Fixed point and common fixed point theorems on ordered cone metric spaces

and Applied Analysis 3 effectively larger than that of the ordinary conemetric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples show the above remarks. Example 7. Let X = {−1, 0, 1}, E = R, andP = {(x, y) : x ≥ 0, y ≥ 0}. Define d : X × X → P by d(x, y) = d(y, x) for all x, y ∈ X, d(x, x) = θ, x ∈ X, and d(−1, 0) = (3, 3), d(−1, 1) = d(0, 1) = (1, 1). Then (X, d) is a complete cone b-metric space but the triangle inequality is not satisfied. Indeed, we have that d(−1, 1) + d(1, 0) = (1, 1) + (1, 1) = (2, 2) ≺ (3, 3) = d(−1, 0). It is not hard to verify that s = 3/2. Example 8. Let X = R, E = R, and P = {(x, y) ∈ E : x ≥ 0, y ≥ 0}. Define d : X × X → E by d(x, y) = (|x − y| 2 , |x − y| 2 ). Then, it is easy to see that (X, d) is a cone b-metric space with the coefficient s = 2. But it is not a cone metric spaces since the triangle inequality is not satisfied. Definition 9 (see [18]). Let (X, d) be a cone b-metric space, {x n } a sequence inX and x ∈ X. (1) For all c ∈ E with θ ≪ c, if there exists a positive integer N such that d(x n , x) ≪ c for all n > N, then x n is said to be convergent and x is the limit of {x n }. One denotes this by x n → x. (2) For all c ∈ E with θ ≪ c, if there exists a positive integer N such that d(x n , x m ) ≪ c for all n,m > N, then {x n } is called a Cauchy sequence inX. (3) A cone metric space (X, d) is called complete if every Cauchy sequence inX is convergent. The following lemma is useful in our work. Lemma 10 (see [24]). (1) If E is a real Banach space with a cone P and a ⪯ λa where a ∈ P and 0 ≤ λ < 1, then a = θ. (2) If c ∈ intP, θ ⪯ a n , and a n → θ, then there exists a positive integerN such that a n ≪ c for all n ≥ N. (3) If a ⪯ b and b ≪ c, then a ≪ c. (4) If θ ⪯ u ≪ c for each θ ≪ c, then u = θ. 2. Fixed Point Results In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma. Lemma 11. Let {x n } be a sequence in a cone b-metric space (X, d) with the coefficient s ≥ 1 relative to a solid cone P such that d (x n , x n+1 ) ⪯ hd (x n−1 , x n ) , (6) where h ∈ [0, 1/s) and n = 1, 2, . . .. Then {x n } is a Cauchy sequence in (X, d). Proof. Letm > n ≥ 1. It follows that d (x n , x m ) ⪯ sd (x n , x n+1 ) + s 2 d (x n+1 , x n+2 ) + ⋅ ⋅ ⋅ + s m−n d (x m−1 , x m ) . (7) Now, (6) and sh < 1 imply that d (x n , x m ) ⪯ sd (x n , x n+1 ) + s 2 d (x n+1 , x n+2 ) + ⋅ ⋅ ⋅ + s m−n d (x m−1 , x m ) ⪯ sh n d (x 0 , x 1 ) + s 2 h n+1 d (x 0 , x 1 ) + ⋅ ⋅ ⋅ + s m−n h m−1 d (x 0 , x 1 ) = (sh n + s 2 h n+1 + ⋅ ⋅ ⋅ + s m−n h m−1 ) d (x 0 , x 1 ) = sh n (1 + sh + (sh) 2 + ⋅ ⋅ ⋅ + (sh) m−n−1 ) d (x 0 , x 1 )