Steepest descent method for set-valued locally accretive mappings

Let E be a real q-uniformly smooth Banach space. Suppose T is a set-valued locally strongly accretive map with open domain D(T) in E and that 0 is an element of Tx has a solution x* in D(T). Then there exists a neighbourhood B in D(T) of x* and a real number r1>0 such that for any r>r1 and some real sequence {cn}, any initial guess x1 is an element of B and any single-valued selection T0 of T, the sequence {xn} generated from x1 by xn+1=xn-cnT0xn, n≥1, remains in D(T) and converges strongly to x* with ||xn-x*|| O(n-(q-1)/q). A related result deals with iterative approximation of a solution of the equation f is an element of x+Ax when A is a locally accretive map. Our theorems generalize important known results and resolve a problem of interest. (author). 39 refs