Abstract
In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.
Highlights
Let X be a Banach space, C be a nonempty subset of X, and T be a mapping from C into itself
The trend constant of the mapping T at a point α ∈ R is given by the formula ια (T ) = sup {∂+ψx, y (α) : x, y ∈ C, x = y}
We provide a few formulas for trend constants of a mapping T : C → C, where C is a nonempty subset of a Banach space X
Summary
Let X be a Banach space, C be a nonempty subset of X, and T be a mapping from C into itself. The mapping T is said to be an initial contraction if ι (T ) < 0, and initially nonexpansive if ι (T ) ≤ 0. The trend constant (more precisely the right trend constant) of the mapping T at a point α ∈ R is given by the formula ια (T ) = sup {∂+ψx, y (α) : x, y ∈ C, x = y}. We say that T : C → C is a pre-initial contraction Pre-initially nonexpansive) if it is a k-Lipschitz mapping, where k > 1, and there exists α∈. The mapping T is said to be firmly nonexpansive if for all x, y ∈ C the function φx, y (t) is nonincreasing on the interval [0, 1]. The mapping T is firmly nonexpansive if and only if τ (T ) ≤ 0
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