Nonlinear Analysis 74 (2011) 2591–2601

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

On cone metric spaces: A surveySlobodanka Janković a, Zoran Kadelburg b, Stojan Radenović c,∗

a Mathematical Institute SANU, Knez Mihailova 36, 11001 Beograd, Serbiab University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Beograd, Serbiac University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Beograd, Serbia

a r t i c l e i n f o

Article history:Received 20 November 2009Accepted 10 December 2010

MSC:47H1054H25

Keywords:Normal coneNon-normal coneSolid coneCone metric spaceFixed pointCommon fixed point

a b s t r a c t

Using an oldM. Krein’s result and a result concerning symmetric spaces from [S. Radenović,Z. Kadelburg, Quasi-contractions on symmetric and cone symmetric spaces, Banach J.Math.Anal. 5 (1) (2011), 38–50], we show in a very short way that all fixed point results incone metric spaces obtained recently, in which the assumption that the underlying cone isnormal and solid is present, can be reduced to the corresponding results in metric spaces.On the other hand, when we deal with non-normal solid cones, this is not possible. Inthe recent paper [M.A. Khamsi, Remarks on cone metric spaces and fixed point theoremsof contractive mappings, Fixed Point Theory Appl. 2010, 7 pages, Article ID 315398,doi:10.1115/2010/315398] the author claims that most of the cone fixed point results aremerely copies of the classical ones and that any extension of known fixed point resultsto cone metric spaces is redundant; also that underlying Banach space and the associatedcone subset are not necessary. In fact, Khamsi’s approach includes a small class of resultsand is very limited since it requires only normal cones, so that all results with non-normalcones (which are proper extensions of the corresponding results for metric spaces) cannotbe dealt with by his approach.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

There aremany generalizations ofmetric spaces:Menger spaces, fuzzymetric spaces, generalizedmetric spaces, abstract(cone) metric spaces or K -metric and K -normed spaces, rectangular metric and rectangular cone metric spaces . . .

In 2007, Huang and Zhang [1] introduced cone metric spaces, being unaware that they already existed under the nameK -metric, and K -normed spaces that were introduced and used in the middle of the 20th century in [2–8]. In both cases theset R of real numbers was replaced by an ordered Banach space E. However, Huang and Zhang went further and definedthe convergence via interior points of the cone by which the order in E is defined. This approach allows the investigation ofcone spaces in the case when the cone is not necessarily normal. And yet, they continued with results concerned with thenormal cones only. One of the main results from [1] is the following Banach Contraction Principle in the setting of normalcone spaces:

Theorem 1.1 ([1]). Let (X, d) be a complete cone metric space over a normal solid cone. Suppose that a mapping T : X → Xsatisfies the contractive condition

d(Tx, Ty) ≼ λ · d(x, y) (1.1)

∗ Corresponding author.E-mail addresses: bobaj@mi.sanu.ac.rs (S. Janković), kadelbur@matf.bg.ac.rs (Z. Kadelburg), radens@beotel.net (S. Radenović).

0362-546X/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2010.12.014

2592 S. Janković et al. / Nonlinear Analysis 74 (2011) 2591–2601

for all x, y ∈ X, where λ ∈ [0, 1) is a constant. Then T has a unique fixed point in X and, for any x ∈ X, the iterative sequenceT nx converges to the fixed point.

Soon after this work many articles concerned with cone spaces with normal cones appeared (see [9–53]). The resultswere focused on fixed points of one or two self-mappings of the cone space, accompaniedmostly with the usual contractiveconditions known from the setting of ordinary metric spaces (Kannan, Zamfirescu, Jungck, Rhoades, . . . ).

The purpose of this paper is to show, using some old and well known results (Lemma 2.6 below) on cones in Banachspaces ([54], 1940), that most of the results of these articles can be reduced to the corresponding results from metric spacetheory. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case whenthe underlying cone is non-normal but just has interior that is nonempty. In this case only, proper generalizations of resultsfrom the ordinary metric spaces can be obtained. We present some examples showing that theorems from ordinary metricspaces cannot be applied in the setting of cone metric spaces, when the cone is non-normal. Such is Theorem 1.1. Namely,applying Lemma 2.6 in the case of a normal cone, this theorem reduces to the classical Banach Contraction Principle, provedin 1922. However, if in Theorem 1.1 only nonemptiness of the interior of cone is assumed, then the obtained result is moregeneral and it is a proper extension of the classical Banach Contraction Principle. Similar approach was applied in [55–83].

We note that there exist other ways of reducing some cone metric results to the respective results in metric spaces,see [84–89], and a remark at the end of the paper.

