In this paper, we proved some fixed point theorems for multivalued contraction maps and the existence

of common fixed points for a pair of occasionally weakly compatible maps satisfying some multivalued contractive

conditions in G-symmetric spaces. The results extend and generalize some results in the literature.

Keywords: Common fixed points; Multivalued contractive mappings; Occasionally weakly compatible maps;

G-symmetric space

Eke, Kanayo Stella

Covenant University Repository

Available online at http://scik.orgAdv. Inequal. Appl. 2015, 2015:13ISSN: 2050-7461FIXED POINT THEOREMS FOR MULTIVALUED CONTRACTIVE MAPPINGS ING-SYMMETRIC SPACESKANAYO STELLA EKEDepartment of Mathematics, Covenant University,Canaanland, KM 10, Idiroko Road, P.M.B. 1023, Ota, Ogun State, NigeriaCopyright c© 2015 Kanayo Stella Eke. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Abstract. In this paper, we proved some fixed point theorems for multivalued contraction maps and the existenceof common fixed points for a pair of occasionally weakly compatible maps satisfying some multivalued contractiveconditions in G-symmetric spaces. The results extend and generalize some results in the literature.Keywords: Common fixed points; Multivalued contractive mappings; Occasionally weakly compatible maps;G-symmetric spaces.2000 AMS Subject Classification: 47H10.1. IntroductionThe fixed point theorem for multivalued contraction mappings which takes each point of ametric space (X ,d) into a closed subset K of X has gained enormous applications in areas suchas control theory, optimization, differential equations and economics. Nadler [18] proved theexistence of the fixed points for multivalued contraction mapping in complete metric spaces.Dugundji and Granas [6] proved that a single-valued weakly contractive mapping of a completemetric space into itself has a unique fixed point. Kaneko [12] gave a partial generalization ofReceived June 12, 201512 KANAYO STELLA EKEthe theorem of Dugundji and Granas to the multivalued mappings. Reich [19] proposed thefollowing problem.Conjecture. Let (X ,d) be a complete metric space. Suppose that T : X → CB(X) satisfiesH(T x,Ty)≤ k(d(x,y))d(x,y), for all x,y∈X , x 6= y, where k : (0,∞)→ [0,∞) and limsupr→t+ k(r)< 1, for all 0 < t < ∞. Then T has a fixed point in X . This conjecture was proved by Mizoguchiand Takahashi [15]. Daffer and Kaneko [5] provided an alternative and somewhat more s-traight forward proof for the theorem of Mizoguchi and Takahashi. Abbas et. al.[1] provedfixed point theorems for multivalued mappings satisfying generalized contractive conditions inordered generalized metric spaces.The notion of metric spaces has been generalized by many authors. In particular, Mustafa andSims [17] generalized the notion of metric spaces. Eke and Olaleru [7] introduced the conceptof G-partial metric spaces which generalized the notion of G-metric spaces. Cartan [3] alsogeneralized the notion of metric spaces by omitting the triangle inequality axiom of the metricspaces. Recently, Eke and Olaleru [8] extend the concept of symmetric spaces to G-symmetricspaces by omitting the rectangle inequality axiom of the G-metric spaces. In the same reference,the existence of common fixed points for singled-valued occasionally weakly compatible map-pings in G-symmetric