We define the notion of
α*-ψ-contractive mappings for cone metric space and
obtain fixed points of multivalued mappings in connection with Hausdorff
distance function for closed bounded subsets of cone metric spaces. We
obtain some recent results of the literature as corollaries of our main
theorem. Moreover, a nontrivial example of α*-ψ-contractive mapping satisfying all conditions of our main result has been
constructed.

1. Introduction

Banach contraction principle is widely recognized as the source of metric fixed point theory. Also, this principle plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions for nonlinear equations, systems of linear equations, and linear integral equations and to prove the convergence of algorithms in computational mathematics. Because of its importance for mathematical theory, Banach contraction principle has been extended in many directions.

In 2007, Huang and zhang [1] introduced cone metric space with normal cone, as a generalization of metric space. Rezapour and Hamlbarani [2] presented the results of [1] for the case of cone metric space without normality in cone. Many authors work out on it (see [3, 4]). Cho and Bae [5] introduced the Hausdorff distance function on cone metric spaces and generalized the result of [6] for multivalued mappings.

In 2012, Samet et al. [7] introduced the concept of α-ψ-contractive type mappings. Their results generalized some ordered fixed point results (see [7]). In [8], Karapinar et al. introduced the notion of a Gm-Meir-Keeler contractive mapping and established some fixed point theorems for the Gm-Meir-Keeler contractive mapping in the setting of G-metric spaces. For more details in fixed point theory related to our paper, we refer to the reader [9–19]. Asl et al. [20] introduced the notion of α*-ψ-contractive mappings and improved the concept of α-ψ-contractive mappings along with some fixed point theorems in metric space. Consequently, Ali et al. [21], Mohammadi et al. [22] and Salimi et al. [23] studied the concept of α*-ψ-contractive mappings for proving fixed point results by using generalized contractive conditions in complete metric spaces.

In this paper, we first define the notion of α*-ψ-contractive mappings for cone metric spaces and then we use it to study fixed point theorems for multivalued mappings satisfying α*-ψ-contractive conditions in a complete cone metric space without the assumption of normality. We also furnish a nontrivial example to support our main result.

2. Preliminaries

In the following, we always suppose that 𝔼 is a real Banach space, P is a cone in 𝔼 with nonempty interior, and ⪯ is the partial ordering with respect to P. By θ, we denote the zero element of 𝔼. A subset P is called a cone if and only if

Pis closed, nonempty, and P={θ};

a,b∈ℝ,a,b≥0,x,y∈P⇒ax+by∈P;

P∩(-P)={θ}.

For a given cone P⊆𝔼, we define a partial ordering ⪯ with respect to P by x⪯y if and only if y-x∈P; x≺y will stand for x⪯y and x≠y, while x≪y stand for y-x∈intP, where intP denotes the interior of P.

Definition 1 (see [<xref ref-type="bibr" rid="B14">1</xref>]).

Let X be a nonempty set. A function d:X×X→𝔼 is said to be a cone metric, if the following conditions hold:

θ⪯d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y;

d(x,y)=d(y,x) for all x,y∈X;

d(x,z)⪯d(x,y)+d(y,z) for all x,y,z∈X.

The pair (X,d) is then called a cone metric space.

Lemma 2 (see [<xref ref-type="bibr" rid="B14">1</xref>]).

Let (X,d) be a cone metric space, x∈X, and let {xn}be a sequence in X. Then

{xn} converges to x whenever for every c∈𝔼 with θ≪c there is a natural number n0 such that d(xn,x)≪c, for all n≥n0. We denote this by limn→∞xn=x;

{xn} is a Cauchy sequence whenever for every c∈𝔼 with θ≪c there is a natural number n0 such that d(xn,xm)≪c, for all n,m≥n0;

(X,d) is complete cone metric if every Cauchy sequence in X is convergent.

Remark 3 (see [<xref ref-type="bibr" rid="B15">3</xref>]).

The results concerning fixed points and other results, in case of cone spaces with nonnormal cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of the lemmas 1–4, in [1] hold. Further, the vector cone metric is not continuous in the general case; that is, from xn→x, yn→y it need not follow that d(xn,yn)→d(x,y).

Let (X,d) be a cone metric space. The following properties of cone metric spaces have been noticed [3].

If u⪯v and v≪w, then u≪w.

If u≪v and v⪯w, then u≪w.

If u≪v and v≪w, then u≪w.

If θ⪯u≪c for each c∈intP, then u=θ.

If a⪯b+c, for each c∈intP, then a⪯b.

