On Fixed Points of α-ψ-Contractive Multivalued Mappings in Cone Metric Spaces
Abstract
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 G m-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
- (i)
P is closed, nonempty, and P = {θ};
- (ii)
a, b ∈ ℝ, a, b ≥ 0, x, y ∈ P⇒ax + by ∈ P;
- (iii)
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 ∈ int P, where int P denotes the interior of P.
Definition 1 (see [1].)Let X be a nonempty set. A function d : X × X → 𝔼 is said to be a cone metric, if the following conditions hold:
-
(C1 θ⪯d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;
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C2 d(x, y) = d(y, x) for all x, y ∈ X;
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C3 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 [1].)Let (X, d) be a cone metric space, x ∈ X, and let {xn} be a sequence in X. Then
- (i)
{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 lim n→∞ xn = x;
- (ii)
{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;
- (iii)
(X, d) is complete cone metric if every Cauchy sequence in X is convergent.
Remark 3 (see [3].)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).
-
(PT1) If u⪯v and v ≪ w, then u ≪ w.
-
(PT2) If u ≪ v and v⪯w, then u ≪ w.
-
(PT3) If u ≪ v and v ≪ w, then u ≪ w.
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(PT4) If θ⪯u ≪ c for each c ∈ int P, then u = θ.
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(PT5) If a⪯b + c, for each c ∈ int P, then a⪯b.
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PT6 {an} be a sequence in 𝔼. If c ∈ int P 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
- (i)
ψ(θ) = θ and θ < ψ(t) < t for t ∈ P∖{θ},
- (ii)
t ∈ IntP implies t − ψ(t) ∈ IntP,
- (iii)
lim n→+∞ ψn(t) = θ for every t ∈ P∖{θ}.
3. Main Result
Lemma 5. Let (X, d) be a cone metric space, and let P be a cone in Banach space 𝔼.
- (1)
Let p, q ∈ 𝔼. If p⪯q, s(q) ⊂ s(p).
- (2)
Let x ∈ X and A ∈ N(X). If θ ∈ s(x, A), then x ∈ A.
- (3)
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.
- (4)
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) = inf s(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
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
If x3 ∈ Fx3, then x3 is a fixed point of F. Assume that x3 ∉ Fx3:
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
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
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
Proof. Take ψ(t) = kt and α*(Fx, Fy) = 1 in the Theorem 8.
Corollary 12 (see [20].)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
By Remark 6, we have the following corollaries.
Corollary 13 (see [20].)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
Proof. Take α*(Fx, Fy) = 1 in the Corollary 12.
Corollary 14 (see [20].)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
Proof. Take ψ(t) = kt in the Corollary 13.
Corollary 15 (see [25].)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
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
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.