Fixed point results on metric-type spaces

Acta Mathematica Scientia 2014,34B(4):1237–1253 http://actams.wipm.ac.cn

FIXED POINT RESULTS ON METRIC-TYPE SPACES∗ Monica COSENTINO1

Peyman SALIMI 2

Pasquale VETRO1

1. Dipartimento di Matematica e Informatica, Universit` a degli Studi di Palermo, Via Archirafi, 34, 90123 Palermo, Italy 2. Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran E-mail : [email protected]; [email protected]; [email protected] Abstract In this paper we obtain fixed point and common fixed point theorems for selfmappings defined on a metric-type space, an ordered metric-type space or a normal cone metric space. Moreover, some examples and an application to integral equations are given to illustrate the usability of the obtained results. Key words

metric-type space; fixed point; common fixed point; Suzuki type mappings; cone metric space; integral equation

2010 MR Subject Classification

1

47H10; 54H25

Introduction

It is well known that the contraction mapping principle, formulated and proved in the Ph.D. dissertation of Banach in 1920, which was published in 1922 [1], is one of the most important theorems in classical functional analysis. Indeed it is widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics. Starting from these considerations, the study of fixed and common fixed points of mappings satisfying a certain metrical contractive condition attracted many researchers, see for example [2, 3]. The reader can also see [4–6], for existence results for fixed points of contractive non-self-mappings. The generalization of the above principle has been a heavily investigated branch of research. In recent years, several authors have obtained fixed and common fixed point results for various classes of mappings on the setting of many generalized metric spaces. Here, we focuses on one of these spaces. Precisely, the concept of the b-metric space appeared in some works, such as Bakhtin [7], Czerwik [8, 9] and several papers deal with the fixed point theory for single-valued and multi-valued operators in b-metric spaces. Seven years later, Khamsi [10] and Khamsi and Hussain [11] reintroduced such spaces under the name of metric-type spaces. They also proved some fixed point theorems in such spaces in the same work. The metric-type space (that is, b-metric space) is a symmetric space with some special properties. A metric-type space can also be regarded as a triplet (X, d, K), where (X, d) is a ∗ Received November 1, 2012; revised December 9, 2013. The third author is supported by Universit` a degli Studi di Palermo (Local University Project ex 60%).

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symmetric space and K ≥ 1 is a real number such that: d(x, z) ≤ K(d(x, y) + d(y, z)) for all x, y, z ∈ X. For some results of fixed and common fixed point in the setting of metric-type spaces see [10–17]. Following this direction of research, in Section 3 of the paper, we will present fixed point theorems of Edelstein type and of Suzuki type for self-mappings on sequential compact metrictype spaces. In Section 4, we will present a common fixed point theorem for self-mappings on sequential compact ordered metric-type spaces. In Sections 5 and 6, we will present fixed point theorems on sequential compact cone metric spaces and on complete metric-type spaces, respectively. Moreover, some examples and an application to integral equations are given to illustrate the usability of the obtained results.

2

Preliminaries The aim of this section is to present some notions and results used in the paper.

Definition 2.1 Let X be a non-empty set and d : X ×X → [0, +∞). (X, d) is a symmetric space (also called E-space) if and only if it satisfies the following conditions: (W1) d(x, y) = 0 if and only if x = y; (W2) d(x, y) = d(y, x) for all x, y ∈ X. Symmetric spaces differ from more convenient metric spaces in the absence of triangle inequality. Nevertheless, many notions can be defined similar to those in metric spaces. For instance, in a symmetric space (X, d) the limit point of a sequence {xn } is defined by lim d(xn , x) = 0

n→+∞

if and only if

lim xn = x.

n→+∞

Also, a sequence {xn } ⊂ X is said to be a Cauchy sequence if, for every given ε > 0, there exists a positive integer n(ε) such that d(xm , xn ) < ε for all m, n ≥ n(ε). A symmetric space (X, d) is said to be complete if and only if each of its Cauchy sequence converges to some x ∈ X. Definition 2.2 Let X be a nonempty set and let K ≥ 1 be a given real number. A function d : X × X → [0, +∞) is said to be a metric-type (or a b-metric) if and only if for all x, y, z ∈ X the following conditions are satisfied: 1. d(x, y) = 0 if and only if x = y; 2. d(x, y) = d(y, x); 3. d(x, z) ≤ K[d(x, y) + d(y, z)]. A triplet (X, d, K), is called a metric-type space. We observe that a metric-type space is included in the class of symmetric spaces. So the notions of convergent sequence, Cauchy sequence and complete space are defined as in symmetric spaces. A metric-type space (X, d, K) is sequentially compact if for any sequence {xn } in X, there exists a subsequence {xnk } of {xn } which converges to a point x ∈ X [11]. Next, we give some examples of metric-type spaces. Example 2.3 Let X = [0, 1] and d : X × X → [0, +∞) be defined by d(x, y) = (x − y)2 , for all x, y ∈ X. Clearly, (X, d, 2) is a metric-type space.

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Example 2.4 Let Cb (X) = {f : X → R : kf k∞ = sup |f (x)| < +∞} and let kf k = x∈X p 3 kf 3 k∞ . The function d : Cb (X) × Cb (X) → [0, +∞) defined by

d(f, g) = kf − gk for all f, g ∈ Cb (X) √ √ is a metric-type with constant K = 3 4 and so (Cb (X), d, 3 4) is a metric-type space. We note that if a, b are two nonnegative real numbers, then √ √ √ 3 3 (a + b)3 ≤ 4(a3 + b3 ) and a + b ≤ 3 a + b. This implies that d(f, g) ≤

√ 3 4(d(f, h) + d(h, g))

for all f, g, h ∈ Cb (X).

