Solvability and Iterative Construction of Solutions for a Class of K- Positive Definite Operator Equations in Uniformly Smooth Banach Spaces

Abstract:

      Let X be a uniformly smooth Banach space and let A:D(A)?X→X be a K-positive definite operator with D(A) = D(K) . Then there exists a constant β>0 such that for every x∈D(A) , ||Ax||≤β||Kx||. Furthermore, the operator A is closed, R(A) = A , and the equation Ax =f , for each f∈X, has a uniqne solution. Let {a_n}_n≥0 be a real sequence in [0,1] satisfying conditions; （i）a_n→0（n→∞）; and （ii）∑_n=0^∞a_n=∞, . Define the sequence {a_n}_n≥0 iteratively by (?) Then the sequence {a_n}_n≥0 defined by (*) converges strongly to the unique solution of the equation Ax=f in X.