International Journal of Analysis and Applications ISSN 2291-8639 Volume 2, Number 2 (2013), 137-146 http://www.etamaths.com FIXED POINT AND COMMON FIXED POINT THEOREMS FOR α−PROPERTY IN CONE BALL-METRIC SPACES RAJESH SHRIVASTAVA1, RAJENDRA KUMAR DUBEY2, PANKAJ TIWARI3, ANIMESH GUPTA4,∗ Abstract. In this paper, we define a new cone ball-metric and get fixed points and common fixed points for the α − property in cone ball-metric spaces. 1. Introduction and preliminaries Throughout this paper, by R+, we denote the set of all non-negative numbers, while N is the set of all natural numbers. Let ♦ : R+ ×R+ → R+ be a binary operation satisfying the following conditions : (i) ♦ is associative and commutative, (ii) ♦ is continuous. Five typical examples of ♦ are : (i) a♦b = max{a,b}, (ii) a♦b = a + b, (iii) a♦b = ab, (iv) a♦b = ab + a + b, (v) a♦b = ab max{a,b,1}. Definition 1.1. The binary operation ♦ is said to satisfy α − property if there exists a positive real number α such that a♦b ≤ α max{a,b} for all a,b ∈ R+. Following five examples are stand for α−property. Example 1.2. If a♦b = a + b, for each a,b ∈ R+, then for α ≥ 2, we have a♦b ≤ α max{a,b}. Example 1.3. If a♦b = ab max{a,b,1}, for each a,b ∈ R +, then for α ≥ 1, we have a♦b ≤ α max{a,b}. Example 1.4. If a♦b = ab, for each a,b ∈ R+, then for α ≥ a, we have a♦b ≤ α max{a,b}. 2010 Mathematics Subject Classification. Primary 47H10; Secondary 54H25, 55M20. Key words and phrases. α − property, cone ball-metric space; common fixed point theorem; fixed point theorem. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 137 138 SHRIVATAVA, DUBEY, TIWARI AND GUPTA Example 1.5. If a♦b = ab + a + b, for each a,b ∈ R+, then for α ≥ a + 2, we have a♦b ≤ α max{a,b}. Example 1.6. If a♦b = max{a,b}, for each a,b ∈ R+, then for α > 1, we have a♦b ≤ α max{a,b}. Huang and Zhang [3] have introduced the concept of the cone metric space, replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive type mappings on cone metric spaces. In 2006, Mustafa and Sims [5] introduced a more appropriate generalization of metric spaces, G-metric spaces. Recently, Beg et al. [2] introduced the notion of generalized cone metric spaces, and proved some fixed point results for mappings satisfying certain contractive conditions. In this paper, we define a new cone ball- metric and get fixed point and common fixed results for the α−property in cone ball-metric spaces. We recall some definitions of the cone metric spaces and some of the properties [3], as follow: Definition 1.7. [3] Let E be a real Banach space and P a subset of E. P is called a cone if and only if: (i) P is nonempty, closed, and P 6= {θ}, (ii) a,b ∈ R+, x,y ∈ P ⇒ ax + by ∈ P , (iii) x ∈ P and −x ∈ P ⇒ x = θ. For given a cone P ⊂ E, we can define a partial ordering with respect to P by x 4 y or x < y if and