2. Preliminaries

2.1. Cones

Let E be a vector space over the real field R. A nonempty convex subset P of E is called a cone if λP ⊂ P for all λ ≥ 0.Clearly a cone P in E determines a transitive and reflexive relation ‘‘≼’’ by

x ≼ y if y − x ∈ P; (2.1)

moreover this relation is compatible with the vector structure, i.e.,(a) if θ ≼ x and θ ≼ y then θ ≼ x + y,(b) if θ ≼ x then θ ≼ λx, for all λ ≥ 0.

The relation ‘‘≼’’ determined by the cone P is called the vector (partial) ordering in E, and the pair (E, P) (or (E, ≼))is referred to as a (partially) ordered vector space. Conversely, if ‘‘≼’’ is a relation in E which is transitive, reflexive, andcompatible with the vector structure of E, and if we define

P = x ∈ E : θ ≼ x, (2.2)

then P is a cone in E, and ‘‘≼’’ is exactly the vector ordering of E induced by P .A cone P is said to be proper if P ∩ (−P) = θ. The vector ordering ‘‘≼’’ of E, induced by P , is antisymmetric if and only

if P is proper.

Example 2.1. (a) The usual ordering in R is the one induced by the proper cone R+ of all nonnegative real numbers.(b) If E is the vector space of real-valued functions defined on a set S, then the usual ordering in E is defined pointwise:

f ≼ g in E ⇔ f (s) ≤ g(s) for all s in S. (2.3)

In particular, if S is a topological space then the ordering in the space of all continuous real-valued functions on S isdefined in this way.

(c) Another example of an ordering defined pointwise is the one in sequence spaces. The usual ordering in the vector spaceω of all sequences λn of real numbers is the one induced by the cone P , consisting of all sequences µn, where eachµn ≥ 0. The following subspaces of ω are endowed with the relative ordering:m: the space of all bounded sequences of real numbers;c: the space of all convergent sequences of real numbers;c0: the space of all null sequences (that is, sequences converging to zero);lp: the space of all lp-summable sequences of real numbers.

(d) Let (Ω, µ) be a measure space and let E be the vector space of all measurable real-valued functions on Ω . As usual,functions that are equal µ-almost everywhere are identified. For f , g in E, define

f ≼ g ⇔ f (x) ≤ g(x) for µ-almost every x in Ω. (2.4)

The subspace of all Lp-summable functions in E is endowedwith the relative ordering, where p is a positive real number.

For a pair of elements x, y in E such that x ≼ y, put

[x, y] = z ∈ E : x ≼ z ≼ y. (2.5)

The sets of the form [x, y] are called order-intervals. It can be easily verified that order-intervals are convex. A subset in (E, P)is said to be order-bounded if it is contained in some order-interval in E. A subset A of E is said to be order-convex (or full) if[a1, a2] ⊂ A, whenever a1, a2 ∈ A and a1 ≼ a2.

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2.2. Ordered topological vector spaces

Let E be a topological vector space, and let P be a nonempty proper cone in E. The pair (E, P) is an ordered topologicalvector space. The underlying cone P of an ordered topological vector space is said to be solid if it has a nonempty interior.We present a result concerning this notion.

Lemma 2.2 ([90]). Let (E, P) be an ordered topological vector space. Then c is an interior point of P if and only if [−c, c] is aneighborhood of θ in E.

For the proof see [90–93].Many important spaces in functional analysis have a cone with the empty interior. These are, for example, c0, lp (p > 0),

Lp (p > 0).Ordered topological vector space (E, P) is order-convex if its base of neighborhoods of zero consists of order-convex

subsets. In this case the cone P is said to be normal, or P-saturated.The following result is of special interest:

Theorem 2.3 ([91]). If the cone P of an ordered topological vector space (E, P) is normal and solid, then (E, P) is an orderednormed space.

Some authors attempt to generalize cone metric spaces by replacing an ordered Banach space (with a normal solid cone)by an ordered topological vector space (also with normal solid cone) (see, e.g., [62]). It follows from Theorem 2.3 that this isnot a generalization at all.

Remark 2.4. There exists an ordered topological vector (order-convex) space that is not normed, but it has a non-normaland a solid cone. Such is the space C1

R[0, 1] equipped with the strongest linear topology, that is, with the strongest locallyconvex topology.

Namely, it is well known that the cone P in C1R[0, 1] is non-normal (see Example 2.7 below), but it has a nonempty

interior. Therefore, it has a nonempty interior with respect to every topology that is finer than the normed topology. Ifit were normal in the strongest linear (locally convex) topology, then the space C1

R[0, 1] would be normed in each of thementioned topologies, which is a contradiction (see [93]—the space with such a topology is not even metrizable).

It follows that it ismeaningful to generalize conemetric spaces by replacing Banach space (with a non-normal solid cone)with an ordered topological vector space (also with a non-normal solid cone) [61].

