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Fixed point results under cdistance in tvscone metric spaces
Fixed Point Theory and Applicationsvolume 2011, Article number: 29 (2011)
Abstract
Fixed point and common fixed point results for mappings in tvscone metric spaces (with the underlying cone which is not normal) under contractive conditions expressed in the terms of cdistance are obtained. Respective results concerning mappings without periodic points are also deduced. Examples are given to distinguish these results from the known ones.
Mathematics Subject Classification (2010)
47H10, 54H25
1 Introduction
Cone metric spaces were considered by Huang and Zhang in [1], who reintroduced the concept which has been known since the middle of 20th century (see, e.g., [2–4]). Topological vector spacevalued version of these spaces was treated in [5–13]; see also [14] for a survey of fixed point results in these spaces.
Fixed point theorems in metric spaces with the socalled wdistance were obtained for the first time by Kada et al. in [15] where nonconvex minimization problems were treated. Further results were given, e.g., in [16–18]. Cone metric version of this notion (usually called a cdistance) was used, e.g., in [19, 20].
In this paper, we consider fixed point and common fixed point results for mappings in tvscone metric spaces (with the underlying cone which is not normal) under contractive conditions expressed in the terms of cdistance. Respective results concerning mappings without periodic points are also deduced. Examples are given to distinguish these results from the known ones.
2 Preliminaries
Let E be a real Hausdorff topological vector space (tvs for short) with the zero vector θ. A proper nonempty and closed subset P of E is called a cone if P + P ⊂ P, λP ⊂ P for λ ≥ 0 and P ∩ (P) = {θ}. We shall always assume that the cone P has a nonempty interior int P (such cones are called solid).
Each cone P induces a partial order ≼ on E by x ≼ y ⇔ y  × ∈ P. x π y will stand for (x ≼ y and x ≠ y), while x ≪ y will stand for y  × ∈ int P. The pair (E, P) is an ordered topological vector space.
For a pair of elements x, y in E such that x ≼ y, put [x, y] = {z ∈ E : x ≼ z ≼ y}. A subset A of E is said to be orderconvex if [x, y] ⊂ A, whenever x, y ∈ A and x ≼ y.
Ordered topological vector space (E, P) is orderconvex if it has a base of neighborhoods of θ consisting of orderconvex subsets. In this case, the cone P is said to be normal. If E is a normed space, this condition means that the unit ball is orderconvex, which is equivalent to the condition that there is a number k such that x, y ∈ E and 0 ≼ x ≼ y implies that x ≤ ky. A proof of the following assertion can be found, e.g., in [2].
Theorem 1 If the underlying cone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.
Note that completions of cone metric spaces in the case of nonnormal underlying cone were treated in [21].
From [1, 5–7], we give the following
Definition 1 Let X be a nonempty set and (E, P) an ordered tvs. A function d : X × X → E is called a tvscone metric and (X, d) is called a tvscone metric space if the following conditions hold:
(c1) θ ≼ d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;
(c2) d(x, y) = d(y, x) for all x, y ∈ X;
(c3) d(x, z) ≼ d(x, y) + d(y, z) for all x, y, z ∈ X.
Taking into account Theorem 1, proper generalizations when passing from normvalued cone metric spaces of [1] to tvscone metric spaces can be obtained only in the case of nonnormal cones. We shall make use of the following properties:
(p_{1}) If u, v, w ∈ E, u ≼ v and v ≪ w then u ≪ w.
(p_{2}) If u ∈ E and θ ≼ u ≪ c for each c ∈ int P then u = θ.
(p_{3}) If u_{ n } , v_{ n } , u, v ∈ E, θ ≼ u_{ n } ≼ v_{ n } for each n ∈ ℕ, and u_{ n } → u, v_{ n } → v (n → ∞), then θ ≼ u ≼ v.
(p_{4}) If x_{ n } , x ∈ X, u_{ n } ∈ E, d(x_{ n } , x) ≼ u_{ n } and u_{ n } → θ (n → ∞), then x_{ n } → x (n → ∞).
(p_{5}) If u ≼ λu, where u ∈ P and 0 ≤ λ < 1, then u = θ.
