Iterative solutions for zeros of multivalued accretive operators

In the sequel, we shall denote the single-valued normalized duality map by j. Let F (T ) = {x ∈ E : Tx = x} denote the set of all fixed point for a mapping T . We write xn ⇀ x (respectively xn ∗ ⇀ x) to indicate that the sequence xn weakly (respectively weak∗) converges to x; as usual xn → x will symbolize strong convergence. A mapping A : D(A) ⊂ E → 2 is called to be accretive if for all x, y ∈ D(A) there exists j(x − y) ∈ J(x− y) such that 〈u− v, j(x− y)〉 ≥ 0, for u ∈ Ax and v ∈ Ay; If E is a Hilbert space, accretive operators are also called monotone. An operator A is called m−accretive if it is accretive and R(I + rA), range of (I + rA), is E for all r > 0; and A is said to satisfy the range condition if D(A) ⊂ R(I + rA),∀r > 0, where I is an identity operator of E and D(A) denotes the closure of the domain of A. Interest in accretive mappings stems mainly from their firm connection with equations of evolution. It is known (see, e.g., [34]) that many physically significant problems can be modeled by initial-value problems of the form x′(t) + Ax(t) = 0, x(0) = x0, (1.1)