spaces is proved. Also, Eke and Olaleru [9] proved existence of commonfixed point for pair of hybrid contractive mappings in G-symmetric spaces. In this work, somefixed point theorems for multivalued contraction mappings in G-symmetric spaces are proved.2. PreliminariesThe following definitions and results are needed in the sequel.Definition 2.1. [17] Let X be a nonempty set, and let G : X × X × X → R+ be a functionsatisfying:(G1) G(x,y,z) = 0 if x = y = z,(G2 ) 0 < G(x,x,y) for all x,y ∈ X with x 6= y,(G3) G(x,x,y)≤ G(x,y,z) for all x,y,z ∈ X with y 6= z,(G4) G(x,y,z) = G(x,z,y) = G(y,z,x) (symmetry in all three variables),FIXED POINT THEOREMS FOR MULTIVALUED CONTRACTIVE MAPPINGS 3(G5) G(x,y,z)≤ G(x,a,a)+G(a,y,z) for all x,y,z,a ∈ X (rectangle inequality).Then, the function G is called a generalised metric, or more specifically a G-metric on X , andthe pair (X ,G) is a G-metric space.Definition 2.2. [3] A symmetric on a set X is a real valued function d on X×X such that(i) d(x,y)≥ 0 and d(x,x) = 0 if and only if x = y; and(ii) d(x,y) = d(y,x).Wilson [21] also gave two more axioms of a symmetric d on X as:(W1) Given {xn}, x and y in X , d(xn,x)→ 0 and d(xn,y)→ 0 imply that x = y;(W2) Given {xn}, {yn} and x ∈ X , d(xn,x)→ 0 and d(xn,yn)→ 0 imply that d(yn,x)→ 0.Definition 2.3. [13] A mapping T : X → 2X is called a multivalued mapping. A point x ∈ X iscalled a fixed point of T if x ∈ T x.Definition 2.4. [11] Let X be a given nonempty set. Assume that g : X → X and T : X → 2X . Ifw = gx ∈ T x for some x ∈ X then x is called a coincidence point of g and T and w is a point ofcoincidence of g and T .Definition 2.5. [11] Maps g : X→X and T : X→ 2X are said to be weakly compatible if gx∈ T xfor each x ∈ X implies gT x⊆ T gx.Definition 2.6. [11] Maps g : X → X and T : X → 2X are said to be occasionally weaklycompatible mappings if and only if there exists some point x in X such that gx ∈ T x and gT x⊆T gx. An occasionally weakly compatible map is weakly compatible but not vice - verse.The following definitions and motivations are found in [8].Definition 2.7. A G-symmetric on a set X is a function Gd : X ×X ×X → R+ such that for allx,y,z ∈ X the following conditions are satisfied:Gd(1) Gd(x,y,z)≥ 0 and Gd(x,y,z) = 0, if x = y = z;Gd(2)0 < Gd(x,x,y) for all x,y ∈ X with x 6= y,Gd(3) Gd(x,x,y)≤ Gd(x,y,z) for all x,y,z ∈ X with y 6= z,Gd(4) Gd(x,y,z) = Gd(y,z,x) = Gd(z,x,y) =,..., (symmetry in all three variables).4 KANAYO STELLA EKEIt should be observed that our notion of a G-symmetric space is the same as that of G-metricspace ( Definition 2.1) without the rectangular property (G5).Example 2.8. Let X = [0,1] equipped with a G-symmetric defined by Gd(x,y,z) = (x− y)2 +(y− z)2 +(z− x)2 for all x,y,z ∈ X . Then (X ,Gd) is a G-symmetric space.This does not satisfy the rectangle inequality property of a G-metric space, hence it is not aG-metric space. The analogue of axioms of Wilson [21] in G-symmetric space is as follows:(W3) Given {xn}, x and y in X ; Gd(xn,x,x)→ 0 and Gd(xn,y,y)→ 0 imply that x = y.(W4) Given {xn}, {yn} and an x in X ; Gd(xn,x,x) → 0 and Gd(xn,yn,yn) → 0 imply thatGd(yn,x,x)→ 0.(W5) Let (X ,Gd) be a complete G-symmetric space. For any sequence {xn} in X,limn,m→∞ Gd(xn,xm,xm) = 0 iff limn→∞ Gd(xn,xn+1,xn+1) = 0.Definition 2.9. Let (X ,Gd) be a G-symmetric space.