2.3. Normality of cones in ordered Banach spaces

Definition 2.5 ([92]). A cone P in a Banach space E is called:1 normal if

inf‖x + y‖ : x, y ∈ P, ‖x‖ = ‖y‖ = 1 > 0; (2.6)2 semi-monotone if there exists K > 0 such that, for all x, y ∈ E,

θ ≼ x ≼ y implies ‖x‖ ≤ K‖y‖; (2.7)3 monotone if, for all x, y ∈ E,

θ ≼ x ≼ y implies ‖x‖ ≤ ‖y‖, (2.8)

i.e. it is semi-monotone with K = 1.

The next lemma contains results on cones in ordered Banach spaces that are rather old (1940, see [54]). It is interestingthat most authors (working with normal cones after 2007) do not use these results, which can be applied to reduce a lot ofresults to the setting of ordinary metric spaces.

Lemma 2.6 ([54]). The following conditions are equivalent for a cone P in the Banach space E with the norm ‖ · ‖:1 P is normal;2 for arbitrary sequences xn, yn, zn in E,

(∀n) xn ≼ yn ≼ zn and limn→∞

xn = limn→∞

zn = x imply limn→∞

yn = x; (2.9)

3 is semi-monotone;4 there exists a norm ‖ · ‖1 on E, equivalent to the given norm ‖ · ‖, such that the cone P is monotone w.r.t. ‖ · ‖1.

Proof of this important lemma can be found in [90,92–96].The minimal constant K satisfying (2.7) will be called the normal constant of P .

Example 2.7 ([91]). Let E = C1R[0, 1], with ‖x‖ = ‖x‖∞ + ‖x′

‖∞, P = x ∈ E : x(t) ≥ 0. This cone is non-normal.Consider, for example, xn(t) =

tnn and yn(t) =

1n . Then θ ≼ xn ≼ yn, and limn→∞ yn = θ , but ‖xn‖ = maxt∈[0,1] |

tnn | +

maxt∈[0,1] |tn−1| =

1n + 1 > 1; hence xn does not converge to zero. It follows by 2 that P is a non-normal cone.

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3. Cone metric spaces

Let E be a Banach space with a solid cone P . We shall write x ≪ y if y − x ∈ int P (this notation was introduced in [97]).

Definition 3.1 ([1]). Let X be a nonempty set. Suppose that a mapping d : X × X → E satisfies the following conditions:

(d1) θ ≼ d(x, y) for all x, y ∈ X , and d(x, y) = θ if and only if x = y;(d2) d(x, y) = d(y, x), for all x, y ∈ X;(d3) 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.

The concept of a conemetric space ismore general than that of ametric space, because eachmetric space is a conemetricspace where E = R and P = [0, +∞).

Example 3.2. 1 [1] Let E = R2, P = (x, y) ∈ R2: x ≥ 0, y ≥ 0, X = R, and let d : X × X → E be defined by

d(x, y) = (|x−y|, α|x−y|), where α ≥ 0 is a constant. Then (X, d) is a conemetric space with the normal cone P , wherethe normal constant K = 1.

2 For other examples of cone metric spaces, i.e. K -metric spaces, one can see [98], pp. 853–854.

Definition 3.3 ([1]). Let (X, d) be a cone metric space. We say that a sequence xn in X is:

(i) a Cauchy sequence if, for every c in E with θ ≪ c , there is an N such that for all n,m > N , d(xn, xm) ≪ c;(ii) a convergent sequence if, for every c in E with θ ≪ c , there is an N such that for all n > N , d(xn, x) ≪ c for some fixed x

in X .

A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X .

Let us mention [1] that if P is a normal solid cone, then xn is a Cauchy sequence if and only if ‖d(xn, xm)‖ → 0,n,m → ∞. Further, xn converges to x ∈ X if and only if ‖d(xn, x)‖ → 0, n → ∞.

4. Symmetric space (X,D) and topologies td and tD

If (X, d) is a cone metric space with a normal solid cone (with the normal constant satisfying K ≥ 1), then D(x, y) =

‖d(x, y)‖ is a symmetric on the set X , that is, a mapping from X × X into [0, +∞) with the following properties:

(s1) D(x, y) ≥ 0 and D(x, y) = 0 if and only if x = y;(s2) D(x, y) = D(y, x).

It satisfies also:(s3) D(x, y) ≤ K(D(x, z) + D(z, y)).

The cone metric d and the associated symmetric D = ‖d‖ in X generate two topologies: td and tD. Their bases ofneighborhoods consist of the sets

Bc(y) = x ∈ X : d(x, y) ≪ c and Bε(y) = x ∈ X : D(x, y) < ε, (4.1)

where y, c, ε are, respectively, a given point from X , a given interior point from P , and a given positive number (for detailssee [45]).