only if y − x ∈ P for all x,y ∈ E. The real Banach space E equipped with the partial ordered induced by P is denoted by (E,4). We shall write x ≺ y to indicate that x 4 y but x 6= y, while x � y will stand for y −x ∈ intP , where intP denotes the interior of P . The cone P is called normal if there exists a real number K > 0 such that for all x,y ∈ E, θ 4 x 4 y ⇒ ‖x‖≤ K‖y‖. The least positive number K satisfying above is called the normal constant of P. The cone P is called regular if every non-decreasing sequence which is bounded from above is convergent, that is, if {xn} is a sequence such that x1 4 x2 4 · · ·4 xn 4 · · ·4 y, for some y ∈ E, then there is x ∈ E such that ‖xn−x‖→ 0 as n →∞. Equivalently, the cone P is regular if and only if every non-increasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone. Moreover, P is called stronger minihedral if every subset of E which is bounded above has a supremum [1]. In the following we always suppose that E is a real Banach space with a stronger minihedral regular cone P and intP 6= φ, and 4 is a partial ordering with respect to P . Metric spaces are playing an important role in mathematics and the applied sciences. In 2003, Mustafa and Sims [5] introduced a more appropriate and robust notion of a generalized metric space as follows. Definition 1.8. [5] Let X be a nonempty set, and let G : X ×X ×X → [0,∞) be a function satisfying the following axioms: FIXED POINT AND COMMON FIXED POINT THEOREMS 139 (1) G(x,y,z) = 0 if and only if x = y = z; (2) G(x,x,y) > 0, for all x 6= y; (3) G(x,y,z) ≥ G(x,x,y), for all x,y,z ∈ X; (4) G(x,y,z) = G(x,z,y) = G(z,y,x) = · · · (symmetric in all three variables); (5) G(x,y,z) ≤ G(x,w,w) + G(w,y,z), for all x,y,z,w ∈ X. Then the function G is called a generalized metric, or, more specifically a G-metric on X, and the pair (X,G) is called a G-metric space. This research subject is interesting and widespread. But is too abstract makes the human difficulty with to understand. So we introduce the concept of cone ball-metric spaces and we prove fixed point results on such spaces for functions satisfying the contractions involving the α−property. In [6] Chen and Tsai introduce the following notion of the cone ball-metric B. Definition 1.9. [6] Let (X,d) be a cone metric space, B : X × X × X → E, x,y,z ∈ X and we denote δ(B) = sup{d(a,b) : a,b ∈ B}, and B(x,y,z) = δ(B), where B = ∩{F ⊂ X|F is a closed ball and {x,y,z} ⊂ F}. Then we call B a ball-metric with respect to the cone metric d, and (X,B) a cone ball-metric space. It is clear that B(x,x,y) = d(x,y). Remark 1.10. It is clear that the cone ball-metric B has the following properties: (1) B(x,y,z) = θ if and only if x = y = z; (2) B(x,x,y) � θ, for all x 6= y; (3) B(x,x,y) 4B(x,y,z), for all x,y,z ∈ X; (4) B(x,y,z) = B(x,z,y) = B(z,y,x) = · · · (symmetric in