Approximate Fixed-Point Theorems for Discontinuous Mappings

Abstract

For a given r ≥ 0, a mapping T : M → M on some convex subset M of a normed linear space X is said to be around r-continuous if for all x ∈ M and ϵ > 0 there exists δ > 0 such that ‖Ty − Tz‖ < r + ϵ holds whenever y, z ∈ M, ‖y − x‖ < δ, and ‖z − x‖ < δ. If δ does not depend on x then T is called uniformly r-continuous. By using the self-Jung constant J s (X) ∈ [1, 2], we state some theorems on approximate fixed points of such mappings. For instance, if M is compact and T is around r-continuous then, for all ρ > 0, there exists x * ∈ M satisfying , where ρ can be replaced by zero under some additional assumptions. This property remains true if M is only relatively compact but T is uniformly r-continuous, or if the relative compactness of M is replaced by the relative compactness of T(M).