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

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