Kannan Fixed Point Theorem on Generalized Metric Space with a Graph

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1 Applied Mathematical Sciences, Vol. 3, 209, no. 6, HIKARI Ltd, Kannan Fixed Point Theorem on Generalized Metric Space with a Graph Karim Chaira Laboratory of Algebra, Analysis and Applications Faculty of Sciences Ben M sik, Hassan II University of Casablanca, Casablanca, Morocco Abderrahim Eladraoui Laboratory of Mathematics and Applications Faculty of Sciences and Technologies Mohammedia University Hassan II Casablanca, Morocco Mustapha Kabil Laboratory of Mathematics and Applications Faculty of Sciences and Technologies Mohammedia University Hassan II Casablanca, Morocco Abdessamad Kamouss Laboratory of Mathematics and Applications Faculty of Sciences and Technologies Mohammedia University Hassan II Casablanca, Morocco This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 209 Hikari Ltd. Abstract In this paper, by using the idea of combining fixed point theory and graph theory, we shall introduce the concept of G-Kannan contraction in a generalized metric space introduced recently by Jleli and Samet, endowed with graph. In this setting, we investigate the existence and uniqueness of the fixed point for mappings satisfying such contraction. This work unifies and generalizes various known comparable results in the literature.

2 264 K. Chaira, A. Eladraoui, M. Kabil and A. Kamouss Mathematics Subject Classification: 47H09; 47H0; 05C20 Keywords: G-Kannan; G-monotone; G-complete; orbitally G-continuous; directed graph; generalized metric space; fixed point Introduction and preliminaries Currently, fixed point theory is a very active area of research because of its applications in multiple fields. It concerns the results which indicate that, under certain conditions, a self-mapping on a set admits a fixed point. Among all the results in metric fixed point theory, the Banach Contraction Principle [2] is the most celebrated one due to its simplicity and ease of application in major areas of mathematics. Subsequently, many authors extend and generalize this principle in different directions. In 969, Kannan [5] proved that a mapping T : X X satisfying a contraction condition dt x, T y k[dx, T x + dy, T y] for all x, y X, where 0 k <, has a 2 unique fixed point in a complete metric space X. The concept of standard metric space is a fundamental tool in topology, functional analysis and nonlinear analysis. In recent years, several generalizations of standard metric space have appeared. In 993, Czerwik [6] introduced the concept of a b-metric spaces. Since then, several works have dealt with fixed point theory in such spaces. In 2000, Hitzler and Seda [3] introduced the notion of dislocated metric spaces in which self-distance of a point need not be equal to zero. Such spaces play a very important role in topology and logical programming. For fixed point theory in dislocated metric spaces, see [4] and references therein. In this work, we present a new generalized metric space introduced by Jleli and Samet in [2] and that recovers a large class of topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces and modular spaces with Fatou property. Also several interesting results about the existence and uniqueness of fixed point were proved in this generalized metric space see [5, 2]. An interesting approch in the theory of fixed point in some general structures was recently given by Jachymski [] in the setting of metric spaces endowed with a graph and Samet and Turinici [9] in the setting of metric spaces endowed with an arbitrary binary relation. In this work, Inspired by the ideas given in [7, 8, 9,, 9], we investigate Kannan fixed point theorem in generalized metric spaces with a graph. As corollary, we obtain Kannan fixed point theorem in the setting of generalized metric spaces. Some examples are provided to illustrate our results. In the following we describe the mathematical back-ground materials which are necessary for establishing the results in this paper. A directed graph or digraph G is determined by a nonempty set V G of its vertices and the set

3 Kannan fixed point theorem 265 EG V G V G of its arcs. Let denote the diagonal of the Cartesian product V G V G. A digraph is said to be reflexive if the set EG of its edges contains all loops, i.e., EG. G is said to be transitive whenever for any x, y, z V G [x, y EG and y, z EG] = x, z EG. We say that a vertex x in V G is isolated if for any vertex y in V G such that x y we have neither x, y EG nor y, x EG. By G we denote the converse of a digraph G, that is, the digraph obtained from G by reversing the direction of arcs. Thus we have EG = {x, y X X : y, x EG}. Also, G denotes the undirected graph obtained from G by ignoring direction of the edges. Thus we have E G = EG EG. The basic concepts related to a graph may be found in any textbook on graph theory, see for example [4, 8]. The concept of G-monotone sequence is introduced in []. A sequence {x n } V G is said to be G-increasing if x n, x n+ EG for all n N, G-decreasing if x n+, x n EG for all n N and G-monotone if it is either G-increasing or G-decreasing. Let X be a nonempty set and D : X X [0,{ + ] be a given mapping. For every x X, let us define the set CD, X, x = {x n } X : lim D x n, x = 0. } n The mapping D is called a generalized metric on X if it satisfies the following conditions: D For every x, y X X, D x, y = 0 = x = y; D 2 For every x, y X X, Dx, y = Dy, x; D 3 There exists C > 0 such that if x, y X X and {x n } CD, X, x, then Dx, y C lim sup Dx n, y. n In this case, we say that the pair X, D is a generalized metric space. Obviously, if the set CD, X, x is empty for every x X, then X, D is a generalized metric space if and only if D and D 2 are satisfied. A sequence {x n } in a generalized metric space X, D is said to be D-convergent to x X if {x n } CD, X, x and a D-Cauchy sequence if lim D x n, x m = 0. Note m,n that in generalized metric spaces, a sequence has at most one limit and a D- convergent sequence may not be D-Cauchy sequence. Moreover, X, D is said to be D-complete if every D-Cauchy sequence in X is D-convergent to some element in X. A generalized metric space X, D endowed with a graph G is said to be G-complete if any D-Cauchy G-monotone sequence {x n } V G is