Theorem 4.1 ([45]). Let (X, d) be a cone metric space with a normal solid cone P and let D be the associated symmetric. Thentd = tD.

In other words, the spaces (X, d) and (X,D) have the same collections of open, closed, bounded and compact sets, andalso the same convergent and Cauchy sequences, and the same continuous functions. Also, the interior of the cone is thesame in both equivalent norms. If the normal constant K = 1, then the symmetric space (X,D) is a metric space.

Thus we obtain the following principal remark: From Lemma 2.6 and Theorem 4.1 it is obvious that, in the investigationof cone metric spaces with normal solid cones, we can assume that the normal constant (often mentioned in the articles [1,9–53]) can be taken to be K = 1. This follows from the fact that we can deal with the space E, equipped with the norm‖ · ‖1, which is equivalent to ‖ · ‖. Taking into account Lemma 2.6 and Theorem 4.1, it follows that results for normal solidcone metric spaces can be derived from the respective results for metric spaces. Namely, if d is a cone metric on X and thenorm ‖ · ‖ is monotone on the cone, then the composition ‖ · ‖ d = D is a usual metric on X . Clearly, non-expansive andcontractive (with respect to d) self-mappings of X are also non-expansive and contractive (respectively) with respect to themetric ‖ · ‖ d = D.

We conclude that Lemma2.6 (the result of Krein from1940) shows thatmany results published after 2007 and concernedwith the normal solid cone spaces, look different now. Especially the results on fixed points with one or twomappings, withnon-expansive or contractive conditions, can be directly reduced to the known results in metric spaces.

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5. Application to the known results

In what follows we present some results from [1,9–53], to which Lemma 2.6 and the metric space (X,D) are applicable.In fact, in every contractive condition the letter ‘‘d’’ becomes ‘‘D’’, while the relation ‘‘≼’’ in the cone P becomes the ordinary‘‘≤’’ in the set [0, +∞). So, instead of the cone metric space (X, d), we are dealing with the metric space (X,D).

Theorem 5.1 ([1]). Let (X, d) be a complete cone metric space over a normal solid cone. Suppose that the mapping T : X → Xsatisfies either of the contractive conditions

d(Tx, Ty) ≼ λ · (d(Tx, x) + d(Ty, y)) (5.1)d(Tx, Ty) ≼ λ · (d(Tx, y) + d(Ty, x)) (5.2)

for all x, y ∈ X, where λ ∈ [0, 12 ) is a constant. Then T has a unique fixed point in X and for any x ∈ X, the iterative sequence

T nx converges to the fixed point.

Theorem 5.2 ([10]). Let (X, d) be a cone metric space over a normal solid cone. Suppose that the mappings f , g : X → X satisfyeither of the following contractive conditions:

d(fx, fy) ≼ λ · d(gx, gy), (5.3)d(fx, fy) ≼ λ · (d(fx, gx) + d(fy, gy)), (5.4)d(fx, fy) ≼ λ · (d(fx, gy) + d(fy, gx)), (5.5)

for all x, y ∈ X, where λ is a constant, λ ∈ [0, 1) in the case (5.3) and λ ∈ [0, 12 ) in the cases (5.4) and (5.5). If the range of g

contains the range of f and g(X) is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover iff and g are weakly compatible, then f and g have a unique common fixed point.

Recall that a point y ∈ X is called a point of coincidence for mappings f , g : X → X if there exists x ∈ X such thatf (x) = g(x) = y. Then, x is called a coincidence point. Mappings f and g are called weakly compatible if they commute attheir coincidence points.

Definition 5.3 ([9]). Let f , g : X → X be mappings. f is a g-weak contraction if

d(f (x), f (y)) ≼ αd(f (x), g(x)) + βd(f (y), g(y)) + γ d(g(x), g(y)), (5.6)

for all x, y ∈ X , where α, β, γ ∈ [0, 1) and α + β + γ < 1.

Theorem 5.4 ([9]). Let (X, d) be a cone metric space over a normal solid cone, f , g : X → X such that f (X) ⊂ g(X). Supposethat f is a g-weak contraction and that f and g are weakly compatible. If f (X) or g(X) is a complete subspace of X, then themappings f and g have a unique common fixed point in X. Moreover, for any x0 ∈ X, the f -g-sequence f (xn) with initial pointx0 converges to the fixed point.

Definition 5.5 ([11,19]). Let (X, d) be a cone metric space, and let g, f : X → X . Then, g is called an f -quasi-contraction(resp. quasi-contraction) if for some constant λ ∈ [0, 1) and for every x, y ∈ X , there exists

u ∈ C(f ; x, y) = d(fx, fy), d(fx, gx), d(fx, gy), d(fy, gy), d(fy, gx), (resp. (5.7)u ∈ C(x, y) = d(x, y), d(x, gx), d(x, gy), d(y, gy), d(y, gx)), (5.8)

such that d(gx, gy) ≼ λ · u.

