there exists a comparison function φ : R + → R + such that.
(b)if an increasing sequence converges to in, then for all ; (v there exists a comparison function such that (3.7).
(i) and are continuous; (ii) is increasing for each ; (iii there exists a comparison function, with for each and any, such that (4.10).
(i) and are continuous; (ii) is increasing for each ; (iii there exists a comparison function, with for each and any, such that (4.2). (iv there exists such that (4.3).
(c2) f is orbitally X≤-continuous and there exists a subsequence of {f n (x0)} such that for any k ∈ N ; (d) there exists a comparison function φ : R + → R + such that .
(i for each with there exists such that and ; (ii) ; (iii)if and, then ; (iv there exists such that ; (v) is orbitally continuous; (vi there exists a comparison function such that, for each one has (3.1).
I propose a type of research that was not cited in Tom Dunkel's article -- probably because it doesn't exist: a comparison study.
A map T is called ( ϕ, L ) -weak contraction if there exist a comparison function ϕ and some L ≥ 0 such that p ( T x, T y ) ≤ ϕ ( p ( x, y ) ) + L p w ( y, T x ).
Suppose T also satisfies the following condition: There exist a comparison function ϕ 1 and some L 1 ≥ 0 such that p ( T x, T y ) ≤ ϕ 1 ( p ( x, y ) ) + L 1 p w ( x, T x ). for all x, y ∈ X.
Also, suppose that there exist a comparison function ϕ 1 and L 1 ≥ 0 such that p ( T x, S y ) ≤ ϕ 1 ( max { p ( x, y ), p ( x, T x ), p ( y, S y ), 1 2 ( p ( T x, y ) + p ( x, S y ) ) } ) + L p ( x, T x ). for all x, y ∈ X.
Also, assume that there exist a comparison function ϕ 1 and L 1 ≥ 0 such that p ( T x, T y ) ≤ ϕ ( max { p ( S x, S y ), p ( S x, T x ), p ( S y, T y ), 1 2 ( p ( T x, S y ) + p ( S x, T y ) ) } ) + L p ( S x, T x ). for all x, y ∈ X.
