Common Fixed-Point Problem for a Family Multivalued Mapping in Banach Space

LetX be a Banach space with dualX∗, and letK be a nonempty subset ofX. A gauge function is a continuous strictly increasing function φ : R → R such that φ 0 0 and limt→∞φ t ∞. The duality mapping Jφ : X → X∗ associated with a gauge function φ is defined by Jφ x : {f ∈ X∗ : 〈x, f〉 ‖x‖‖f‖, ‖f‖ φ ‖x‖ }, x ∈ X, where 〈·, ·〉 denotes the generalized duality pairing. In the particular case φ t t, the duality map J Jφ is called the normalized duality map. We note that Jφ x φ ‖x‖ /‖x‖ J x . It is known that if X is smooth, then Jφ is single valued and norm to weak∗ continuous see 1 . When {xn} is a sequence in X, then xn → x xn ⇀ x, xn ⇁ x will denote strong weak, weak∗ convergence of the sequence {xn} to x. s Following Browder 2 , we say that a Banach spaceX has the weakly continuous duality mapping if there exists a gauge function φ for which the duality map Jφ is single valued and weak to weak∗ sequentially continuous, that is, if {xn} is a sequence in X weakly convergent to a point x, then the sequence {Jφ xn } converges weak∗ to Jφ x . It is known that lp 1 < p < 1 spaces have a weakly continuous duality mapping Jφ with a gauge φ t tp−1. Setting