Theorem 5.6. (a) [11] Let (X, d) be a complete cone metric space over a normal solid cone and let g, f : X → X. Let g be anf -quasi-contraction, f commutes with g, f or g is continuous and g(X) ⊂ f (X). Then f and g have a unique common fixedpoint in X.

(b) [19] Let (X, d) be a complete cone metric space over a normal solid cone. Suppose the mapping f : X → X is a quasi-contraction. Then f has a unique fixed point in X and for any x ∈ X, iterative sequence f nx converges to the fixed point.

Theorem 5.7 ([14]). Let (X, d) be a complete conemetric space over a normal solid cone. Suppose that f and g are two self-mapsof X satisfying

d(fx, gy) ≼ αd(x, y) + β[d(x, fx) + d(y, gy)] + γ [d(x, gy) + d(y, fx)] (5.9)

for all x, y ∈ X, where α, β, γ ≥ 0 and α + 2β + 2γ < 1. Then f and g have a unique common fixed point in X. Moreover, anyfixed point of f is a fixed point of g, and conversely.

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Theorem 5.8 ([13]). Let (X, d) be a complete cone metric space over a normal solid cone, β ∈ [0, 1), and let f : X → X be acontinuous function such that for every x ∈ X, there is an n(x) ∈ N such that

d(f n(x)x, f n(x)y) ≼ βd(x, y), (5.10)

for every y ∈ X. Then f has a unique fixed point u in X and limn→∞ f n(x0) = u, for every x0 ∈ X.

Theorem 5.9 ([46]). Let (X, d) be a complete cone metric space over a regular cone P and let f : X → X be such that for every0 = c ∈ P, there exists 0 ≪ d such that d(fx, fy) ≺ c, for all x, y ∈ X with d(x, y) ≺ c + d. Then f has a unique fixed point.

Recall that the cone P is called regular if every increasing sequence in E is convergent, provided it is bounded from above.Each regular cone is normal, but the converse is not true.

Theorem 5.10 ([16]). Let (X, ⊑) be a partially ordered set and suppose that there exists a cone metric d on X such that the conemetric space (X, d) over a normal solid cone is complete. Let f : X → X be a continuous and nondecreasing mapping w.r.t. ⊑.Suppose that the following two assertions hold:

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

Then f has a fixed point in X.

Denote by N(X) the collection of all nonempty subsets of X , by C(X) the collection of all nonempty closed subsets of Xand by K(X) the collection of all nonempty sequentially compact subsets of X . An element x ∈ X is said to be a fixed pointof a set-valued map T : X → N(X), if x ∈ Tx. Put Fix(T ) = x ∈ X : x ∈ Tx. An element x ∈ X is said to be an endpointof T , if Tx = x. We denote by End(T ) the set of all endpoints of T . Also, for x ∈ X , put R(x, Tx) = d(x, z) : z ∈ Tx , andS(x, Tx) = u ∈ R(x, Tx) : ‖u‖ = infv∈S(x,Tx) ‖v‖.

Theorem 5.11 ([27]). Let (X, d) be a complete cone metric space over a normal solid cone and let the function I : X → R,defined by I(x) = infy∈Tx ‖d(x, y)‖, x ∈ X, be lower semi-continuous.

(a) Let T : X → C(X). If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that

∀x ∈ X∃y ∈ Tx∃v ∈ R(y, Ty)∀u ∈ R(x, Tx)[bd(x, y) ≼ u] ∧ [v ≼ λd(x, y)], (5.11)

then Fix(T ) = ∅.(b) Let T : X → K(X). If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that

∀x ∈ X∃y ∈ Tx∃v ∈ S(y, Ty)∀u ∈ S(x, Tx)[bd(x, y) ≼ u] ∧ [v ≼ λd(x, y)], (5.12)

then Fix(T ) = ∅.(c) Let T : X → K(X). If there exist λ ∈ [0, 1), b ∈ (λ, 1] such that

∀x ∈ X∃y ∈ Tx∃v ∈ S(y, Ty)∀u ∈ S(x, Tx)[bd(x, y) ≼ u] ∧ [v ≼ λd(x, y)], (5.13)

then Fix(T )End(T ) = ∅.

Theorem 5.12 ([22]). Let (X, d) be a complete cone metric space over a regular solid cone P, and let T : X → C(X). Assumethat there exist ϕ : P → [0, 1) and a dynamic process xn ∈ Din(T , x0) for some point x0 ∈ X such that

(i) for any sequence rn ⊂ P, lim supn→∞ ϕ(rn) < 1, provided rn+1 ≼ rn, n ∈ N,(ii) d(xn, xn+1) ≼ ϕ(d(xn−1, xn))d(xn−1, xn) for all n ∈ N.

