Some fixed point results in ordered partial metric spaces

J. Nonlinear Sci. Appl. 4 (2011), no. 3, 210–217

The Journal of Nonlinear Sciences and Applications http://www.tjnsa.com

SOME FIXED POINT RESULTS IN ORDERED PARTIAL METRIC SPACES HASSEN AYDI Abstract. In this paper, we establish some fixed point theorems in ordered partial metric spaces. An example is given to illustrate our obtained results.

1. Introduction and preliminaries When fixed point problems in partially ordered metric spaces are concerned, first results were obtained by Ran and Reurings [17], and then by Nieto and L´opez [11]. The following fixed point theorem was proved in these papers. Theorem 1.1. [11, 17] Let (X, ≤X ) be a partially ordered set and let d be a metric on X such that (X, d) is a complete metric space. Let f : X → X be a non-decreasing map with respect to ≤X . Suppose that the following conditions hold: (i) ∃0 ≤ c < 1, d(f x, f y) ≤ cd(x, y) for any y ≤X x (ii) ∃x0 ∈ X such that x0 ≤X f x0 (iii) f is continuous in (X, d), or (iii’) if a non-decreasing sequence {xn } converges to x ∈ X, then xn ≤X x for all n. Then f has a fixed point u ∈ X. Results on weakly contractive mappings in such spaces were obtained by Harjani and Sadarangani in [8]. An extension of the previous result is the following Date: Revised: 9 Aut, 2011. ∗ Corresponding author c 2011 N.A.G. ⃝ 2000 Mathematics Subject Classification. Primary 47H10; Secondary 54H25. Key words and phrases. Fixed point, partial metric space, ordered set. 210

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Theorem 1.2. [8] Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f : X → X be a non-decreasing mapping with respect to ≤X such that d(f x, f y) ≤ d(x, y) − ψ(d(x, y)), for y ≤X x, where ψ : [0, +∞[→ [0, +∞[ is a continuous and non-decreasing function such that it is positive in ]0, +∞[, ψ(0) = 0 and lim ψ(t) = +∞. t→+∞

Assume that (i) f is continuous in (X, d), or (ii) if a non-decreasing sequence {xn } converges to x ∈ X, then xn ≤X x for all n. If there exists x0 ∈ X with x0 ≤X f x0 , then f has a fixed point. Many other results on the existence of fixed points or common fixed points in ordered metric spaces were given, we can cite for example [1, 3, 5, 6, 7, 9, 12, 15, 21, 23] and the references therein. In this paper we extend the results of Harjani and Sadarangani [8] to the case of partial metric spaces. An example is considered to illustrate our obtained results. First, we start with some preliminaries on partial metric spaces. For more details, we refer the reader to [2, 4, 10, 13, 14, 16, 18, 19, 20, 22, 23]. Definition 1.3. Let X be a nonempty set. A partial metric on X is a function p : X × X −→ R+ such that for all x, y, z ∈ X: (p1) x = y ⇐⇒ p(x, x) = p(x, y) = p(y, y), (p2) p(x, x) ≤ p(x, y), (p3) p(x, y) = p(y, x), (p4) p(x, y) ≤ p(x, z) + p(z, y) − p(z, z). A partial metric space is a pair (X, p) such that X is a nonempty set and p is a partial metric on X. If p is a partial metric on X, then the function ps : X × X −→ R+ given by ps (x, y) = 2p(x, y) − p(x, x) − p(y, y), is a metric on X. Definition 1.4. Let (X, p) be a partial metric space. Then: (i) a sequence {xn } in a partial metric space (X, p) converges to a point x ∈ X if and only if p(x, x) = lim p(x, xn ); n−→+∞

(ii) a sequence {xn } in a partial metric space (X, p) converges properly to a point x ∈ X if and only if p(x, x) = lim p(xn , xn ) = lim p(x, xn ), if and only if lim ps (x, xn ) = 0;

n−→+∞

n−→+∞

n−→+∞

(iii) A sequence {xn } in a partial metric space (X, p) is called a Cauchy sequence if there exists (and is finite) lim p(xn , xm ); n,m−→+∞

