and Y be two topological spaces. Then a multivalued map T:X→𝒫(Y) is said to be upper semicontinuous (lower semicontinuous) if the inverse image of a closed set (open set) is closed (open). A multivalued map is continuous if it is upper as well as lower semicontinuous.Lemma 22 (see [21]).
Let(X,d) be a metric space, and let T:X→𝒫(X) be an upper semicontinuous map such that Tx is closed for all x∈X. If xn→x0,yn→y0 and yn∈Txn, then y0∈Tx0.Theorem 23.
Let(X,d) be a complete metric space and let T:X→CB(X) be an α-admissible multivalued mapping such that
(16)α(x,y)H(Tx,Ty)≤ψ(d(x,y))+Lmin{D(x,Ty),D(y,Tx)}
for all x,y∈X, where ψ∈Ψ is strictly increasing and L≥0. Suppose that there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1. If T is upper semicontinuous or α has (B) property, then T is an MPP operator.Proof.
Letx0 and x1 be as mentioned in the hypotheses. If x0=x1 or x1∈Tx1, then the proof is complete. Let x0≠x1 and x1∉Tx1, then
(17)0<D(x1,Tx1)≤α(x0,x1)H(Tx0,Tx1)<qα(x0,x1)H(Tx0,Tx1),
where q>1 is a constant. Therefore, there exists x2∈Tx1 such that
(18)0<d(x1,x2)<qα(x0,x1)H(Tx0,Tx1)≤qψ(d(x0,x1))+qLmin{D(x0,Tx1),D(x1,Tx0)}=qψ(d(x0,x1)).
Also, since T is α-admissible, x1∈Tx0, and α(x0,x1)≥1, then α(x1,u)≥1 for all u∈Tx1 and so α(x1,x2)≥1. Since ψ is strictly increasing, we have
(19)0<ψ(d(x1,x2))<ψ(qψ(d(x0,x1))).
Get q1=ψ(qψ(d(x0,x1)))/ψ(d(x1,x2)). Then q1>1. If x2∈Tx2, then the proof is complete. Let x2∉Tx2. Then
(20)0<D(x2,Tx2)≤α(x1,x2)H(Tx1,Tx2)<q1α(x1,x2)H(Tx1,Tx2).
Therefore, there exists x3∈Tx2 such that
(21)0<d(x2,x3)<q1α(x1,x2)H(Tx1,Tx2)≤q1ψ(d(x1,x2))+q1Lmin{D(x1,Tx2),D(x2,Tx1)}=q1ψ(d(x1,x2))=ψ(qψ(d(x0,x1))).
Since T is α-admissible, x2∈Tx1, and α(x1,x2)≥1, then α(x2,u)≥1 for all u∈Tx2. Thus, α(x2,x3)≥1 since x3∈Tx2. Since ψ is strictly increasing, we have
(22)0<ψ(d(x2,x3))<ψ2(qψ(d(x0,x1))).
Get q2=ψ2(qψ(d(x0,x1)))/ψ(d(x1,x2)). Then q2>1. If x3∈Tx3, then the proof is complete. By the way, we can construct a sequence {xn} in X such that xn+1∈Txn,xn≠xn+1,α(xn,xn+1)≥1, and
(23)d(xn,xn+1)≤ψn(qψ(d(x0,x1)))
for all n∈ℕ. Now, for each m,n∈ℕ,m>n, we have
(24)d(xn,xm)≤∑k=nm-1‍d(xk,xk+1)≤∑k=nm-1‍ψk(qψ(d(x0,x1))).
Therefore, {xn} is a Cauchy sequence in X. Since X is complete, there exists z∈X such that limn→∞xn=z. If T is upper semicontinuous, then from Lemma 22, we have z∈Tz. Now assume that α has (B) property. Then α(xn,z)≥1 for all n∈ℕ. Also, since limn→∞d(xn,z)=0 and ψ is continuous at 0, then
(25)D(xn+1,Tz)≤H(Txn,Tz)≤α(xn,z)H(Txn,Tz)≤ψ(d(xn,z))+Lmin{D(xn,Tz),D(z,Txn)}≤ψ(d(xn,z))+Lmin{D(xn,Tz),d(z,xn+1)}⟶0.
Therefore, we have D(z,Tz)=0 and so z∈Tz.Althoughα*-admissibility implies α-admissibility of T, we will give the following theorem because the contractive condition is slightly different from (16).Theorem 24.