4 266 K. Chaira, A. Eladraoui, M. Kabil and A. Kamouss D-convergent to a point in V G. The G-completeness is finer than the usual completeness as shown by Example 3.3. in []. Also, the digraph G is said to satisfy the property P, if for any G-monotone increasing resp. decreasing sequence {x n } which D-converges to some x V G, we have x n, x EG resp. x, x n EG for any n N. Now, we recall some useful types of continuity of mappings wich are well known and often used in metric fixed point theory. Definition.. A self-mapping T on X is called i weak continuous if the following condition holds : if {x n } X is D- convergent to x X, then there exists a subsequence {x nq } of {x n } such that {T x nq } D-converges to T x as q. ii orbitally G-continuous see [] if for all x, y V G and any sequence {k n } of positive integers, {T kn x} D-converges to y and T kn x, T k n+ x E G imply {T T kn x} D-converges to T y. Motivated by [3, 9, ], we introduce G-Kannan mappings in a generalized metric space X, D with a digraph G. A mapping T : X X is said to be a G-Kannan mapping if the following conditions are satisfied: i T is G-monotone, that is, for all x, y X, x, y EG = T x, T y EG, for every x, y X X; ii there exists k [0, such that for every x, y V G 2 x, y EG = DT x, T y k DT x, x + Dy, T y Remark.2. It follows immediately from the above definition that :. If T is a G-Kannan mapping, then T is both a G -Kannan and G- Kannan mapping. 2. Any Kannan mapping is a G 0 -Kannan mapping, where the complete graph G 0 is defined by V G 0 = X and EG 0 =X X. The following example shows that a G-Kannan mapping is not necessarily a Kannan mapping. Example.3. Let X = {0,, 2, 3}. Consider the function D defined on X by Dx, y = x y 2. We can show that D is a generalized metric with constant C 2. Define the mapping f : X X by: f0 =, f = f2 = 0 and f3 =.

5 Kannan fixed point theorem Figure : The loops and not represented. Since D f0, f = and D f0, 0 + D, f = 2, then f is not a Kannan contraction mapping. But by considering the digraph G = X, E represented in the following figure: we can show easily that f is a G-Kannan mapping with constant k 2 [ 5, 2. 2 Main results Throughout this section, let X, D be a generalized metric space. Consider a reflexive digraph such that V G = X. Let T : X X be a mapping. Denote for any x 0 X the complete subgraph G[O T x 0 ] induced by the orbit O T x 0 := {T n x 0 : n N}. For the proof of our main result, we need the following technical lemma. Lemma 2.. Let T : X X be a G-monotone mapping and suppose there exists x 0 X such that x 0, T x 0 EG respectively, T x 0, x 0 EG and the complete subgraph G[O T x 0 ] is transitive, then {T n x 0 } is a G-increasing respectively, G-decreasing sequence and T m x 0, T n x 0 EG respectively, T n x 0, T m x 0 EG for any m, n N such that m n. Proof. Without loss of generality, we assume that x 0, T x 0 EG. Since T is G-monotone, we obtain T x 0, T 2 x 0 EG. By induction on n, we get T n x 0, T n+ x 0 EG for all n N. Therefore {T n x 0 } is a G-monotone increasing sequence. Since T m x 0, T m+ x 0, T m+ x 0, T m+2 x 0,..., T n x 0, T n x 0 EG and G[O T x 0 ] is transitive, then T m x 0, T n x 0 EG. The following notation is useful in the sequel : δd, T, x 0 := sup { DT i x 0, x 0 : i N }.

