EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, 2016, 443-451 ISSN 1307-5543 – www.ejpam.com Generalized Fixed Point Theorems in Partial Metric Spaces Mehmet Kir 1,∗and Hukmi Kiziltunc 2 1 Department of Civil Engineering, Faculty of Engineering Şırnak University, Şırnak, Turkey 2 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkey Abstract. This paper consist of some generalized fixed point theorems in partial metric spaces. The concept of TF -contractive mappings are introduced in partial metric space and thus, a generalization of Banach’s, Kannan’s, Chatterjea’s, Bianciari’s fixed point theorems are established for concept of partial metric space. 2010 Mathematics Subject Classifications: 47H10, 54H25 Key Words and Phrases: Fixed point, Kannan fixed point theorem, Chatterjea fixed point theorem, Contraction mappings, TF -contraction, Partial metric space 1. Introduction and Preliminaries The notion of partial metric space was introduced by Matthews in 1992 [2]. A partial metric is a extension of metric by replacing the condition d (x , x) = 0 of the (usual) metric with the inequality d (x , x) ≤ d � x , y � for all x , y . Also, this concept provide to study denotational semantics of dataflow networks [1–4]. Matthews gave some basic definitions and properties on partial metric space such as Cauchy sequence, Convergent sequence etc. One of the most interesting properties of this space is the self-distance (p (x , x)) for any point may not be zero. He also introduced the first fixed point theorem that re-named partial contraction mapping theorem. Due to importance of the fixed point theory it is very interesting to study fixed point the- orems on different concepts. Recently, many mathematicians have studied generalized fixed point theorems that arising from concept of partial metric space and the authors obtained some useful results [8, 9]. Now, we give some basic structures and results on the concept of partial metric space. Definition 1 ([1]). Let X be a nonempty set and p : X × X → [0,∞) be such that for all x , y, z ∈ X the followings are satisfied: ∗Corresponding author. Email addresses: mehmetkir04@gmail.com (M. Kir), hukmu@atauni.edu.tr (H. Kiziltunc) http://www.ejpam.com 443 c© 2016 EJPAM All rights reserved. M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 444 P1) x = y if and only if p (x , x) = p � y, y � = p � x , 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). Then, p is called partial metric on X and the pair � X , p � is called partial metric space. Remark 1. It is clear that if p (x , x) = 0, then x = y. But, on the contrary p (x , x) need not be zero. Example 1. [[8]] a) Let X = {[a, b] : a, b ∈ R, a ≤ b} and define p ([a, b] , [c, d]) =max {b, d}−min {a, c}. Then, � X , p � is a partial metric space. b) Let X = [0,∞) and define p � x , y � =max � x , y . Then, � X , p � is a partial metric space. Note that each partial metric p that defined on X generates a T0 topology τ that has a base the family of open balls {B (x ,ε) : x ∈ X ,ε > 0} such that B (x ,ε) = � y ∈ X : p � x , y � < ε+ p (x , x) Remark 2. [9] Let � X , p � be a partial metric space. M1) The function ds : X → X defined by ds � x , y � = 2p � x , y � − p (x , x)− p � y, y � is a (usual) metric on X and (X , ds) is a (usual) metric space. M2) The function dM : X → X defined by dM � x , y � =max � p(x , y)− p(x , x), p(x , y)− p(y, y) is a (usual) metric on X and � X , dM � is a (usual) metric space. Corollary 1 ([9]). Let � X , p � be a