Abstract
We modify and apply new property obtained recently in (Udo-utun, Fixed Point Theory and Applications 2014, 2014:65) and results in (Berinde, Carpath. J. Math. 19(1):7-22, 2003; Nonlinear Anal. Forum 9(1):43- 53, 2004) on (δ, k)−weak contractions to obtain asymptotic fixed point theorems for bi-Lipschitz mappings and Lipschitz quotient mappings in Banach spaces. Our results complement and improve several fixed point theorems for Lipschitzian mappings.
Highlights
Studies of Affine localization has been intensive in connection with linear approximations of Lipschitzian mappings [1, 6, 8], differentiability of continuous operators [8] and in the quest for linear isomorphisms on - and linear quotients of Banach spaces concerning Lipschitz quotient mappings [1, 6]
We have proved that every Lipschitzian mapping which admit affine quasi-localization in the orbits of certain points x0 in a closed convex set of a Banach space satisfies (1), and has a fixed point that depends on convergence of the Krasnoselskii scheme
Uniqueness of fixed points is not guaranteed except in the cases where condition (6) of Theorem 2 is satisfied by Sλ i.e; Sλ x − Sλ y ≤ θ x − y + k1 x − Sλ x, for all x, y ∈ {T nx0}∞n=0 and for any x0 ∈ K where θ ∈ (0, 1) and k1 ≥ 0
Summary
Studies of Affine localization has been intensive in connection with linear approximations of Lipschitzian mappings [1, 6, 8], differentiability of continuous operators [8] and in the quest for linear isomorphisms on - and linear quotients of Banach spaces concerning Lipschitz quotient mappings [1, 6]. We have proved that every Lipschitzian mapping which admit affine quasi-localization in the orbits of certain points x0 in a closed convex set of a Banach space satisfies (1), and has a fixed point that depends on convergence of the Krasnoselskii scheme. [9] Let K be a convex set, Y a vector space, a mapping A : K → Y is called an affine if it satisfies A[(1 − λ )x + λ y] = (1 − λ )Ax + λ Ay for all x, y ∈ K and λ ∈ (0, 1) By this definition the identity operator is an example of affines.
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