Fixed-Point Properties of Roughly Contractive Mappings

For given k ∈ (0, 1) and r > 0, a self-mapping T : M → M is said to be r-roughly k-contractive provided ‖Tx− Ty‖ ≤ k ‖x− y‖+ r (x, y ∈ M). To state fixed-point properties of such a mapping, the self-Jung constant Js(X) is used, which is defined as the supremum of the ratio 2 rconv S(S)/ diam S over all non-empty, non-singleton and bounded subsets S of some normed linear space X, where rconv S(S) = infx∈conv S supy∈S ‖x − y‖ is the self-radius of S and diam S is its diameter. If M is a closed and convex subset of some finite-dimensional normed space X and if T : M → M is r-roughly k-contractive, then for all ε > 0 there exists x∗ ∈ M such that ‖x∗ − Tx∗‖ < 1 2 Js(X) r + ε. If dim X = 1, or X is some two-dimensional strictly convex normed space, or X is some Euclidean space, then there is x∗ ∈ M satisfying ‖x∗ − Tx∗‖ ≤ 1 2 Js(X) r.