We introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings in complete metric spaces.
1. Introduction
Let T be a mapping on a complete (or compact) metric space (X,d). We do not assume richer structure such as convex metric spaces and Banach spaces. There are thousands of theorems which assure the existence of a fixed point of T. We can categorize these theorems into the following four types.
Leader type [1]: T has a unique fixed point and {Tnx} converges to the fixed point for all x∈X. Such a mapping is called a Picard operator in [2].
Unnamed type: T has a unique fixed point and {Tnx} does not necessarily converge to the fixed point.
Subrahmanyam type [3]: T may have more than one fixed point and {Tnx} converges to a fixed point for all x∈X. Such a mapping is called a weakly Picard operator in [3, 4].
Caristi type [5, 6]: T may have more than one fixed point and {Tnx} does not necessarily converge to a fixed point.
We know that most of the theorems such as Banach’s [7], Ćirić’s [8], Kannan’s [9], Kirk’s [10], Matkowski’s [11], Meir and Keeler’s [12], and Suzuki’s [13, 14] belong to (T1). Also, very recently, Suzuki [15] characterized (T1). Subrahmanyam’s theorem [3] belongs to (T3), and Caristi’s theorem [5, 6] and its generalizations [15–17] belong to (T4). On the other hand, as far as the authors do know, there are no theorems belonging to (T2); see Kirk’s survey [18]. Also, recently many interesting fixed point theorems are proved in the framework of ordered metric spaces; see [18–35] and others.
In this paper, motivated by the above, we introduce two new types of fixed point theorems in the collection of multivalued and single-valued mappings and will prove them, which belong to (T3).
Let (X,d) be a metric space, and let Pcl,bd(X) denote the class of all nonempty, closed, and bounded subsets of X. Let T:X→Pcl,bd(X) be a multivalued mapping on X. A point x∈X is called a fixed point of T if x∈Tx. Set Fix(T)={x∈X:x∈Tx}.
A famous theorem on multivalued mappings is due to Nadler [36], which extended the Banach contraction principle to multivalued mappings. Many authors have studied the existence and uniqueness of strict fixed points for multivalued mappings in metric spaces; see, for example, [37–44] and references therein.
Let H be the Hausdorff metric on Pcl,bd(X) induced by d; that is,
(1)H(A,B)∶=max{supx∈Bd(x,A),supx∈Ad(x,B)},sssssssssssssssssssssssssssssssA,B∈Pcl,bd(X).
Denote δ(x,A)=sup{d(x,y):y∈A} and D(x,A)=inf{d(x,y):y∈A}, where A∈Pcl,bd(X).
2. Main Results
The following is the first our main results.
Theorem 1.
Let (X,d) be a complete metric space and let T be a mapping from X into itself. Suppose that T satisfies the following condition:
(2)d(Tx,Ty)≤(d(x,Ty)+d(y,Tx)d(x,Tx)+d(y,Ty)+1)d(x,y),
for all x,y∈X. Then
T has at least one fixed point x˙∈X;
{Tnx} converges to a fixed point, for all x∈X;
if x˙,y˙ are two distinct fixed points of T, then d(x˙,y˙)≥1/2.
Proof.
Let x0∈X be arbitrary and choose a sequence {xn} such that xn+1=Txn. We have
(3)d(xn+1,xn)=d(Txn,Txn-1)≤(d(xn,xn)+d(xn-1,xn+1)d(xn,xn+1)+d(xn-1,xn)+1)d(xn,xn-1)=(d(xn-1,xn+1)d(xn,xn+1)+d(xn-1,xn)+1)d(xn,xn-1)≤(d(xn-1,xn)+d(xn,xn+1)d(xn,xn+1)+d(xn-1,xn)+1)d(xn,xn-1).
Given
(4)βn=d(xn-1,xn)+d(xn,xn+1)d(xn,xn+1)+d(xn-1,xn)+1,
we have
(5)d(xn+1,xn)≤βnd(xn,xn-1)≤(βnβn-1)d(xn-1,xn-2)⋮≤(βnβn-1⋯β1)d(x1,x0).
Observe that (βn) is nonincreasing, with positive terms. So β1⋯βn≤β1n and β1n→0. It follows that
(6)limn→∞(β1β2⋯βn)=0.
Thus, it is verified that
(7)limn→∞d(xn+1,xn)=0.
Now for all m,n∈N we have
(8)d(xm,xn)≤d(xn,xn+1)+d(xn+1,xn+2)+⋯+d(xm-1,xm)≤[(βnβn-1⋯β1)+(βn+1βn⋯β1)+⋯+(βm-1βm-2⋯β1)]d(x1,x0)=∑k=nm-1(βkβk-1⋯β1)d(x1,x0).
Suppose that ak=(βkβk-1⋯β1). Since
(9)limk→∞ak+1ak=0∑k=1∞ak<∞. It means that
(10)∑k=nm-1(βkβk-1⋯β1)⟶0,
as m,n→∞. In other words, {xn} is a Cauchy sequence and so converges to x˙∈X.
We claim that x˙ is a fixed point.
Note that
(11)d(xn+1,Tx˙)≤(d(xn,Tx˙)+d(x˙,Txn)d(x˙,Tx˙)+d(xn,Txn)+1)d(xn,x˙).
On taking limit on both sides of (11), we have d(x˙,Tx˙)=0. Thus, Tx˙=x˙.
If there exist two distinct fixed points x˙,y˙∈X, then
(12)d(x˙,y˙)=d(Tx˙,Ty˙)≤[d(x˙,Ty˙)+d(Tx˙,y˙)]d(x˙,y˙)=2[d(x˙,y˙)]2.
Therefore, d(x˙,y˙)≥1/2 and we find the desired results.
