We mainly study fixed point theorem for multivalued mappings with δ-distance using Wardowski’s technique on complete metric space. Let (X,d) be a metric space and let B(X) be a family of all nonempty bounded subsets of X. Define δ:B(X)×B(X)→R by δ(A,B)=supd(a,b):a∈A,b∈B. Considering δ-distance, it is proved that if (X,d) is a complete metric space and T:X→B(X) is a multivalued certain contraction, then T has a fixed point.

1. Introduction

Fixed point theory concern itself with a very basic mathematical setting. It is also well known that one of the fundamental and most useful results in fixed point theory is Banach fixed point theorem. This result has been extended in many directions for single and multivalued cases on a metric space X (see [1–9]). Fixed point theory for multivalued mappings is studied by both Pompeiu-Hausdorff metric H [10, 11], which is defined on CB(X) (the family of all nonempty, closed, and bounded subsets of X), and δ-distance, which is defined on B(X) (the family of all nonempty and bounded subsets of X). Using Pompeiu-Hausdorff metric, Nadler [12] introduced the concept of multivalued contraction mapping and show that such mapping has a fixed point on complete metric space. Then many authors focused on this direction [13–18]. On the other hand, Fisher [19] obtained different type of multivalued fixed point theorems defining δ-distance between two bounded subsets of a metric space X. We can find some results about this way in [20–23].

In this paper, we give some new multivalued fixed point results by considering the δ-distance. For this we use the recent technique, which was given by Wardowski [24]. For the sake of completeness,we will discuss its basic lines. Let F be the set of all functions F:(0,∞)→R satisfying the following conditions:

F is strictly increasing; that is, for all α,β∈(0,∞) such that α<β, F(α)<F(β).

For each sequence {an} of positive numbers limn→∞an=0 if and only if limn→∞F(an)=-∞.

There exists k∈(0,1) such that limα→0+αkF(α)=0.

Definition 1 (see [<xref ref-type="bibr" rid="B29">24</xref>]).

Let (X,d) be a metric space and let T:X→X be a mapping. Given F∈F, we say that T is F-contraction, if there exists τ>0 such that
(1)x,y∈X,d(Tx,Ty)>0⟹τ+F(d(Tx,Ty))≤F(d(x,y)).

Taking different functions F∈F in (1), one gets a variety of F-contractions, some of them being already known in the literature. The following examples will certify this assertion.

Example 2 (see [<xref ref-type="bibr" rid="B29">24</xref>]).

Let F1:(0,∞)→R be given by the formulae F1(α)=lnα. It is clear that F1∈F. Then each self-mapping T on a metric space (X,d) satisfying (1) is an F1-contraction such that
(2)d(Tx,Ty)≤e-τd(x,y),∀x,y∈X,Tx≠Ty.

It is clear that for x, y∈X such that Tx=Ty the inequality d(Tx,Ty)≤e-τd(x,y) also holds. Therefore T satisfies Banach contraction with L=e-τ; thus T is a contraction.

Example 3 (see [<xref ref-type="bibr" rid="B29">24</xref>]).

Let F2:(0,∞)→R be given by the formulae F2(α)=α+lnα. It is clear that F2∈F. Then each self-mapping T on a metric space (X,d) satisfying (1) is an F2-contraction such that
(3)d(Tx,Ty)d(x,y)ed(Tx,Ty)-d(x,y)≤e-τ,∀x,y∈X,Tx≠Ty.

We can find some different examples for the function F belonging to F in [24]. In addition, Wardowski concluded that every F-contraction T is a contractive mapping, that is,
(4)d(Tx,Ty)<d(x,y),∀x,y∈X,Tx≠Ty.
Thus, every F-contraction is a continuous mapping.

Also, Wardowski concluded that if F1, F2∈F with F1(α)≤F2(α) for all α>0 and G=F2-F1 is nondecreasing, then every F1-contraction T is an F2-contraction.

He noted that, for the mappings F1(α)=lnα and F2(α)=α+lnα, F1<F2 and a mapping F2-F1 is strictly increasing. Hence, it was obtained that every Banach contraction satisfies the contractive condition (3). On the other side, [24, Example 2.5] shows that the mapping T is not an F1-contraction (Banach contraction) but still is an F2-contraction. Thus, the following theorem, which was given by Wardowski, is a proper generalization of Banach Contraction Principle.