6 268 K. Chaira, A. Eladraoui, M. Kabil and A. Kamouss Lemma 2.2. Under the assumptions of Lemma 2., if T is a G-Kannan mapping with constant k [0,, then 2. for every n N, we have DT n x 0, T n x 0 δ 0 β n 2 2. for every m, n N N, we have DT n x 0, T m x 0 k δ 0 β n + β m 3 where β = k k and δ 0 = δd, T, x 0. Proof.. The inequality 2 holds trivially for n =. Assume that n 2. Since T is a G-Kannan mapping and T n x 0, T n 2 x 0 E G, then DT n x 0, T n x 0 kdt n x 0, T n x 0 + kdt n x 0, T n 2 x 0 k k DT n x 0, T n 2 x 0. By induction on n, we prove that k n DT DT n x 0, T n x 0 x0, x 0 k δ 0 β n. 2. Let n, m N. Since T is a G-Kannan mapping and T n x 0, T m x 0 E G, then DT n x 0, T m x 0 k DT n x 0, T n x 0 + DT m x 0, T m x 0. By using the inequality 2, we get DT n x 0, T m x 0 k δ 0 β n + β m. Theorem 2.3. Let X, D be a generalized G-complete metric space endowed with a reflexive digraph G such that VG=X and T : X X a G-Kannan mapping with constant k [0, inf {, } 2 C. Suppose that there exists x0 X such that δd, T, x 0 <, x 0, T x 0 E G and the subgraph G[O T x 0 ] is transitive, then the sequence {T n x 0 } converges to some ω X. Moreover, if one of the following conditions holds:

7 Kannan fixed point theorem 269. T is weak continuous; 2. T is orbitally G-continuous; 3. G satisfies the Property P and Dx 0, T ω <. Then ω is a fixed point of T. Proof. Without loss of generality we assume that x 0, T x 0 EG. Let m, n N N such that m n, from Lemma 2. we have T m x 0, T n x 0 EG. If T is a G-Kannan mapping, then by Lemma 2.2 we get for every m, n N N, where β = sequence. DT n x 0, T m x 0 k δ 0 β n + β m, k 0,. Thus {T n x k 0 } is a D-Cauchy In both cases, from the G-completeness of X, D the sequence {T n x 0 } D-converges to some ω X.. Assume that T is weak continuous, then there exists a subsequence {T nq x 0 } such that {T nq+ x 0 } D-converges to T ω when n q. Using the uniqueness of the limit, we get T ω = ω. 2. Assume that T is orbitally G-continuous. Since {T n x 0 } D-converges to ω and T n x 0, T n+ x 0 EG, then T T n x 0 T ω. This yields ω = T ω since, simultaneously, T T n x 0 = T n+ x 0 ω. 3. Assume that G satisfies the Property P and Dx 0, T ω <. Since {T n x 0 } is a G-monotone increasing sequence which D-converges to ω X, we have T n x 0, ω EG for any n N. Let n N such that n. If T is G-Kannan mapping, then DT n x 0, T ω k DT n x 0, T n x 0 + k DT ω, ω. Since {T p x 0 } p N D-converges to ω, by D 3 and Lemma 2.2 we have DT n x 0, T ω k δ 0 β n + k C lim sup DT ω, T p x 0 p Taking limit superior as n, we get lim sup DT n x 0, T ω lim sup k δ 0 β n + k C lim sup DT p x 0, T ω. n n p Thus k C lim sup n DT n x 0, T ω lim sup k δ 0 β n. n Since k < inf { 2, C } and β 0,, then DT n x 0, T ω 0.

8 270 K. Chaira, A. Eladraoui, M. Kabil and A. Kamouss Then {T n x 0 } D-converges to T ω. T ω = ω. By the uniqueness of the limit, we get Proposition 2.4. Suppose that T is a G-Kannan mapping. If ω X is a fixed point of T satisfying D ω, ω <, then D ω, ω = 0. Proof. Let ω X be a fixed point of T such that Dω, ω <. Since ω, ω EG and T is a G-Kannan mapping, we have Dω, ω = DT ω, T ω k DT ω, ω + Dω, T ω, which implies hence Thus, Dω, ω = 0. Dω, ω 2k Dω, ω, 2k Dω, ω 0. Proposition 2.5. Suppose that T is a G-Kannan mapping. If T has two fixed points ω and ω in X such that Dω, ω < and ω, ω EG, then ω = ω. Proof. Suppose that ω, ω X are two fixed points of T such that D ω, ω <. If T is a G-Kannan mapping, we have which implies that Dω, ω = DT ω, T ω k DT ω, ω + Dω, T ω, Dω, ω k Dω, ω + Dω, ω. Since ω, ω are two fixed points of T, then by Proposition 2.4 we have Dω, ω = 0 and Dω, ω = 0. Which implies that Dω, ω = 0. Then ω = ω. In the following we give an example to illustrate Theorem 2.3. Example 2.6. Let X = [0, ]. Consider the generalized distance function D defined on X by Dx, y = x y 2 and a self mapping T on X defined by { } x if x {0} 3 3 : n N ; n T x = otherwise 2 Consider the graph G on X consisting of the transitive closure of the graph represented in figure 2.