partial metric space. Then, ds and dM are equivalent metric on X . Furthermore, if we take � X , p � as in Example 1, part b), we obtain the following equality; dM � x , y � = ds � x , y � = � �x − y � � . Definition 2 ([8]). Let � X , p � be a partial metric space. (i) A sequence � xn in � X , p � converges to x ∈ X if and only if p (x , x) = lim n→∞ p � xn, x � . M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 445 (ii) A sequence � xn in � X , p � is called a Cauchy sequence if and only if lim n,m→∞ p � xn, xm � exists (and finite). (iii) A partial metric space is called complete if every Cauchy sequence � xn in X converges, with respect to τ, to a point x ∈ X such that lim n,m→∞ p � xn, xm � = p (x , x). Lemma 1. [[8]] Let � X , p � be a partial metric space. a) � xn is a Cauchy sequence in � X , p � if and only if it is Cauchy sequence in (X , ds). b) � X , p � is complete if and only if (X , ds) is complete. Moreover, lim n→∞ ds � xn, x � = 0 if and only if lim n→∞ p � xn, x � = lim n→∞ p � xn, xm � = p (x , x) . Moradi and Beiranvand [12] introduced TF - type contraction mappings as follows: Definition 3 ([12]). Let (X , d) be a metric space. A mapping T : X → X is said to be graph closed if for every sequence � xn such that lim n→∞ T xn = a then for some b ∈ X , T b = a. Definition 4 ([12]). Let (X , d) be a metric space and f , T : X → X be two functions. The mapping f is said to be a TF -contraction if there exists α ∈ [0, 1) such that for all x , y ∈ X F � d � T f x , T f y �� ≤ αF � d � T x , T y �� (1) where 1) F : [0,∞)→ [0,∞), F is nondecreasing continuous from the right and F−1 (0) = {0}. 2) T is one to one and graph closed. Theorem 1 ([12]). Let (X , d) be a complete metric space. If f : X → X is a TF -contraction mapping then f has a unique fixed point in complete metric space (X , d). Kir and Kiziltunc [13] introduced TF -contractive conditions for Kannan fixed point theo- rem and Chatterjea fixed point theorem. Definition 5 ([10, 11]). Let (X , d) be a metric space. (i) A mapping T : X → X is said to be sequentially convergent if we have, for every sequence � yn , if � T yn is convergent then � yn is also convergent. (ii) T is said to be subsequentially convergent if we have, for every sequence � yn , if � T yn is convergence then � yn has a convergent subsequence. For instance, the mappings T x = x , T x = ln x(x > 0) are sequentially convergent on the metric space (R, |·|). The mapping T x = x2 is not sequentially convergent on the metric space (R, |·|) but it is subsequentially convergent. In this study, we aim to introduce TF type fixed point theorems in partial metric space. M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 446 2. TF− Type Contractive Conditions for Banach’s, Kannan’s and Chatterjea’s Fixed Point Theorems Theorem 2. Let � X , p � be a complete partial metric space and T, f : X → X be mappings such that T is one to one and subsequentially convergent. If for all k ∈ [0, 1) and x , y ∈ X F � p � T f x , T f y �� ≤ kF � p � T x , T y �� (2) where F : [0,∞)→ [0,∞) is nondecreasing continuous and F (t) = 0 if and only if t = 0. Then f has a unique fixed point in X . Proof. Let x0 ∈ X be an arbitrary point and xn = f xn−1 = f n x0, n= 1, 2,3, · · · F � p � T xn, T xn+1 �� =F � p � T f xn−1, T f xn �� ≤kF � p � T xn−1, T xn �� ... ≤kn−1F � p � T x0, T x1 �� . (3) Also, for all m, n ∈ N, for m> n, we have F � p � T xn, T xm �� =F � p � T f n x0, T f m x0 �� ≤knF � p � T x0, T f m−n x0 �� . (4) Let m, n→∞ in (4), we obtain F � p � T xn, T xm �� → 0+ as m, n→∞. As F is continuous, we obtain lim m,n→∞ p � T xn, T xm � = 0. (5) Thus, we see that � T xn is a Cauchy sequence in � X , p � . From Lemma 1, we get that � T xn is Cauchy sequence in (X , ds). Since � X , p � is a complete partial metric space then (X , ds) is also complete metric space and there exists v ∈ X such that � T xn converges to v ∈ X . Note that T is subsequentially convergent, then there exists an u ∈ X such that lim k→∞ p � xn(k), u � = lim k→∞ p (u, u) . Also, T is continuous and xn(k)→ u, therefore lim k→∞ T xn(k) = Tu and lim k→∞ p � T xn(k), Tu � = p (Tu, Tu) . Since, � T xn(k) is a subsequence of � T xn , so we obtain Tu= v. M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 447 Also, ds � T xn, Tu � = 2p � T xn, Tu � − p � T xn, T xn � − p (Tu, Tu) . (6) Let n→∞ in (6), we have lim n→∞ ds � T xn, Tu � = 0. Consider Lemma 1/part b) and (5) we hold lim n→∞ p � T xn, Tu � = lim m,n→∞ p � T xn, T xm � = p (Tu, Tu) = 0. Now, we will show that u ∈ X is a fixed point of f . Indeed, as F is continuous F � p � T f u, T xn+1 �� =F � p � T f u, T f xn �� ≤kF � p � Tu, T xn �� . (7) Let n→∞ in (7), we obtain F(p � T f u, Tu � )≤ 0 this implies that p � Tu, T f u � = 0 and hence Tu = T f u. Also, T is one to one, we obtain f u= u. Now, we show that the fixed point is unique. Assume u′ is an other fixed point of f then, we have f u′ = u′ and F(p � Tu, Tu′ � ) =F(p � T f u, T f u′ � ) ≤kF � p � Tu, Tu′ �� (8) The inequality (8) is contradiction unless p � Tu, Tu′ � = 0. Thus, Tu = Tu′ with consideration T is one to one, we obtain the fixed point is unique. Also, if we take T is sequentially convergent, by replacing {n} with {n (k)} we conclude that lim n→∞ xn = u this shows that � xn converges to the fixed point of f . In Theorem 2, if we consider F and T as identity, we get the following result given by Matthews [2]. Corollary 2. Let � X , p � be a complete partial metric space and f : X → X be mapping. If α ∈ [0,1) and x , y ∈ X , p � f x , f y � ≤ αp � x , y � (9) then, f has a unique fixed point. M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 448 Theorem 3. Let � X , p � be a complete partial metric space and T, f : X → X be mappings such that T is one to one, continuous and subsequentially convergent (or graph closed). If for each β ∈ � 0, 1 2 � and x , y ∈ X , we have F � p � T f x , T f y �� ≤ β � F � p � T x , T f x �� + F � p � T y, T f y ��� (10) where F : [0,∞)→ [0,∞) is nondecreasing continuous and F (t) = 0 if and only if t = 0. Then f has a unique fixed point in X . Proof. Let x0 ∈ X be an arbitrary point and xn = f xn−1 = f n x0, n= 1, 2,3, . . . F � p � T xn, T xn+1 �� =F � p � T f xn−1, T f xn �� ≤β � F � p � T xn−1, T xn �� + F � p � T xn, T xn+1 ��� therefore, we have F � p � T xn, T xn+1 �� ≤ β 1− β F � p � T xn−1, T xn �� . Also, we obtain that F � p � T xn, T xn+1 �� ≤ � β 1− β �n F � p � T x0, T x1 �� . (11) Let n→∞ in (11), we obtain that F � p � T xn, T xn+1 �� → 0+ as n→∞. Again using (11), for all m, n ∈ N, taking m> n, we have F � p � T xn, T xm �� ≤ � β 1− β �n F � p � T x0, T f m−n x0 �� (12) Letting m, n→∞ in (12), we have F � p � T xn, T xm �� → 0+ as m, n→∞. So, we have p � T xn, T xm � → 0 as m, n→∞. In the next stage, by using similar methods in Theorem 2, we obtain that � T xn is Cauchy sequence in complete partial metric space � X , p � and there exist u ∈ X such that � T xn con- verges to Tu ∈ X and xn(k)→ u, such that lim k→∞ T xn(k) = Tu