While refusing to second-guess the review, Thompson says: "If we are going to make a change it's going to be contraction rather than expansion.
A mapping (T:X rightarrow X) is said to be contraction if there is (0< k<1) such that (d(Tx,Ty) leq k d x,y)) for all (x,y in X).
Let ω be modular on a set X. A mapping T : X ω → X ω is said to be contraction [[2], Definition 3.1] if there exists k ∈ [ 0, 1 ) such that ω λ ( T x, T y ) ≤ k ω λ ( x, y ) (1). for all λ > 0 and x, y ∈ X ω. Recently, Mongkolkeha et al. [2] proved the following theorems.
A mapping f from C into C is said to be contraction, if there exists a constant k ∈ [ 0, 1 ) such that ∥ f ( x ) − f ( y ) ∥ ≤ k ∥ x − y ∥, for all x, y ∈ C. In 2000, Moudafi [2] introduced the following viscosity approximation methods: x 1 ∈ C and x n + 1 = α n f ( x n ) + ( 1 − α n ) S x n, n ∈ N, where f is a contraction on closed convex subset of a real Hilbert space.
T is said to be nonexpansive if ∥ T x − T y ∥ ≤ ∥ x − y ∥, ∀ x, y ∈ C. Let D be a nonempty subset of C. Let Q : C → D. Q is said to be contraction if Q 2 = Q ; sunny if for each x ∈ C and t ∈ ( 0, 1 ), we have Q ( t x + ( 1 − t ) Q x ) = Q x ; sunny nonexpansive retraction if Q is sunny, nonexpansive, and contraction.
A mapping T: X → X is said to be contraction if there exists r ∈ [0, 1) such that d ( T ( x ), T ( y ) ) ≤ r d ( x, y ), ∀ x, y ∈ X. (1.1). In 1922, Banach [1] proved that if (X, d) is a complete metric space and the mapping T satisfies (1.1), then T has a unique fixed point, that is T u) = u for some u ∈ X.
A mapping is said to be -contraction if there exists a constant such that (22).
A mapping f : S → S is said to be -contraction if there exists a constant α ∈ ( 0, 1 ) such that for x ∈ S and y ∈ N x we have F f x, f y ( α t ) ≥ F x, y ( t ) for all t > 0. (3.11).
A mapping is said to be -weak contraction or -weak contraction if there exist two constants and such that (2.3).
If we now choose A to be a contraction without a non-trivial invariant subspace, then (D_{1}) (along with A) is a proper contraction [5].
There was bound to be market contraction regardless.
