Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. We prove (under certain assumptions on f) that the iterates of a generic mapping in these spaces converge strongly to a retract...

The objective of this paper is to study the iterative solutions of a class of generalized co-complementarity problems in p-uniformly smooth Banach spaces, with the devotion of sunny retraction mapping, p-strongly accretive, p-relaxed accretive and Lipschitzian (or more generally uniformly continuous) mappings. Our results are new and represents a significant improvement of previously known resu...

In this paper we propose a new Riemannian conjugate gradient method for optimization on the Stiefel manifold. We introduce two novel vector transports associated with the retraction constructed by the Cayley transform. Both of them satisfy the Ring-Wirth nonexpansive condition, which is fundamental for convergence analysis of Riemannian conjugate gradient methods, and one of them is also isomet...

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)= {z ∈H : infs∈G supt∈G ‖Ttsx−z‖ = inf t∈G ‖Ttx−z‖} for each x ∈ C and L( )= ⋂x∈C L(x). In this paper, we prove that ⋂s∈G conv{Ttsx : t ∈ G}⋂L( ) is nonempty for each x ∈ C if and only if there exist...

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be an asymptotically nonexpansive mapping with {kn} ⊂ [1,∞) such that ∑∞ n=1(kn − 1) <∞ and F(T) is nonempty, where F(T) denotes the fixed points set of T . Let {αn}, {α′n}, and {α′′ n } be real sequences in (0,1) and ≤ αn,α′n,α′′ n ≤ 1− for all...

and Applied Analysis 3 or the modified Mann iterative sequence method xn 1 QG ( 1 − αn xn αnT ΠGT xn ) , n 1, 2, . . . , x1 ∈ G, 1.6 where QG : B → G is a sunny nonexpansive retraction. So, in some ways, our results extend and improve some results of other authors such as, see 1–5, 7, 9–13 , from self mappings to non-self-mappings, from Hilbert spaces to Banach spaces. 2. Preliminaries In the s...

Let K be a nonempty closed convex nonexpansive retract of a uniformly convex Banach space E with P as a nonexpansive retraction. Let T : K → E be non-self asymptotically nonexpansive in the intermediate sense mapping with F (T ) 6= φ. Let {α n }, {β n } and {γ n } are sequences in [0, 1] with α (i) n + β (i) n + γ (i) n = 1 for all i = 1, 2, . . . , N . From arbitrary x1 ∈ K, define the sequenc...

and Applied Analysis 3 2. Preliminaries Let E be a real Banach space with dual space E∗. When {xn} is a sequence in E, we denote strong convergence of {xn} to x ∈ E by xn → x and weak convergence by xn ⇀ x, respectively. As usual, we denote the duality pairing of E∗ by 〈x, x∗〉, when x∗ ∈ E∗ and x ∈ E, and the closed unit ball by UE, and denote by R and N the set of all real numbers and the set ...

and Applied Analysis 3 It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [7–9]. Very recently, Cai and Bu [10] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding (x∗,...