On Fixed Point Theory of Monotone Mappings with Respect to a Partial Order Introduced by a Vector Functional in Cone Metric Spaces

and Applied Analysis 3 By (1) we have for allm ∈ N such thatm ≥ n, d (x α n , x α m ) ⪯ φ (x α n ) − φ (x α m ) . (7) Let n → ∞, by (6) we have lim m,n→∞ [φ(x α n ) − φ(x α m )] = θ and hence lim m,n→∞ d(x α n , x α m ) = θ by the normality of P. Moreover by Remark 2, {x α n } is a Cauchy sequence in X. Therefore by the completeness of X, there exists some x ∈ X such that