The following result is useful for some of the proofs in the paper. Lemma 2.5 Let (X, d, K) be a metric-type space and let {xn } be a sequence in X. If xn → y and xn → z, then y = z. Lemma 2.6 Let (X, d, K) be a metric-type space and let {xk }nk=0 ⊂ X. Then: d(xn , x0 ) ≤ Kd(x0 , x1 ) + · · · + K n−1 d(xn−2 , xn−1 ) + K n−1 d(xn−1 , xn ). From Lemma 2.6, we deduce the following lemma. Lemma 2.7 Let {yn } be a sequence in a metric-type space (X, d, K) such that d(yn , yn+1 ) ≤ λd(yn−1 , yn ) for some λ, 0 < λ < 1/K, and each n ∈ N. Then {yn } is a Cauchy sequence in X. Let X be a non-empty set. If (X, d, K) is a metric-type space and (X, ) is a partially ordered set, then (X, d, K, ) is called an ordered metric-type space. Then x, y ∈ X are called comparable if x  y or y  x holds. Let (X, ) be a partially ordered set. A self-mappings f is said to be dominated if f x  x for all x ∈ X and dominating if x  f x for all x ∈ X. An ordered metric-type space (X, d, K, ) has a sequential limit comparison property if for every decreasing sequence {xn } in X such that xn → x ∈ X, we have x ≺ xn .

3

Fixed Points for Self-Mappings on Sequentially Compact MetricType Spaces

In this section, we prove fixed point theorems of Edelstein type and of Suzuki type for self-mappings on the setting of sequential compact metric-type spaces. Theorem 3.1 Let (X, d, K) be a sequentially compact metric-type space and let f : X → X be such that δ d(f x, f y) < αd(x, y) + βd(x, f x) + γd(y, f y) + d(x, f y) + Ld(y, f x) (3.1) K for all x, y ∈ X, x 6= y, where α + β + γ + 2δ = 1, γ 6= 1 and L ≥ 0. If d and f are continuous, δ then f has a fixed point. Moreover, if α + K + L ≤ 1, then the fixed point of f is unique. Proof Let x0 ∈ X be an arbitrary point, and let {xn } be the Picard sequence of initial point x0 , i.e., xn = f n x0 = f xn−1 . If xn = xn−1 for some n ∈ N, then xn is a fixed point of f .

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Now, let dn = d(xn , xn+1 ) for all n ∈ N ∪ {0}. If xn 6= xn−1 for all n ∈ N, using the contractive condition (3.1) with x = xn−1 and y = xn , we get dn = d(xn , xn+1 ) = d(f xn−1 , f xn ) < αd(xn−1 , xn ) + βd(xn−1 , f xn−1 ) + γd(xn , f xn ) + = αd(xn−1 , xn ) + βd(xn−1 , xn ) + γd(xn , xn+1 ) +

δ d(xn−1 , f xn ) + Ld(xn , f xn−1 ) K

δ d(xn−1 , xn+1 ) + Ld(xn , xn ) K

δ d(xn−1 , xn+1 ) K ≤ (α + β)dn−1 + γdn + δ[dn−1 + dn ].

= αdn−1 + βdn−1 + γdn +

This implies that (1 − γ − δ)dn < (α + β + δ)dn−1 . From α + β + γ + 2δ = 1 and γ 6= 1, we deduce that 1 − γ − δ > 0 and so dn <

α+β+δ dn−1 = dn−1 . 1−γ−δ

Consequently, {dn } is a decreasing sequence of positive real numbers and hence there exists d∗ ≥ 0 such that limn→+∞ dn = d∗ . Using the sequentially compactness of X, there exists a subsequence {xni } of {xn } such that xni → x∗ ∈ X as i → +∞. Using the continuity of d and f , we deduce that dni = d(xni , xni +1 ) = d(xni , f xni ) → d(x∗ , f x∗ )

as i → +∞.

Since dni → d∗ , we obtain d∗ = d(x∗ , f x∗ ). Similarly, we deduce that dni +1 = d(xni +1 , xni +2 ) = d(f xni , f f xni ) → d(f x∗ , f f x∗ ) = d∗

as i → +∞.

If x∗ = f x∗ , then f has a fixed point. Assume that x∗ 6= f x∗ , that is d∗ > 0, then taking x = x∗ and y = f x∗ in (3.1), we have d∗ = d(f x∗ , f f x∗ ) < αd(x∗ , f x∗ ) + βd(x∗ , f x∗ ) + γd(f x∗ , f f x∗ ) +

δ d(x∗ , f f x∗ ) + Ld(f x∗ , f x∗ ) K

δ d(x∗ , f f x∗ ) K ≤ (α + β + γ)d∗ + δ[d(x∗ , f x∗ ) + d(f x∗ , f f x∗ )]

≤ (α + β + γ)d∗ +

= (α + β + γ + 2δ)d∗ = d∗ , which is a contradiction and hence d∗ = 0, that is, x∗ = f x∗ . Now, we prove the uniqueness of the fixed point. Assume that z ∈ X is a fixed point of f , different from x∗ . This means that d(z, x∗ ) > 0. Taking x = z and y = x∗ in (3.1), we have d(z, x∗ ) = d(f z, f x∗ ) < αd(z, x∗ ) + βd(z, f z) + γd(x∗ , f x∗ ) + = (α +

δ d(z, f x∗ ) + Ld(x∗ , f z) K

δ + L)d(z, x∗ ) ≤ d(z, x∗ ), K

which is a contradiction and hence z = x∗ .



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Example 3.2 Let X and d as in the Example 2.3. Define f : X → X by f (x) = 4(x21+1) . Clearly, (X, d, 2) is a sequentially compact metric-type space. Since 2 x+y d(f x, f y) = |x − y|2 < |x − y|2 = d(x, y) for all x, y ∈ X, x 6= y, 2 2 4(x + 1)(y + 1)

we deduce that all the hypotheses of the Theorem 3.1 are satisfied when α = 1, β = γ = δ = L = 0 and hence f has a unique fixed point. As a first corollary of Theorem 3.1, taking α = 1 and β = γ = δ = L = 0, we obtain Edelstein’s Theorem [18]. Further, putting α = δ = L = 0 and β + γ = 1 and β 6= 0, we obtain the following version of Kannan’s result. Its non-compact version can be found in [19]. Corollary 3.3 Let (X, d, K) be a sequentially compact metric-type space and let f : X → X be such that d(f x, f y) < βd(x, f x) + γd(y, f y) (3.2) holds for all x, y ∈ X, x 6= y, where nonnegative constants β and γ satisfy β + γ = 1, β 6= 0. If d and f are continuous, then f has a unique fixed point in X. An Edelstein version of the Chatterjea [20] fixed point theorem is obtained from the Theorem 3.1 putting α = β = γ = 0 and δ = 1/2. Corollary 3.4 Let (X, d, K) be a sequentially compact metric-type space and f : X → X such that 1 d(f x, f y) < d(x, f y) + Ld(y, f x) (3.3) 2K holds for all x, y ∈ X, x 6= y. If d and f are continuous, then f has a fixed point in X. If L ≤ 1/2, then the fixed point of f is unique. The following theorem is a Suzuki type result [21] in metric-type spaces. Theorem 3.5 Let (X, d, K) be a sequentially compact metric-type space and let f : X → X be such that 1 d(x, f x) < d(x, y) implies d(f x, f y) < d(x, y) (3.4) 2K for all x, y ∈ X. If d is continuous, then f has a unique fixed point. Proof

