Volume 2013, Issue 1 815289

Research Article

Open Access

Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

Corresponding Author

Sahar Mohammad Abusalim

[email protected]

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia ukm.my

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Mohd Salmi Md Noorani,

Mohd Salmi Md Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia ukm.my

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Sahar Mohammad Abusalim,

Corresponding Author

Sahar Mohammad Abusalim

[email protected]

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia ukm.my

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Mohd Salmi Md Noorani,

Mohd Salmi Md Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia ukm.my

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First published: 13 May 2013

https://doi.org/10.1155/2013/815289

Citations: 4

Academic Editor: Abdelghani Bellouquid

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Abstract

The concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.

1. Introduction

Fixed point theory has attracted many researchers since 1922 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied dilemma in mathematical sciences and engineering. A large literature on this subject exists, and this is a very active area of research at present. Banach contraction principle has been generalized in dissimilar directions in different spaces by mathematicians over the years; for more details on this and related topics, we refer to [1–6] and references therein.

In contemporary time, fixed point theory has evolved speedily in partially ordered cone metric spaces; that is, cone metric spaces equipped with a partial ordering, for some new results in ordered metric spaces see [7]. A coming early result in this bearing was constituted by Altun and Durmaz [8] under the condition of normality for cones. Then, Altun et al. [9] generalized the results of Altun and Durmaz [8] by omitting the assumption of normality condition for cones. Afterward, several authors have studied fixed point and common fixed point problems in ordered cone metric spaces; for more details see [10–17].

In 2011, Hussain and Shah [18] presented cone b-metric spaces as a generalization of b-metric spaces and cone metric spaces; for some new results in b-metric spaces see [19]. They not only constructed some topological properties in such spaces but also ameliorated some current results about KKM mappings in the setting of a cone b-metric space. After some time, many authors have been motivated to demonstrate fixed point theorems as well as common fixed point theorems for two or more mappings on cone b-metric spaces by the incipient work of Hussain and Shah [18] (see [20–23] and the references therein).

In [8], Altun and Durmaz proved the following results under the condition of normality for cones.

Theorem 1 (see [8].)Let (X, ⊑) be a partially ordered set, suppose that there exists a cone metric d in X such that the cone metric space (X, d) is complete, and let P be a normal cone with normal constant K. Let f : X → X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold:

(i) there exists k ∈ [0,1) such that d(fx, fy)⪯kd(x, y) for all x, y ∈ X with y⊑x;
(ii) there exists x₀ ∈ X such that x₀⊑fx₀.

Then f has a fixed point in X.

In [9], Altun et al. generalized the above theorem and proved it without normality condition for cones.

Theorem 2 (see [9].)Let (X, ⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X, d) is complete over a solid cone P. Let f : X → X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following two assertions hold:

(i) there exist k, l, r ∈ [0,1) with k + 2l + 2r < 1 such that
(1)
for all x, y ∈ X with y⊑x;
(ii) there exists x₀ ∈ X such that x₀⊑fx₀.

Then f has a fixed point in X.

Theorem 3 (see [9].)Let (X, ⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X, d) is complete over a solid cone P. Let f : X → X be a nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist k, l, r ∈ [0,1) with k + 2l + 2r < 1 such that
(2)
for all x, y ∈ X with y⊑x;
(ii) there exists x₀ ∈ X such that x₀⊑fx₀;
(iii) if an increasing sequence {x_n} converges to x in X, then x_n⊑x for all n.

Then f has a fixed point in X.

In the same paper, they also presented the following two common fixed point results in ordered cone metric spaces.

Theorem 4 (see [9].)Let (X, ⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X, d) is complete over a solid cone P. Let f, g : X → X be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist k, l, r ∈ [0,1) with k + 2l + 2r < 1 such that
(3)
for all comparative x, y ∈ X;
(ii) f or g is continuous.

Then f and g have a common fixed point x^* ∈ X.

Theorem 5 (see [9].)Let (X, ⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X, d) is complete over a solid cone P. Let f, g : X → X be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist k, l, r ∈ [0,1) with k + 2l + 2r < 1 such that
(4)
for all comparative x, y ∈ X;
(ii) if an increasing sequence {x_n} converges to x in X, then x_n⊑x for all n.

Then f and g have a common fixed point x^* ∈ X.

In this paper, we prove some fixed point and common fixed point theorems on ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the cone P is solid, that is, int P ≠ ∅.

The following definitions and results shall be needed in the sequel.