(i) (X ,Gd) is Gd- complete if for every Gd- Cauchy sequence {xn}, there exists x in X withlimn→ ∞ Gd(xn,x,x) = 0.(ii) f : X→ X is Gd-continuous if limn→ ∞ Gd(xn,x,x) = 0 implies limn→ ∞ Gd( f xn, f x, f x) = 0.Let (X ,Gd) be a G-symmetric space. We denote the family of all nonempty closed and boundedsubsets of X by CB(X).Definition 2.10. Two mappings T,S : X→CB(X) are said to be weak-contraction if there exists0 < α < 1 such thatHGd(T x,Sy,Sy)≤ αM(x,y,y)andHGd(T x,T x,Sy)≤ αM(x,x,y)for all x,y ∈ X , whereM(x,y,y) = max{Gd(x,y,y),Gd(x,T x,T x),Gd(y,Sy,Sy),Gd(x,Sy,Sy),Gd(y,T x,T x)}andM(x,x,y) = max{Gd(x,x,y),Gd(x,x,T x),Gd(y,y,Sy),Gd(x,x,Sy),Gd(y,y,T x)}.Definition 2.11. A mapping T : X → CB(X) are said to be weak-contraction if there exists0 < α < 1 such thatFIXED POINT THEOREMS FOR MULTIVALUED CONTRACTIVE MAPPINGS 5HGd(T x,Ty,Ty)≤ αN(x,y,y)andHGd(T x,T x,Ty)≤ αN(x,x,y)for all x,y ∈ X , whereN(x,y,y) = max{Gd(x,y,y),Gd(x,T x,T x),Gd(y,Ty,Ty),Gd(x,Ty,Ty),Gd(y,T x,T x)}andN(x,x,y) = max{Gd(x,x,y),Gd(x,x,T x),Gd(y,y,Ty),Gd(x,x,Ty),Gd(y,y,T x)}where HGd denotes the Hausdorff G-symmetric on CB(X) induced by Gd , that isHGd(A,B,B) = max{Supx∈AGd(x,B,B),Supy∈BGd(y,A,A)},for all A,B ∈CB(X).Moutawakil [16] gave a generalization of the well-known Nadler multivalued contractionfixed point to the setting of symmetric spaces. Chandra and Bhatt [4] proved a fixed pointtheorem for generalized contractions under restrictive conditions in symmetric spaces. Joshiet al. [10] proved some fixed point theorems for multivalued contractive mappings in sym-metric spaces. Bhatt et al. [2] established some fixed point theorems for occasionally weaklycompatible mappings in symmetric spaces. Jungck and Rhoades [11] proved some fixed pointtheorems for occasionally weakly compatible maps satisfying certain contractive conditions insymmetric spaces. Rouhani and Moradi [20] established the existence and uniqueness of fixedand coincidence points of multivalued generalized ϕ-weak contractive mappings on completemetric spaces. The aim of this paper, is to prove some existence and uniqueness of fixed point ofmultivalued contraction maps and common fixed theorems for occasionally weakly compatiblemappings satisfying multivalued weak - contractive conditions in G-symmetric spaces.3. Main resultsTheorem 3.1. Let (X ,Gd) be a Gd-complete G-symmetric space which satisfies W4 and W5 suchthat(i) the map f : X → R defined by f (x) = Gd(x,T x,T x), x ∈ X, is lower semi-continuous;(ii) T : X →C(X) a multivalued mapping such that :6 KANAYO STELLA EKED(T x,Ty,Ty)≤ kGd(x,y,y), k∈ [0,1), (3.1)for all x,y ∈ X. Then there exists u ∈ X such that u ∈ Tu.Proof. Let x1 ∈ X and α ∈ (k,1). Since T x1 is nonempty, there exists x2 ∈ T x1 such thatGd(x1,x2,x2) = Gd(x1,T x1,T x1). In view of (3.1) we obtain,Gd(x2,T x2,T x2)≤ D(T x1,T x2,T x2)≤ kGd(x1,x2,x2)< αGd(x1,x2,x2).Similarly there exists x3 ∈ T x2 such that Gd(x2,x3,x3) = Gd(x2,T x2,T x2) andGd(x3,T x3,T x3)≤ DGd(T x2,T x3,T x3)≤ kGd(x2,x3,x3)< αGd(x2,x3,x3).Consequently, we obtain a sequence {xn}∞≥1 in X satisfying xn+1 ∈T xn such that Gd(xn,xn+1,xn+1)=Gd(xn,T xn,T xn) and haveGd(xn,T xn,T xn)≤ DGd(T xn−1,T xn,T xn)≤ kGd(xn−1,xn,xn)< αGd(xn−1,xn,xn)< (α)n−1Gd(x1,x2,x2).This implies that limn→∞ Gd(xn,T xn,T xn) = 0. Since Gd(xn,xn+1,xn+1) = Gd(xn,T