We put r = inf{d(x, f x) : x ∈ X}. We choose a sequence {xn } such that lim d(xn , f xn ) = r.

n→+∞

(3.5)

As X is sequentially compact, we may assume that xn → u and f xn → v with u, v ∈ X. Now, we prove that r = 0. Assume on the contrary that r > 0. Using the continuity of d, we get lim d(xn , v) = d(u, v) = lim d(xn , f xn ) = r n→+∞

n→+∞

and lim d(u, f xn ) = d(u, v) = lim d(xn , f xn ) = r.

n→+∞

n→+∞

Consequently, there exists n1 ∈ N such that 2 4 r < d(xn , v) and d(xn , f xn ) < r for all n ≥ n1 . 3 3 Then, for all n ≥ n1 , we have 1 1 4 1 2 1 d(xn , f xn ) < r= r < d(xn , v) ≤ d(xn , v), 2K 2K 3 K3 K

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and by condition (3.4), we obtain d(f xn , f v) < d(xn , v). From the previous inequality, as n → +∞, we get d(v, f v) = lim d(f xn , f v) ≤ lim d(xn , v) = d(u, v) = r. n→+∞

n→+∞

From the definition of r, we deduce that d(v, f v) = r. Since r > 0, v 6= f v. So 1 d(v, f v) < d(v, f v) 2K and by condition (3.4), we obtain d(f v, f f v) < d(v, f v) = r, which is a contradiction with respect to the definition of r. Thus r = 0 and hence u = v. Now, arguing by contradiction, we will prove that f has a fixed point. Assume on the contrary that f does not have fixed points. Since 1 d(xn , f xn ) < d(xn , f xn ) 2K by condition (3.4), we have d(f xn , f f xn ) < d(xn , f xn )

for all n ∈ N,

for all n ∈ N.

(3.6)

From d(u, f f xn ) ≤ K[d(u, f xn ) + d(f xn , f f xn )] ≤ K[d(u, f xn ) + d(xn , f xn )], as n → +∞, we deduce that f 2 xn → u, note that f xn → u. Now, suppose that there exists n ∈ N such that 1 1 d(xn , f xn ) ≥ d(xn , u), and d(f xn , f f xn ) ≥ d(f xn , u), 2K 2K then by (3.6) we get d(xn , f xn ) ≤ K[d(xn , u) + d(u, f xn )] 1 1 ≤K d(xn , f xn ) + K d(f xn , f f xn ) 2K 2K 1 1 < d(xn , f xn ) + d(xn , f xn ) 2 2 = d(xn , f xn ), which is a contradiction. Hence, for every n ∈ N, we have 1 d(xn , f xn ) < d(xn , u) 2K Thus, by (3.4), for each n ∈ N, d(f xn , f u) < d(xn , u)

or

or

1 d(f xn , f f xn ) < d(f xn , u). 2K

d(f f xn , f u) < d(f xn , u)

holds. Assume that the first inequality holds for every n ∈ J ⊂ N. If J is an infinite set, then d(u, f u) =

lim

n→+∞,n∈J

d(f xn , f u) ≤

lim

n→+∞,n∈J

d(xn , u) = 0

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and hence u = f u. The same conclusion holds if N \ J is an infinite set, in this case we use the second inequality. In both cases, we have shown that u is a fixed point of f . Now, we prove that f has a unique fixed point. Arguing by contradiction, we assume that z, w ∈ X are fixed points of f with z 6= w. Using condition (3.4) with x = z and y = w, we get d(z, w) = d(f z, f w) < d(z, w), which is a contradiction and hence z = w.



Theorem 3.5 is a proper extension of the following recent result of Hussain et al. [13]. Theorem 3.6 ([13], Theorem 4) Let (X, d, K) be a compact metric-type space, where the function d is continuous. Let T : X → X be a self-mapping, satisfying for all x, y ∈ X, x 6= y the condition 1 1 d(x, T x) < d(x, y) ⇒ d(T x, T y) < d(x, y). 1+K K Then T has a unique fixed point in X. Example 3.7 Let X and d as in the Example 2.3. Define f : X → X by f (x) = ax where 0 < a < 1. Since d(f x, f y) = a2 (x − y)2 < (x − y)2 = d(x, y)

for all x, y ∈ X, x 6= y,

and 14 d(x, f x) ≥ 0 = d(x, x) for all x ∈ X, then we get 1 d(x, f x) < d(x, y) implies d(f x, f y) < d(x, y) 4 for all x, y ∈ X. Thus all the hypotheses of the Theorem 3.5 are satisfied and hence f has a √ unique fixed point. Note that if a ≥ 1/ K, we cannot use the Theorem 3.6 to get that f has a fixed point Definition 3.8 The function F : [0, +∞) × [0, +∞) → [0, +∞) is called upper semicontinuous from the right if for each sequence {(xn , yn )} ⊂ [0, +∞) × [0, +∞) such that lim xn = x+ and lim yn = y + , it follows that n→+∞

n→+∞

lim sup F (xn , yn ) ≤ F (x, y). n→+∞

We denote by Ψ the set of all the functions ψ : [0, +∞) × [0, +∞) → [0, +∞) satisfying the following conditions: (ψ1) ψ is upper semi-continuous from the right; (ψ2) ψ(t, 0) ≤ t for all t ≥ 0. Let (X, d, K) be a metric-type space. We denote also by ΨL the set of all the functions αL : X × X → [0, +∞) satisfying the following conditions: (α1) if {xn } and {yn } are two sequences in (X, d, K) such that xn → x and yn → y, then lim sup αL (xn , yn ) ≤ αL (x, y); n→+∞

(α2) αL (x, y) = 0 when x = y. Theorem 3.9 Let (X, d, K) be a sequentially compact metric-type space and let f : X → X be such that
1 d(x, f x) < d(x, y) implies d(f x, f y) < ψ d(x, y), d(y, f x) + αL (y, f x) (3.7) 2K for all x, y ∈ X. If d is continuous, then f has a fixed point.

