This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Fixed point of generalized weakly contractive mappings in ordered partial metric spaces Fixed Point Theory and Applications 2012, 2012:1 doi:10.1186/1687-1812-2012-1 Mujahid Abbas (mujahid@lums.edu.pk) Talat Nazir (talat@lums.edu.pk) ISSN 1687-1812 Article type Research Submission date 6 July 2011 Acceptance date 2 January 2012 Publication date 2 January 2012 Article URL http://www.fixedpointtheoryandapplications.com/content/2012/1/1 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Fixed point of generalized weakly contractive mappings in ordered partial metric spaces Mujahid Abbas ∗ and Talat Nazir Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan ∗ Corresponding author: mujahid@lums.edu.pk Email address: TN: talat@lums.edu.pk Abstract In this article, we prove some fixed point results for generalized weakly contractive mappings defined on a partial metric space. We provide some examples to validate our results. These results unify, generalize and complement various known comparable results from the current literature. AMS Classification 2010: 47H10; 54H25; 54E50. Keywords: partial metric space; weakly contractive condition; non- decreasing map; fixed point; partially ordered set. 1 Introduction and preliminaries In the past years, the extension of the theory of fixed point to generalized structures as cone metrics, partial metric spaces and quasi-metric spaces has received much attention (see, for instance, [1–7] and references therein). Partial metric space is generalized metric space in which each object does not necessarily have to have a zero distance from itself [8]. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle, more suitable in this context [8,9]. Salvador and Schellekens [10] have shown that the dual complexity space can be modelled as stable partial monoids. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions ( e.g., [1,2,11–18]). 1 Existence of fixed points in ordered metric spaces has been initiated in 2004 by Ran and Reurings [19], and further studied by Nieto and Lopez [20]. Subse- quently, several interesting and valuable results have appeared in this direction [21–28]. The aim of this article is to study the necessary conditions for existence of common fixed points of four maps satisfying generalized weak contractive conditions in the framework of complete partial metric spaces endowed with a partial order. Our results extend and strengthen various known results [8, 29–32]. In the sequel, the letters R, R + , ω and N will denote the set of real numbers, the set of nonnegative real numbers, the set of nonnegative integer numbers and the set of positive integer numbers, respectively. The usual order on R (respectively, on R + ) will be indistinctly denoted by ≤ or by ≥ . Consistent with [8,12], the following definitions and results will be needed in the sequel. Definition 1.1. Let X be a nonempty set. A mapping p : X × X → R + is said to be a partial metric on X if for any x, y, z ∈ X, the following conditions hold true: (P 1 ) p(x, x) = p(y, y) = p(x, y) if and only if x = y; (P 2 ) p(x, x) ≤ p(x, y); (P 3 ) p(x, y) = p(y, x); (P 4 ) p(x, z) ≤ p(x, y) + p(y, z) − p(y, y). The pair (X, p) is then called a partial metric space. Throughout this article, X will denote a partial metric space equipped with a partial metric p unless or otherwise stated. If p(x, y) = 0, then (P 1 ) and (P 2 ) imply that x = y. But converse does not hold always. A trivial example of a partial metric space is the pair (R + , p) , where p : R + × R + → R + is defined as p(x, y) = max{x, y}. Example 1.2. [8] If X = {[a, b] : a, b ∈ R, a ≤ b} then p([a, b], [c, d]) = max{b, d} − min{a, c} defines a partial metric p on X. For some more examples of partial metric spaces, we refer to [12,13,16,17]. Each partial metric p on X generates a T 0 topology τ p on X which has as a base 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. Observe (see [8, p. 187]) that a sequence {x n } in X converges to a point x ∈ X, with respect to τ p , if and only if p(x, x) = lim n→∞ p(x, x n ). If p is a partial metric on X, then the function p S : X × X → R + given by p S (x, y) = 2p(x, y) − p(x, x) − p(y, y), defines a metric on X. Furthermore, a sequence {x n } converges in (X, p S ) to a point x ∈ X if and only if lim n,m→∞ p(x n , x m ) = lim n→∞ p(x n , x) = p(x, x). (1.1) 2 Definition 1.3. [8] Let X be a partial metric space. (a) A sequence {x n } in X is said to be a Cauchy sequence if lim n,m→∞ p(x n , x m ) exists and is finite. (b) X 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 lim n→∞ p(x, x n ) = p(x, x). In this case, we say that the partial metric p is complete. Lemma 1.4. [8,12] Let X be a partial metric space. Then: (a) A sequence {x n } in X is a Cauchy sequence in X if and only if it is a Cauchy sequence in metric space (X, p S ). (b) A partial metric space X is complete if and only if the metric space (X, p S ) is complete. Definition 1.5. A mapping f : X → X is said to be a weakly contractive if d(fx, fy) ≤ d(x, y) − ϕ(d(x, y)), for all x, y ∈ X, (1.2) In 1997, Alber and Guerre-Delabriere [33] proved that weakly contractive mapping defined on a Hilbert space is a Picard operator. Rhoades [34] proved that the corresponding result is also valid when Hilbert space is replaced by a complete metric space. Dutta et al. [35] generalized the weak contractive con- dition and proved a fixed point theorem for a selfmap, which in turn generalizes Theorem 1 in [34] and the corresponding result in [33]. Recently, Aydi [29] obtained the following result in partial metric spaces. Theorem 1.6. 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 nondecreasing map with respect to ≤ X . Suppose that the following conditions hold: for y ≤ x, we have p(fx, fy) ≤ p(x, y) − ϕ(p(x, y)), (1.3) 1. where ϕ : [0, +∞[→ [0, +∞[ is a continuous and non-decreasing function such that it is positive in ]0, +∞[, ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞; (i) (ii) there exist x 0 ∈ X such that x 0 ≤ X fx 0 ; (iii) f is continuous in (X, p), or; (iii’) if a non-decreasing sequence {x n } converges to x ∈ X, then x n ≤ X x for all n. Then f has a fixed point u ∈ X. Moreover, p(u, u) = 0. A nonempty subset W of a partially ordered set X is said to be well ordered if every two elements of W are comparable. 3 2 Fixed point results In this section, we obtain several fixed point results for selfmaps satisfying gener- alized weakly contractive conditions defined on an ordered partial metric space, i.e., a (partially) ordered set endowed with a complete partial metric. We start with the following result. Theorem 2.1. Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X. Supp ose that for every two elements x, y ∈ X with y  x, we have ψ(p(fx, fy)) ≤ ψ(M(x, y)) − φ(M(x, y)), (2.1) where M(x, y) = max{p(x, y), p(f x, x), p(fy, y), p(x, fy) + p(y, fx) 2 }, ψ, φ : R + → R + , ψ is continuous and nondecreasing, φ is a lower semicontinuous, and ψ(t) = φ(t) = 0 if and only if t = 0. If there exists x 0 ∈ X with x 0  f x 0 and one of the following two conditions is satisfied: (a) f is continuous self map on (X, p S ); (b) for any nondecreasing sequence {x n } in (X, ) with lim n→∞ p S (z, x n ) = 0 it follows x n  z for all n ∈ N, then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point. Proof. Note that if f has a fixed point u, then p(u, u) = 0. Indeed, assume that p(u, u) > 0. Then from (2.1) with x = y = u, we have ψ(p(u, u)) = ψ(p(fu, fu)) ≤ ψ(M (u, u)) − φ(M(u, u)), (2.2) where M(u, u) = max{p(u, u), p(f u, u), p(fu, u), p(u, fu) + p(u, fu) 2 } = max{p(u, u), p(u, u), p(u, u), p(u, u) + p(u, u) 2 } = p(u, u). Now we have: ψ(p(u, u)) = ψ(p(fu, fu)) ≤ ψ(p(u, u)) − φ(p(u, u)), φ(p(u, u)) ≤ 0, a contradiction. Hence p(u, u) = 0. Now we shall prove that there exists a nondecreasing sequence {x n } in (X, ) with fx n = x n+1 for all n ∈ N, and lim n→∞ p(x n , x n+1 ) = 0. For this, let x 0 be an arbitrary point of X. Since f is nondecreasing, and x 0  fx 0 , we have x 1 = fx 0  f 2 x 0   f n x 0  f n+1 x 0  . 4 Define a sequence {x n } in X with x n = f n x 0 and so x n+1 = fx n for n ∈ N. We may assume that M (x n+1 , x n ) > 0, for all n ∈ N. If not, then it is clear that x k = x k+1 for some k, so f x k = x k+1 = x k , and thus x k is a fixed point of f. Now, by taking M (x n+1 , x n ) > 0 for all n ∈ N, consider ψ(p(x n+2 , x n+1 )) = ψ(p(f x n+1 , fx n )) ≤ ψ(M(x n+1 , x n )) − φ(M (x n+1 , x n )), (2.3) where M(x n+1 , x n ) = max{p(x n+1 , x n ), p(fx n+1 , x n+1 ), p(fx n , x n ), p(x n+1 , fx n ) + p(x n , fx n+1 ) 2 } = max{p(x n+1 , x n ), p(x n+2 , x n+1 ), p(x n+1 , x n ), p(x n+1 , x n+1 ) + p(x n , x n+2 ) 2 } ≤ max{p(x n+1 , x n ), p(x n+2 , x n+1 ), p(x n , x n+1 ) + p(x n+1 , x n+2 ) 2 } = max{p(x n+1 , x n ), p(x n+2 , x n+1 )}. Suppose that max{p(x k+1 , x k ), p(x k+2 , x k+1 )} = p(x k+2 , x k+1 ) for some k ∈ N. Then ψ(p(x k+2 , x k+1 )) ≤ ψ(p(x k+2 , x k+1 ))−φ(M(x k+1 , x k )) implies φ(M (x k+1 , x k )) ≤ 0, a contradiction. Consequently ψ(p(x n+2 , x n+1 )) ≤ ψ(p(x n+1 , x n )) − φ(M (x n+1 , x n )) ≤ ψ(p(x n+1 , x n )), for all n ∈ N. Since ψ is nondecreasing, so the sequence of p ositive real numbers {p(x n+1 , x n )} is nonincreasing, therefore {p(x n+1 , x n )} converges to a c ≥ 0. Suppose that c > 0. Then ψ(p(x n+2 , x n+1 )) ≤ ψ(M(x n+1 , x n )) − φ(M (x n+1 , x n )), and lower semicontinuity of φ gives that lim sup n→∞ ψ(p(x n+2 , x n+1 )) ≤ limsup n→∞ ψ(M(x n+1 , x n )) − lim inf n→∞ φ(M(x n+1 , x n )), which implies that ψ(c) ≤ ψ(c) − φ(c), a contradiction. Therefore c = 0, i.e., lim n→∞ p(x n+1 , x n ) = 0. Now, we prove that lim n,m→∞ p(x n , x m ) = 0. If not, then there exists ε > 0 and sequences {n k }, {m k } in N, with n k > m k ≥ k, and such that p(x n k , x m k ) ≥ ε for all k ∈ N. We can suppose, without loss of generality that p(x n k , x m k −1 ) < ε. So ε ≤ p(x m k , x n k ) ≤ p(x n k , x m k −1 ) + p(x m k −1 , x m k ) − p(x m k −1 , x m k −1 ) 5 implies that lim k→∞ p(x m k , x n k ) = ε. (2.4) Also (2.4) and inequality p(x m k , x n k ) ≤ p(x m k , x m k −1 )+p(x m k −1 , x n k )−p(x m k −1 , x m k −1 ) gives that ε ≤ lim k→∞ p(x m k −1 , x n k ), while (2.4) and inequality p(x m k −1 , x n k ) ≤ p(x m k −1 , x