Let E be a real Banach space and θ denotes the zero element in E. A cone P is a subset of E such that

(1) P is nonempty closed set and P ≠ {θ};
(2) if a, b are nonnegative real numbers and x, y ∈ P, then ax + by ∈ P;
(3) x ∈ P and −x ∈ P imply x = θ.

For any cone P ⊂ E, the partial ordering ⪯ with respect to P is defined by x⪯y if and only if y − x ∈ P. The notation of ≺ stands for x⪯y but x ≠ y. Also, we use x ≪ y to indicate that y − x ∈ int P, where int P denotes the interior of P. A cone P is called normal if there exists the number K such that

(5)

for all x, y ∈ E. The least positive number K satisfying the above condition is called the normal constant of P.

Definition 6 (see [18].)Let X be a nonempty set and E a real Banach space equipped with the partial ordering ⪯ with respect to the cone P. A vector-valued function d : X × X → E is said to be a cone b-metric function on X with the constant s ≥ 1 if the following conditions are satisfied:

(1) θ⪯d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;
(2) d(x, y) = d(y, x) for all x, y ∈ X;
(3) d(x, y)⪯s(d(x, y) + d(y, z)) for all x, y, z ∈ X.

Then pair (X, d) is called a cone b-metric space (or a cone metric type space); we shall use the first mentioned term.

Observe that if s = 1, then the ordinary triangle inequality in a cone metric space is satisfied; however, it does not hold true when s > 1. Thus the class of cone b-metric spaces is effectively larger than that of the ordinary cone metric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples show the above remarks.

Example 7. Let X = {−1,0, 1}, E = ℝ², and P = {(x, y) : x ≥ 0, y ≥ 0}. Define d : X × X → P by d(x, y) = d(y, x) for all x, y ∈ X, d(x, x) = θ, x ∈ X, and d(−1,0) = (3,3), d(−1,1) = d(0,1) = (1,1). Then (X, d) is a complete cone b-metric space but the triangle inequality is not satisfied. Indeed, we have that d(−1,1) + d(1,0) = (1,1) + (1,1) = (2,2)≺(3,3) = d(−1,0). It is not hard to verify that s = 3/2.

Example 8. Let X = ℝ, E = ℝ², and P = {(x, y) ∈ E : x ≥ 0, y ≥ 0}. Define d : X × X → E by d(x, y) = (|x−y|², |x−y|²). Then, it is easy to see that (X, d) is a cone b-metric space with the coefficient s = 2. But it is not a cone metric spaces since the triangle inequality is not satisfied.

Definition 9 (see [18].)Let (X, d) be a cone b-metric space, {x_n} a sequence in X and x ∈ X.

(1) For all c ∈ E with θ ≪ c, if there exists a positive integer N such that d(x_n, x) ≪ c for all n > N, then x_n is said to be convergent and x is the limit of {x_n}. One denotes this by x_n → x.
(2) For all c ∈ E with θ ≪ c, if there exists a positive integer N such that d(x_n, x_m) ≪ c for all n, m > N, then {x_n} is called a Cauchy sequence in X.
(3) A cone metric space (X, d) is called complete if every Cauchy sequence in X is convergent.

The following lemma is useful in our work.

Lemma 10 (see [24].)

(1) If E is a real Banach space with a cone P and a⪯λa where a ∈ P and 0 ≤ λ < 1, then a = θ.
(2) If c ∈ int P, θ⪯a_n, and a_n → θ, then there exists a positive integer N such that a_n ≪ c for all n ≥ N.
(3) If a⪯b and b ≪ c, then a ≪ c.
(4) If θ⪯u ≪ c for each θ ≪ c, then u = θ.

2. Fixed Point Results

In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma.

Lemma 11. Let {x_n} be a sequence in a cone b-metric space (X, d) with the coefficient s ≥ 1 relative to a solid cone P such that

(6)

where h ∈ [0, 1/s) and n = 1,2, …. Then {x_n} is a Cauchy sequence in (X, d).

Proof. Let m > n ≥ 1. It follows that

(7)

Now, (6) and sh < 1 imply that

(8)

According to Lemma 10(2), and for any c ∈ E with c ≫ θ, there exists N₀ ∈ ℕ such that for any n > N₀, (shⁿ/(1 − sh))d(x₀, x₁) ≪ c. Furthermore, from (8) and for any m > n > N₀, Lemma 10(3) shows that

(9)

Hence, by Definition 9(2) {x_n} is a Cauchy sequence in X.