{an} be a sequence in 𝔼. If c∈intP and θ⪯an→θ (as n→∞), then there exists n0∈N such that for all n≥n0, we have an≪c.

With some modifications, we have the following definition from [24].

Definition 4.

Let Ψ be a family of nondecreasing functions, ψ:P→P such that

ψ(θ)=θ and θ<ψ(t)<t for t∈P∖{θ},

t∈IntP implies t-ψ(t)∈IntP,

limn→+∞ψn(t)=θ for every t∈P∖{θ}.

3. Main Result

For a cone metric space (X,d), denote (see [5])
(1)N(X)={A:AisnonemptysubsetofX}CB(X)={A:Aisnonempty,closedandboundedsubsetofX},s(p)={q∈𝔼:p⪯q}forq∈𝔼,s(a,B)=⋃b∈Bs(d(a,b))=⋃b∈B{x∈𝔼:d(a,b)⪯x}fora∈XandB∈CB(X).

For A,B∈CB(X) we denote
(2)s(A,B)=(⋂a∈As(a,B))∩(⋂b∈Bs(b,A)).

Lemma 5.

Let (X,d) be a cone metric space, and let P be a cone in Banach space 𝔼.

Let p,q∈𝔼. If p⪯q, s(q)⊂s(p).

Let x∈X and A∈N(X). If θ∈s(x,A), then x∈A.

Let q∈P and let A,B∈CB(X) and a∈A. If q∈s(A,B), then q∈s(a,B) for all a∈A or q∈s(A,b) for all b∈B.

Let q∈P and let λ≥0, then λs(q)⊆s(λq).

Remark 6.

Let (X,d) be a cone metric space. If 𝔼=R and P=[0,+∞), then (X,d) is a metric space. Moreover, for A,B∈CB(X), H(A,B)=infs(A,B) is the Hausdorff distance induced by d.

Definition 7.

Let (X,d) be a complete cone metric space with cone P,ψ∈Ψ, α:X×X→[0,+∞), and F:X→CB(X) is known as α*-ψ-contractive multivalued mapping whenever
(3)ψ(d(x,y))∈α*(Fx,Fy)s(Fx,Fy)
for all x,y∈X, where α*(A,B)=inf{α(a,b):a∈A,b∈B}. Also, we say that F is α*-admissible whenever α(x,y)≥1 implies α*(Fx,Fy)≥1.

Note that an α*-ψ-contractive multivalued mappings for cone metric space is generalized α*-ψ-contractive. When ψ∈Ψ is a strictly increasing mapping, α*-ψ-contractive is called strictly generalized α*-ψ-contractive.

Theorem 8.

Let (X,d) be a complete cone metric space with cone P,α:X×X→[0,+∞) be a function, ψ∈Ψ be a strictly increasing map and F:X→CB(X), and F be α*-admissible and α*-ψ-contractive multivalued mapping on X. Suppose that there exist x0∈X,x1∈Fx0 such that α(x0,x1)≥1. Assume that if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n and xn→u as n→+∞ then α(xn,u)≥1 for all n. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

We may suppose that x0≠x1. Then α*(Fx0,Fx1)≥1 and
(4)ψ(d(x0,x1))∈α*(Fx0,Fx1)s(Fx0,Fx1).
By Lemma 5(3), we have
(5)ψ(d(x0,x1))∈α*(Fx0,Fx1)s(x1,Fx1).
By definition, we can take x2∈Fx1 such that
(6)ψ(d(x0,x1))∈α*(Fx0,Fx1)s(d(x1,x2)).
By Lemma 5(4), we have
(7)ψ(d(x0,x1))∈α*(Fx0,Fx1)s(d(x1,x2))⊆s(α*(Fx0,Fx1)d(x1,x2)).
So,
(8)α*(Fx0,Fx1)d(x1,x2)⪯ψ(d(x0,x1)).
Hence,
(9)0≺d(x1,x2)⪯α*(Fx0,Fx1)d(x1,x2)⪯ψ(d(x0,x1))x1≠x2 and α(x1,x2)≥1. Thus α*(Fx1,Fx2)≥1 and d(x1,x2)⪯ψ(d(x0,x1)). If x2∈Fx2, then x2 is a fixed point of F. Assume that x2∉Fx2. Then
(10)ψ(d(x1,x2))∈α*(Fx1,Fx2)s(Fx1,Fx2).
By Lemma 5(3), we have
(11)ψ(d(x1,x2))∈α*(Fx1,Fx2)s(x2,Fx2).
By definition, we can take x3∈Fx2 such that
(12)ψ(d(x1,x2))∈α*(Fx1,Fx2)s(d(x2,x3)).
By Lemma 5(4), we have
(13)ψ(d(x1,x2))∈α*(Fx1,Fx2)s(d(x2,x3))⊆s(α*(Fx1,Fx2)d(x2,x3)).
So,
(14)α*(Fx1,Fx2)d(x2,x3)⪯ψ(d(x1,x2)).
Hence,
(15)0≺d(x2,x3)⪯α*(Fx1,Fx2)d(x2,x3)⪯ψ(d(x1,x2)).
It is clear that x2≠x3 and α(x2,x3)≥1. Thus, α*(Fx2,Fx3)≥1 and d(x2,x3)<ψ2(ψ(d(x0,x1))).