9 Kannan fixed point theorem n+2 3 n+ 3 n Figure 2: All loops and isolated vertices are not represented. Note that EG = { 0, 3 n } { } : n N 3, : n, m N and n m. n 3 m One can see that T 0, T 3 = 0, EG for any n N, n 3 n+ T, T 3 n 3 = m 3, EG for any n, m N such that n m, n+ 3 m+ then T is G-monotone. For x 0 =, we have T x 0, x 0 EG, G[O T x 0 ] is transitive and δt, D, x 0 = sup {D } 3, : i N = <. i Let x, y X such that x, y EG. If x = y X, then DT x, T x = 0 kdt x, x + Dy, T y. If x, y = 0,, then 3 n D T 0, T 3 n If x, y = 3 n, D T 3 n, T 3 m = 4k 32n+ 3 = k 2n+ 3 m, then = 3n 3 m 2 4k 32n + 3 2m 3 2n+m+ 3 2n+m+ = k D D 0, T 0 + D T 3, 3. n n T 3 n, 3 n + D 3, T. m 3 m

10 272 K. Chaira, A. Eladraoui, M. Kabil and A. Kamouss In all cases, DT x, T y kdt x, x + Dy, T y, proving that T is a G- Kannan mapping with constant k [,. 4 2 The sequence {T n x 0 } = { } is G-decreasing and D-convergent to 0 and 3 n 0, 3 EG, then G has the P Property. From Theorem 2.3 T has a n fixed point which is 0. Remark 2.7. Theorem 2.3 extends main fixed point theorems of [3] and [9]. Let X, D, be a generalized metric space endowed with a partial order. We define the directed graph G on X as follows : V G = X and EG = {x, y X X : x y}. In this setting, we say that T : X X is a monotone Kannan mapping if it is a G -Kannan mapping. We also say that T is orbitally monotone continuous if T is orbitally G -continuous. The generalized metric space X, D, satisfies the P Property if {x n } is a decreasing respetively increasing sequence such that x n x in X, then for all n N, x x n respectivelly x n x. We can now get a version of Theorem??TH in a partially ordered generalized metric space. Theorem 2.8. Let X, D, be a generalized D-complete metric space endowed with a partial order and T : X X be a monotone Kannan mapping with constant k [ 0, inf {, } 2 C. Suppose that there exists x0 X such that δd, T, x 0 < and x 0 T x 0 or T x 0 x 0, then the sequence {T n x 0 } converges to some ω X. Moreover, assume that one of the following conditions holds:. T is weak continuous; 2. T is orbitally monotone continuous; 3. X satisfies the Property P and Dx 0, T ω <. Then ω is a fixed point of T. Proof. Since the subgraph G [O T x 0 ] is transitive, we obtain Theorem 2.8 as a corollary of Theorem 2.3. If we remove the ordering, we obtain the following result. Theorem 2.9. Let X, D be a D-complete generalized metric space and T : X X be a Kannan contraction with constant 0 k < inf {, } 2 C. Suppose that there exists x 0 X such that δd, T, x 0 <. Then, {T n x 0 } converges to some ω X. If Dx 0, T ω <, then ω is a fixed point of T with Dω, ω = 0. Moreover, if ω X is another fixed point of T such that Dω, ω <, then ω = ω.

11 Kannan fixed point theorem 273 Proof. Taking G = G 0, where G 0 is the complete graph, i.e., V G 0 = X and EG 0 = X X, the proof of Theorem 2.9 follows from Theorem 2.3 and Propositions 2.4 and 2.5. Remark 2.0. Theorem 2.9 generalizes all classical versions of Kannan fixed point theorems in standard metric spaces, dislocated metric spaces and b-metric spaces see for example [6, 0]. References [] M.R. Alfuraidana, M. Bacharb, M.A. Khamsi, Almost monotone contractions on weighted graphs, The Journal of Nonlinear Science and Applications, 9 206, [2] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 922, [3] F. Bojor, fixed points of kannan mappings in metric spaces endowed with a graph, Analele Universitatii Ovidius Constanta-Seria Matematica, , [4] J.A. Bondy, U.S.R. Murty, Graph Theory, vol. 244, Graduate texts in mathematics, [5] K. Chaira, A. Eladraoui and M. Kabil, An extension of Fisher fixed point theorem in partially ordered generalized metric spaces, Malaya Journal of Mathematics, 5, no.4 207, [6] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 993, 5-. [7] F. Echenique, A short and constructive proof of Tarskis fixed-point theorem, International Journal of Game Theory, , [8] R. Espinola, W.A. Kirk, Fixed point theorems in r-trees with applications to graph theory, Topology and its Applications, , [9] K. Fallahi, A. Aghanians, Fixed points for Chatterjea contractions on a metric space with a graph, Int. J. Nonlinear Anal. Appl., ,

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