© Hindawi Publishing Corp. ITERATIVE SOLUTIONS OF K-POSITIVE DEFINITE OPERATOR EQUATIONS IN REAL UNIFORMLY SMOOTH BANACH SPACES

Abstract. Let X be a real uniformly smooth Banach space and let T: D(T) ⊆ X → X be a K-positive definite operator. Under suitable conditions we establish that the iterative method by Bai (1999) converges strongly to the unique solution of the equation Tx = f, f ∈ X. The results presented in this paper generalize the corresponding results of Bai (1999), Chidume and Aneke (1993), and Chidume and Osilike (1997). 2000 Mathematics Subject Classification. 47H06, 47H07, 47H14. 1. Introduction and preliminaries. Let X be a real Banach space with a dual space X∗. The normalized duality mapping J:X → 2X ∗ is defined by J(x) = {f ∈X ∗ : 〈x,f 〉 = ‖x‖2 = ‖f‖2}, x ∈X. (1.1) It is known that X is uniformly smooth (equivalently, X ∗ is uniformly convex) if and only if J is single-valued and uniformly continuous on any bounded subset of X