We prove strong convergence theorem for infinite family of uniformly L−Lipschitzian total quasi-φ-asymptotically nonexpansive multi-valued mappings using a generalized f−projection operator in a real uniformly convex and uniformly smooth Banach space. The result presented in this paper improve and unify important recent results announced by many authors.

In general, the Gelfand widths cn(T ) of a map T between Banach spaces X and Y are not equivalent to the Gelfand numbers cn(T ) of T . We show that cn(T ) = cn(T ) (n ∈ N) provided that X and Y are uniformly convex and uniformly smooth, and T has trivial kernel and dense range. c ⃝ 2012 Elsevier Inc. All rights reserved.

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure. A Banach space is said to have the fixed point property (FPP) if for every nonempty bounded closed convex C ⊂ X and every nonexpansive self-mapping T : C → C there is a fixed point of T in C. A Banach space is said to have the weak fixed point prope...

This paper discusses the existence and decay of solutions u(t) of the variational inequality of parabolic type: >_ 0 for v e LP([O,=);V (p>2) with v(t) e K a.e. in [0,=), where K is a closed convex set of a separable uniformly convex Banach space V, A is a nonlinear monotone operator from V to V* and B is a nonlinear operator from Banach space W to W*. V ...

and Applied Analysis 3 Theorem 1.3. Let E be a real q-uniformly smooth Banach space which is also uniformly convex, let C be a nonempty closed convex subset of E, let T : C → C be a (λ, {kn})-strictly asymptotically pseudocontractive mapping such that ∑∞ n 1 k 2 n − 1 < ∞, and let F T / ∅. Let {αn} ⊂ 0, 1 be a real sequence satisfying the following condition: 0 < a ≤ αn ≤ b < q 1 − k 2cq 1 L − ...

We study the asymptotic behavior of the sequence of the iterates for a nonexpansive mapping, defined on a suitable subset of a Banach space with Opial's condition. Some results are stated also for semigroups of nonexpansive mappings and for mappings of asymptotically nonexpansive type in uniformly convex Banach spaces with Opial's condition.

and Applied Analysis 3 nonexpansive mapping such that F S ∩ A−10/ ∅. Let {xn} be a sequence generated by x0 x ∈ C and un J−1 ( βnJxn ( 1 − βn ) JSJrnxn ) , Cn { z ∈ C : φ z, un ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0 1.6 for all n ∈ N ∪ {0}, where J is the duality mapping on E, {βn} ⊂ 0, 1 , and {rn} ⊂ a,∞ for some a > 0. If lim infn→∞ 1 − βn > 0, then {xn} converges s...

Let E be a real q-uniformly smooth Banach space, which is also uniformly convex (for example, Lp or p spaces, 1 < p <∞), and C be a nonempty bounded closed convex subset of E. Let T : C→ C be a k-strictly asymptotically pseudocontractive map with a nonempty fixed point set. A hybrid algorithm is constructed to approximate fixed points of such maps. Furthermore, strong convergence of the propose...

The implicit midpoint rule (IMR) for nonexpansive mappings is established in Banach spaces. The IMR generates a sequence by an implicit algorithm. Weak convergence of this algorithm is proved in a uniformly convex Banach space which either satisfies Opial’s property or has a Fréchet differentiable norm. Consequently, this algorithm applies in both `p and Lp for 1 < p < ∞.