In the following, two examples of such type of mappings, which satisfy (2), are given.
Example 2.
Let X={0,1/2,1} and let d:X×X→[0,∞) be defined by
(13)d(0,12)=2,d(1,12)=52,d(0,1)=3,d(0,0)=d(12,12)=d(1,1)=0,d(a,b)=d(b,a),∀a,b∈X.(X,d) is a complete metric space. Let T:X→X be defined by
(14)T(0)=0,T(12)=12,T(1)=0d(T0,T1)=d(0,0)=0,d(T0,T(12))=d(0,12)=2,d(T1,T(12))=d(0,12)=2,
and we have
(15)d(T0,T(12))=d(0,12)=2≤(d(0,T(1/2))+d(1/2,T(0))d(0,T0)+d(1/2,T(1/2))+1)×d(0,12)=8
and also
(16)d(T1,T(12))=d(0,12)=2≤(d(1,T(1/2))+d(1/2,T(1))d(1,T1)+d(1/2,T(1/2))+1)×d(1,12)=(5/2+24)×52=4516.
Therefore, T satisfies all the conditions of Theorem 1. Also, T has two distinct fixed points {0,1/2} and d(0,1/2)=2≥1/2.
Example 3.
Let X=[0,2-3] be endowed with Euclidean metric and let T:X→X be defined by
(17)Tx={00≤x<2-32-3x=2-3.
Then we claim that T satisfies all the conditions of Theorem 1.
If x=2-3 and 0≤y<2-3, we have
(18)|Tx-Ty|(|x-Tx|+|y-Ty|+1)=(2-3)(|y|+1)=(2-3)(y+1)≤(2-3-y)2-(2-3)(2-3-y)=(|x-Ty|+|y-Tx|)|x-y|.
Thus,
(19)|Tx-Ty|≤(|x-Ty|+|y-Tx||x-Tx|+|y-Ty|+1)|x-y|.
Similar argument holds for the other conditions.
Remark 4.
Note that in (2) the ratio
(20)d(x,Ty)+d(y,Tx)d(x,Tx)+d(y,Ty)+1
might be greater or less than 1 and has not introduced an upper bound. Note that if, for every x,y∈X, d(x,y)<1/2, then we have
(21)d(x,Ty)+d(y,Tx)≤2d(x,y)+d(x,Tx)+d(y,Ty)<d(x,Tx)+d(y,Ty)+1.
It means that
(22)d(x,Ty)+d(y,Tx)d(x,Tx)+d(y,Ty)+1<1,
and thus Theorem 1 is a special case of Banach contraction principle. Therefore, when (X,d) is a complete metric space such that, for all x,y∈X, d(x,y)≥1/2, Theorem 1 is valuable because (20) might be greater than 1. Example 2 shows this note precisely.
The following is the second in our main results.
Theorem 5.
Let (X,d) be a complete metric space and let T be a multivalued mapping from X into Pcl,bd(X). Let T satisfy the following:
(23)H(Tx,Ty)≤(D(x,Ty)+D(y,Tx)δ(x,Tx)+δ(y,Ty)+1)d(x,y),
for all x,y∈X. Then T has a fixed point x˙∈X.
Proof.
Let x0∈X and x1∈Tx0. For each 0<h1<1 one can choose x2∈Tx1 such that
(24)d(x1,x2)<H(Tx0,Tx1)+(1-1h1)H(Tx0,Tx1)=1h1H(Tx0,Tx1).
For each 0<hn<1 we can choose xn+1∈Txn such that
(25)d(xn,xn+1)<H(Txn-1,Txn)+(1-1hn)H(Tx0,Tx1)=1hnH(Tx0,Tx1).
Specifically if
(26)hn=d(xn-1+xn+1)d(xn-1+xn)+d(xn+xn+1)+1=βn,
then
(27)d(xn,xn+1)≤βnd(xn-1,xn)≤βnd(xn-1,xn).
Therefore,
(28)d(xn+1,xn)≤βnd(xn,xn-1)≤(βnβn-1)d(xn-1,xn-2)⋮≤(βnβn-1⋯β1)d(x1,x0).
It can easily be seen that
(29)limn→∞(β1β2⋯βn)=0.
Thus, it is easily verified that
(30)limn→∞d(xn+1,xn)=0.
Now for all m,n∈N we have
(31)d(xm,xn)≤d(xn,xn+1)+d(xn,xn+1)+⋯+d(xm-1,xm)≤[(βnβn-1⋯β1)+(βn+1βn⋯β1)+⋯+(βm-1βm-2⋯β1)]d(x1,x0)=∑k=nm-1(βkβk-1⋯β1)d(x1,x0).
Suppose that ak=(βkβk-1⋯β1). Since
(32)limk→∞ak+1ak=0,∑k=1∞ak<∞. It means that
(33)∑k=nm-1(βkβk-1⋯β1)⟶0,
as m,n→∞. In other words, {xn} is a Cauchy sequence and so converges to x˙∈X. We claim that x˙ is a fixed point. Consider
(34)D(x˙,Tx˙)≤d(x˙,xn+1)+D(xn+1,Tx˙)≤H(Txn,Tx˙)+d(x˙,xn+1)≤(D(x˙,Txn)+D(xn,Tx˙)δ(x˙,Tx˙)+δ(xn,Txn)+1)×d(xn,x˙)+d(x˙,xn+1)≤[D(x˙,xn+1)+D(xn,Tx˙)]×d(xn,x˙)+d(x˙,xn+1).
On taking limit on both sides of (31) we have D(x˙,Tx˙)=0. It means that x˙∈Tx˙.
Remark 6.
Note that Theorem 5 is a generalization of Theorem 1 because by taking Fx={Tx} and applying Theorem 5 for F we obtain Theorem 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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