Banach Contraction Principle for Cyclical Mappings on Partial Metric Spaces

The widespread applications of the notion of partial metric spaces in programming theory have attracted the attention of many authors who recently published important results in the direction of generalizing this principle; see for instance [11,12,13,14]. The contraction type conditions used in these generalizations, however, do not apparently reflect the structure of partial metric spaces. In the remarkable paper [15], the authors proved more appropriate contraction principle in partial metric spaces. Indeed, it is more convenient to call the contraction type condition used in this paper by partial contractive condition.

In this paper, we prove that the Banach contraction principle obtained in [1] on 0-complete partial metric spaces can be extended to cyclical mappings. However, the generalized contraction principle proved in [15] on complete partial metric spaces can not be extended to cyclical mappings. Some examples are given to illustrate the effectiveness of our results. In addition to this, we generalize some of the results obtained in [16]. Finally, an Edelstein’s type theorem is also extended in case one of the sets in the cyclic decomposition is 0-compact.

We recall some definitions of partial metric spaces and state some of their properties. A partial metric space (PMS) is a pair (X, p : X × X → R + ) (where R + denotes the set of all non negative real numbers) such that (P1) p(x, y) = p(y, x) (symmetry);

(P2) If 0 ≤ p(x, x) = p(x, y) = p(y, y) then x = y (equality); (P3) p(x, x) ≤ p(x, y) (small self-distances);

(P4) p(x, z) + p(y, y) ≤ p(x, y) + p(y, z) (triangularity); for all x, y, z ∈ X.

For a partial metric p on X, the function p s : X × X → R + given by

is a (usual) metric on X. Each partial metric p on X generates a T 0 topology τ p on X with a base of the family of open p-balls {B p (x, ε) : x ∈ X, ε > 0}, where B p (x, ε) = {y ∈ X : p(x, y) < p(x, x) + ε} for all x ∈ X and ε > 0.

(i) A sequence {x n } in a PMS (X, p) converges to x ∈ X if and only if p(x, x) = lim n→∞ p(x, x n ).

(ii) A sequence {x n } in a PMS (X, p) is called a Cauchy if and only if lim n,m→∞ p(x n , x m ) exists (and finite).

(iii) A PMS (X, p) is said to be complete if every Cauchy sequence {x n } in X converges, with respect to τ p , to a point x ∈ X such that p(x, x) = lim n,m→∞ p(x n , x m ).

(iv) A mapping f : X → X is said to be continuous at

(a2) A PMS (X, p) is complete if and only if the metric space (X, p s ) is complete. Moreover,

Lemma 2. Let (X, p) be a partial metric space and let T : X → X be a continuous self-mapping. Assume {x n } ∈ X such that x n → z as n → ∞. Then

Proof. Let ǫ > 0 be given. Since T is continuous at z find δ > 0 such that T (B p (z, δ)) ⊆ B p (T z, ǫ). Since x n → z then lim n→∞ p(x n , z) = p(z, z) and hence find

This shows our claim.

A sequence {x n } is called 0-Cauchy if lim m,n→∞ p(x n , x m ) = 0. The partial metric space (X, p) is called 0-complete if every 0-Cauchy sequence in x converges to a point x ∈ X with respect to p and p(x, x) = 0. Clearly, every complete partial metric space is complete. The converse need not be true; see [17] for more details.

with the partial metric p(x, y) = max{x, y} where Q is the set of rationals. Then (X, p) is a 0-complete partial metric space which is not complete.

Theorem 1. [1,17] Let(X, p) be a 0-complete partial metric space and f : X → X be such that p(f (x), f (y)) ≤ αp(x, y)∀x, y ∈ Xand α ∈ [0, 1)

there exists a unique u ∈ X such that u = f (u) and p(u, u) = 0.

Let ρ p = inf{p(x, y) : x, y ∈ X} and define X p = {x ∈ X : p(x, x) = ρ p }.

Theorem 2. [15] Let (X, p) be a complete metric space, α ∈ [0, 1) and T : X → X a given mapping. Suppose that for each x, y ∈ X the following condition holds p(x, y) ≤ max{αp(x, y), p(x, x), p(y, y)}.

(1) the set X p is nonempty;

(2) there is a unique u ∈ X p such that T u = u;

(3) for each x ∈ X p the sequence {T n x} n≥1 converges with respect to the metric p s to u.

Definition 2. Let A and B be two nonempty closed subsets of a complete partial metric space (X, p) such that

If (C2) in Definition 2 is replaced by the condition

Then T is called a partial cyclical contraction.

Remark 1. The partial cyclical contractions reflects the real structure of partial metric space.

The proof of the following lemma can be easily achieved by using the partial metric topology. Lemma 3. A subset A of a partial metric space is closed if and only if x ∈ A whenever x n ∈ A satis