Then:

(a) there exists x∗∈ X such that limn→∞ xn = x∗,

(b) x∗ is a fixed point of T , provided the mapping f (x) = infy∈Tx ‖d(x, y)‖, x ∈ X, is Din(T , x0)-dynamic lower semi-continuousat x∗,

(c) x∗ is a unique fixed point of T provided

n∈N T n(X) is a singleton.

According to Lemma 2.6, Theorem 4.1 and our principal remark from Section 4, the results mentioned above can bereduced to the well known results from the setting of metric spaces. As an illustration, let us prove Theorem 5.9 usingthese observations.

Proof. Let ε > 0. Then there exists c ∈ int P such that ‖c‖ < ε. Now, we have that for δ = ‖d‖,

D(x, y) = ‖d(x, y)‖ ≤ ‖c + d‖ ≤ ‖c‖ + ‖d‖ < ε + δ

⇒ D(fx, fy) = ‖d(fx, fy)‖ ≤ ‖c‖ < ε. (5.14)

Thus we obtain that the mapping f : X → X satisfies the Meir–Keeler condition in the metric space (X,D). Using theclassical result from [99] the proof of the theorem is completed.

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Remark 5.13. It follows from the preceding proof that the regularity condition in Theorem 5.9 is superfluous.

Remark 5.14. Some of the mentioned theorems have long and complicated proofs (for instance, Theorems 5.6, 5.8 and5.9). In almost all cited works concerned with the normal solid cone, the authors use neither Sandwich Theorem, nor thecontinuity of the vector metric d, thus making the proofs several steps longer. Using Lemma 2.6 and Theorem 4.1, we can,shortly and elegantly, reduce these proofs to some known ones (Jungck, Das-Naik, Sehgal-Gusseman, Boyd-Wong, . . .) fromthe setting of metric spaces. It turns out that, in the frame of the cone metric spaces, it makes sense to investigate fixedpoints under various contractive conditions in the case of a cone that is only solid.

5.1. T-stability in cone metric spaces

Let (X, d) be a cone metric space and let T be a self-map on X . Let x0 be a point in X , and assume that xn+1 = f (T , xn) isan iteration procedure, involving T , which yields to a sequence xn of points from X .

Lemma 5.15 ([17]). Let P be a normal cone, and let an and bn be sequences in E satisfying the following inequality an+1 ≼

han + bn, where h ∈ (0, 1) and bn → θ as n → ∞. Then limn→∞ an = θ .

Definition 5.16 ([17]). The iteration procedure xn+1 = f (T , xn) is said to be T -stable if xn converges to a fixed point x∗ ofT and, whenever yn is a sequence in X with limn→∞ d(yn+1, f (T , yn)) = 0, we have limn→∞ yn = x∗.

Theorem 5.17 ([17]).

(a) Let (X, d) be a cone metric over a normal solid cone and let T : X → X with Fix(T ) = ∅. If there exist numbers a ≥ 0 and0 ≤ b < 1, such that

d(Tx, x∗) ≼ ad(x, Tx) + bd(x, x∗) (5.15)

for each x ∈ X, x∗∈ Fix(T ), then Picard’s iteration is T -stable.

(b) Let (X, d) be a cone metric space over a normal solid cone and let T be a quasi-contraction on X with some λ ∈ (0, 12 ). Then

Picard’s iteration is T -stable.

Let us analyze these results from [17]. We present first the following results from [100]:

Lemma 5.18 ([100]). Let xn, εn be nonnegative sequences satisfying xn+1 ≤ hxn + εn, for all n ∈ N, 0 ≤ h < 1,limn→∞ εn = 0. Then limn→∞ xn = 0.

Theorem 5.19 ([100]). Let (X, d) be a complete metric space and T a self-map on X with Fix(T ) = ∅. If there exist numbersL ≥ 0, 0 ≤ h < 1, such that

d(Tx, x∗) ≤ Ld(x, Tx) + hd(x, x∗) (5.16)

for each x ∈ X, x∗∈ Fix(T ), and, in addition, limn→∞ d(yn, Tyn) = 0, then Picard’s iteration is T -stable.

Corollary 5.20 ([100]). Let (X, d) be a complete metric space and T a self-map on X satisfying the following: there exists0 ≤ h < 1, such that, for each x, y ∈ X,

d(Tx, Ty) ≤ hmaxd(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx). (5.17)

Then, Picard’s iteration is T -stable.