(p_{6}) If c ≫ θ and u_{ n } ∈ E, u_{ n } → θ (n → ∞), then there exists n_{0} such that u_{ n } ≪ c for all n ≥ n_{0}.
In the sequel, E will always denote a topological vector space, with the zero vector θ and with order relation ≼, generated by a solid cone P. For notions such as convergent and Cauchy sequences, completeness, continuity etc. in (tvs)cone metric spaces, we refer to [1, 7, 14] and references therein.
Kada et al. [15] introduced the notion of wdistance in metric spaces and proved some fixed point results using this notion (see also [16–18]). Cho et al. [19] transferred it to the setting of cone metric spaces (see also [20]).
Definition 2[19] Let (X, d) be a tvscone metric space. A function q : X × X → E is called a cdistance in X if:
(q1) θ ≼ q(x, y) for all x, y ∈ X;
(q2) q(x, z) ≼ q(x, y) + q(y, z) for all x, y, z ∈ X;
(q3) If a sequence {y_{ n } } in X converges to a point y ∈ X, and for some x ∈ X and u = u_{ x } ∈ P, q(x, y_{ n } ) ≼ u holds for each n ∈ ℕ, then q(x, y) ≼ u;
(q4) For each c ∈ E with θ ≪ c, there exists e ∈ E with θ ≼ e, such that q(z, x) ≪ e and q(z, y) ≪ e implies d(x, y) ≪ c.
Each wdistance q in a metric space (X, d) (in the sense of [15]) is a cdistance in the tvscone metric space (X, d) (with E = ℝ and P = [0, +∞)). Indeed, only property (q3) has to be checked. Let y_{ n }∈ X, y_{ n }→ y in the cone metric d (n → ∞), and let q(x, y_{ n } ) ≤ u_{ x } ∈ [0, +∞). Since q is (as a wdistance) lower semicontinuous, we have that q(x, y) ≤ lim inf_{n → ∞}q(x, y_{ n }) ≤ lim inf_{n → ∞}u_{ x }= u_{ x } , i.e., q(x, y) ≤ u_{ x } holds true.
The first two of the following examples are variations of [[19], Examples 2.7, 2.8], adjusted to the case of a tvscone metric.
Example 1 Let (X, d) be a tvscone metric space such that the metric d(·,·) is a continuous function in second variable. Then, q(x, y) = d(x, y) is a cdistance. Indeed, only property (q3) is nontrivial and it follows from q(x, y_{ n } ) = d(x, y_{ n } ) ≼ u, passing to the limit when n → ∞ and using continuity of d.
Example 2 Let (X, d) be a tvscone metric space, and let u ∈ X be fixed. Then, q(x, y) = d(u, y) defines a cdistance on X. Indeed, (q1) and (q3) are clear. (q2) follows from q(x, z) = d(u, z) ≼ d(u, y) + d(u, z) = q(x, y) + q(y, z). Finally, (q4) is obtained by taking e = c/ 2.
Example 3 Consider the Banach space E = C[0, 1] of realvalued continuous functions with the maxnorm and ordered by the cone P = {f ∈ E : f(t) ≥ 0 for t ∈ [0, 1]}. This cone is normal in the Banachspace topology on E. Let τ* be the strongest locally convex topology on the vector space E. Then, the cone P is solid, but it is not normal in the topology τ*. Indeed, if this were the case, Theorem 1 would imply that the topology τ* is normed, which is impossible since an infinite dimensional space with the strongest locally convex topology cannot be metrizable (see, e.g., [14]).
Let now X = [0, + ∞) and d : X × X → (E, τ*) be defined by d(x, y)(t) = x  yφ(t) for a fixed element φ ∈ P. Then, (X, d) is a tvscone metric space which is not a cone metric space in the sense of [1]. We can introduce two cdistances on this space:
They are the examples of cdistances in tvscone metric spaces which are not cdistances in cone metric spaces of [19, 20].
These examples show, among other things, that for a cdistance q:

1.
q(x, y) = q(y, x) does not necessarily hold for all x, y ∈ X;

2.
q(x, y) = θ is not necessarily equivalent to x = y.