If x3∈Fx3, then x3 is a fixed point of F. Assume that x3∉Fx3:
(16)ψ(d(x2,x3))∈α*(Fx2,Fx3)s(Fx2,Fx3).
By Lemma 5(3), we have
(17)ψ(d(x2,x3))∈α*(Fx2,Fx3)s(x3,Fx3).
By definition, we can take x4∈Fx3 such that
(18)ψ(d(x2,x3))∈α*(Fx2,Fx3)s(d(x3,x4)).
By Lemma 5(4), we have
(19)ψ(d(x2,x3))∈α*(Fx2,Fx3)s(d(x3,x4))⊆s(α*(Fx2,Fx3)d(x3,x4)).
So
(20)α*(Fx2,Fx3)d(x3,x4)⪯ψ(d(x2,x3)).
Hence
(21)0≺d(x3,x4)⪯α*(Fx2,Fx3)d(x3,x4)⪯ψ(d(x2,x3)).
It is clear that x3≠x4 and α(x3,x4)≥1. Thus, α*(Fx3,Fx4)≥1 and d(x3,x4)<ψ3(ψ(d(x0,x1))). By continuing this process, we obtain a sequence {xn} in X such that xn∈Txn-1, xn≠xn-1, α(xn,xn+1)≥1 and
(22)d(xn,xn+1)⪯ψ(d(xn-1,xn))⪯⋯⪯ψn(d(x0,x1))
for all n.

Fix θ≪c. We choose a positive real number δ such that (c-ψ(c))+I(θ,δ)⊂IntP, where I(θ,δ)={y∈E:∥y∥<δ}. By (iii) of Definition 4, there exists a natural number N such that ψm(d(x0,x1))∈I(θ,δ), for all m≥N. Then
(23)ψm(d(x0,x1))≪c-ψ(c),
for all m≥N. Consequently, d(xm,xm+1)≪c-ψ(c), for all m≥N. Fix m≥N. Now we prove
(24)d(xm,xn+1)≪c

for all n≥m. Note that (24) holds when n=m. Assume that (24) holds for some n≥m. Then, we have
(25)d(xm,xn+1)≪c.

Now by (22), we have
(26)d(xm,xn+2)≤d(xm,xm+1)+d(xm+1,xn+2)≪c-ψ(c)+ψd(xm,xn+1)≪c-ψ(c)+ψ(c)=c.
Therefore, (24) holds when n=n+1. By induction, we deduce that (24) holds for all n≥m. This is sufficient to conclude that {xn} is a Cauchy sequence. Choose u∈X such that xn→u. Since α(xn,u)≥1 for all n and F is α*-admissible, so α*(Fxn,Fu)≥1 for all n. From (3), we have
(27)ψ(d(xn,u))∈α*(Fxn,Fu)s(Fxn,Fu)

for all n∈ℕ. By Lemma 5(3), we have
(28)ψ(d(xn,u))∈α*(Fxn,Fu)s(xn+1,Fu).

By definition, we can take vn∈Fu such that
(29)ψ(d(xn,u))∈α*(Fxn,Fu)s(d(xn+1,vn)).

By Lemma 5(4), we have
(30)ψ(d(xn,u))∈α*(Fxn,Fu)s(d(xn+1,vn))⊆s(α*(Fxn,Fu)d(xn+1,vn)).

Moreover, for a given c∈IntP, we have (33)d(u,vn)⪯d(u,xn+1)+d(xn+1,vn)⪯d(u,xn+1)+α*(Fxn,Fu)d(xn+1,vn)⪯d(u,xn+1)+ψ(d(xn,u))⪯d(u,xn+1)+d(xn,u)≪c2+c2=c,forn≥k=k(c).
Hence, according to Lemma 2(i), we have limn→∞vn=u. Since Fu is closed, u∈Fu.