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We put r = inf{d(x, f x) : x ∈ X}. We choose a sequence {xn } such that

Proof

lim d(xn , f xn ) = r.

(3.8)

n→+∞

As X is sequentially compact, we may assume that xn → u and f xn → v with u, v ∈ X. Now, we prove that r = 0. Assume on the contrary that r > 0. Using the continuity of d, we get lim d(xn , v) = d(u, v) = lim d(xn , f xn ) = r n→+∞

n→+∞

and lim d(u, f xn ) = d(u, v) = lim d(xn , f xn ) = r.

n→+∞

n→+∞

Consequently, there exists n1 ∈ N such that 2 r < d(xn , v) 3 Then, for all n ≥ n1 , we have

and

d(xn , f xn ) <

4 r 3

for all n ≥ n1 .

1 1 4 1 2 1 d(xn , f xn ) < r= r < d(xn , v) ≤ d(xn , v), 2K 2K 3 K3 K and by condition (3.7), we obtain
d(f xn , f v) < ψ d(xn , v), d(v, f xn ) + αL (v, f xn ).

This implies


d(v, f v) = lim sup d(f xn , f v) ≤ lim sup ψ d(xn , v), d(v, f xn ) + lim sup αL (v, f xn ) n→+∞ n→+∞ n→+∞
≤ ψ d(u, v), 0 + αL (v, v) ≤ d(u, v) = r.

From the definition of r, we deduce that d(v, f v) = r. Since r > 0, v 6= f v. So 1 d(v, f v) < d(v, f v) 2K and by condition (3.7), we obtain

that is,


d(f v, f f v) < ψ d(v, f v), d(f v, f v) + αL (f v, f v), d(f v, f f v) < d(v, f v) = r,

which is a contradiction with respect to the definition of r. Thus r = 0 and hence u = v. Now, arguing by contradiction, we will prove that f has a fixed point. Assume on the contrary that f does not have fixed points. Since 1 d(xn , f xn ) < d(xn , f xn ) 2K by condition (3.7), we have

that is,

for all n ∈ N,


d(f xn , f f xn ) < ψ d(xn , f xn ), d(f xn , f xn ) + αL (f xn , f xn ), d(f xn , f f xn ) < d(xn , f xn )

for all n ∈ N.

From d(u, f f xn ) ≤ K[d(u, f xn ) + d(f xn , f f xn )] ≤ K[d(u, f xn ) + d(xn , f xn )],

(3.9)

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as n → +∞, we deduce that f 2 xn → u, note that f xn → u. Suppose that there exists n ∈ N such that 1 1 d(xn , f xn ) ≥ d(xn , u), and d(f xn , f f xn ) ≥ d(f xn , u), 2K 2K then by (3.9), we get d(xn , f xn ) ≤ K[d(xn , u) + d(u, f xn )] 1 1 ≤K d(xn , f xn ) + K d(f xn , f f xn ) 2K 2K 1 1 < d(xn , f xn ) + d(xn , f xn ) 2 2 = d(xn , f xn ), which is a contradiction. Hence, for every n ∈ N, we have 1 d(xn , f xn ) < d(xn , u), 2K Then, by (3.7), for each n ∈ N,

or

or

1 d(f xn , f f xn ) < d(f xn , u). 2K


d(f xn , f u) < ψ d(xn , u), d(u, f xn ) + αL (u, f xn )
d(f f xn , f u) < ψ d(f xn , u), d(u, f f xn ) + αL (u, f f xn )

holds. Assume that the first inequality holds for every n ∈ J ⊂ N. If J is an infinite set, then d(u, f u) =

lim sup d(f xn , f u) n→+∞,n∈J

≤


lim sup ψ d(xn , u), d(u, f xn ) + lim sup αL (u, f xn )

n→+∞,n∈J

≤0

n→+∞,n∈J

and hence u = f u. The same conclusion holds if N \ J is an infinite set, in this case we use the second inequality. In both cases, we have shown that u is a fixed point of f .  If in the above theorem we take αL (y, f x) = L min{d(y, f x), d(x, f x), d(y, f y)} with L ≥ 0 and ψ(t1 , t2 ) = t1 , then we have the following corollary. Corollary 3.10 Let (X, d, K) be a sequentially compact metric-type space and let f : X → X be such that 1 d(x, f x) < d(x, y) implies d(f x, f y) < d(x, y) + L min{d(y, f x), d(x, f x), d(y, f y)} 2K for all x, y ∈ X. If d is continuous, then f has a fixed point.

4

Fixed Points in Ordered Metric-Type Spaces

The existence of fixed points of self-mappings defined on certain type of ordered sets plays an important role in the order theoretic approach. It has been initiated in 2004 by Ran and Reurings [22], and further studied by Nieto and Rodr´ıguez-Lopez [23]. Then, several interesting and valuable results have appeared in this direction [24–28].

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Theorem 4.1 Let (X, d, K, ) be an ordered metric-type space such that d is continuous. Let f, g : X → X be such that f (X) ⊂ g(X), g(X) is a sequentially compact subspace of X, f is a dominated mapping and g is a dominating mapping. Suppose that d(f x, f y) < αd(gx, gy) + βd(gx, f x) + γd(gy, f y) +

δ d(gx, f y) + Ld(gy, f x) K

(4.1)

for every comparable elements x, y ∈ X, gx 6= gy, where α + β + γ + 2δ = 1, γ 6= 1 and L ≥ 0. If the following condition is satisfied: (1) X has a sequential limit comparison property, δ then f and g have a coincidence point in X. Moreover, if α + K + L ≤ 1, then the set of points of coincidence of f and g is well ordered if and only if f and g have one and only one point of coincidence. Proof

Let x0 ∈ X be an arbitrary point and let {xn } be defined as follows gxn+1 = f xn

for all n ∈ N ∪ {0}.