(iv) A partial metric space (X, p) is said to be complete if every Cauchy sequence {xn } in X converges to a point x ∈ X, that is p(x, x) = lim p(xn , xm ). n,m−→+∞

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Lemma 1.5. Let (X, p) be a partial metric space. (a) {xn } is a Cauchy sequence in (X, p) if and only if it is a Cauchy sequence in the metric space (X, ps ); (b) A partial metric space (X, p) is complete if and only if the metric space (X, ps ) is complete. Furthermore, lim ps (xn , x) = 0 if and only if n−→+∞

p(x, x) =

lim p(xn , x) =

n−→+∞

lim

n,m−→+∞

p(xn , xm ).

Definition 1.6. Suppose that (X1 , p1 ) and (X2 , p2 ) are partial metrics. Denote τ (p1 ) and τ (p2 ) their respective topologies. We say T : (X1 , p1 ) → (X2 , p2 ) is continuous if both T : (X1 , τ (p1 )) → (X2 , τ (p2 )) and T : (X1 , τ (ps1 )) → (X2 , τ (ps2 )) are continuous. Proposition 1.7. Let (X, p) be a partial metric space, partially ordered and T : X → X be a given mapping. We say that T is continuous in x0 ∈ X if for every sequence {xn } is X, we have (a) xn converges to x0 in (X, p) implies T xn converges to T x0 in (X, p). (b) xn converges properly to x0 in (X, p) implies T xn converges properly to T x0 in (X, p). If T is continuous on each point x0 ∈ X, then we say that T is continuous on X. Definition 1.8. If (X, ≤X ) is a partially ordered set and f : X → X, we say that f is monotone nondecreasing if x, y ∈ X, x ≤X y implies f x ≤X f y. 2. Main results Our first result is the following theorem Theorem 2.1. Let (X, ≤X ) be a partially ordered set and let p be a partial metric on X such that (X, p) is complete. Let f : X → X be a non-decreasing map with respect to ≤X . Suppose that the following conditions hold: for y ≤ x, we have (i) p(f x, f y) ≤ p(x, y) − ψ(p(x, y)), (2.1) where ψ : [0, +∞[→ [0, +∞[ is a continuous and non-decreasing function such that it is positive in ]0, +∞[, ψ(0) = 0 and lim ψ(t) = +∞; t→+∞

(ii) ∃x0 ∈ X such that x0 ≤X f x0 ; (iii) f is continuous in (X, p), or; (iii’) if a non-decreasing sequence {xn } converges to x ∈ X, then xn ≤X x for all n. Then f has a fixed point u ∈ X. Moreover, p(u, u) = 0. Proof. Let x0 ∈ X be such that x0 ≤X f x0 . As f is monotone non-decreasing, then x0 ≤X f x0 ≤X f 2 x0 ≤X f 3 x0 ≤X ... ≤X f n x0 ≤X f n+1 x0 ≤X ... Put xn = f n x0 , then for any n ∈ N∗ , we have xn−1 ≤X xn . Then for each integer n ≥ 1, from (2.1) and, as the elements xn and xn−1 are comparable, we get p(xn+1 , xn ) = p(f xn , f xn−1 ) ≤ p(xn , xn−1 ) − ψ(p(xn , xn−1 )),

(2.2)

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If there exists n1 ∈ N∗ such that p(xn1 , xn1 −1 ) = 0, then xn1 −1 = xn1 = f xn1 −1 and xn1 −1 is a fixed point of f and the proof is finished. In other case, suppose that p(xn+1 , xn ) ̸= 0 for all n ∈ N. Then, using an assumption on ψ in (2.2) we have for n ∈ N∗ p(xn+1 , xn ) ≤ p(xn , xn−1 ) − ψ(p(xn , xn−1 )) < p(xn , xn−1 ). Put ρn = p(xn+1 , xn ), then we have ρn ≤ ρn−1 − ψ(ρn−1 ) < ρn−1 .

