Convergence theorems for multivalued Φ-hemicontractive operators and Φ-strongly accretive operators

Suppose E is a uniformly smooth Banach space and T : E → 2E is a multivalued φ-hemicontractive operator with bounded range. Suppose {an}, {bn}, {cn} and {an′}, {bn′} {cn′} are real sequences in [0, 1] satisfying the following conditions: (i) an + bn + cn = a′n + b′n + c′n = 1, for all n ε ; (ii) limn→∞ bn = limn→∞b′n = limn→∞ cn = 0; (iii) ∑∞n=1 bn = ∞; (iv) cn = o(bn). For arbitrary xi, u1, v1 ε E, define the sequence {xn}n=1∞ by xn+1 = anxn+bnηn+cnun, ηn ε Tyn, n ε ; yn = a′nxn + b′nξn + c′nvn ξn ε Txn, n ε , where {un}n=1∞ {vn}n=1∞ are arbitrary bounded sequences in E. Then {xn}n=1∞ converges strongly to the unique fixed point of T. Related results deal with the iterative solutions of nonlinear multivalued φ-accretive operator equation f ε Tx.