xn,T xn)then we have limn→∞ Gd(xn,xn+1,xn+1) = 0 and with (W5), {xn} is a Cauchy sequence. Hencethere exists u ∈ X such that xn → u. Also by (i) where f (u) = Gd(u,Tu,Tu) we obtain 0 ≤Gd(u,Tu,Tu) ≤ limn→∞ Gd(xn,T xn,T xn) = 0. Since T is closed and (X ,Gd) is complete thenu ∈ Tu . Thus u is a fixed point of T .Remark 3.2. Theorem 2.1 is an extension of Theorem 2.2 in [16] in the G-symmetric spaces.Corollary 3.3. Let (X ,Gd) be a Gd-bounded and Gd-complete G-symmetric space which satis-fies W4 and T : X →C(X) be a multivalued mapping such that:FIXED POINT THEOREMS FOR MULTIVALUED CONTRACTIVE MAPPINGS 7Gd(T x,Ty,Ty)≤ kGd(x,y,y), (3.2)for all x,y ∈ X and k ∈ [0,1). Then there exist u ∈ X such that u ∈ Tu.The proof of corollary 3.2 follows from Theorem 2.1. The following Lemma is needed in theproof of the next theorem and we refer to Nadler [18] for its proof.Lemma 3.4. If A,B ∈CB(X) and a ∈ A, then for each ε > 0, there exists b ∈ B such thatGd(a,b,b)≤ HGd(A,B,B)+ ε .Theorem 3.5. Let T and S be multivalued weak-contraction mapping of a G-symmetric spaceX such that the pair {T,S} are occasionally weakly compatible. If for all x,y ∈ XHGd(T x,Sy,Sy)≤ αM(x,y,y) (3.3)andHGd(T x,T x,Sy)≤ αM(x,x,y) (3.4)where 0 ≤ α < 1 and β = α + ε < 1. Then there exists a point x ∈ X such that x ∈ T x andx ∈ Sx (i.e. T and S have a common fixed point in X).Proof. Since {T,S} are occasionally weakly compatible then there exist x∈X such that T x∈ Sx.Suppose there exist x1, x2 ∈ X such that x1 ∈ T x1 and x2 ∈ Sx2 with T Sx2 ⊆ ST x1 then using(3.3) and Lemma 3.4 we show that x1 = x2. On the contrary we assume that x1 6= x2 and have,Gd(x1,x2,x2)≤ HGd(T x1,Sx2,Sx2)+ εM(x1,x2,x2)≤ αM(x1,x2,x2)+ εM(x1,x2,x2)= β M(x1,x2,x2)= β max{Gd(x1,x2,x2),Gd(x1,T x1,T x1),Gd(x2,Sx2,Sx2),Gd(x1,Sx2,Sx2),Gd(x2,T x1,T x1)}≤ β max{Gd(x1,x2,x2),Gd(x1,x1,x1),Gd(x2,x2,x2),Gd(x1,x2,x2),Gd(x2,x1,x1)}≤ β max{Gd(x1,x2,x2),Gd(x2,x1,x1)}.Case (i)Suppose max{Gd(x1,x2,x2),Gd(x2,x1,x1)}=Gd(x1,x2,x2). Then Gd(x1,x2,x2)≤ β Gd(x1,x2,x2).8 KANAYO STELLA EKECase (ii)If max{Gd(x1,x2,x2),Gd(x2,x1,x1)}= Gd(x2,x1,x1). ThenGd(x1,x2,x2)≤ β Gd(x2,x1,x1) (3.5)Similarly, using (3.4), we obtainGd(x1,x1,x2)≤ HGd(T x1,T x1,Sx2)+ εM(x1,x1,x2)≤ αM(x1,x1,x2)+ εM(x1,x1,x2)= β M(x1,x1,x2)= β max{Gd(x1,x1,x2),Gd(x1,x1,T x1),Gd(x2,x2,Sx2),Gd(x1,x1,Sx2), Gd(x2,x2,T x1)}≤ β max{Gd(x1,x1,x2),Gd(x1,x1,x1),Gd(x2,x2,x2),Gd(x1,x1,x2),Gd(x2,x2,x1)}≤ β max{Gd(x1,x1,x2),Gd(x2,x2,x1)}Case (i)Suppose max {Gd(x1,x1,x2),Gd(x2,x2,x1)}= Gd(x1,x1,x2) thenGd(x1,x1,x2)≤ βGd(x1,x1,x2)Case(ii)If max {Gd(x1,x1,x2),Gd(x2,x2,x1)}= Gd(x2,x2,x1) thenGd(x1,x1,x2)≤ β Gd(x2,x2,x1) (3.6)Combining (3.5) and (3.6) with Gd(4) yieldsGd(x1,x2,x2)≤ βGd(x1,x2,x2)Since β < 1 then x1 = x2. This implies that x1 ∈ T x1 and x1 ∈ Sx1. Hence T and S have acommon fixed point.Remarks 3.6. Theorem 2.5 is an analogy result of Theorem 3.1 in [20] in the setting of G-symmetric spaces.If T = S in Theorem 2.5, then we have the following corollary.FIXED POINT THEOREMS FOR MULTIVALUED CONTRACTIVE MAPPINGS 9Corollary 3.7. Let (X ,Gd) be a Gd-complete G-symmetric space and let T : X →CB(X) be amultivalued weak-contraction mapping such that for all x,y ∈ XHGd(T x,Ty,Ty)≤αN(x,y,y), (3.7)where 0≤ α < 1. Then there exists a point x ∈ X such that x ∈ T x.Example 3.8. Let X = [0,1] be endowed with the G-symmetric . Let S,T : X → CB(X) bedefined by T x = [0, x6 ] and Sy = [y6 ].HGd(T x,Sy,Sy) = max{(y6− x6)2,(y6)2}≤ 13max{(y− x)2,(y− y6)2}≤ 13max{Gd(x,y,y),Gd(y,Sy,Sy)}≤ 13M(x,y,y).So, T and S have a common fixed point (x = 0).Conflict of InterestsThe authors declare that there is no conflict of interests.REFERENCES[1] M. Abbas, A. R. Khan and T. Nazir, Common fixed point of multivalued mappings in ordered generalizedmetric spaces, Filomat 26 (2012), 1045-1053.