This can be done as the range of g contains the range of f . If d(gxn , gxn+1 ) = 0 for some n ∈ N ∪ {0}, then gxn = gxn+1 = f xn and so xn is a coincidence point of f and g. Assume that d(gxn , gxn+1 ) > 0 for all n ∈ N ∪ {0}. Using the property of the mappings f and g, we deduce xn+1  gxn+1 = f xn  xn  gxn

for all n ∈ N ∪ {0}.

Then xn and xn+1 are comparable for all n ∈ N ∪ 0. Since d(gxn , gxn+1 ) > 0, we obtain that gxn+1 ≺ gxn , for all n ∈ N ∪ {0}. Thus {gxn } is a decreasing sequence. Using the hypothesis that g(X) is a sequentially compact subspace of X, we can assume that gxn → gu for some u ∈ X. Now, condition (1) ensures that gu ≺ gxn for all n ∈ N ∪ {0}. We prove that f u = gu. We have d(gu, f u) = lim d(gxn+1 , f u) = lim d(f xn , f u) n→+∞

n→+∞

≤ lim [αd(gxn , gu) + βd(gxn , f xn ) + γd(gu, f u) + n→+∞

= γd(gu, f u) +

δ d(gxn , f u) + Ld(gu, f xn )] K

δ d(gu, f u) K

δ )d(gu, f u) K < d(gu, f u),

= (γ +

which is a contradiction and hence f u = gu. Therefore, u is a coincidence point of f and g. Now, suppose that the set of points of coincidence of f and g is well ordered. We claim that the point of coincidence of f and g is unique. Assume on the contrary that there exists another point v in X such that f v = gv with gu 6= gv. Assume that gu ≺ gv, then u  gu ≺ gv = f v  v and u, v are comparable. Now, using the condition (4.1), we get d(f u, f v) < αd(gu, gv) + βd(gu, f u) + γd(gv, f v) + δ + L)d(f u, f v) K ≤ d(f u, f v),

= (α +

δ d(gu, f v) + Ld(gv, f u) K

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which is a contradiction and hence gu = gv. The same holds if gv ≺ gu. Therefore f u = gu = z is the unique point of coincidence of f and g in X. Conversely, if f and g have one and only one point of coincidence, then the set of points of coincidence of f and g being singleton is well ordered.  Theorem 4.2 Adding to the hypotheses of the Theorem 4.1 the following conditions: (ii) if {gxn } is a decreasing sequence that converges to gu for some u ∈ X, then ggu  gu; (iii) f and g are weakly compatible; then f and g have a common fixed point in X. Moreover, f and g have a unique common fixed point in X if the set of points of coincidence of f and g is well ordered. Proof

Let x0 ∈ X be an arbitrary point and let {xn } be defined as follows gxn+1 = f xn

for all n ∈ N ∪ {0}.

Proceeding as in the proof of the Theorem 4.1, we deduce that {gxn } is a decreasing sequence that converges to gu for some u ∈ X and gu = f u = z. Using condition (ii), we have gz  gu. Since, the mappings f and g are weakly compatible we obtain that f z = f gu = gf u = gz. If gz = gu = z, then z is a common fixed point of f and g. If gz ≺ gu, then u, z are comparable and using the condition (4.1), we get gu = gz. So z is a common fixed point of f and g. If the set of points of coincidence of f and g is well ordered, then f and g have a unique point of coincidence and so z is a unique common fixed point of f and g. 

5

Fixed Points in Cone Metric Spaces

The fixed point theory in K-metric and K-normed spaces was developed by Perov and his consortiums ([29–31]). The main idea consists in using an ordered Banach space instead of the set of real numbers as the codomain for a metric. For more details on fixed point theory in K-metric and K-normed spaces, we refer the reader to [32]. Huang and Zhang [33] recently considered cone metric spaces, where the set of real numbers is replaced by an ordered Banach space E with a class of convergent sequences. They have established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, other authors [10, 24, 30, 34–38] generalized the results of Huang and Zhang and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. We recall the definition of cone metric space and the notion of convergence [33]. Let E be a real Banach space with θ as the zero element and P be a subset of E. The subset P is called an order cone if it has the following properties: (i) P is non-empty, closed and P 6= {θ}; (ii) 0 ≤ a, b ∈ R and x, y ∈ P ⇒ ax + by ∈ P ; (iii) P ∩ (−P ) = {θ}. For a given cone P ⊂ E, we can define a partial ordering  on E with respect to P by x  y if and only if y − x ∈ P . We shall write x ≺ y if x  y and x 6= y, while x ≪ y will stands for y − x ∈ IntP , where IntP denotes the interior of P. The cone P is called normal if there is a number K ≥ 1 such that for all x, y ∈ E θ  x  y ⇒ kxk ≤ K kyk .

(5.1)

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The least number K ≥ 1 satisfying (5.1) is called the normal constant of P. In the following we always suppose that E is a real Banach space and P is an order cone in E with IntP 6= ∅ and  is the partial ordering with respect to P. Definition 5.1 Let X be a non-empty set. Suppose that the mapping d : X × X → E satisfies the following conditions: (i) θ  d(x, y), for all x, y ∈ X, and d(x, y) = θ if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y)  d(x, z) + d(z, y), for all x, y, z ∈ X. Then d is called a cone metric on X, and (X, d) is called a cone metric space. Let {xn } be a sequence in X and x ∈ X. If for every c ∈ E, with θ ≪ c there is n0 ∈ N such that for all n ≥ n0 we have d(xn , x) ≪ c, then {xn } is said to be convergent to x and x is the limit of {xn } . We denote this by lim xn = x, or xn → x, as n → +∞. If for every c ∈ E n→+∞