all three variables); (5) B(x,y,z) 4B(x,w,w) + B(w,y,z), for all x,y,z,w ∈ X; (6) B(x,y,z) 4B(x,w,w) + B(y,w,w) + B(z,w,w), for all x,y,z,w ∈ X. Definition 1.11. [6] Let (X,B) be a cone ball-metric space and {xn} be a sequence in X. We say that {xn} is (1) Cauchy sequence if for every ε ∈ E with θ � ε, there exists n0 ∈ N such that for all n,m,l > n0, B(xn,xm,xl) � ε. (2) Convergent sequence if for every ε ∈ E with θ � ε, there exists n0 ∈ N such that for all n,m > n0, B(xn,xm,x) � ε for some x ∈ X. Here x is called the limit of the sequence {xn} and is denoted by limn→∞xn = x or xn → x as n →∞. Definition 1.12. [6] Let (X,B) be a cone ball-metric space. Then X is said to be complete if every Cauchy sequence is convergent in X. Proposition 1.13. [6] Let (X,B) be a cone ball-metric space and {xn} be a se- quence in X. Then the following are equivalent: (i) {xn} converges to x; (ii) B(xn,xn,x) → θ as n →∞; (iii) B(xn,x,x) → θ as n →∞; (iv) B(xn,xm,x) → θ as n,m →∞. 140 SHRIVATAVA, DUBEY, TIWARI AND GUPTA Proposition 1.14. [6] Let (X,B) be a cone ball-metric space and {xn} be a se- quence in X, x,y ∈ X. If xn → x and xn → y as n →∞, then x = y. Proof. Let ε ∈ E with θ � ε be given. Since xn → x and xn → y as n →∞, there exists n0 ∈ N such that for all m,n > n0, B(xn,xm,x) � ε 3 and B(xn,xm,y) � ε 3 . Therefore, B(x,x,y) 4 B(x,xn,xn) + B(xn,x,y) = B(x,xn,xn) + B(y,xn,x) 4 B(x,xn,xn) + B(y,xm,xm) + B(xm,xn,x) � ε 3 + ε 3 + ε 3 = ε. Hence, B(x,x,y) � ε α for all α ≥ 1, and so ε α −B(x,x,y) ∈ P for all α ≥ 1. Since ε α → θ as α →∞ and P is closed, we have that −B(x,x,y) ∈ P . This implies that B(x,x,y) = θ, since B(x,x,y) ∈ P . So x = y. � Proposition 1.15. [6] Let (X,B) be a cone ball-metric space and {xn},{ym},{zl} be three sequences in X. If xn → x, ym → y, zl → z as n →∞, then B(xn,ym,zl) → B(x,y,z) as n →∞. Proof. Let ε ∈ E with θ � ε be given. Since xn → x, ym → y, zl → z as n → ∞, there exists n0 ∈ N such that for all n,m,l > n0, B(xn,x,x) � ε 3 , B(ym,y,y) � ε 3 , B(zl,z,z) � ε 3 , Therefore, B(xn,ym,zl) 4 B(xn,x,x) + B(x,ym,zl) 4 � B(xn,x,x) + B(ym,y,y) + B(y,x,zl) 4 B(xn,x,x) + B(ym,y,y) + B(zl,z,z) + B(z,x,y) � ε 3 + ε 3 + ε 3 + B(x,y,z), that is, B(xn,ym,zl) −B(x,y,z) � ε. Similarly, B(x,y,z) −B(xn,ym,zl) � ε. Therefore, for all α ≥ 1, we have B(xn,ym,zl) −B(x,y,z) � ε α , and B(x,y,z) −B(xn,ym,zl) � ε α . These imply that ε α −B(xn,ym,zl) + B(x,y,z) ∈ P, ε α + B(xn,ym,zl) −B(x,y,z) ∈ P. Since P is closed and ε α → θ as α →∞, we have that lim n,m,l→∞ [−B(xn,ym,zl) + B(x,y,z)] ∈ P, FIXED POINT AND COMMON FIXED POINT THEOREMS 141 lim n,m,l→∞ [B(xn,ym,zl) −B(x,y,z)] ∈ P. These show that lim n,m,l→∞ B(xn,ym,zl) = B(x,y,z). So we complete the proof. � 2. Main results Let (X,B) be a cone ball-metric space with P , and let ♦ : R+ ×R+ → R+. We now state our main common fixed point result for