[2] A. Bhatt, H. Chandra and D. R. Sahu, Common fixed point theorems for occasionally weakly compatiblemappings under relaxed conditions, Nonlinear Anal. 73 (2010),176-182.[3] E. Cartan, Sur la determination d’un systeme orthogonal complet dans un espace de Riemman symetriqueclos, Rend. Ciric. Mat. Palermo 53 (1929), 217-252.[4] H. Chandra and A. Bhatt, Some fixed point theorems for set valued maps in symmetric spaces, Int. J. Math.Anal. 3 (2009), 839-846.[5] P. Z. Daffer, Fixed points of generalised contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995),655-666.[6] J.Dugundji and A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. GreekMath. Soc. 19 (1978), 141-151.10 KANAYO STELLA EKE[7] K. S. Eke and J. O. Olaleru, Some fixed point results on ordered G-partial metric spaces, ICASTOR J. Mathe.Sci. 7 (2013), 65-78 .[8] K. S. Eke and J. O. Olaleru, Common fixed point results for occasionally weakly compatible maps in G-symmetric spaces, Appl. Math. 5 (2014), 744-752.[9] K. S. Eke and J. O. Olaleru, Common fixed point results for pairs of hybrid maps in G-symmetric spaces, Int.J. Pure Appl. Math. 101 (2015), 117-132.[10] L. K. Joshi, M. C. Joshi and A. Kumar, Fixed point theorems for multivalued mappings in symmetric spaces,Adv. Inequal. Appl. 2014 (2014), Article ID 31.[11] G. Jungck and B. E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, FixedPoint Theory, 7 (2006), 286-296.[12] H. Kaneko, Generalized contractive multivalued mappings and their fixed points, Math. Japon. 33 (1988),57-64.[13] M. S. Khan, Common fixed point theorems for multivalued mappings, Pacific J. Math. 95 (1981), 337-347.[14] D. Klim and D. Wardowski, Fixed point theorem for set valued contractions in complete metric spaces, J.Math. Anal. Appl. 334 (2007), 132-139.[15] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces,J. Math. Anal. Appl. 141 (1989), 177-188.[16] D. El Moutawakil, A fixed point theorem for multivalued maps in symmetric spaces, Appl. Math. E-Notes 4(2004), 26-32.[17] Z. Mustafa and B. Sims, A New Approach To Generalised Metric Spaces, J. Nonlinear Convex Anal. 7(2006), 289-297.[18] S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 1969.[19] S. Reich, Some problems and results in fixed point theory, Contemp. Math. 21 (1983), 179-187.[20] B. D. Rouhani and S. Moradi, Common fixed point of multivalued generalized ψ-weak contractive mappings,Fixed Point Theory Appl. 2010 (2010), Article ID 708984.[21] W. A. Wilson, On semi-metric spaces, Amer. J. Math. 53 (1931), 361-373.

Fixed Point Theorems For Multivalued Contractive Mappings In

G-Symmetric Spaces

http://eprints.covenantuniversity.edu.ng/6196/1/multivalued%20maps.pdf

Fixed Point Theorems For Multivalued Contractive Mappings In G-Symmetric Spaces

Abstract

Similar works

Full text

Available Versions

Covenant University Repository