with θ ≪ c there is n0 ∈ N such that for all n, m ≥ n0 we have d(xn , xm ) ≪ c, then {xn } is called a Cauchy sequence in X. If every Cauchy sequence is convergent in X, then X is called a complete cone metric space. Remark 5.2 (see, [33]) Let (X, d) be a cone metric space with respect to a normal cone P , then (i) {xn } ⊂ X converges to x ∈ X if and only if d(xn , x) → θ as n → +∞. (ii) {xn } is a Cauchy sequence if and only if d(xn , xm ) → θ as n, m → +∞. (iii) Let {xn } and {yn } be two sequences in X, x, y ∈ X and d(xn , x) → θ, d(yn , y) → θ as n → +∞. Then d(xn , yn ) → d(x, y) as n → +∞. The following lemma is a well-known result. Lemma 5.3 Let (X, d) be a cone metric space with respect to a normal cone P with normal constant K. Then θ  u ≪ v implies kuk < Kkvk. The following theorem is a Suzuki type result in normal cone metric spaces. Theorem 5.4 Let (X, d) be a sequentially compact cone metric space with respect to a normal cone P with normal constant K and let f : X → X be such that 1 d(x, f x) − d(x, y) ∈ / P implies 2K 2 for all x, y ∈ X. Then f has a unique fixed point.

d(f x, f y) ≪

1 d(x, y) K

(5.2)

Proof Define D(x, y) = kd(x, y)k. Since the cone P is normal, we say that D is a metrictype and (X, D, K) is a sequentially compact metric-type space. Let us prove that the mapping f satisfies the condition 1 D(x, f x) < D(x, y) 2K

implies

D(f x, f y) < D(x, y)

(5.3)

for all x, y ∈ X. 1 1 Suppose that 2K D(x, f x) < D(x, y) where x 6= y and we prove that 2K / 2 d(x, f x)− d(x, y) ∈ 1 1 P . Assume on the contrary that 2K 2 d(x, f x) − d(x, y) ∈ P . From d(x, y)  2K 2 d(x, f x), 1 1 we get kd(x, y)k ≤ 2K kd(x, f x)k, that is, D(x, y) ≤ 2k D(x, f x) which is a contradiction. 1 The assumption (5.2) implies that d(f x, f y) ≪ K d(x, y) which, by Lemma 5.3, means that 1 D(f x, f y) < D(x, y). On the other hand 2K D(x, f x) ≥ 0 = D(x, y) when x = y. Hence, the

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condition (3.4) is satisfied, using Theorem 3.5 the conclusion follows since D is continuous by the statement (iii) of Remark 5.2.  Proceeding as in the proof of Theorem 5.4, we deduce the following theorem. Theorem 5.5 Let (X, d) be a sequentially compact cone metric space with respect to a 1 normal cone P with normal constant K and let f : X → X be such that 2K / 2 d(x, f x)− d(x, y) ∈ P implies 1 d(x, y) + Lp min{kd(y, f x)k, kd(x, f x)k, kd(y, f y)k} K for all x, y ∈ X and Lp ∈ P . Then f has a fixed point. d(f x, f y) ≪

6

Fixed Points in Complete Metric-Type Spaces

In this section, we give some fixed point results in complete metric-type spaces for selfmappings that are Cα -Cβ -adimissible. Definition 6.1 Let f : X → X, α, β : X × X → [0, +∞) and Cα > 0, Cβ ≥ 0. We say that f is a Cα -Cβ -admissible mapping with respect to K ≥ 1 if the following conditions hold: (i) α(x, y) ≥ Cα implies α(f x, f y) ≥ Cα , x, y ∈ X; (ii) β(x, y) ≤ Cβ implies β(f x, f y) ≤ Cβ , x, y ∈ X; (iii) 0 ≤ Cβ /Cα < 1/K. Theorem 6.2 Let (X, d, K) be a complete metric-type space and let f : X → X be a Cα -Cβ -admissible mapping with respect to K ≥ 1. Assume that α(x, y)d(f x, f y) ≤ β(x, y)d(x, y)

(6.1)

for all x, y ∈ X. If the following conditions hold: (i) f is continuous; (ii) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ Cα and β(x0 , f x0 ) ≤ Cβ ; then f has a fixed point in X. Proof Let x0 ∈ X be such that α(x0 , f x0 ) ≥ Cα and β(x0 , f x0 ) ≤ Cβ . Define a sequence {xn } in X by xn = f n x0 = f xn−1 for all n ∈ N. If xn+1 = xn for some n ∈ N, then x = xn is a fixed point for f and the result is proved. Hence, we suppose that xn+1 6= xn for all n ∈ N. Since f is a Cα -Cβ -admissible mapping with respect to K ≥ 1 and α(x0 , f x0 ) = α(x0 , x1 ) ≥ Cα , we deduce that α(x1 , x2 ) = α(f x0 , f x1 ) ≥ Cα . By continuing this process, we get α(xn , xn+1 ) ≥ Cα for all n ∈ N ∪ {0}. Similarly, β(xn , xn+1 ) ≤ Cβ for all n ∈ N ∪ {0}. Now, using (6.1) with x = xn−1 and y = xn , we get Cα d(xn , xn+1 ) ≤ α(xn−1 , xn )d(xn , xn+1 ) ≤ β(xn−1 , xn )d(xn−1 , xn ) ≤ Cβ d(xn−1 , xn ), and hence d(xn , xn+1 ) ≤ C

Cβ d(xn−1 , xn ) Cα

for all n ∈ N.