Theorem 12. Let (X, ⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X, d) is complete with the coefficient s ≥ 1 relative to a solid cone P. Let f : X → X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist a_i, i = 1, …, 5, such that 2sa₁ + (s + 1)(a₂ + a₃)+(s² + s)(a₄ + a₅) < 2 with ,
(10)
for all x, y ∈ X with y⊑x;
(ii) there exists x₀ ∈ X such that x₀⊑fx₀.

Then f has a fixed point x^* ∈ X.

Proof. If x₀ = fx₀, then the proof is finished. Suppose that x₀ ≠ fx₀. Since x₀⊑fx₀ and f is nondecreasing with respect to ⊑, we obtain by induction that x₀⊑fx₀ = x₁⊑f¹x₀ = x₂⊑⋯⊑fⁿ⁻¹x₀ = x_n⊑fⁿx₀ = x_n+1⊑⋯. Then we have,

(11)

Then, one can assert that

(12)

On the other hand, we have

(13)

Then, one can assert that

(14)

Adding (12) and (14), we get

(15)

where λ = (2a₁ + a₂ + a₃ + sa₄ + sa₅)/(2 − (a₂ + a₃ + sa₄ + sa₅)) < 1/s. According to Lemma 11, we have {x_n} is a Cauchy sequence in X. Since X is complete, there exists x^* ∈ X such that x_n → x^*. Since f is continuous, then x^* = lim x_n+1 = lim fⁿx₀ = lim f(fⁿ⁻¹x₀) = f(lim fⁿ⁻¹x₀) = f(lim x_n) = f(x^*). Therefore, x^* is a fixed point of f.

If we use condition (iii) instead of the continuity of f in Theorem 12, we have the following result.

Theorem 13. Let (X, ⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X, d) is complete with the coefficient s ≥ 1 relative to a solid cone P. Let f : X → X be a nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist a_i, i = 1, …, 5, such that 2sa₁ + (s + 1)(a₂ + a₃)+(s² + s)(a₄ + a₅) < 2 with ,
(16)
for all x, y ∈ X with y⊑x;
(ii) there exists x₀ ∈ X such that x₀⊑fx₀;
(iii) if an increasing sequence {x_n} converges to x in X, then x_n⊑x for all n.

Then f has a fixed point x^* ∈ X.

Proof. As in the Theorem 12, we can construct an increasing sequence {x_n} and prove that there exists x^* ∈ X such that x_n → x^*. Now, condition (iii) implies x_n⊑x^* for all n. Therefore, we can use condition (i) and so

(17)

Taking n → ∞, we have d(x^*, fx^*)⪯(a₃ + a₅)d(x^*, fx^*)d(x^*, fx^*). Since (a₃ + a₅) < 1, Lemma 10(1) shows that d(x^*, fx^*) = θ; that is, x^* = fx^*. Therefore x^* is a fixed point of f.

3. Common Fixed Point Results

Now, we give two common fixed point theorems on ordered cone b-metric spaces. We need the following definition.

Definition 14 (see [9].)Let (X, ⊑) be a partially ordered set. Two mappings f, g : X → X are said to be weakly increasing if fx⊑gfx and gx⊑fgx hold for all x ∈ X.

Theorem 15. Let (X, ⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X, d) is complete with the coefficient s ≥ 1 relative to a solid cone P. Let f, g : X → X be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist a_i, i = 1, …, 5, such that 2sa₁ + (s + 1)(a₂ + a₃)+(s² + s)(a₄ + a₅) < 2 with ,
(18)
for all comparative x, y ∈ X;
(ii) f or g is continuous.

Then f and g have a common fixed point x^* ∈ X.

Proof. Let x₀ be an arbitrary point of X and define a sequence {x_n} in X as follows: x_2n+1 = fx_2n and x_2n+2 = gx_2n+1 for all n > 0. Note that, since f and g are weakly increasing, we have x₁ = fx₀⊑gfx₀ = gx₁ = x₂ and x₂ = gx₁⊑fgx₁ = fx₂ = x₃, and continuing this process we have x₁⊑x₂⊑⋯⊑x_n⊑x_n+1⊑⋯. That is, the sequence {x_n} is nondecreasing. Now, since x_2n and x_2n+1 are comparative, we can use the inequality (18), and then we have

(19)

Hence,

(20)

On the other hand and by symmetry we have

(21)

Hence,

(22)

Adding inequalities (20) and (22), we get

(23)

where λ = (2a₁ + a₂ + a₃ + sa₄ + sa₅)/(2 − (a₂ + a₃ + sa₄ + sa₅)) < 1/s. Similarly, it can be shown that