Theorem 4 (see [<xref ref-type="bibr" rid="B29">24</xref>]).

Let (X,d) be a complete metric space and let T:X→X be an F-contraction. Then T has a unique fixed point in X.

Following Wardowski, Mınak et al. [25] introduced the concept of Ćirić type generalized F-contraction. Let (X,d) be a metric space and let T:X→X be a mapping. Given F∈F, we say that T is a Ćirić type generalized F-contraction if there exists τ>0 such that
(5)x,y∈X,d(Tx,Ty)>0⟹τ+F(d(Tx,Ty))≤F(m(x,y)),
where
(6)m(x,y)=max{12d(x,y),d(x,Tx),d(y,Ty),12[d(x,Ty)+d(y,Tx)]}.
Then the following theorem was given.

Theorem 5.

Let (X,d) be a complete metric space and let T:X→X be a Ćirić type generalized F-contraction. If T or F is continuous, then T has a unique fixed point in X.

Considering the Pompeiu-Hausdorff metric H, both Theorems 4 and 5 were extended to multivalued cases in [26] and [27], respectively (see also [28, 29]). In this work, we give a fixed point result for multivalued mappings using the δ-distance. First recall some definitions and notations which are used in this paper.

Let (X,d) be a metric space. For A, B∈B(X) we define
(7)δ(A,B)=sup{d(a,b):a∈A,b∈B},D(a,B)=inf{d(a,b):b∈B}.
If A={a} we write δ(A,B)=δ(a,B) and also if B={b}, then δ(a,B)=d(a,b). It is easy to prove that for A,B,C∈B(X)(8)δ(A,B)=δ(B,A)≥0,δ(A,B)≤δ(A,C)+δ(C,B),δ(A,A)=sup{d(a,b):a,b∈A}=diamA,δ(A,B)=0,impliesthatA=B={a}.
If {An} is a sequence in B(X), we say that {An} converges to A⊆X and write An→A if and only if

a∈A implies that an→a for some sequence {an} with an∈An for n∈N,

for any ɛ>0, ∃m∈N such that An⊆Aɛ for n>m, where
(9)Aɛ={x∈X:d(x,a)<ɛforsomea∈A}.

Lemma 6 (see [<xref ref-type="bibr" rid="B4">20</xref>]).

Suppose {An} and {Bn} are sequences in B(X) and (X,d) is a complete metric space. If An→A∈B(X) and Bn→B∈B(X) then δ(An,Bn)→δ(A,B).

Lemma 7 (see [<xref ref-type="bibr" rid="B4">20</xref>]).

If {An} is a sequence of nonempty bounded subsets in the complete metric space (X,d) and if δ(An,y)→0 for some y∈X, then An→{y}.

2. Main Result

In this section, we prove a fixed point theorem for multivalued mappings with δ-distance and give an illustrative example.

Definition 8.

Let (X,d) be a metric space and let T:X→B(X) be a mapping. Then T is said to be a generalized multivalued F-contraction if F∈F and there exists τ>0 such that
(10)τ+F(δ(Tx,Ty))≤F(M(x,y)),
for all x,y∈X with min{δ(Tx,Ty),d(x,y)}>0, where
(11)M(x,y)=max{12d(x,y),D(x,Tx),D(y,Ty),12[D(x,Ty)+D(y,Tx)]}.

Theorem 9.

Let (X,d) be a complete metric space and let T:X→B(X) be a multivalued F-contraction. If F is continuous and Tx is closed for all x∈X, then T has a fixed point in X.

Proof.

Let x0∈X be an arbitrary point and define a sequence {xn} in X as xn+1∈Txn for all n≥0. If there exists n0∈N∪{0} for which xn0=xn0+1, then xn0 is a fixed point of T and so the proof is completed. Thus, suppose that, for every n∈N∪{0}, xn≠xn+1. So d(xn,xn+1)>0 and δ(Txn-1,Txn)>0 for all n∈N. Then, we have from (10)
(12)τ+F(d(xn,xn+1))≤τ+F(δ(Txn-1,Txn))≤F(M(xn-1,xn))=F(max{d(xn-1,xn),D(xn-1,Txn-1),D(xn,Txn),12[D(xn-1,Txn)+D(xn,Txn-1)]})≤F(max{d(xn-1,xn),d(xn,xn+1)})=F(d(xn-1,xn)),
and so
(13)F(d(xn,xn+1))≤F(d(xn-1,xn))-τ≤F(d(xn-2,xn-1))-2τ⋮≤F(d(x0,x1))-nτ.

