We prove the following theorem: Let T be an order preserving nonexpansive operator on L1 (μ) (or L(Formula presented.)) of a σ-finite measure, which also decreases the L∞-norm, and let S=tI+(1−t) T for 0< t<1. Then for every f ∈ Lp (1< p<∞), the sequence Snf converges weakly in Lp. (The assumptions do not imply that T is nonexpansive in Lp for any p>1, even if μ is finite.) For the proof we show that ∥Sn+1f−Snf∥p → 0 for every f ∈Lp, 1< p<∞, and apply to S the following theorem: Let T be order preserving and nonexpansive in L(Formula presented.), and assume that T decreases the L∞-norm. Then for g ∈Lp (1< p<∞) Tng is weakly almost convergent. If for f ∈ Lp we have Tn+1f−Tnf → 0 weakly, then Tnf converges weakly in Lp (1< p<∞).