m k ) + p(x m k , x n k ) − p(x m k , x m k ) yields lim k→∞ p(x m k −1 , x n k ) ≤ ε, and hence lim k→∞ p(x m k −1 , x n k ) = ε. (2.5) Also (2.5) and inequality p(x m k −1 , x n k ) ≤ p(x m k −1 , x n k +1 ) + p(x n k +1 , x n k ) − p(x n k +1 , x n k +1 ) implies that ε ≤ lim k→∞ p(x m k −1 , x n k +1 ), while inequality p(x m k −1 , x n k +1 ) ≤ p(x m k −1 , x n k )+p(x n k , x n k +1 )−p(x n k , x n k ) yields lim k→∞ p(x m k −1 , x n k +1 ) ≤ ε, and hence lim k→∞ p(x m k −1 , x n k +1 ) = ε. (2.6) Finally p(x m k , x n k ) ≤ p(x m k , x n k +1 ) + p(x n k +1 , x n k ) − p(x n k +1 , x n k +1 ) gives that ε ≤ lim k→∞ p(x m k , x n k +1 ), and the inequality p(x m k , x n k +1 ) ≤ p(x m k , x n k ) + p(x n k , x n k +1 ) − p(x n k , x n k ) gives lim k→∞ p(x m k , x n k +1 ) ≤ ε, and hence lim k→∞ p(x m k , x n k +1 ) = ε. (2.7) As M(x n k , x m k −1 ) = max{p(x n k , x m k −1 ), p(fx n k , x n k ), p(fx m k −1 , x m k −1 ), p(x n k , fx m k −1 ) + p(x m k −1 , fx n k ) 2 } = max{p(x n k , x m k −1 ), p(x n k +1 , x n k ), p(x m k , x m k −1 ), p(x n k , x m k ) + p(x m k −1 , x n k +1 ) 2 }, therefore lim k→∞ M(x n k , x m k −1 ) = max{ε, 0, 0, ε} = ε. From (2.1), we obtain ψ(p(x n k +1 , x m k )) = ψ(p(fx n k , fx m k −1 )) ≤ ψ(M(x n k , x m k −1 ))−φ(M(x n k , x m k −1 )). Taking upper limit as k → ∞ implies that ψ(ε) ≤ ψ(ε) − φ(ε), which is a con- tradiction as ε > 0. Thus, we obtain that lim n,m→∞ p(x n , x m ) = 0, i.e., {x n } is a Cauchy sequence in (X, p), and hence in the metric space (X, p S ) by Lemma 1.4. Finally, we prove that f has a fixed point. Indeed, since (X, p) is complete, then from Lemma 1.4, (X, p S ) is also complete, so the sequence {x n } is con- vergent in the metric space (X, p S ). Therefore, there exists u ∈ X such that lim n→∞ p S (u, x n ) = 0, equivalently, lim n,m→∞ p(x n , x m ) = lim n→∞ p(x n , u) = p(u, u) = 0, (2.8) 6 because lim n,m→∞ p(x n , x m ) = 0. If f is continuous self map on (X, p S ), then it is clear that fu = u. If f is not continuous, we have, by our hypothesis, that x n  u for all n ∈ N, because {x n } is a nondecreasing sequence with lim n→∞ p S (u, x n ) = 0. Now from the following inequalities p(fu, u) ≤ M(u, x n ) = max{p(u, x n ), p(fu, u), p(fx n , x n ), p(u, fx n ) + p(x n , fu) 2 } = max{p(u, x n ), p(fu, u), p(x n+1 , x n ), p(u, x n+1 ) + p(x n , fu) 2 } ≤ max{p(u, x n ), p(fu, u), p(x n+1 , x n ), p(x n+1 , u) + p(x n , u) + p(u, fu) − p(u, u) 2 }, we deduce, taking limit as n → ∞, that lim n→∞ M(u, x n ) = p(f u, u). Hence, ψ(p(fu, fx n+1 )) ≤ ψ(M(u, x n )) − φ(M (u, x n )). (2.9) On taking upper limit as n → ∞, we have ψ(p(fu, u)) ≤ ψ(p(fu, u)) − φ(p(fu, u)), and fu = u. Finally, suppose that set of fixed points of f is well ordered. We prove that fixed point of f is unique. Assume on contrary that fv = v and fw = w but v = w. Hence ψ(p(v, w)) = ψ(p(fv, fw)) ≤ ψ(M(v, w)) − φ(M (v, w)), (2.10) where M(v, w) = max{p(v, w), p(fv, v), p(f w, w), p(v, fw) + p(w, fv) 2 } = max{p(v, w), p(v, v), p(w, w), p(v, w) + p(w, v) 2 } = p(v, w), that is, by (2.10), ψ(p(v, w)) = ψ(p(fv, fw)) ≤ ψ(p(v, w)) − φ(p(v, w)), a contradiction because p(v, w) > 0. Hence v = w. Conversely, if f has only one fixed point then the set of fixed point of f being singleton is well ordered.  