An alternating procedure for operators on uniformly convex and uniformly smooth banach spaces

Let A" be a real uniformly convex and uniformly smooth Banach space For any 1 < p < ∞ J_p, J*_p respectively denote the duality mapping with gauge function φ(t) = t^{p − 1} from X onto X* and X* onto X. If T: X → X is a bounded linear operator, then M(T): X → X is the mapping defined by M(T) = J*_pT* J_p T, where T*: X* → X* is the adjoint of T and q = (p − l)^−lp. It is proved that if T_n is a sequence of operators on X such that T_n ≤ 1 for all n, then M(T_n,…, T₁)x strongly converges in X for any x ∈ X, with an estimate of the rate of convergence:M(T_n,…, T₁)x − M(x) ≤ σ(x)xψ(l − {m(x)lT_n,…, T₁)⁾, and σ: X → R⁺, ψ: R⁺ → R⁺ are definite, strictly increasing positive functions.The result obtained generalizes and improves on the theorem offered recently by Akcoglu and Sucheston [1].