3 Results
3.1 Fixed point and common fixed point results under cdistance
We will call a sequence {u_{ n } } in P a csequence if for each c ≫ θ there exists n_{0} ∈ N such that u_{ n } ≪ c for n ≥ n_{0}. It is easy to show that if {u_{ n } } and {v_{ n } } are csequences in E and α, β > 0, then { αu_{ n } + βv_{ n } } is a csequence.
Note that in the case that the cone P is normal, a sequence in E is a csequence iff it is a θ sequence (see property (p_{6})). However, when the cone is not normal, a csequence need not be a θ sequence (see [7, 14]). Also, from [7], we know that the cone metric d need not be a continuous function.
The following lemma is a tvscone metric version of lemmas from [15, 19].
Lemma 1 Let (X, d) be a tvscone metric space and let q be a cdistance on X. Let {x_{ n } } and {y_{ n } } be sequences in × and x, y, z ∈ X. Suppose that {u_{ n } } and {v_{ n } } are csequences in P. Then the following hold:
(1) If q(x_{ n } , y) ≼ u_{ n } and q(x_{ n } , z) ≼ v_{ n } for n ∈ ℕ, then y = z. In particular, if q(x, y) = θ and q(x, z) = θ, then y = z.
(2) If q(x_{ n } , y_{ n } ) ≼ u_{ n } and q(x_{ n } , z) ≼ v_{ n } for n ∈ ℕ, then {y_{ n } } converges to z.
(3) If q(x_{ n } , x_{ m } ) ≼ u_{ n } for m > n > n_{0}, then {x_{ n } } is a Cauchy sequence in X.
(4) If q(y, x_{ n } ) ≼ u_{ n } for n ∈ ℕ, then {x_{ n } } is a Cauchy sequence in X.
Proof We will prove assertions (1) and (2). Proofs of the other two are similar.

(1)
In order to prove that y = z, according to (p_{2}), it is enough to show that d(y, z) ≪ c for each c ≫ θ. For the given c choose e ≫ θ such that property (q4) is satisfied. Choose then n _{0} ∈ ℕ such that u_{ n } ≪ e and v_{ n } ≪ e for n ≥ n _{0}. Then, by property (p_{1}), we get that q(x_{ n } , y) ≪ e and q(x_{ n } , z) ≪ e and (q4) imply that d(y, z) ≪ c.

(2)
Let again c ≪ θ be arbitrary and choose a corresponding e ≫ θ satisfying property (q4). If n _{0} ∈ N is such that u_{ n } ≪ e and v_{ n } ≪ e for n ≥ n _{0}, then (p_{1}) implies that q(x_{ n } , y_{ n } ) ≪ e and q(x_{ n } , z) ≪ e for n ≥ n _{0}. Then, by (q4), d(y_{ n } , z) ≪ c and y_{ n } → z (n → ∞). ■
Our first result is the following theorem of HardyRogers type.
Theorem 2 Let (X, d) be a complete tvscone metric space and let q be a cdistance on X. Suppose that a continuous selfmap f : X → X satisfies the following two conditions:
for all x, y ∈ X, where A, B, C, D, E are nonnegative constants such that A + B + C + 2D + 2E < 1. Then f has a fixed point in X. If fu = u, then q(u, u) = θ.
Proof Let x_{0} ∈ X be arbitrary and form the sequence {x_{ n } } with x_{ n } = f^{n}x_{0}. In order to prove that it is a Cauchy sequence, put x = x_{ n } and y = x_{n  1}in (3.1) to get
Similarly, putting y = x_{n  1}and x = x_{ n } in (3.2), one obtains
Denote u_{ n } = q(x_{n+1}, x_{ n })+ q(x_{ n } , x_{n+1}). Adding up (3.3) and (3.4), we get that
i.e. u_{ n } ≼ hu_{n  1}with
since A + B + C + 2D + 2E < 1 and, e.g., A + C + E > 0.
By induction, u_{ n } ≼ h^{n}u_{0} and q(x_{ n }, x_{n+1}) ≼ u_{ n } ≼ h^{n} (q(x_{1}, x_{0}) + q(x_{0}, x_{1})). In the usual way, it follows that
for m > n, where {v_{ n } } is a csequence. Lemma 1.(3) implies that {x_{ n } } is a Cauchy sequence in X and, since X is complete, x_{ n } → x* ∈ X (n → ∞). Continuity of f implies that x_{n+1}= fx_{ n } → fx*, and since the limit of a sequence in tvscone metric space in unique, we get that fx* = x*.