Denote an=d(xn,xn+1), for n=0,1,2,…. Then, an>0 for all n and, using (10), the following holds:
(14)F(an)≤F(an-1)-τ≤F(an-2)-2τ≤⋯≤F(a0)-nτ.
From (14), we get limn→∞F(an)=-∞. Thus, from (F2), we have
(15)limn→∞an=0.
From (F3) there exists k∈(0,1) such that
(16)limn→∞ankF(an)=0.
By (14), the following holds for all n∈N:
(17)ankF(an)-ankF(a0)≤-anknτ≤0.
Letting n→∞ in (17), we obtain that
(18)limn→∞nank=0.
From (18), there exits n1∈N such that nank≤1 for all n≥n1. So we have
(19)an≤1n1/k,
for all n≥n1. In order to show that {xn} is a Cauchy sequence consider m,n∈N such that m>n≥n1. Using the triangular inequality for the metric and from (19), we have
(20)d(xn,xm)≤d(xn,xn+1)+d(xn+1,xn+2)+⋯+d(xm-1,xm)=an+an+1+⋯+am-1=∑i=nm-1ai≤∑i=n∞ai≤∑i=n∞1i1/k.
By the convergence of the series ∑i=1∞(1/i1/k), we get d(xn,xm)→0 as n→∞. 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. Now, suppose F is continuous. In this case, we claim that z∈Tz. Assume the contrary; that is, z∉Tz. In this case, there exist an n0∈N and a subsequence {xnk} of {xn} such that D(xnk+1,Tz)>0 for all nk≥n0. (Otherwise, there exists n1∈N such that xn∈Tz for all n≥n1, which implies that z∈Tz. This is a contradiction, since z∉Tz.) Since D(xnk+1,Tz)>0 for all nk≥n0, then we have
(21)τ+F(D(xnk+1,Tz))≤τ+F(δ(Txnk,Tz))≤F(M(xnk,z))≤F(max{12d(xnk,z),d(xnk,xnk+1),D(z,Tz),12[D(xnk,Tz)+d(z,xnk+1)]}).
Taking the limit k→∞ and using the continuity of F, we have τ+F(D(z,Tz))≤F(D(z,Tz)), which is a contradiction. Thus, we get z∈Tz¯=Tz. This completes the proof.

Example 10.

Let X={0,1,2,3,…} and d(x,y)={0;x=yx+y;x≠y. Then (X,d) is a complete metric space. Define T:X→B(X) by
(22)Tx={{0};x=0{0,1,2,3,…,x-1};x≠0.
We claim that T is multivalued F-contraction with respect to F(α)=α+lnα and τ=1. Because of the min{δ(Tx,Ty), d(x,y)}>0, we can consider the following cases while x≠y and {x,y}∩{0,1} is empty or singleton.

Case 1. For y=0 and x>1, we have
(23)δ(Tx,Ty)M(x,y)eδ(Tx,Ty)-M(x,y)=x-1xex-1-x=x-1xe-1<e-1.

Case 2. For y=1 and x>1, we have
(24)δ(Tx,Ty)M(x,y)eδ(Tx,Ty)-M(x,y)=x-1xex-1-x=x-1xe-1<e-1.

Case 3. For x>y>1, we have
(25)δ(Tx,Ty)M(x,y)eδ(Tx,Ty)-M(x,y)=x+y-2x+yex+y-2-x-y=x+y-2x+ye-2<e-1.
This shows that T is multivalued F-contraction; therefore, all conditions of theorem are satisfied and so T has a fixed point in X.

On the other hand, for y=0 and x≠0, since δ(Tx,Ty)=x-1 and d(x,y)=x, we get
(26)limn→∞δ(Tx,Ty)M(x,y)=limn→∞x-1x=1;
then T does not satisfy
(27)δ(Tx,Ty)≤λM(x,y),
for λ∈[0,1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to the referees because their suggestions contributed to improving the paper.

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