Theorems for Boyd-Wong-Type Contractions in Ordered Metric Spaces

and Applied Analysis 3 The purpose of this paper is to generalize the above results using an ICS mapping T : X → X and involving some generalized weak contractions of Boyd-Wong-type 31 . Also, some examples are presented to show that our results are effective. 2. Main Result First, denote by Φ the set of functions φ : 0, ∞ → 0, ∞ satisfying a φ t < t for all t > 0, b φ is upper semicontinuous from the right i.e., for any sequence {tn} in 0,∞ such that tn → t, tn > t as n → ∞, we have lim supn→∞φ tn ≤ φ t . Now we prove our first result. Theorem 2.1. Let X,≤ be a partially ordered set. Suppose there exists a metric d such that X, d is a complete metric space. Let f, T : X → X be such that T is an ICS mapping and f a nondecreasing mapping satisfying d ( Tfx, Tfy ) ≤ φMx, y 2.1 for all distinct x, y ∈ X with x ≤ y where φ ∈ Φ and M ( x, y ) max { d ( Tx, Tfx ) d ( Ty, Tfy ) d ( Tx, Ty ) , d ( Tx, Ty ) } . 2.2 Also, assume either i f is continuous or; ii if {xn} is a nondecreasing sequence in X such that xn → x, then x sup{xn}. If there exists x0 ∈ X such that x0 ≤ fx0, then f has a fixed point. Proof. Given x1 fx0, define a sequence {xn} in X as follows: xn fxn−1 for n ≥ 1. 2.3 Since f is a nondecreasing mapping, together with x0 ≤ x1 fx0, we have x1 fx0 ≤ fx1 x2. Inductively, we obtain x0 ≤ x1 ≤ x2 ≤ · · · ≤ xn−1 ≤ xn ≤ xn 1 ≤ · · · . 2.4 Assume that there exists n0 such that xn0 xn0 1. Since xn0 xn0 1 fxn0 , then f has a fixed point which ends the proof. Suppose that xn / xn 1 for all n ∈ N. Thus, by 2.4 we have x0 < x1 < x2 < · · · < xn−1 < xn < xn 1 < · · · . 2.5 4 Abstract and Applied Analysis Since 2.5 holds, then condition 2.1 implies that d Txn, Txn 1 d ( Tfxn−1, Tfxn ) ≤ φ M xn−1, xn , 2.6