the α−property in a cone ball-metric space (X,B), as follows: Theorem 2.1. Let (X,B) be a complete cone ball-metric space, P be a regular cone in E and f,g be two self mappings of X such that f(X) ⊂ g(X). Suppose that ♦ satisfies α−property with α > 0 such that B(fx,fy,fz) 4 k1[B(gx,gy,gz)♦B(gx,fx,fx)] + k2[B(gx,gy,gz)♦B(gy,fy,fy)] + k3[B(gx,gy,gz)♦B(gz,fz,fz)],(2.1) where k1,k2,k3 > 0 and 0 < α(k1 + k2 + k3) < 1. If g(X) is closed, then f and g have a coincidence point in X. Moreover, if f and g commute at their coincidence points, then f and g have a unique common fixed point in X Proof. Given x0 ∈ X. Since f(X) ⊂ g(X), we can choose x1 ∈ X such that gx1 = fx0. Continuing this process, we define the sequence {xn} in X recursively as follows: fxn = gxn+1 for each n ∈ N∪{0}. In what follows we will suppose that fxn+1 6= fxn for all n ∈ N, since if fxn+1 = fxn for some n, then fxn+1 = gxn+1, that is , f,g have a coincidence point xn+1, and so we complete the proof. By (2.1), we have B(fxn,fxn+1,fxn+1) 4 k1[B(gxn,gxn+1,gxn+1)♦B(gxn,fxn,fxn)] + k2[B(gxn,gxn+1,gxn+1)♦B(gxn+1,fxn+1,fxn+1)] + k3[B(gxn,gxn+1,gxn+1)♦B(gxn+1,fxn+1,fxn+1)]. Therefore, by the condition of α−property, we conclude that for each n ∈ N, B(fxn,fxn+1,fxn+1) 4 k1α max{B(fxn−1,fxn,fxn),B(fxn−1,fxn,fxn)} + k2α max{B(fxn−1,fxn,fxn),B(fxn,fxn+1,fxn+1)} + k3α max{B(fxn−1,fxn,fxn),B(fxn,fxn+1,fxn+1)}. If B(fxn,fxn+1,fxn+1) > B(fxn−1,fxn,fxn), we obtain B(fxn,fxn+1,fxn+1) � α(k1 + k2 + k3)B(fxn,fxn+1,fxn+1), which contradiction. Hence B(fxn,fxn+1,fxn+1) ≤ B(fxn−1,fxn,fxn). simi- larly it is easy to see that B(fxn+1,fxn+2,fxn+2) ≤B(fxn,fxn+1,fxn+1) 142 SHRIVATAVA, DUBEY, TIWARI AND GUPTA B(fxn,fxn+1,fxn+1) � α(k1 + k2 + k3)B(fxn−1,fxn,fxn), and B(fxn,fxn+1,fxn+1) 4 δB(fxn−1,fxn,fxn) 4 · · · 4 δnB(fx0,fx1,fx1) where α(k1 + k2 + k3) = δ < 1. So we have B(fxn,fxn+1,fxn+1) 4 δnB(fx0,fx1,fx1) → θ as n →∞. Next, we claim that the sequence {fxn} is a Cauchy sequence. Suppose that {fxn} is not a Cauchy sequence. Then there exists γ ∈ E with θ � γ such that for all k ∈ N, there are mk,nk ∈ N with mk > nk ≥ k satisfying: (1) mk is even and nk is odd, (2) B(fxnk,fxmk,fxmk ) < γ, and (3) mk is the smallest even number such that the conditions (1), (2) hold. Since limn→∞B(fxn,fxn+1,fxn+1) = θ and by (2), (3), we have that γ 4 B(fxnk,fxmk,fxmk ) 4 B(fxnk,fmk−1,fmk−1) + B(fxmk−1,fxmk,fxmk ) 4 B(fxnk,fxmk−2,fxmk−2) + B(fxmk−2,fxmk−1,fxmk−1) +B(fxmk−1,fmk,fmk ) 4 γ + B(fxmk−2,fxmk−1,fxmk−1) + B(fxmk−1,fxmk,fxmk ). Taking limk→∞, we deduce lim k→∞ B(fxnk,fxmk,fxmk ) = γ. Since B(fxnk−1,fxmk−1,fxmk−1) 4 B(fxnk−1,fxnk,fxnk ) + B(fxnk,fxmk,fxmk ) +B(fxmk,fxmk−1,fxmk−1). Taking limk→∞, we deduce lim k→∞ B(fxnk−1,fxmk−1,fxmk−1) 4 γ.(2.2) On the other hand, γ 4 B(fxnk,fxmk,fxmk ) 4 B(fxnk,fxnk−1,fxnk−1) + B(fxnk−1,fxmk,fxmk ) 4 B(fxnk,fxnk−1,fxnk−1) + B(fxnk−1,fxmk−1,fxmk−1) +B(fxmk−1,fxmk,fxmk ). Taking limk→∞, we also deduce γ 4 lim k→∞ B(fxnk−1,fxmk−1,fxmk−1).