(24)

Therefore,

(25)

According to Lemma 11, we have {x_n} is a Cauchy sequence in X. Since X is complete, there exists x^* ∈ X such that x_n → x^*. Suppose that f is continuous. Then, x^* = lim x_n+1 = lim fⁿx₀ = lim f(fⁿ⁻¹x₀) = f(lim fⁿ⁻¹x₀) = f(lim x_n) = f(x^*). Therefore, x^* is a fixed point of f. Now, we need to show that x^* is a fixed point of g. Since x^*⊑x^*, we can use the inequality (18) for x = y = x^*. Then we have

(26)

Hence,

(27)

Since (a₃ + a₅) < 1, Lemma 10(1) shows that d(x^*, gx^*) = θ; that is, x^* = gx^*. Therefore x^* is a fixed point of g. Therefore, f and g have a common fixed point. The proof is similar when g is a continuous mapping.

Theorem 16. Let (X, ⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X, d) is complete with the coefficient s ≥ 1 relative to a solid cone P. Let f, g : X → X be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:

(i) there exist a_i, i = 1, …, 5 such that 2sa₁ + (s + 1)(a₂ + a₃)+(s² + s)(a₄ + a₅) < 2 with ,
(28)
for all comparative x, y ∈ X;
(ii) if an increasing sequence {x_n} converges to x in X, then x_n⊑x for all n.

Then f and g have a common fixed point x^* ∈ X.

Proof. As in Theorem 15, we can construct an increasing sequence {x_n} and prove that there exists x^* ∈ X such that x_n → x^*, also; by the construction of x_n, gx_n → x^*. Now, condition (iii) implies x_n⊑x^* for all n. Putting x = x^* and y = x_n in (28), we get

(29)

Hence,

(30)

Since x_n → x^* and gx_n → x^*, then by Definition 9(1) and for c ≫ θ there exists N₀ ∈ ℕ such that for all n > N₀, d(x_n, x^*) ≪ c(1 − (a₂ + a₄))/2(a₁ + a₃ + a₄), and d(gx_n, x^*) ≪ c(1 − (a₂ + a₄))/2(a₂ + a₃ + a₄ + a₅). Then we have

(31)

Now again, according to Definition 9(1) it follows that gx_n → fx^*. It follows that fx^* = x^*. In a similar way and using that x^*⊑x^*, we can prove that gx^* = x^*. Therefore, f and g have a common fixed point.

Now, we present two examples to illustrate our results. In the first example (the case of a normal cone), the conditions of Theorem 12 are fulfilled, but Theorem 2 of Altun et al. [9, Theorem 12] cannot be applied. In the second example (the case of a nonnormal cone), the conditions of Theorem 12 are fulfilled, but Theorem 3 of Altun et al. [9, Theorem 13] cannot be applied.

Example 17. Let X = [0,1] endowed with the standard order and E = ℝ² and let P = {(x, y) : x, y ≥ 0}. Define d : X × X → E as in Example 8. Define f : X → X by f(x) = x²/6. Then f is a continuous and nondecreasing mapping with respect to ⊑. Then we have

(32)

where a₁ = 4/36, a₂ = a₃ = a₄ = a₅ = 0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, f has a fixed point x = 0.

Example 18. Let X = [0, ∞) endowed with the standard order and with and let P = {u ∈ E : u(t) ≥ 0 on [0,1]}. It is well known that this cone is solid, but it is not normal. Define a cone metric d : X × X → E by d(x, y)(t) = |x−y|²e^t. Then (X, d) is a complete cone b-metric space with the coefficient s = 2. Let us define f : X → X by f(x) = x/2. Then f is a continuous and nondecreasing mapping with respect to ⊑. Then we have f is an increasing mapping; also we have

(33)

where a₁ = 1/4, a₂ = 1/5, a₃ = a₄ = a₅ = 0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, f has a fixed point x = 0.

Acknowledgments

The authors would like to acknowledge the financial support received from Ministry of Higher Education, Malaysia, under the Research Grant no. ERGS/1/2011/STG/UKM/01/13. The authors thank the referee for his/her careful reading of the paper and useful suggestions.

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Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

Abstract

1. Introduction

2. Fixed Point Results

3. Common Fixed Point Results

Acknowledgments

References

Citing Literature

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Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

Abstract

1. Introduction

2. Fixed Point Results

3. Common Fixed Point Results

Acknowledgments

References

Citing Literature

References

Related

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