7 Consistent with the terminology in [36], we denote Υ the set of all functions ϕ : R + → R + , where ϕ is a Lebesgue integrable mapping with finite integral on each compact subset of R + , nonnegative, and for each ε > 0,  ε 0 ϕ(t)dt > 0 (see also, [37]). As a consequence of Theorem 2.1, we obtain following fixed point result for a mapping satisfying contractive conditions of integral type in a complete partial metric space X. Corollary 2.2. Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X. Supp ose that for every two elements x, y ∈ X with y  x, we have  ψ (p(f x,f y)) 0 ϕ(t)dt ≤  ψ ( M (x,y)) 0 ϕ(t)dt −  φ(M(x,y)) 0 ϕ(t)dt, (2.11) where ϕ ∈ Υ, M(x, y) = max{p(x, y), p(f x, x), p(fy, y), p(x, fy) + p(y, fx) 2 }, ψ, φ : R + → R + , ψ is continuous and nondecreasing, φ is a lower semicontinuous, and ψ(t) = φ(t) = 0 if and only if t = 0. If there exists x 0 ∈ X with x 0  f x 0 and one of the following two conditions is satisfied: (a) f is continuous self map on (X, p S ); (b) for any nondecreasing sequence {x n } in (X, ) with lim n→∞ p S (z, x n ) = 0 it follows x n  z for all n ∈ N, then f has a fixed point. Proof. Define Ψ : [0, ∞) → [0, ∞) by Ψ(x) =  x 0 ϕ(t)dt, then from (2.11), we have Ψ (ψ(p(fx, fy))) ≤ Ψ (ψ(M(x, y))) − Ψ (φ(M(x, y))) , (2.12) which can be written as ψ 1 (p(fx, fy)) ≤ ψ 1 (M(x, y)) − φ 1 (M(x, y)), (2.13) where ψ 1 = Ψ ◦ ψ and φ 1 = Ψ ◦ φ. Clearly, ψ 1 , φ 1 : R + → R + , ψ 1 is continuous and nondecreasing, φ 1 is a lower semicontinuous, and ψ 1 (t) = φ 1 (t) = 0 if and only if t = 0. Hence by Theorem 2.1, f has a fixed point.  If we take ψ(t) = t in Theorem 2.1, we have the following corollary. Corollary 2.3. Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X. Supp ose that for every two elements x, y ∈ X with y  x, we have p(fx, fy) ≤ M (x, y) − φ(M(x, y)), (2.14) where M(x, y) = max  p(x, y), p(fx, x), p(f y, y), p(x, fy) + p(y, fx) 2  , 8 φ : R + → R + is a lower semicontinuous and φ(t) = 0 if and only if t = 0. If there exists x 0 ∈ X with x 0  fx 0 and one of the following two conditions is satisfied: (a) f is continuous self map on (X, p S ); (b) for any nondecreasing sequence {x n } in (X, ) with lim n→∞ p S (z, x n ) = 0 it follows x n  z for all n ∈ N, then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point. If we take φ(t) = (1−k)t for k ∈ [0, 1) in Corollary 2.3, we have the following corollary. Corollary 2.4. Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X. Supp ose that for every two elements x, y ∈ X with y  x, we have p(fx, fy) ≤ kM(x, y), (2.15) where M(x, y) = max  p(x, y), p(fx, x), p(f y, y), p(x, fy) + p(y, fx) 2  , and k ∈ [0, 1). If there exists x 0 ∈ X with x 0  fx 0 and one of the following two conditions is satisfied: (a) f is continuous self map on (X, p S ); (b) for any nondecreasing sequence {x n } in (X, ) with lim n→∞ p S (z, x n ) = 0 it follows x n  z for all n ∈ N, then f has a fixed point. Moreover, the set of fixed points of f is well ordered if and only if f has one and only one fixed point. Corollary 2.5. Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X. Supp ose that for every two elements x, y ∈ X with y  x, we have ψ(p(fx, fy)) ≤ p(x, y) − φ(p(x, y)), (2.16) where φ : R + → R + is a lower semicontinuous, and φ(t) = 0 if and only if t = 0. If there exists x 0 ∈ X with x 0  fx 0 and one of the following two conditions is satisfied: (a) f is continuous self map on (X, p S ); (b) for any nondecreasing sequence {x n } in (X, ) with lim n→∞ p S (z, x n ) = 0 it follows x n  z for all n ∈ N, 9 [...]