(2.3)

Therefore {ρn } is a nonnegative non-increasing sequence and hence possesses a limit ρ∗ . From (2.3), taking limit when n → +∞, we get ρ∗ ≤ ρ∗ − ψ(ρ∗ ), and, consequently, ψ(ρ∗ ) = 0. By our assumptions on ψ, we conclude ρ∗ = 0, that is, lim p(xn , xn+1 ) = 0. (2.4) n→+∞

In what follows we shall show that {xn } is a Cauchy sequence in the partial metric space (X, p). Fix ε > 0, as ρn = p(xn+1 , xn ) → 0, there exists n0 ∈ N such that ε ε p(xn0 +1 , xn0 ) ≤ min{ , ψ( )}. (2.5) 2 2 We claim that if z ∈ X verifies p(z, xn0 ) ≤ ε and xn0 ≤X z, we get p(f z, xn0 ) ≤ ε. Indeed, to do this we distinguish two cases : Case 1. p(z, xn0 ) ≤ 2ε . In this case, as z and xn0 are comparable, we have p(f z, xn0 ) ≤p(f z, f xn0 ) + p(f xn0 , xn0 ) =p(f z, f xn0 ) + p(xn0 +1 , xn0 ) ≤p(z, xn0 ) − ψ(p(z, xn0 )) + p(xn0 +1 , xn0 ) ≤p(z, xn0 ) + p(xn0 +1 , xn0 ) ε ε ≤ + = ε. 2 2 ε Case 2. 2 ≤ p(z, xn0 ) ≤ ε. In this case, as ψ is a non-decreasing function, ψ( 2ε ) ≤ ψ(p(z, xn0 )). Therefore, from (2.5) we have p(f z, xn0 ) ≤p(f z, f xn0 ) + p(f xn0 , xn0 ) =p(f z, f xn0 ) + p(xn0 +1 , xn0 ) ≤p(z, xn0 ) − ψ(p(z, xn0 )) + p(xn0 +1 , xn0 ) ε ≤p(z, xn0 ) − ψ( ) + p(xn0 +1 , xn0 ) 2 ε ε ≤p(z, xn0 ) − ψ( ) + ψ( ) 2 2 ≤p(z, xn0 ) ≤ ε. This proves the claim. As xn0 +1 verifies p(xn0 +1 , xn0 ) ≤ ε and xn0 ≤X xn0 +1 , the claim gives us that xn0 +2 = f xn0 +1 verifies p(xn0 +2 , xn0 ) ≤ ε. We repeat this process to get p(xn , xn0 ) ≤ ε for any n ≥ n0 .

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It follows that p(xn , xm ) ≤ p(xn , xn0 ) + p(xn0 , xm ) ≤ ε + ε = 2ε for any n, m ≥ n0 . Since ε is arbitrary, then

lim

n,m→+∞

p(xn , xm ) = 0. Thus, {xn } is a Cauchy sequence

in (X, p), so by Lemma 1.5, {xn } is a Cauchy sequence in the metric space (X, ps ). Since (X, p) is complete, hence (X, ps ) is complete, so there exists u ∈ X such that lim ps (xn , u) = 0. (2.6) n→+∞

Thus, by Lemma 1.5, from the condition (p2) and (2.4), we get p(u, u) = lim p(xn , u) = lim p(xn , xn ) = 0. n→+∞

n→+∞

(2.7)