Suppose that fu = u. Then, (3.1) implies that
which is, by property (p_{5}) and A + B + C + D + E < A + B + C +2D +2E < 1, possible only if q(u, u) = θ. ■
Some special cases of the previous theorem, for example Banachtype and Kannantype fixed point results, need only one condition:
and
respectively.
Remark 1 If the underlying cone P of the given tvscone metric space (X, d) is normal (and, hence, this space is a cone metric space in the sense of [1], see Theorem 2.1), then continuity of f in Theorem 2 can be replaced by the condition
It may be of interest to note that in this case, property (q3) of cdistance has to be used in the course of the proof (see, e.g., the respective procedure in ordered cone metric spaces in [19]), while in our case (when f is continuous), this property is not needed.
The next is a result including two mappings and the existence of their common fixed point.
Theorem 3 Let (X, d) be a complete tvscone metric space and let q be a cdistance on X. Suppose that continuous selfmaps f, g : X → X satisfy the following two conditions:
for all x, y ∈ X, where A, B, D are nonnegative constants, such that A + 2B + 4D < 1. Then f and g have a common fixed point in X. If fu = gu = u, then q(u, u) = θ.
Proof Let x_{0} ∈ X be arbitrary and form the sequence {x_{ n } } such that x_{2n+1}= fx_{2n}and x_{2n+2}= gx_{2n+1}for n ≥ 0. Denote u_{ n } = q(x_{2n}, x_{2n+1})+q(x_{2n+1}, x_{2n}) and v_{ n } = q(x_{2n+1}, x_{2n+2}) + q(x_{2n+2}, x_{2n+1}).
Putting x = x_{2n+2}, y = x_{2n+1}in (3.5) we obtain that
Similarly, putting the same values for x, y in (3.6), we get
It follows by adding up (3.7) and (3.8) that
i.e.,
where , since A + B + D > 0 and A + 2B + 4D < 1.
By a similar procedure, starting with x = x_{2n}and y = x_{2n+1}, one can get
Combining the last two inequalities, it follows that
and we get that {u_{ n } } and {v_{ n } } are csequences. We have that q(x_{2n}, x_{2n+1}) ≼ u_{ n } , q(x_{2n+1}, x_{2n+2}) ≼ v_{ n } and it follows that q(x_{ n } , x_{n+1}) ≼ u_{ n } + v_{ n } , where u_{ n } + v_{ n } is a csequence. Using Lemma 1.(3), we obtain that {x_{ n } } is a Cauchy sequence in X. Hence, x_{ n } → x* ∈ X (n → ∞). Since f and g are continuous, it easily follows from the definition of {x_{ n } } that fx* = gx* = x*.
Thus, mappings f and g have a common fixed point. Suppose that u ∈ X is any point satisfying fu = gu = u. Then, (3.5) implies that
and, since 0 < A + 2B + 2D < A + 2B + 4D < 1, property (p_{5}) implies that q(u, u) = θ. ■
As corollaries, we obtain, for example, common fixed point result for selfmaps f and g satisfying
or for a selfmap f satisfying
where m, n ∈ ℕ, A + 2B + 4D < 1.
Remark 2 Similarly as in Remark 1, we note that if the cone P is normal, then continuity of mappings f and g in Theorem 3 can be replaced by conditions
Example 4 Let E = ℝ and P = [0, +∞). Let X = [0, +∞), d(x, y) = x  y and define q(x, y) = x. It is easy to check that q is a cdistance on a cone metric space (X, d).
Take functions f, g : X → X defined by and . If x = 5, , then and . Hence, there is no A ∈ (0, 1) (and hence no triple (A, B, D)) such that d(fx, gy) ≤ Ad(x, y) for each x, y ∈ [0, +∞), i.e., the existence of a common fixed point of f and g cannot be deduced from the wellknown metric version of Theorem 3.
However, conditions of the cdistance version (Theorem 3) are satisfied. Indeed, take arbitrary A, and B = D = 0. Then, for each x, y ∈ [0, +∞), and (see (3.9)). Note that f and g have a (trivial) common fixed point u = 0 and that q(u, u) = q(0, 0) = 0.