Comment. Since we can assume monotonicity of the cone P (Lemma 2.6), then Lemma 5.15 follows directly fromLemma 5.18, by putting xn = ‖an‖, εn = ‖bn‖. Theorem 5.17(a) is, for the same reason, a direct consequence of Lemma 2.6.Bearing in mind Corollary 5.20, Theorem 5.17(b) is true for every λ ∈ (0, 1), that is, without the restriction λ ∈ (0, 1

2 ).Therefore, our approach results in a proper generalization of the results from [17].

It follows from the preceding comment that results on T -stability of Picard’s iteration, in the frame of the normal solidcone space, does not, in fact, differ from the corresponding results for metric spaces. As we have showed in this article, thesame holds for the majority of the remaining results.

6. Non-normal solid cones

Results concerning fixed points and other results, in the case of cone spaces with non-normal solid cones, cannot beproved by reducing to metric spaces, because in this case neither of the conditions from Lemma 2.6, 1–4 hold. Further, thevector conemetric is not continuous in the general case, i.e., from xn → x, yn → y it need not follow that d(xn, yn) → d(x, y)(see Example 6.1. below).

2598 S. Janković et al. / Nonlinear Analysis 74 (2011) 2591–2601

To replace standard properties of a metric, the following properties of cone metrics are often useful while dealing withcone metrics when the cone is not normal:

(p1) If u ≼ v and v ≪ w, then u ≪ w.(p2) If u ≪ v and v ≼ w, then u ≪ w.(p3) If u ≪ v and v ≪ w, then u ≪ w.(p4) If θ ≼ u ≪ c for each c ∈ int P , then u = θ .(p5) If a ≼ b + c , for each c ∈ int P , then a ≼ b.(p6) For each θ ≪ c1 and c2 ∈ P , there is an element θ ≪ d such that c1 ≪ d, c2 ≪ d(d = 2c1 + c2).(p7) For each θ ≪ c1 and θ ≪ c2, there is an element θ ≪ e such that e ≪ c1, e ≪ c2 (e =

1N (c1 + c2), N is large enough).

(p8) If θ ≼ d(xn, y) ≼ bn and bn → θ , then d(xn, x) ≪ c where xn, x are, respectively, a sequence and a given point in X .(p9) If E is a real Banach space with a cone P , and if a ≼ λa, where a ∈ P and 0 ≤ λ < 1, then a = θ .

(p10) If c ∈ int P , θ ≼ an, and an → θ , then there exists an n0 such that, for all n > n0, we have an ≪ c .

It follows from (p10) that the sequence xn converges to x ∈ X if d(xn, x) → θ , as n → ∞, and xn is a Cauchy sequence ifd(xn, xm) → θ as n,m → ∞. In the case when the cone is not necessarily normal, we only have one half of the statementsof Lemmas 1 and 4 from [1].

For example, for Theorem 1.1, without the normality property, we give the following proof:

Proof. Choose x0 ∈ X . Set x1 = Tx0, . . ., xn+1 = Txn = T n+1x0. We have

d(xn+1, xn) = d(Txn, Txn−1) ≼ kd(xn, xn−1)

≼ k2d(xn−1, xn−2) ≼ . . . ≼ knd(x1, x0). (6.1)

So for n > m,

d(xn, xm) ≼ d(xn, xn−1) + d(xn−1, xn−2) + · · · + d(xm+1, xm)

≼ (kn−1+ kn−2

+ · · · + km)d(x1, x0)

≼km

1 − kd(x1, x0) → 0, asm → ∞. (6.2)

According to (p10) and (p1) it follows that xn is a Cauchy sequence. By the completeness of X , there is x∗∈ X such that

xn → x∗. Since T is continuous then Txn → Tx∗, that is Tx∗= x∗. (Here we have to be careful in the definition of continuity:

sequential.)That Tx∗

= x∗ can be proved in the following way:

d(Tx∗, x∗) ≼ d(Tx∗, Txn−1) + d(Txn−1, x∗) = d(Tx∗, Txn−1) + d(xn, x∗)

≼ kd(x∗, xn−1) + d(xn, x∗) ≪ kc2k

+c2

= c, (6.3)

(given c ∈ int P , there exists a natural number n0 such that for n > n0 we have: d(x∗, xn−1) ≪c2k and d(xn, x∗) ≪

c2 ).

The proofs in the case of non-normal solid cone use in the first place ‘‘c–d’’, i.e., ‘‘c–n0’’ calculus (like in the real analysis)as a replacement for Sandwich Theorem and the continuity of vector metric d. It is important to know that, in such a cone,from un → θ (in the Banach space norm) it follows that un ≪ c for n0 big enough, where c is a fixed interior point. However,the converse is not true, i.e., from un ≪ c it does not necessarily follow that un → θ .