and lim k→∞ p � T xn(k), Tu � = p (Tu, Tu) = 0. Now, we will show that u ∈ X is a fixed point of f . Indeed, we have F � p � T f u, T xn+1 �� =F � p � T f u, T f xn �� M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 449 ≤β � F � p � Tu, T f u �� + F � p � T xn, T xn+1 ��� . (13) Let n→∞ in (13), we have F(p � T f u, Tu � )≤ βF � p � Tu, T f u �� . (14) The inequality (14) is contradiction unless p � Tu, T f u � = 0. Thus, Tu = T f u. Also, T is one to one, we obtain f u= u. Thus we provide u ∈ X is a fixed point of f . The uniqueness of the fixed point can be shown easily. Some results of the Theorem 3 are following. Corollary 3. Let � X , p � be a complete partial metric space and T, f : X → X be mappings such that T is one to one, continuous and subsequentially convergent (or graph closed). If for β ∈ � 0, 1 2 � and for x , y ∈ X , p � T f x , T f y � ≤ β � p � T x , T f x � + p � T y, T f y �� . Then, f has a unique fixed point in � X , p � . Corollary 4. Let � X , p � be a complete partial metric space and f : X → X be a mapping. If for β ∈ � 0, 1 2 � and for x , y ∈ X , F � p � f x , f y �� ≤ β � F � p � x , f x �� + F � p � y, f y ��� where F : [0,∞) → [0,∞), F is nondecreasing continuous from the right and F−1 (0) = {0}. Then f has a unique fixed point. Corollary 5. Let � X , p � be a complete partial metric space and f : X → X be mapping. If β ∈ � 0, 1 2 � and x , y ∈ X . p � f x , f y � ≤ β � p � x , f x � + p � y, f y �� Then, f has a unique fixed point in � X , p � . Theorem 4. Let � X , p � be a complete partial metric space and T, f : X → X be mappings such that T is one to one, continuous and subsequentially convergent (or graph closed). If λ ∈ � 0, 1 2 � and for each x , y ∈ X F � p � T f x , T f y �� ≤ λ � F � p � T x , T f y �� + F � p � T y, T f x ��� (15) where F : [0,∞)→ [0,∞) is nondecreasing continuous from the right and F−1 (0) = {0}. Then, f has a unique fixed point in X . Proof. Let x0 ∈ X be an arbitrary point and xn = f xn−1 = f n x0. Also consider p � T xn, T xn � ≤ p � T xn, T xn+1 � F(p � T xn, T xn+1 � ) =F(p � T f xn−1, T f xn � ) M. Kir, H. Kiziltunc / Eur. J. Pure Appl. Math, 9 (2016), 443-451 450 ≤λF(p � T xn−1, T xn+1 � ) +λF(p � T xn, T xn � ) ≤λF(p � T xn−1, T xn+1 � ) +λF(p � T xn+1, T xn � ) therefore, we have F(p � T xn, T xn+1 � )≤ λ 1−λ F(p � T xn−1, T xn+1 � ). Also, for all m (k) , n (k) ∈ N, taking m (k)> n (k), we have F(p � T xm(k), T xn(k) � ≤ � λ 1−λ �n(k) F(p � T xm(k)−n(k), T xn(k) � ). (16) Note that T is subsequentially convergent, then there exists u ∈ X such that lim k→∞ p � xn(k), u � = lim k→∞ p (u, u) . Let k→∞ in (16), we obtain that F(p � T xm(k), T xn(k) � → 0+ as k→∞. (17) The inequality (17) implies that p � T xm(k), T xn(k) � = 0 Hence, we obtain that � T xn is Cauchy sequence in complete partial metric space � X , p � and there exist a point u ∈ X such that this point the unique fixed point of f . Corollary 6. Let � X , p � be a complete partial metric space and T, f : X → X be mappings such that T is one to one, continuous and subsequentially convergent (or graph closed). If λ ∈ � 0, 1 2 � and for each x , y ∈ X , p � T f x , T f y � ≤ λ � p � T x , T f y � + p � T y, T f x �� then f has a unique fixed point. Also, if T is sequentially convergent then for every x0 ∈ X the sequence of iterates � f n x0 converges to the fixed point. Corollary 7. Let � X , p � be a complete partial metric space and f : X → X be a mapping. 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