for all x ∈ E, where 〈·,·〉 denotes the generalized duality pairing. It is well known that if E is a uniformly smooth Banach space, then J is single valued and such that J(−x) =−J(x), J(tx) = tJ(x) for all x ∈ E and t ≥ 0; and J is uniformly continuous on any bounded subset of E. In the sequel, we shall denote single-valued normalized duality mapping by j by means of the normalized duality mappi...

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge φ. Let f be an α-contraction and {Tn} a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes xn+1 = αnf(xn) + (1− αn)Tnxn (1) with a general theorem a...

where E∗ denotes the dual space of E and 〈·, ·〉 denotes the generalized duality pairing. In the sequel, we denote a single-valued normalized duality mapping by j. Throughout this paper, we use F T to denote the set of fixed points of the mapping T . ⇀ and → denote weak and strong convergence, respectively. Let K be a nonempty subset of E. For a given sequence {xn} ⊂ K, let ωω xn denote the weak...

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 ...

Throughout this paper, we assume that X is a uniformly convex Banach space and X∗ is the dual space of X. Let J denote the normalized duality mapping form X into 2 ∗ given by J x {f ∈ X∗ : 〈x, f〉 ‖x‖2 ‖f‖2} for all x ∈ X, where 〈·, ·〉 denotes the generalized duality pairing. It is well known that if X is uniformly smooth, then J is single valued and is norm to norm uniformly continuous on any b...

for all x ∈ E, where 〈·,·〉 denotes the generalized duality pairing. It is well known that if E is a uniformly smooth Banach space, then J is single valued such that J(−x) = −J(x), J(tx) = tJ(x) for all t ≥ 0, x ∈ E; and J is uniformly continuous on any bounded subset of E. In the sequel we will denote single-valued normalized duality mapping by j. In the following we give some concepts. Let T :...

Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For examp...

Let E be a 2-uniformly convex real Banach space with uniformly Gâteaux differentiable norm, and [Formula: see text] its dual space. Let [Formula: see text] be a bounded strongly monotone mapping such that [Formula: see text] For given [Formula: see text] let [Formula: see text] be generated by the algorithm: [Formula: see text]where J is the normalized duality mapping from E into [Formula: see ...