Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

and Applied Analysis 3 where R is the set of all real numbers; in particular, J(−x) = −J(x) for all x ∈ E ([28]). We say that a Banach space E has a weakly sequentially continuous duality mapping if there exists a gauge function φ such that the duality mapping Jφ is single valued and continuous from the weak topology to the weak topology; that is, for any {xn} ∈ E with xn ⇀ x, Jφ(xn) ∗ ⇀ Jφ(x). For example, every l space (1 < p < ∞) has a weakly continuous duality mapping with gauge function φ(t) = t. It is well known that ifE is a Banach space having aweakly sequentially continuous duality mapping Jφ with gauge function φ, then E has the opial condition [29]; this is, whenever a sequence {xn} in E converges weakly to x ∈ E, then lim sup n→∞ 󵄩󵄩󵄩󵄩xn − x 󵄩󵄩󵄩󵄩 < lim sup n→∞ 󵄩󵄩󵄩󵄩xn − y 󵄩󵄩󵄩󵄩 ∀y ∈ E, y ̸ = x. (15) AmappingT : C → CB(E) is ∗-nonexpansive [30] if, for all x, y ∈ C and ux ∈ Tx with ‖x − ux‖ = inf{‖x − z‖ : z ∈ Tx}, there exists uy ∈ Ty with ‖y − uy‖ = inf{‖y − w‖ : w ∈ Ty} such that 󵄩󵄩󵄩󵄩 ux − uy 󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 . (16) It is known that ∗-nonexpansiveness is different from nonexpansiveness for multivalued mappings. There are some ∗-nonexpansiveness multivalued mappings which are not nonexpansive and some nonexpansivemultivaluedmappings which are not ∗-nonexpansive [31]. We introduce some terminology for boundary conditions for nonself-mappings. The inward set of C at x is defined by IC (x) = {z ∈ E : z = x + λ (y − x) : y ∈ C, λ ≥ 0} . (17) Let IC(x) = x + TC(x) with TC (x) = {y ∈ E : lim inf λ→0+ d (x + λy, C)