Now, 0 ≤ Cαβ < 1/K, by the Lemma 2.7, implies that {xn } is a Cauchy sequence. Since X is complete, then there is z ∈ X such that d(xn , z) → 0 as n → +∞. Now, using the continuity

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of the mapping f , we deduce that d(xn+1 , f z) = d(f xn , f z) → 0 and, by Lemma 2.5, we get z = f z. Thus z is a fixed point of f .  Theorem 6.3 Let (X, d, K) be a complete metric-type space and f : X → X be a Cα -Cβ -admissible mapping with respect to K ≥ 1. Assume that α(x, y)d(f x, f y) ≤ β(x, y)d(x, y)

(6.2)

for all x, y ∈ X. If the following conditions hold: (i) there exists x0 ∈ X such that α(x0 , f x0 ) ≥ Cα and β(x0 , f x0 ) ≤ Cβ ; (ii) if {xn } is a sequence in X such that α(xn , xn+1 ) ≥ Cα and β(xn , xn+1 ) ≤ Cβ for all n ∈ N ∪ {0} and xn → x as n → +∞, then α(xn , x) ≥ Cα and β(xn , x) ≤ Cβ for all n ∈ N ∪ {0}; then f has a fixed point in X. Proof Let x0 ∈ X be such that α(x0 , f x0 ) ≥ Cα and β(x0 , f x0 ) ≤ Cβ . Define a sequence {xn } in X by xn = f n x0 = f xn−1 for all n ∈ N. Following the proof of the Theorem 6.2, we say that {xn } is a Cauchy sequence such that α(xn , xn+1 ) ≥ Cα and β(xn , xn+1 ) ≤ Cβ for all n ∈ N ∪ {0}. Since X is complete, then there is z ∈ X such that the sequence {xn } converges to z. Now, using the contractive condition (6.2) and condition (ii), we deduce that Cα d(xn+1 , f z)] Cα K ≤ Kd(z, xn+1 ) + α(xn , z)d(f xn , f z) Cα K ≤ Kd(z, xn+1 ) + β(xn , z)d(xn , z) Cα KCβ d(xn , z). ≤ Kd(z, xn+1 ) + Cα

d(z, f z) ≤ K[d(z, xn+1 ) +

Letting n → +∞, we obtain that d(z, f z) ≤ 0, that is, z = f z. Hence, f has a fixed point.  Example 6.4 Let X = [0, +∞) and d : X ×X → [0, +∞) be defined by d(x, y) = (x−y)2 . Clearly (X, d, 2) is a complete metric-type space. Let f : X → X be defined by   1 (x + x2 ) if x ∈ [0, 1] , f (x) = 6  esin x if x ∈ (1, +∞), and α, β : X × X → [0, +∞) be defined by   1 if x, y ∈ [0, 1], α(x, y) = 2  0 otherwise,

and

β(x, y) =

1 . 8

Now, we prove that all the hypotheses of the Theorem 6.3 are satisfied and hence f has a fixed point. Proof Let x, y ∈ X, if α(x, y) ≥ 21 then x, y ∈ [0, 1]. On the other hand for all w ∈ [0, 1], we have f w ≤ 1 and hence α(f x, f y) ≥ 21 . Since, β(x, y) ≤ 81 for all x, y ∈ [0, +∞), we have β(f x, f y) ≤ 81 . This implies that f is a Cα -Cβ -admissible mapping with respect to K = 2 for Cα = 1/2 and Cβ = 1/8. Clearly, α(0, f 0) ≥ 12 and β(0, f 0) ≤ 18 . Now, if {xn } is a sequence in X such that α(xn , xn+1 ) ≥ 21 for all n ∈ N ∪ {0} and xn → x as n → +∞, then {xn } ⊂ [0, 1] and hence x ∈ [0, 1]. This implies that α(xn , x) ≥ 12 and β(xn , x) ≤ 81 for all n ∈ N ∪ {0}.

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Let x, y ∈ [0, 1], then 1 1 |f x − f y|2 = |x − y|2 |1 + x + y|2 2 2 × 36 1 ≤ |x − y|2 = β(x, y)d(x, y). 8

α(x, y)d(f x, f y) =

Otherwise, α(x, y) = 0 and so 0 = α(x, y)d(f x, f y) ≤ β(x, y)d(x, y). Then, all the hypotheses of the Theorem 6.3 are satisfied and hence f has a fixed point.

7



Application to Existence of Solutions of Integral Equations

Let X = C([0, T ], R) be the set of real continuous functions defined on [0, T ] and d : X × X → [0, +∞) be defined by d(x, y) = k(x − y)2 k∞ for all x, y ∈ X. Then (X, d, 2) is a complete metric-type space. Consider the integral equation Z T x(t) = p(t) + S(t, s)f (s, x(s))ds

(7.1)

0

and let F : X → X be defined by F (x)(t) = p(t) +

Z

T

S(t, s)f (s, x(s))ds.

(7.2)

0

We assume that (A) f : [0, T ] × R → R is continuous; (B) p : [0, T ] → R is continuous; (C) S : [0, T ] × [0, T ] → [0, +∞) is continuous; (D) there exist η : X × X → [0, +∞) and θ : X × X → R such that if θ(x, y) ≥ 0 for x, y ∈ X, then for every s ∈ [0, T ] we have 0 ≤ |f (s, x(s)) − f (s, y(s))| ≤ η(x, y)|x(s) − y(s)|; (E) there exists x0 ∈ X and λ0 ∈ [0, 1/2) such that θ x0 , F (x0 )) ≥ 0 and

Z T

p



≤ λ0 ; S(t, s)η(x , F x )ds 0 0

0

∞

(F) if θ(x, y) ≥ 0, x, y ∈ X, then θ(F x, F y) ≥ 0 and if λ ∈ [0, 1/2)

Z T

Z T

√



≤ λ⇒

S(t, s)η(x, y)ds S(t, s)η(F x, F y)ds



0

∞

0

≤

√ λ;

∞

(G) if {xn } is a sequence in X such that θ(xn , xn+1 ) ≥ 0 for all n ∈ N ∪ {0} and xn → x as n → +∞, then θ(xn , x) ≥ 0 for all n ∈ N ∪ {0}. Theorem 7.1 Under the assumptions (A)–(G), the integral equation (7.1) has a solution in X = C([0, T ], R).

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Consider the mapping F : X → X defined by (7.2). By the condition (D), we 2  Z T |F (x)(t) − F (y)(t)| = S(t, s)[f (s, x(s)) − f (s, y(s))]ds 0 Z T 2 ≤ S(t, s)|f (s, x(s)) − f (s, y(s))|ds 2

0

≤

Z

≤

Z

T

p S(t, s)η(x, y) |x(s) − y(s)|2 ds

0 T 0

p S(t, s)η(x, y) k(x − y)2 k∞ ds 2

= k(x − y) k∞

Z

T

Z



T

S(t, s)η(x, y)ds

0

Then 2

2

k(F x − F y) k∞ ≤ k(x − y) k∞

0

Now, define α : X × X → [0, +∞) by   1 if θ(x, y) ≥ 0 α(x, y) = and  0 otherwise

Then for all x, y ∈ X, we have

2

2

2

.