... fixed point theory Nonlinear Anal Theory 69, 126–139 (2008) [4] Karapinar, E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces Comput Math Appl 59, 3656–3668 (2010) [5] Karapinar, E: Generalizations of Caristi Kirk’s theorem on partial metric spaces Fixed Point Theory Appl 2011(4) (2011) doi:10.1186/1687-18122011-4 [6] Latif, A, Al-Mezel, SA: Fixed point results in quasi metric. .. O’Regan, D, Petru¸el, A: Fixed point theorems for generalized contractions s in ordered metric spaces J Math Anal Appl 341, 1241–1242 (2008) [28] Shatanawi, W, Samet, B, Abbas, M: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces Math Comput Model (2011) doi:10.1016/j.mcm.2011.08.042 [29] Aydi, H: Some fixed point results in ordered partial metric spaces arxiv: 1103.3680v1... arxiv: 1103.3680v1 [math.GN] 1–8 (2011) [30] Nashine, HK, Samet B: Fixed point results for mappings satisfying (ψ, φ )weakly contractive condition in partially ordered metric spaces Nonlinear Anal Theory 74, 2201–2209 (2011) [31] Popescu, O: Fixed points for (ψ, φ)-weak contractions Appl Math Lett 24, 1–4 (2011) [32] Zhang, Q, Song, Y: Fixed point theory for generalized ϕ-weak contractions Appl Math Lett... u = v Conversely, if f has only one fixed point then the set of fixed point of f being singleton is well ordered Following similar arguments to those given in Corollary 2.2, we obtain following corollary as an application of Theorem 2.6 Corollary 2.7 Let (X, ) be a partially ordered set such that there exist a complete partial metric p on X and f a nondecreasing selfmap on X Suppose that for every two... Reuring, MC: A fixed point theorem in partially ordered sets and some applications to matrix equations Proc Am Math Soc 132, 1435–1443 (2004) [20] Nieto, JJ, Rodr´ ıguez-L´pez, R: Contractive mapping theorems in partially o ordered sets and applications to ordinary differential equations Order 22, 223–239 (2005) [21] Abbas, M, Nazir, T, Radenovi´, S: Common fixed points of four maps in c partially ordered. .. Pouso, RL, Rodr´ ıguez-L´pez, R: Fixed point theorems in ordered o abstract sets Proc Am Math Soc 135, 2505–2517 (2007) [25] Nieto, JJ, Rodr´ ıguez-L´pez, R: Existence and uniqueness of fixed points in o partially ordered sets and applications to ordinary differential equations Acta Math Sin (Engl Ser.), 23, 2205–2212 (2007) [26] Petru¸el, A, Rus, I: Fixed point theorems in ordered L-spaces Proc Am s Math... λ) : Bp (x0 , r) → Bp (x0 , r) Taking V = U and applying Corollary 2.5, we obtain that H(., λ) has a fixed point in U But this fixed point must be in U in the presence of assumption (a) Thus λ ∈ A for any λ ∈ (λ0 − ε, λ0 + ε), therefore A in open in [0, 1] Similarly, the reverse implication follows 19 Competing interests The authors declare that they have no competing interests Authors’ contributions... 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Fixed point of generalized weakly contractive mappings in ordered partial metric spaces Fixed Point Theory and Applications 2012, 2012:1 doi:10.1186/1687-1812-2012-1 Mujahid. distribution, and reproduction in any medium, provided the original work is properly cited. Fixed point of generalized weakly contractive mappings in ordered partial metric spaces Mujahid Abbas ∗ and. 54E50. Keywords: partial metric space; weakly contractive condition; non- decreasing map; fixed point; partially ordered set. 1 Introduction and preliminaries In the past years, the extension of the theory of