We prove now that f u = u. We shall distinguish the cases (iii) and (iii’) of the Theorem 2.1. Case 1. Suppose that the mapping f is continuous. In particular, thanks to condition (b) in proposition 1.7, we have f xn converges properly to f u in (X, p), that is ps (f xn , f u) −→ 0, since ps (xn , u) −→ 0, i.e, {xn } converges properly to u in (X, p). Hence we have {f xn } converges to f u in (X, ps ). On the other hand, {f xn = xn+1 } converges to u in (X, ps ) because of (2.6). By uniqueness of the limit in metric space (X, ps ), we deduce that f u = u. Case 2. Suppose now that the condition (iii′ ) of the theorem holds. The sequence {xn } is non-decreasing with respect to ≤X , and it follows that xn ≤X u. Take x = xn and y = u (which are comparable) in (2.1) to obtain that p(u, f u) ≤ p(u, xn+1 ) + p(xn+1 , f u) ≤ p(u, xn+1 ) + p(u, xn ) − ψ(p(u, xn )). (2.8) Letting n → +∞ in (2.8) we find using (2.7) and the properties of ψ that p(f u, u) ≤ 0 − ψ(0) = 0, hence p(f u, u) = 0, so f u = u. This completes the proof of Theorem 2.1.



Corollary 2.2. Let (X, ≤X ) be a partially ordered set and let p be a partial metric on X such that (X, p) is complete. Let f : X → X be a non-decreasing map with respect to ≤X . Suppose that the following conditions hold: (i) ∃0 ≤ c < 1 such that p(f x, f y) ≤ cp(x, y)

for any y ≤X x.

(2.9)

(ii) ∃x0 ∈ X such that x0 ≤X f x0 ; (iii) f is continuous in (X, p), or; (iii’) if a non-decreasing sequence {xn } converges to x ∈ X, then xn ≤X x for all n. Then f has a fixed point u ∈ X. Moreover, p(u, u) = 0. Proof. We take ψ(t) = (1 − c)t in Theorem 2.1. Next theorem gives a sufficient condition for the uniqueness of the fixed point. Theorem 2.3. Let all the conditions of Theorem 2.1 be fulfilled and let the following condition be satisfied: for arbitrary two points x, y ∈ X there exists z ∈ X which is comparable with both x and y. Then the fixed point of f is unique.

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Proof. Let u and v be two fixed points of f , i.e., f u = u and f v = v. Consider the following two cases: Case 1. u and v are comparable. Then we can apply condition (2.1) and obtain that p(u, v) = p(f u, f v) ≤ p(u, v) − ψ(p(u, v)), hence ψ(p(u, v)) ≤ 0, i,e, p(u, v) = 0, so u = v, that is the uniqueness of the fixed point of f . Case 2. Suppose now that u and v are not comparable. Choose an element w ∈ X comparable with both of them. Then also, u = f n u is comparable with f n w for each n (since f is non-decreasing). Applying (2.1), one obtains for n ∈ N∗ that p(u, f n w) = p(f f n−1 u, f f n−1 w) ≤ p(f n−1 u, f n−1 w) − ψ(p(f n−1 u, f n−1 w)) ≤ p(f n−1 u, f n−1 w) = p(u, f n−1 w). It follows that the sequence {p(u, f n w)} is non-increasing and it has a limit l ≥ 0. Assuming that l > 0 and passing to the limit in the relation p(u, f n w) ≤ p(u, f n−1 w) − ψ(p(u, f n−1 w)), one obtains that l = 0, a contradiction. In the same way it can be deduced that p(v, f n w) → 0 as n → +∞. Now, passing to the limit in p(u, v) ≤ p(u, f n w) + p(f n w, v), it follows that p(u, v) = 0, so u = v, and the uniqueness of the fixed point is proved.  Example 2.4. Let X = [0, +∞[ endowed with the usual partial metric p defined by p : X × X → [0, +∞[ with p(x, y) = max{x, y}. We give the partial order on X by x ≤X y ⇐⇒ p(x, x) = p(x, y) ⇐⇒ x = max{x, y} ⇐= y ≤ x. It is clear that (X, ≤X ) is totally ordered. The partial metric space (X, p) is complete because (X, ps ) is complete. Indeed, for any x, y ∈ X, ps (x, y) = 2p(x, y) − p(x, x) − p(y, y) =2 max{x, y} − (x + y) =|x − y|, Thus, (X, ps ) = ([0, +∞[, |.|) is the usual metric space, which is complete. Again, we define 1 f (t) = t, if t ≥ 0. 2 The function f is continuous on (X, p). Indeed, let {xn } be a sequence converging to x in (X, p), then lim max{xn , x} = lim p(xn , x) = p(x, x) = x

