SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE ...

f is called ordered Kannan type contraction if for all x, y ∈ X with x ⊑ y, there exists λ ∈ [0, 1 2 )such that(2)d(fx, fy) ≼ λ[d(x, fx) + d(y, fy)].If (2) is satisfied for all x, y ∈ X, then f is called Kannan contraction.f is called ordered Reich type contraction if for all x, y ∈ X with x ⊑ y, λ, µ, δ ∈ [0, 1) suchthat λ + µ + δ < 1 and(3)d(fx, fy) ≼ λd(x, y) + µd(x, fx) + δd(y, fy).If (3) is satisfied for all x, y ∈ X, then f is called Reich contraction.Kannan showed that the conditions (1) and (2) are independent of each other (see [16, 17])and Reich showed that the condition (3) is a proper generalization of (1) and (2) (see [24]). Notethat Reich type contraction turns into Banach and Kannan type contractions with µ = δ = 0 andλ = 0, µ = δ, respectively.Definition 5. Let X be a nonempty set equipped with partial order “ ⊑ ”. A nonempty subsetA of X is said to be well ordered if every two elements of A are comparable with respect to “ ⊑ ”.Now we can state our main results.◭◭ ◭ ◮ ◮◮Go backFull ScreenCloseQuit2. Main ResultsTheorem 1. Let (X, ⊑, d) be an ordered complete cone rectangular metric space and f : X → Xbe a mapping. Suppose that the following conditions hold(I) f is an ordered Reich type contraction, i.e., it satisfies (3);(II) there exists x 0 ∈ X such that x 0 ⊑ fx 0 ;(III) f is nondecreasing with respect to “ ⊑ ”;(IV) if {x n } is a nondecreasing sequence in X and converging to some z, then x n ⊑ z.