This example can be easily modified to the tvscone metric case. It is enough to define tvscone metric on X by d(x, y)(t) = x  yφ(t) with fixed φ ∈ P = {f ∈ C[0, 1] : f(t) ≥ 0 for t ∈ [0, 1]} and take cdistance q_{1}(x, y)(t) = x · φ (t) (see Example 3).
3.2 Mappings without periodic points
The first part of the following result was given with an incorrect proof in [20] (using lim inf which may not be defined in the case of an arbitrary cone metric space).
Recall that a map f : X → X is said to have property (P) if it satisfies F(f) = F(f^{n} ) for each n ∈ ℕ, where F(f) stands for the set of all fixed points of f[22].
Theorem 4 Let (X, d) be a tvscone metric space and q : X × X → E be a cdistance on X. Suppose that a continuous selfmap f : X → X satisfies
for some λ ∈ (0, 1) and each × ∈ X. Then:
1. f has a fixed point and if fu = u, then q(u, u) = θ;
2. f has property (P).
Proof (1) Let x_{0} ∈ X and x_{n+1}= fx_{ n } , n ≥ 0. If for some n_{0} ∈ ℕ_{0}, then is a fixed point of f. Otherwise, we get from (3.10) that
Using Lemma 1.(3) again, one obtains that {x_{ n } } is a Cauchy sequence in X.
Hence, x_{ n } → x*, and continuity of f implies that x_{n+1}= fx_{ n } → fx* and fx* = x*.

(2)
Obviously, F(f) ⊆ F(f^{n} ) for each n ∈ ℕ. Let u ∈ F(f^{n} ), i.e., f^{n}u = u. Then, (3.10) implies that
By property (p_{5}), it follows that q(u, fu) = θ.
Now, for arbitrary k ∈ {1, 2,..., n}, we have that q(f^{k}u, f^{k+1}u) ≼ λ^{k}q(u, fu) θ and so q(f^{k}u, f^{k+1}u) = θ. It follows that q(u, f^{2}u) ≼ q(u, fu)+q(fu, f^{2}u) = θ, i.e., q(u, f^{2}u) = θ and, similarly,
From q(u, fu) = θ = q(u, f^{n}u) = q(u, u) and Lemma 1.(1), we conclude that fu = u, i.e., u ∈ F(f). ■
Another way to obtain property (P) is the following.
Theorem 5 Let (X, d) be a tvscone metric space and q : X × X → E be a cdistance on X. Suppose that a continuous selfmap f : X → X satisfies
for some λ ∈ (0, 1) and each × ∈ X. Then f has property (P).
Proof Denote z_{1}(x) = q(x, fx)+ q(fx, x) and z_{2}(x) = q(fx, f^{2}x)+ q(f^{2}x, fx).
Then, the given condition is written as z_{2}(x) ≼ λz_{1}(x) for each x ∈ X. Suppose that f^{n}u = u. Then,
Since 0 < λ^{n} < 1, property (p_{5}) implies that z_{1}(u) = q(u, fu) + q(fu, u) = θ. Again, the triangle inequality (q2) implies that q(u, u) = q(fu, fu) = θ, and by Lemma 1.(1), we get that fu = u. ■
Corollary 1 Let q be a cdistance on a tvscone metric space (X, d) and let f : X → X be continuous and such that for some nonnegative A, B, C, D, E such that A + B + C + 2D + 2E < 1, inequalities (3.1) and (3.2) hold for all x, y ∈ X. Then f has property (P).
Proof Putting x = x and y = fx in conditions (3.1) and (3.2) leads to the following inequalities:
Adding up, one obtains inequality (3.11) with , since A + B + C + 2D + 2E < 1. ■
Similar results concerning property (Q) of two selfmappings f and g (i.e., property that F(f) ∩ F(g) = F(f^{n} ) ∩ F(g^{n} ) for each n ∈ ℕ) can be obtained.
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The authors thank the referees for their valuable comments that helped to improve the text. The authors are thankful to the Ministry of Science and Technological Development of Serbia.
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Keywords
 Tvscone metric space
 cDistance
 Common fixed point