Let(X,d) be a complete metric space, and let T:X→CB(X) be an α*-admissible multivalued mapping such that
(26)α*(Tx,Ty)H(Tx,Ty)≤ψ(d(x,y))+Lmin{D(x,Ty),D(y,Tx)}
for all x,y∈X, where ψ∈Ψ is strictly increasing and L≥0. Suppose that there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1. If T is upper semicontinuous or α has (B) property, then T is an MPP operator.Proof.
Letx0 and x1 be as mentioned in the hypotheses, then α*(Tx0,Tx1)≥1 since T is α*-admissible. If x0=x1 or x1∈Tx1, then the proof is complete. Let x0≠x1 and x1∉Tx1, then
(27)0<D(x1,Tx1)≤α*(Tx0,Tx1)H(Tx0,Tx1)<qα*(Tx0,Tx1)H(Tx0,Tx1),
where q>1 is a constant. Therefore, there exists x2∈Tx1 such that
(28)0<d(x1,x2)<qα*(Tx0,Tx1)H(Tx0,Tx1)≤qψ(d(x0,x1))+qLmin{D(x0,Tx1),D(x1,Tx0)}=qψ(d(x0,x1)).
Since α(x1,x2)≥α*(Tx0,Tx1)≥1, then α*(Tx1,Tx2)≥1. Since ψ is strictly increasing,
(29)0<ψ(d(x1,x2))<ψ(qψ(d(x0,x1))).
Get q1=ψ(qψ(d(x0,x1)))/ψ(d(x1,x2)). Then q1>1. If x2∈Tx2, then the proof is complete. Let x2∉Tx2. Then
(30)0<D(x2,Tx2)≤α*(Tx1,Tx2)H(Tx1,Tx2)<q1α*(Tx1,Tx2)H(Tx1,Tx2).
Therefore, there exists x3∈Tx2 such that
(31)0<d(x2,x3)<q1α*(Tx1,Tx2)H(Tx1,Tx2)≤q1ψ(d(x1,x2))+q1Lmin{D(x1,Tx2),D(x2,Tx1)}=q1ψ(d(x1,x2))=ψ(qψ(d(x0,x1))).
Since α(x2,x3)≥α*(Tx1,Tx2)≥1, then α*(Tx2,Tx3)≥1. By the way, we can construct a sequence {xn} in X such that xn+1∈Txn,xn≠xn+1,α(xn,xn+1)≥1, and
(32)d(xn,xn+1)≤ψn(qψ(d(x0,x1)))
for all n∈ℕ. Now, for each m,n∈ℕ,m>n, we have
(33)d(xn,xm)≤∑k=nm-1d(xk,xk+1)≤∑k=nm-1‍ψk(qψ(d(x0,x1))).
Therefore, {xn} is a Cauchy sequence in X. Since X is complete, there exists z∈X such that limn→∞xn=z. If T is upper semicontinuous, then from Lemma 22, we have z∈Tz. Now assume that α has (B) property. Then α(xn,z)≥1 for all n∈ℕ. Since T is α*-admissible, α*(Txn,Tz)≥1. Therefore,
(34)D(xn+1,Tz)≤H(Txn,Tz)≤α*(Txn,Tz)H(Txn,Tz)≤ψ(d(xn,z))+Lmin{D(xn,Tz),D(z,Txn)}≤ψ(d(xn,z))+Lmin{D(xn,Tz),d(z,xn+1)},
and taking limit n→∞ we have D(z,Tz)=0 and so z∈Tz.Remark 25.
If we takeα:X×X→[0,∞) by α(x,y)=1, then any multivalued mappings T:X→CB(X) are α-admissible as well as α*-admissible. Therefore, Theorem 3 is a special case of Theorems 23 and 24.Remark 26.
If we takeL=0 in Theorems 23 and 24, then we have Theorems 16 and 17, respectively.Now we give an example to illustrate our main results. Note that both Theorems3 and 16 cannot be applied to this example.Example 27.
LetX=[0,1]∪{2,3} and d(x,y)=|x-y|. Define T:X→CB(X) by
(35)Tx={[0,x2],x∈[0,12][14,3x-12],x∈(12,1]{x},x∈{2,3}
and α:X×X→[0,∞) by
(36)α(x,y)={1,x,y∈[0,12]∪{2,3}0,otherwise.
Then T is an α-admissible, and condition (16) is satisfied for ψ(t)=t/2 and L=1. Indeed, first, we show that T is an α-admissible. Letting x∈X and y∈Tx with α(x,y)≥1, then it should be x∈[0,1/2]∪{2,3}. Thus, y∈Tx⊂[0,1/2]∪{2,3}, and hence α(y,z)≥1 for all z∈Ty⊂[0,1/2]∪{2,3}. Therefore, T is an α-admissible.