Theorem 9.

Let (X,d) be a complete cone metric space with cone P,α:X×X→[0,+∞) be a function, and F:X→CB(X) be α*-admissible. If there exists a constant k∈[0,1) such that
(34)kd(x,y)∈α*(Fx,Fy)s(Fx,Fy)
for all x,y∈X. Suppose that there exist x0∈X,x1∈Fx0 such that α(x0,x1)≥1. Assume that if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n and xn→u as n→+∞; then α(xn,u)≥1 for all n. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take ψ(t)=kt in Theorem 8.

Theorem 10.

Let (X,d) be a complete cone metric space with cone P,ψ∈Ψ be a strictly increasing map, and F:X→CB(X) be multivalued mapping such that
(35)ψ(d(x,y))∈s(Fx,Fy)
for all x,y∈X. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take α*(Fx,Fy)=1 in the Theorem 8.

Corollary 11.

Let (X,d) be a complete cone metric space with cone P and let F:X→CB(X) be a multivalued mapping. If there exists a constant k∈[0,1) such that
(36)kd(x,y)∈s(Fx,Fy)
for all x,y∈X, then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take ψ(t)=kt and α*(Fx,Fy)=1 in the Theorem 8.

Corollary 12 (see [<xref ref-type="bibr" rid="B3">20</xref>]).

Let (X,d) be a complete metric space,α:X×X→[0,+∞) be a function, ψ∈Ψ be a strictly increasing map, and F:X→CB(X) be α*-admissible such that
(37)α*(Fx,Fy)H(Fx,Fy)≤ψ(d(x,y))
for all x,y∈X. Suppose that there exist x0∈X such that α(x0,Fx0)≥1. Assume that if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n and xn→u as n→+∞ then α(xn,u)≥1 for all n. Then, there exists a point x* in X such that x*∈Fx*.

By Remark 6, we have the following corollaries.

Corollary 13 (see [<xref ref-type="bibr" rid="B3">20</xref>]).

Let (X,d) be a complete metric space,ψ∈Ψ be a strictly increasing map, and F:X→CB(X) be a multivalued mapping such that
(38)H(Fx,Fy)≤ψ(d(x,y))
for all x,y∈X. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take α*(Fx,Fy)=1 in the Corollary 12.

Corollary 14 (see [<xref ref-type="bibr" rid="B3">20</xref>]).

Let (X,d) be a complete metric space, α:X×X→[0,+∞) be a function, and F:X→CB(X) be α*-admissible. If there exists a constant k∈[0,1) such that
(39)α*(Fx,Fy)H(Fx,Fy)≤kd(x,y)
for all x,y∈X. Suppose that there exist x0∈X such that α(x0,Fx0)≥1. Assume that if {xn} is a sequence in X such that α(xn,xn+1)≥1 for all n and xn→u as n→+∞ then α(xn,u)≥1 for all n. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take ψ(t)=kt in the Corollary 13.

Corollary 15 (see [<xref ref-type="bibr" rid="B6">25</xref>]).

Let (X,d) be a complete metric space and let F:X→CB(X) be a multivalued mapping. If there exists a constant k∈[0,1) such that
(40)H(Fx,Fy)≤kd(x,y)
for all x,y∈X. Then, there exists a point x* in X such that x*∈Fx*.

Proof.

Take α*(Fx,Fy)=1 in the Corollary 14.

Example 16.

Let X=[0,1], E=C1[0,1], P={θ⪯x(t):t∈X}, where θ(t)=0 for all t∈X. Define d:X×X→E by
(41)d(x,y)=|x-y|et
and F:X→2X by Fx=[0,(x/12)] for all x∈X and
(42)α(x,y)={1|x-y|ifx≠y1ifx=y.
Then, α(x,y)≥1⇒α*(Fx,Fy)=inf{α(x,y):x∈Fx,y∈Fy}≥1. Then, clearly F is α*-admissible. Now for x,y and x≤y, we get
(43)ψ(d(x,y))=15|x-y|et⪰112|x-y|et=α*(Fx,Fy)s(Fx,Fy)
which implies that
(44)ψ(d(x,y))∈α*(Fx,Fy)s(Fx,Fy).
So F is α*-ψ-contractive multivalued mapping on X where ψ(t)=t/5. Thus, all the conditions of main result are satisfied to obtain the fixed point of F.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

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