2

S(t, s)η(x, y)ds

. ∞

Z

β(x, y) =

0

T

2

S(t, s)η(x, y)ds

. ∞

α(x, y)d(F (x), F (y)) ≤ β(x, y)d(x, y). Now, if we choose Cα = 1 and Cβ = λ0 , then it is easy to show that all the hypotheses of Theorem 6.3 are satisfied and hence the mapping F has a fixed point that is a solution in X = C([0, T ], R) of the integral equation (7.1).  References [1] Banach S. Sur les op´ erations dans les ensembles abstraits et leur application aux ´ equations int´ egrales. Fund Math, 1922, 3: 133–181 [2] Rus I A. Generalized Contractions and Applications. Cluj-Napoca: Cluj University Press, 2001 [3] Rus I A, Petru¸sel A, Petru¸sel G. Fixed Point Theory. Cluj-Napoca: Cluj University Press, 2008 [4] Reem D, Reich S, Zaslavski A J. Two results in metric fixed point theory. J Fixed Point Theory Appl, 2007, 1: 149–157 [5] Reich S, Zaslavski A J. A fixed point theorem for Matkowski contractions. Fixed Point Theory, 2007 8: 303–307 [6] Reich S, Zaslavski A J. A note on Rakotch contraction. Fixed Point Theory, 2008, 9: 267–273 [7] Bakhtin I A. The contraction mapping principle in quasimetric spaces (Russian). Func An, Gos Ped Inst Unianowsk, 1989, 30: 26–37 [8] Czerwik S. Contraction mappings in b-metric spaces. Acta Math Inform Univ Ostraviensis, 1993, 1: 5–11 [9] Czerwik S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem Mat Univ Modena, 1998, 46: 263–276 [10] Khamsi M A. Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl, 2010, 2010: Article ID 315398, 7 pages [11] Khamsi M A, Hussain N. KKM mappings in metric type spaces. Nonlinear Anal, 2010, 73: 3123–3129 [12] Cicchese M. Questioni di completezza e contrazioni in spazi metrici generalizzati. Boll Un Mat Ital, 1976, 5: 175–179

No.4

M. Cosentino et al: FIXED POINT ON METRIC-TYPE SPACES

1253

[13] Hussain N, Dori´ c D, Kadelburg Z, Radenovi´ c S. Suzuki-type fixed point results in metric type spaces. Fixed Point Theory Appl, 2012, 2012:126 [14] Hussain N, Shah M H. KKM mappings in cone b-metric spaces. Comput Math Appl, 2011, 62: 1677–1684 [15] Jovanovic M, Kadelburg Z, Radenovic S. Common fixed point results in metric-type spaces. Fixed Point Theory Appl, 2010, 2010: Article ID 978121, 15 pages [16] Shah M H, Hussain N. Nonlinear contractions in partially ordered quasi b-metric spaces. Commun Korean Math Soc, 2012, 27: 117–128 [17] Shah M H, Simic S, Hussain N, Sretenovic A, Radenovi´ c S. Common fixed points theorems for occasionally weakly compatible pairs on cone metric type spaces. J Comput Anal Appl, 2012, 14: 290–297 [18] Edelstein M. On fixed and periodic points under contractive mappings. J London Math Soc, 1962, 37:74–79 [19] Kannan R. Some results on fixed points. Bull Calcutta Math Soc, 1968, 60: 71–76 [20] Chatterjea S K. Fixed-point theorems. C R Acad Bulgare Sci, 1972, 25: 727–730 [21] Suzuki T. A new type of fixed point theorem in metric spaces. Nonlinear Anal, 2009 71: 5313–5317 [22] Ran A C M, Reurings M C. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc Amer Math Soc, 2004, 132: 1435–1443 [23] Nieto J J, Rodr´ıguez-L´ opez R. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order, 2005, 22: 223–239 [24] Altun I, Durmaz G. Some fixed point theorems on ordered cone metric spaces. Rend Circ Mat Palermo, 2009, 58: 319–325 [25] Nieto J J, Rodr´ıguez-L´ opez R. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math Sin (English Ser), 2007, 23: 2205–2212 [26] Nieto J J, Pouso R L, Rodr´ıguez-L´ opez R. Fixed point theorems in ordered abstract spaces. Proc Amer Math Soc, 2007, 132: 2505–2517 [27] Paesano D, Vetro P. Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topology Appl, 2012, 159: 911–920 [28] Saadati R, Vaezpour S M, Vetro P, Rhoades B E. Fixed point theorems in generalized partially ordered G-metric spaces. Math Comput Modelling, 2010, 52: 797–801 [29] Kvedaras B V, Kibenko A V, Perov A I. On some boundary value problems. Litov Matem Sbornik, 1965, 5: 69–84 [30] Perov A I. The Cauchy problem for systems of ordinary differential equations//Approximate Methods of Solving Differential Equations (Russian). Kiev: Naukova Dumka, 1964: 115–134 [31] Perov A I, Kibenko A V. An approach to studying boundary value problems (Russian). Izvestija AN SSSR, Seria Math, 1966, 30: 249–264 [32] Zabrejko P P. K-metric and K-normed linear spaces: Survey. Collect Math, 1997, 48: 825–859 [33] Huang L-G, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl, 2007, 332: 1468–1476 [34] Arshad M, Azam A, Vetro P. Some common fixed point results in cone metric spaces. Fixed Point Theory Appl, 2009, 2009: Article ID 493965, 11 pages [35] Azam A, Arshad M, Beg I. Common fixed points of two maps in cone metric spaces. Rend Circ Mat Palermo, 2008, 57: 433–441 [36] Di Bari C, Vetro P. ϕ-pairs and common fixed points in cone metric spaces. Rend Circ Mat Palermo, 2008, 57: 279–285 [37] Di Bari C, Vetro P. Weakly ϕ-pairs and common fixed points in cone metric spaces. Rend Circ Mat Palermo, 2009, 58: 125–132 [38] Vetro P. Common fixed points in cone metric spaces. Rend Circ Mat Palermo, 2007, 56: 464–468