@ Appl. Gen. Topol. 18, no. 1 (2017), 45-52 doi:10.4995/agt.2017.4573 c© AGT, UPV, 2017 Common fixed point theorems for mappings satisfying (E.A)-property via C-class functions in b-metric spaces Vildan Ozturk a and Arslan Hojat Ansari b a Department of Mathematics and Science Education,Faculty Of Education, Artvin Coruh Uni- versity, Artvin, Turkey (vildanozturk84@gmail.com) b Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran (analsisamirmath2@gmail.com) Communicated by M. Abbas Abstract In this paper, we consider and generalize recent b-(E.A)-property re- sults in [11] via the concepts of C-class functions in b- metric spaces. A example is given to support the result. 2010 MSC: 47H10; 54H25. Keywords: Common fixed point; (E.A)-property; b-metric space; C-class function. 1. Introduction and preliminaries Bakhtin in [5] introduced the consept of b−metric space and prove the Ba- nach fixed point theorem in the setting of b−metric spaces. Since then many authors have obtain various generalizations of fixed point theorems in b−metric spaces. On the other hand, Aamri and Moutaawakil in [1] introduced the idea of (E.A)−property in metric spaces. Later on some authors employed this con- cept to obtain some new fixed point results. See ([6, 10]). In this paper, we prove common fixed point results for two pairs of mappings which satisfy the b− (E.A)-property using the concept of C-class functions in b−metric spaces. Received 19 January 2016 – Accepted 28 January 2017 http://dx.doi.org/10.4995/agt.2017.4573 V. Ozturk and A. H. Ansari Definition 1.1 ([5]). Let X be a nonempty set and s ≥ 1 be a given real number. A function d : X ×X → [0,∞) is a b-metric if, for all x,y,z ∈ X, the following conditions are satisfied: (b1) d (x,y) = 0 if and only if x = y, (b2) d (x,y) = d (y,x) , (b3) d (x,z) ≤ s [d (x,y) + d (y,z)] . In this case, the pair (X,d) is called a b-metric space. It should be noted that, the class of b-metric spaces is effectively larger than that of metric spaces, every metric is a b-metric with s = 1. However, if (X,d) is a metric space, then (X,ρ) is not necessarily a metric space. Definition 1.2 ([7]). Let {xn} be a sequence in a b-metric space (X,d). (a) {xn} is called b−convergent if and only if there is x ∈ X such that d (xn,x) → 0 as n →∞. (b) {xn} is a b−Cauchy sequence if and only if d (xn,xm) → 0 as n,m → ∞. A b-metric space is said to be complete if and only if each b−Cauchy sequence in this space is b−convergent. Proposition 1.3 ([7]). In a b−metric space (X,d), the following assertions hold: (p1) A b−convergent sequence has a unique limit. (p2) Each b−convergent sequence is b−Cauchy. (p3) In general, a b−metric is not continuous. Definition 1.4 ([7]). Let (X,d) be a b−metric space. A subset Y ⊂ X is called closed if and only if for each sequence {xn} in Y is b−convergent and converges to an element x. Definition 1.5 ([11]). Let (X,d) be a b−metric space and f and g be self- mappings on X. (i) f and g are said to compatible if whenever a sequence {xn} in X is such that {fxn} and {gxn} are b−convergent to some t ∈ X, then limn→∞d (fgxn,gfxn) = 0. (ii) f and g are said to noncompatible if there exists at least one sequence {xn} in X is such that {fxn} and {gxn} are b−convergent to some t ∈ X, but limn→∞d (fgxn,gfxn) does not exist. (iii) f and g are said to satisfy the b − (E.A)-property if there exists a sequence {xn} such that limn→∞fxn = limn→∞gxn = t, for some t ∈ X. Remark 1.6 ([11]). Noncompatibility implies property (E.A). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 46 Common fixed point in b-metric spaces Example 1.7 ([11]). X = [0, 2] and define d : X ×X → [0,∞) as follows d (x,y) = (x−y)2 . Let f,g : X → X be defined by f (x) = { 1,x ∈ [0, 1] x+1 8 ,x ∈ (1, 2] g (x) = { 3−x 2 ,x ∈ [0, 1] x 4 ,x ∈ (1, 2] For a sequence {xn} in X such that xn = 1 + 1n+2, n = 0, 1, 2, ..., limn→∞fxn = limn→∞gxn = 1 4 . So f and g are satisfy the b− (E.A)-property. But limn→∞d (fgxn,gfxn) 6= 0. Thus f and g are noncompatible. Definition 1.8 ([8]). Let f and g be given self-mappings on a set X. The pair (f,g) is said to be weakly compatible if f and g commute at their coincidence points (i.e. fgx = gfx whenever fx = gx). In 2014, Ansari [3] introduced the concept of C-class functions. See also [4] Definition 1.9. A mapping F : [0,∞)2 → R is called C-class function if it is continuous and satisfies following axioms: (i) F(s,t) ≤ s; (ii) F(s,t) = s implies that either s = 0 or t = 0; for all s,t ∈ [0,∞). Note for some F we have that F(0, 0) = 0. We denote C-class functions as C. Example 1.10. The following functions F : [0,∞)2 → R are elements of C, for all s,t ∈ [0,∞): (1) F(s,t) = s− t, F(s,t) = s ⇒ t = 0; (2) F(s,t) = ms, 0 1, F(s,t) = s ⇒ s = 0 or t = 0; (5) F(s,t) = ln(1 + as)/2, a > e, F(s, 1) = s ⇒ s = 0; (6) F(s,t) = (s + l)(1/(1+t) r) − l, l > 1,r ∈ (0,∞), F(s,t) = s ⇒ t = 0; (7) F(s,t) = s logt+a a, a > 1, F(s,t) = s ⇒ s = 0 or t = 0; (8) F(s,t) = s− ( 1+s 2+s )( t 1+t ), F(s,t) = s ⇒ t = 0; (9) F(s,t) = sβ(s), β : [0,∞) → (0, 1), and is continuous, F(s,t) = s ⇒ s = 0; (10) F(s,t) = s− t k+t ,F(s,t) = s ⇒ t = 0; (11) F(s,t) = s − ϕ(s),F(s,t) = s ⇒ s = 0, here ϕ : [0,∞) → [0,∞) is a continuous function such that ϕ(t) = 0 ⇔ t = 0; (12) F(s,t) = sh(s,t),F(s,t) = s ⇒ s = 0,here h : [0,∞) × [0,∞) → [0,∞)is a continuous function such that h(t,s) < 1 for all t,s > 0; (13) F(s,t) = s− ( 2+t 1+t )t, F(s,t) = s ⇒ t = 0. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 47 V. Ozturk and A. H. Ansari (14) F(s,t) = n √ ln(1 + sn), F(s,t) = s ⇒ s = 0. (15) F(s,t) = φ(s),F(s,t) = s ⇒ s = 0,here φ : [0,∞) → [0,∞) is a upper semicontinuous function such that φ(0) = 0, and φ(t) < t for t > 0, (16) F(s,t) = s (1+s)r ; r ∈ (0,∞), F(s,t) = s ⇒ s = 0. Definition 1.11 ([9]). A function ψ : [0,∞) → [0,∞) is called an altering distance function if the following properties are satisfied: (i) ψ is non-decreasing and continuous, (ii) ψ (t) = 0 if and only if t = 0. See also [2] and [12]. Definition 1.12 ([3]). An ultra altering distance function is a continuous, nondecreasing mapping ϕ : [0,∞) → [0,∞) such that ϕ(t) > 0 , t > 0 and ϕ(0) ≥ 0 2. Main results Through out this section, we assume ψ is altering distance function, ϕ is ultra altering distance function and F is a C-class function. We shall start the following theorem. Theorem 2.1. Let (X,d) be a b−metric space and f,g,S,T : X → X be mappings with f (X) ⊆ T (X) and g (X) ⊆ S (X) such that (2.1) ψ(d (fx,gy)) ≤ F(ψ(Ms (x,y)),ϕ(Ms (x,y))), for all x,y ∈ X where, Ms (x,y) = max { d (Sx,Ty) ,d (fx,Sx) ,d (gy,Ty) , d (fx,Ty) + d (Sx,gy) 2s } . Suppose that one of the pairs (f,S) and (g,T) satisfy the b − (E.A)-property and that one of the subspaces f (X) ,g (X) ,S (X) and T (X) is closed in X. Then the pairs (f,S) and (g,T) have a point of coincidence in X. Moreover, if the pairs (f,S) and (g,T) are weakly compatible, then f,g,S and T have a unique common fixed point. Proof. If the pairs (f,S) satisfies the b − (E.A)-property, then there exists a sequence {xn} in X satisfying limn→∞fxn = limn→∞Sxn = q, for some q ∈ X. As f (X) ⊆ T (X) there exists a sequence {yn} in X such that fxn = Tyn. Hence limn→∞Tyn = q. Let us show that limn→∞gyn = q. By (2.1), (2.2) ψ (d (fxn,gyn)) ≤ F(ψ (Ms (xn,yn)) ,ϕ (Ms (xn,yn))) ≤ ψ (Ms (xn,yn)) c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 48 Common fixed point in b-metric spaces where Ms (xn,yn) = max { d (Sxn,Tyn) ,d (fxn,Sxn) ,d (Tyn,gyn) , d(Sxn,gyn)+d(fxn,T yn) 2s } = max { d (Sxn,fxn) ,d (fxn,gyn) , d(Sxn,gyn)+d(fxn,fxn) 2s } ≤ max { d (Sxn,fxn) ,d (fxn,gyn) , s[d(Sxn,fxn),d(fxn,gyn)] 2s } . In (2.2), on taking limit, ψ (limn→∞d (q,gyn)) ≤ F(ψ (limn→∞d (q,gyn)) ,ϕ (limn→∞d (q,gyn))). So, ψ (limn→∞d (q,gyn)) = 0, or ,ϕ (limn→∞d (q,gyn)) = 0. Thus limn→∞d (q,gyn) = 0. Hence limn→∞gyn = q. If T (X) is closed subspace of X, then there exists a r ∈ X, such that Tr = q. By (2.1), (2.3) ψ (d (fxn,gr)) ≤ F(ψ (Ms (xn,r)) ,ϕ (Ms (xn,r))) where Ms (xn,r) = max { d (Sxn,Tr) ,d (fxn,Sxn) ,d (Tr,gr) , d(fxn,T r)+d(Sxn,gr) 2s } = max { d (Sxn,q) ,d (fxn,Sxn) ,d (q,gr) , d(fxn,q)+d(Sxn,gr) 2s } . Letting n →∞, limn→∞Ms (xn,r) = max { d (q,q) ,d (q,q) ,d (q,gr) , d (q,q) + d (q,gr) 2s } = d (q,gr) . Now, (2.3) and definition of ψ and ϕ, as n →∞, ψ(d (q,gr) ≤ F(ψ(d (q,gr)),ϕ(d(q,gr))) which implies ψ(d (q,gr)) = 0 or ϕ(d(q,gr)) = 0 gives gr = q. Thus r is a coincidence point of the pair (g,T). As g (X) ⊆ S (X) , there exists a point z ∈ X such that q = Sz. We claim that Sz = fz. By (2.1), we have (2.4) ψ(d (fz,gr)) ≤ F(ψ(Ms (z,r)),ϕ(Ms(z,r))) c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 49 V. Ozturk and A. H. Ansari where Ms (z,r) = max { d (Sz,Tr) ,d (fz,Sz) ,d (Tr,gr) , d (fz,Tr) + d (Sz,gr) 2s } = max { d (q,q) ,d (fz,q) ,d (q,q) , d (fz,q) + d (q,q) 2s } ≤ max { d (fz,q) , d (fz,q) 2s } = d (fz,q) . Thus from (2.4), ψ(d (fz,gr)) = ψ(d (fz,q)) ≤ F(ψ(d (fz,q)),ϕ(d (fz,q))) implies that ψ(d (fz,q)) = 0, or ,ϕ(d (fz,q)) = 0. Therefore Sz = fz = q. Hence z is a coincidence point of the pair (f,S) . Thus fz = Sz = gr = Tr = q. By weak compatibility of the pairs (f,S) and (g,T ), we deduce thatfq = Sq and gq = Tq. We will show that q is a common fixed point of f,g,S and T . From (2.1) , (2.5) ψ (d (fq,q)) = ψ(d(fq,gr)) ≤ F(ψ (Ms (q,r)) ,ϕ (Ms (q,r))) where, Ms (q,r) = max { d (Sq,Tr) ,d (fq,Sq) ,d (Tr,gr) , d (fq,Tr) + d (Sq,gr) 2s } = max { d (fq,q) ,d (fq,fq) ,d (q,q) , d (fq,q) + d (fq,q) 2s } = d (fq,q) . By (2.5) ψ (d (fq,q)) ≤ F(ψ(d (fq,q)),ϕ (d (fq,q))). So fq = Sq = q. Similarly, it can be shown gq = Tq = q. To prove the uniqueness of the fixed point of f,g,S and T . Suppose for contradiction that p is another fixed point of f,g,S and T. By (2.1), we obtain ψ (d (q,p)) = ψ(d (fq,gp)) ≤ F(ψ (Ms (q,p)) ,ϕ (Ms (q,p))) and Ms (q,p) = max { d (Sq,Tp) ,d (fq,Sq) ,d (Tp,gp) , d (fq,Tp) + d (Sq,gp) 2s } = max { d (q,p) ,d (q,q) ,d (p,p) , d (q,p) + d (q,p) 2s } = d (q,p) . Hence we have ψ (d (q,p)) ≤ F(ψ (d (q,p)) ,ϕ (d (q,p))), which implies that ψ (d (q,p)) = 0 or ϕ (d (q,p)) = 0. So q = p. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 50 Common fixed point in b-metric spaces Corollary 2.2. Let (X,d) be a b−metric space and f,g,S,T : X → X be mappings with f (X) ⊆ T (X) and g (X) ⊆ S (X) such that d (fx,gy) ≤ F(Ms (x,y) ,ϕ(Ms (x,y))), for all x,y ∈ X, where Ms (x,y) = max { d (Sx,Ty) ,d (fx,Sx) ,d (gy,Ty) , d (fx,Ty) + d (Sx,gy) 2s } . Suppose that one of the pairs (f,S) and (g,T) satisfy the b − (E.A)-property and that one of the subspaces f (X) ,g (X) ,S (X) and T (X) is closed in X. Then the pairs (f,S) and (g,T) have a point of coincidence in X. Moreover, if the pairs (f,S) and (g,T) are weakly compatible, then f,g,S and T have a unique common fixed point. Corollary 2.3. Let (X,d) be a b−metric space and f,T : X → X be mappings such that ψ(d (fx,fy)) ≤ F(ψ(Ms (x,y)),ϕ(Ms (x,y))), for all x,y ∈ X, where Ms (x,y) = max { d (Tx,Ty) ,d (fx,Tx) ,d (fy,Ty) , d (fx,Ty) + d (Tx,fy) 2s } . Suppose that the pair (f,T) satisfies the b−(E.A)-property and T (X) is closed in X. Then the pair (f,T) has a common point of coincidence in X. Moreover, if the pair (f,T) is weakly compatible, then f and T have a unique common fixed point. Example 2.4. Let F(s,t) = 99 100 s , X = [0, 1] and define d : X ×X → [0,∞) as follows d (x,y) = { 0,x = y (x + y) 2 ,x 6= y Then (X,d) is a b−metric space with constant s = 2. Let f,g,S,T : X → X be defined by f (x) = x 4 , g (x) = { 0,x 6= 1 2 1 8 ,x = 1 2 , S (x) = { 2x, 0 ≤ x < 1 2 1 8 , 1 2 ≤ x ≤ 1 and T (x) = { x, 0 ≤ x < 1 2 1 2 , 1 2 ≤ x ≤ 1 . Clearly, f (X) is closed and f (X) ⊆ T (X) and g (X) ⊆ S (X). The sequence {xn} , xn = 12 + 1 n , is in X such that limn→∞fxn = limn→∞Sxn = 1 8 . So that the pair (f,S) satisfies the b − (E.A)−property. But the pair (f,S) is noncompatible for limn→∞d (fSxn,Sfxn) 6= 0. The altering functions ψ,ϕ : [0,∞) → [0,∞) are defined by ψ (t) = √ t . To check the contractive condition (2.1), for all x,y ∈ X, if x = 0 or x = 1 2 , then (2.1) is satisfied. if x ∈ ( 0, 1 2 ) , then c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 51 V. Ozturk and A. H. Ansari ψ (d (fx,gy)) = x 4 ≤ 99 100 9x 4 = 99 100 d (fx,Sx) ≤ 99 100 ψ(Ms (x,y)). If x ∈ ( 1 2 , 1 ] , then ψ (d (fx,gy)) = x 4 ≤ 99 100 ( x 4 + 1 8 ) = 99 100 d (fx,Sx) ≤ 99 100 ψ(Ms (x,y)). Then (2.1) is satisfied for all x,y ∈ X. The pairs (f,S) and (g,T) are weakly compatible. Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover 0 is the unique common fixed point of f,g,S and T. Acknowledgements. The authors would like to thank the referee for useful comments. References [1] M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002), 181–188. [2] M. Abbas, N. Saleem and M. De la Sen, Optimal coincidence point results in partially or- dered non-Archimedean fuzzy metric spaces, Fixed Point Theory and Appl. 2016 (2016), Article ID 44. [3] A. H. 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