Such examples for the convex function of two and three variables are shown in [[2], Example 1,2].
Note that in [[2], Example 2.1], Choudhury considered the function ψ ( a, b ) = 1 2 min { a, b } for all a, b ∈ [ 0, ∞ ). Unfortunately, for this function, we have ψ ( 0, 1 ) = 0, which is a contradiction to the condition that ψ ( s, t ) = 0 if and only if s = t = 0 in [[2], Definition 1.3], also in Definition 2.1.
In particular if d1 x, y):= |x − y| and d2 x, y):= α|x − y|, where α ≥ 0 then D is the same famous cone metric which has been introduced in [[2], Example 1] and its equivalent metric is d ( x, y ) = 1 + α 2 | x - y |.
Remark 7 In [[2], Example 3.7], the authors mentioned that 'Thus, T is not a contraction mapping and then the Banach contraction mapping cannot be applied to this example.' It is true that T is not contraction with the Euclidean metric, but one can easily verify that d ω ∘ ( T x, T y ) ≤ 3 2 d ω ∘ ( x, y ).
(b) It was proved in [36, Example 2.7].
Let f and φ be as in [[11], Example 2].
Example 3 (Band noise covariance matrix using model in [16, Example ]).
In fact it is shown in [7], Example 2, that the maximum improvement is lnn.
Now we consider the following Fredholm integral equation appearing in [5, Example (11.2.1)]: (3.3).
It is proved in [[20], Example 1] that both and are total asymptotically nonexpansive mappings with,,.
It is proved in [[19], Example 3.1] that both and are asymptotically nonexpansive mappings with,.