Because of that, we have here a proper extension of the results concerned with ordinary metric spaces. In fact, accordingto our previous proof, it is easy to find an example of a contraction in the case of a non-normal solid cone, that has a uniquefixed point, but Banach theorem cannot be applied in this case (because the cone is non-normal). For details see our papers[64–69].

The following example is a very important one:

Example 6.1. Let E = C1R[0, 1] with the norm ‖x‖ = ‖x‖∞ + ‖x′

‖∞ and P = x ∈ E : x(t) ≥ 0 on [0, 1] that is anon-normal solid cone ([92], p. 221). Consider,

xn(t) =1 − sin ntn + 2

, yn(t) =1 + sin ntn + 2

so θ ≼ xn ≼ xn + yn → θ in the Banach space and ‖xn‖ = ‖yn‖ = 1. Therefore, xn 9 θ in E. Hence, according to (p10) and(p1) we have that xn ≪ c for n large enough, but xn 9 θ in the Banach space E.

Further, define the cone metric d : P × P → E by d(x, y) = x + y, x = y, d(x, x) = θ . This cone metric is not continuous.Namely, since, θ ≼ xn ≪ c we have d(xn, θ) ≪ c , but d(xn, x) = xn 9 θ = d(θ, θ) in E.

Now, consider the mapping f : P → P , defined in the following way: f (x) =13x, for all x ∈ P . This is an example of

Banach type contraction mapping on non-normal solid cone metric space (E, P). According to our proof, the mapping f hasa unique fixed point in P , but Banach theorem and the method of the proof from [1] cannot be applied (because the cone isnon-normal).

S. Janković et al. / Nonlinear Analysis 74 (2011) 2591–2601 2599

It is interesting that, unlike Kantorovich, Zabreiko and others, Huang and Zhangwent further by defining the convergenceand Cauchy sequences in terms of interior points of the cone and the relation ‘‘≪’’, but in their assumptions and results theyremainedwithin the normal solid cones approach. It should bementioned that Rezapour and Hamlbarani [57] were the firstto obtain proper generalizations of results concerning ordinary metric spaces, i.e., to obtain results were the existing resultsfrommetric spaces were not applicable. However, this article has no examples confirming that these are the cases of properextensions.

We finish the present article with the following problem, to which we have no answer yet:Problem: Let (X, d) be a solid cone metric space and let f : X → X be a quasi-contraction in the sense of Definition 5.5(see [19], too), where λ ∈ [

12 , 1) is a constant. Does f have a unique fixed point in X? If λ ∈ [0, 1

2 ), the response isaffirmative [67].

Remark 6.2 (Added in Proof). While the manuscript of this article was under review, an affirmative answer to the previousquestion was published in [80]. Also, new articles that appeared meanwhile, are included in the bibliography.

Remark 6.3 (Added in Proof). It has been shown in the recent papers by Du [84,85] and Amini-Harandi and Fakhar [86] thatfixed point results in the setting of cone metric spaces in which linear contractive conditions appear, can be reduced tothe respective results in the metric setting. In [84,85] the so-called scalarization functions have been used that had beenintroduced in the optimization theory, while in [86] properties of basis and dual basis in Banach spaces have been applied.

Similar results are obtained in paper [87] by simple application of the Minkowski functional of the zero-neighborhoodgenerated by an interior point of the cone. However, none of these approaches can be applied in most of the cases whennonlinear contractive conditions are given. In particular, this is true for important fixed point results from [98].

Remark 6.4 (Added in Proof). In the recent papers [39,40], Khamsi claims thatmost of the cone fixed point results aremerelycopies of the classical ones and that any extension of known fixed point result to cone metric spaces is redundant; also thatunderlying Banach space and the associated cone subset are not necessary. In fact, Khamsi’s approach includes a very smallclass of results and is very limited since it requires only normal cones, so that all results with non-normal cones cannot bedealt with by his approach.

Examples 2.7 and 6.1 of the present paper, as well as our proof of Theorem 1.1 and the approach in all the papers [55–83],where the only assumption on the cone is that it has a nonempty interior, show how to obtain results without usingcontinuity of the metric and Sandwich Theorem. In these cases fixed point theorems and some other results are properextensions of the corresponding results for metric spaces. This shows again that Khamsi’s claim is false.

Moreover, some results with normal cones (even with the normal constant K = 1) are not equivalent to their respectivecounterparts. An example was already given in the original paper [1] (see Remark 1 at the end of that paper) where a cone-contraction is presented which is not a Euclidean contraction, and so the existence of a fixed point cannot be obtained fromthe classical result.

Acknowledgements

The authors are very grateful to the referees for the valuable suggestions and comments that enabled us to revise thispaper. The authors are thankful to the Ministry of Science and Technological Development of Serbia.

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