n→+∞

n→+∞

hence by definition of f , we have 1 1 max{xn , x} = x = p(f x, f x), n→+∞ n→+∞ n→+∞ 2 2 (2.10) that is {f (xn )} converges to f (x) in (X, p). On the other hand, if {xn } converges properly to x in X, hence lim ps (xn , x) = 0. lim p(f xn , f x) = lim max{f xn , f x} = lim

n→+∞

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Thus, by definition of ps and f , one can find lim ps (f xn , f x) = 0.

n→+∞

(2.11)

Both convergences (2.10)-(2.11) yield that f is continuous on (X, p). Any x, y ∈ X are comparable, so for example we take x ≤X y, and then p(x, x) = p(x, y), so y ≤ x. Since f (y) ≤ f (x), so f (x) ≤X f (y), giving that f is monotone nondecreasing with respect to ≤X . In particular, for any x ≤X y, we have 1 (2.12) p(x, y) = x, p(f x, f y) = f (x) = x. 2 Let us take ψ : [0, +∞[→ [0, +∞[ such that ψ(t) = 14 t. We have for any x ∈ X, 1 x ≤ x − 14 x. Consequently, we get for any x ≤X y, thanks to this and (2.12) 2 p(f x, f y) ≤ p(x, y) − ψ(p(x, y), that is (2.1) holds. All the hypotheses of Theorem 2.1 are satisfied, so f has a unique fixed point in X, which is u = 0. Acknowledgements: The author thanks the referees for their kind comments and suggestions to improve this paper. References 1. I. Altun and G. Durmaz, Some fixed point theorems on ordered cone metric spaces, Rend. Circ. Mat. Palermo. 58 (2009), 319325. 1 2. I. Altun and A. Erduran, Fixed point theorems for monontone mappings on partial metric spaces, Fixed Point Theory Appl. 2011 (2011), Article ID 508730. 1 3. I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010 (2010), Article ID 621469. 1 4. I. Altun, F. Sola and H. Simsek, Generalized contractions on partial metric spaces, Topology Appl. 157 (18) (2010), 2278-2785. 1 5. I. Beg and A. R. Butt, Coupled fixed points of set valued mappings in partially ordered metric spaces, J. Nonlinear Sci. Appl. 3 (3) (2010), 179-185. 1 6. T. Gnana Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379-1393. 1 ´ c, N. Caki´c, M. Rajovi´c and J. Sheok Ume, Monotone generalized nonlinear con7. Lj. Ciri´ tractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 131294. 1 8. J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (2009), 3403-3410. 1, 1.2, 1 ´ c, Coupled fixed point theorems for nonlinear contractions 9. V. Lakshmikantham and Lj. Ciri´ in partially ordered metric spaces, Nonlinear Anal. 70 (2009), 4341-4349. 1 10. S. G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci. 728 (1994), 183-197. 1 11. J. J. Nieto and R. R. L´opez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order. 22 (2005), 223-239. 1, 1.1 12. H. K. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ, φ)weakly contraction in partially ordered metri spaces, Nonlinear Analysis (2010), doi: 10.1016/j.na.2010.11.024. 1 13. S. J. O’Neill, Two topologies are better than one, Tech. report, University of Warwick, Coventry, UK, http://www.dcs.warwick.ac.uk/reports/283.html, 1995. 1

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