Now we consider the following cases.
Case
1. Letting x,y∈X with {x,y}∩(1/2,1]≠∅, then α(x,y)=0. Thus, (16) is satisfied. Also note that if x=y, then H(Tx,Ty)=0 and so (16) is satisfied. Therefore, we will consider x≠y in the following.
Case
2. Letting x,y∈X with x,y∈[0,1/2]∪{2,3}, then α(x,y)=1. There are some subcases as follows.
Subcase
1. Consider that x,y∈[0,1/2], and suppose that (without loss of generality) x>y; then
(37)H(Tx,Ty)=H([0,x2],[0,y2])=12(x-y)≤12d(x,y)+min{D(x,Ty),D(y,Tx)}
and so again (16) is satisfied.
Subcase
2. Consider that x∈[0,1/2] and y∈{2,3}; then
(38)H(Tx,Ty)=H([0,x2],{y})=y-x2≤32(y-x)=12(y-x)+min{y-x,y-x2}=12d(x,y)+min{D(x,Ty),D(y,Tx)}.
Subcase
3. If x=2 and y=3, then
(39)H(Tx,Ty)=1=d(x,y)≤12d(x,y)+min{D(x,Ty),D(y,Tx)}.
Thus, (16) is satisfied.
Finally,α has (B) property, then by Theorem 23, T is an MPP operator.
Note that sinceH(T1,T(3/4))=3/8, d(1,3/4)=1/4, and D(3/4,T1)=0, then condition (6) is not satisfied. Therefore, Theorem 3 cannot be applied to this example.
Also, note that, sinceα(2,3)=1, H(T2,T3)=1, and d(2,3)=1, T is not multivalued α-ψ-contractive mapping. Therefore, Theorem 16 cannot be applied to this example.


## 3. Applications

Our results can be applied to some ordered fixed point results. First we recall some ordered notions. LetX be a nonempty set and ⪯ be a partial order on X.Definition 28 (see [22]).
LetA,B be two nonempty subsets of X; the relations between A and B are defined as follows.(r1)
If for everya∈A, there exists b∈B such that a⪯b, then A≺1B.(r2)
If for everyb∈B, there exists a∈A such that a⪯b, then A≺2B.(r3)
IfA≺1B and A≺2B, then A≺B.Remark 29 (see [22]).
≺
1 and ≺2 are different relations between A and B. For example, let X=ℝ, A=[1/2,1], B=[0,1], and ⪯ be a usual order on X; then A≺1B, but A⊀2B; if A=[0,1], B=[0,1/2], then A≺2B, while A⊀1B.Remark 30 (see [22]).
≺
1, ≺2, and ≺ are reflexive and transitive but are not antisymmetric. For instance, let X=ℝ, A=[0,3], B=[0,1]∪2,3], and ⪯ be a usual order on X; then A≺B and B≺A, but A≠B. Hence, they are not partial orders.Theorem 31.
Let(X,⪯) be a partially ordered set, and suppose that there exists a metric d in X such that (X,d) is complete metric space. Let T:X→CB(X) be a multivalued mapping such that
(40)H(Tx,Ty)≤ψ(d(x,y))+Lmin{D(x,Ty),D(y,Tx)}
for all x,y∈X with x⪯y, where ψ∈Ψ is strictly increasing and L≥0. Suppose that there exists x0∈X such that {x0}≺1Tx0. Assume that for each x∈X and y∈Tx with x⪯y, are has y⪯z for all z∈Ty. If T is upper semicontinuous or X satisfies the following condition that
(41){xn}⊂Xisanondecreasingsequencewithxn⟶xinX,thenxn⪯x∀n,
then T has a fixed point.Proof.
Define the mappingα:X×X→[0,∞) by
(42)α(x,y)={1,x⪯y0,otherwise.
Then we have
(43)α(x,y)H(Tx,Ty)≤ψ(d(x,y))+Lmin{D(x,Ty),D(y,Tx)}
for all x,y∈X. Also, since {x0}≺1Tx0, then there exists x1∈Tx0 such that x0⪯x1 and so α(x0,x1)≥1. Now letting x∈X and y∈Tx with α(x,y)≥1, then x⪯y, and so by the hypotheses we have y⪯z for all z∈Ty. Therefore, α(y,z)≥1 for all z∈Ty. This shows that T is α-admissible. Finally, if T is upper semicontinuous or X satisfies (41), then T is upper semicontinuous or α has (B) property. Therefore, from Theorem 23, T has a fixed point.Remark 32.
We can give a similar result using≺2 instead of ≺1.



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