(2.3) By (2.2) and (2.3), we get lim k→∞ B(fxnk−1,fxmk−1,fxmk−1) = γ. FIXED POINT AND COMMON FIXED POINT THEOREMS 143 And, by (2.1), we have that B(fxnk,fxmk,fxmk ) 4 ψ(L(xnk,xmk,xmk )) where B(fxnk,fxmk,fxmk ) 4 k1[B(gxnk,gxmk,gxmk )♦B(gxnk,fxnk,fxnk )] + k2[B(gxnk,gxmk,gxmk )♦B(gxmk,fxmk,fxmk )] + k3[B(gxnk,gxmk,gxmk )♦B(gxmk,fxmk,fxmk )]. B(fxnk,fxmk,fxmk ) 4 k1[B(fxnk−1,fxmk−1,fxmk−1)♦B(fxnk−1,fxnk,fxnk )] + k2[B(fxnk−1,fxmk−1,fxmk−1)♦B(fxmk−1,fxmk,fxmk )] + k3[B(fxnk−1,fxmk−1,fxmk−1)♦B(fxmk−1,fxmk,fxmk )]. B(fxnk,fxmk,fxmk ) 4 αk1 max{B(fxnk−1,fxmk−1,fxmk−1),B(fxnk−1,fxnk,fxnk )} + αk2 max{B(fxnk−1,fxmk−1,fxmk−1),B(fxmk−1,fxmk,fxmk )} + αk3 max{B(fxnk−1,fxmk−1,fxmk−1),B(fxmk−1,fxmk,fxmk )}. (I) If B(fxnk−1,fxmk−1,fxmk−1), is maximum then taking limk→∞, we deduce lim k→∞ B(fxnk−1,fxmk−1,fxmk−1) = γ, and γ 4 lim k→∞ B(fxnk,fxmk,fxmk ) � γ, a contradiction. (II) If B(fxnk−1,fxnk,fxnk ), or B(fxmk−1,fxmk,fxmk ), is maximum then taking limk→∞, we deduce lim k→∞ B(fxnk−1,fxnk,fxnk ) = θ, lim k→∞ B(fxmk−1,fxmk,fxmk ) = θ, and γ 4 lim k→∞ B(fxnk,fxmk,fxmk ) 4 θ, a contradiction. Follow (I) and (II), we get the sequence {fxn} is a Cauchy sequence. Since X is complete and g(X) is closed, there exist ν,µ ∈ X such that lim n→∞ g(xn) = lim n→∞ f(xn) = g(µ) = ν. We shall show that µ is a coincidence point of f and g, that is, we claim that B(gµ,fµ,fµ) = θ. 144 SHRIVATAVA, DUBEY, TIWARI AND GUPTA If not, assume that B(gµ,fµ,fµ) 6= θ, then by (2.1), we have B(gµ,fµ,fµ) 4 B(gµ,fxn,fxn) + B(fxn,fµ,fµ) 4 B(gµ,fxn,fxn) + k1[B(gxn,gµ,gµ)♦B(gxn,fxn,fxn)] +k2[B(gxn,gµ,gµ)♦B(gµ,fµ,fµ)] +k3[B(gxn,gµ,gµ)♦B(gµ,fµ,fµ)], 4 B(gµ,fxn,fxn) +α(k1 + k2 + k3) max{B(gxn,gµ,gµ),B(gxn,fxn,fxn),B(gµ,fµ,fµ)}, (III) If max{B(gxn,gµ,gµ),B(gxn,fxn,fxn),B(gµ,fµ,fµ)} = B(gxn,gµ,gµ), then taking limn→∞, we deduce lim n→∞ B(gxn,gµ,gµ) = B(gµ,gµ,gµ) = θ, and B(gµ,fµ,fµ) = lim n→∞ B(gµ,fxn,fxn) + α(k1 + k2 + k3) lim n→∞ B(gxn,gµ,gµ) 4 θ, a contradiction. (IV) If max{B(gxn,gµ,gµ),B(gxn,fxn,fxn),B(gµ,fµ,fµ)} = B(gxn,fxn,fxn), then taking limn→∞, we deduce lim n→∞ B(gxn,fxn,fxn) = B(gµ,gµ,gµ) = θ, and B(gµ,fµ,fµ) = lim n→∞ B(gµ,fxn,fxn) + α(k1 + k2 + k3) lim n→∞ B(gxn,fxn,fxn) 4 θ, a contradiction. (V) If max{B(gxn,gµ,gµ),B(gxn,fxn,fxn),B(gµ,fµ,fµ)} = B(gµ,fµ,fµ), then B(gµ,fµ,fµ) = α(k1 + k2 + k3)B(gµ,fµ,fµ) �B(gµ,fµ,fµ), a contradiction. Follow (III)-(V), we obtain that B(gµ,fµ,fµ) = θ, that is, gµ = fµ = ν, and so µ is a coincidence point of f and g. Suppose that f and g commute at µ. Then fν = fgµ = gfµ = gν. Later, we claim that B(fµ,fν,fν) = θ. By (2.1), we have FIXED POINT AND COMMON FIXED POINT THEOREMS 145 B(fµ,fν,fν) 4 k1[B(gµ,gν,gν)♦B(gµ,fµ,fµ)] + k2[B(gµ,gν,gν)♦B(gν,fν,fν)] + k3[B(gµ,gν,gν)♦B(gν,fν,fν)] = α(k1 + k2 + k3) max{B(fµ,fν,fν),B(fµ,fµ,fµ),B(fν,fν,fν)} = α(k1 + k2 + k3) max{B(fµ,fν,fν),θ}. Therefore, if B(fµ,fν,fν) 4 α(k1 + k2 + k3)B(fµ,fν,fν) �B(fµ,fν,fν), then we get a contradition, which implies that B(fµ,fν,fν) = θ, B(ν,fν,fν) = θ, that is, ν = fν = gν. So ν is a common fixed point of f and g. Let ν be another common fixed point of f and g. By (2.1), B(ν,ν,ν) = B(fν,fν,fν), where B(fν,fν,fν) 4 k1[B(gν,gν,gν)♦B(gνfν,fν)] + k2[B(gν,gν,gν)♦B(gν,fν,fν)] + k3[B(gν,gν,gν)♦B(gν,fν,fν)] = α(k1 + k2 + k3) max{B(fν,fν,fν),B(fν,fν,fν),B(fν,fν,fν)} = α(k1 + k2 + k3) max{B(fν,fν,fν),θ} = α(k1 + k2 + k3) max{B(ν,ν,ν),θ} 4 B(ν,ν,ν). Therefore, we also conclude that B(ν,ν,ν) = θ, that is ν = ν. So we show that ν is the unique common fixed point of g and f. � Corollary 2.2. Let (X,B) be a complete cone ball-metric space, P be a regular cone in E and f,g be two self mappings of X such that f(X) ⊂ g(X). Suppose that B(fx,fy,fz) 4 α max{B(gx,gy,gz),B(gx,fx,fx), B(gy,fy,fy),B(gz,fz,fz)}(2.4) where α ∈ [0, 1) . If g(X) is closed, then f and g have a coincidence point in X. Moreover, if f and g commute at their coincidence points, then f and g have a unique common fixed point in X Proof. It is sufficient if we take a♦b = α max{a,b} for α ∈ [0, 1) in Theorem 2.1 then we get the result. � Corollary 2.3. Let (X,B) be a complete cone ball-metric space, P be a regular cone in E and f be a self mapping of X. Suppose that ♦ satisfies α−property with α > 0 such that B(fx,fy,fz) 4 k1[B(x,y,z)♦B(x,fx,fx)] + k2[B(x,y,z)♦B(y,fy,fy)] + k3[B(x,y,z)♦B(z,fz,fz)],(2.5) where k1,k2,k3 > 0 and 0 < α(k1 + k2 + k3) < 1. Then f has a fixed point in X. 146 SHRIVATAVA, DUBEY, TIWARI AND GUPTA Proof. It is sufficient if we take g = I (identity mapping) in Theorem 2.1 then we get the result. � References [1] C.D. Aliprantis and R. Tourky, Cones and Duality, in: Graduate studies in Mathematics, Amer. Math. Soc. 84 (2007). 215–240. [2] I. Beg, M. Abbas and T. Nazir, Generalized cone metric spaces, J. Nonlinear Sci. Appl. 3 (2010), no. 1, 23–31. [3] L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 322 (2007), 1468–1476. [4] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326–329. [5] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear and Convex Anal. 7 (2006), no. 2, 289–297. [6] C. Chen and P. Tsai, Fixed point and common fixed point theorems for the Meir-Keeler type functions in cone ball-metric spaces, Ann. Funct. Anal. 3 (2012), no. 2, 155–169. 1Department of Mathematics, Govt. Science & Commerce College, Benazir Bhopal- India 2Department of Mathematics, Govt. Science P.G. College, Reewa -India 3Department of Applied Mathematics, Vidhyapeeth Institute of Science & Technol- ogy, Bhopal- India 4Department of Applied Mathematics, Sagar Institute of Science Technology & Re- search, Ratibad Bhopal- India ∗Corresponding author