To be more precise, we give two examples for X such that ((mathcal M_{X}, d_{L})) is not separable: (i) Infinite unions of shrinking closed intervals with zero { 0 } cup bigcup_{n=1}^{infty}left[frac{1}{2^{n}}, frac{1}{2^{n}}+frac{1}{2^{n+1}}right]; (ii) Closed interval [0,1].
Since f ( x ) is continuous, f ( x n ) belongs to another bounded closed interval, so, f ( x n ) is bounded.
Similarly, we can show that F 2 : Ω 2 → X is equicontinuous on a finite closed interval on ( 0, ∞ ).
But it is well known that the boundary of a nonempty (closed) interval cannot be its retract (see [19]).
Further note that Fix(g) is a nonempty closed interval if g is a nonexpansive self-map of [0,1].
Let (I subset mathbb {R}) be a nonempty closed interval and ([alpha,beta]) be a nonempty functional interval in (AC(I)).
Let I be a finite closed interval [ m, M ] on R and C ( I ) the space of all continuous real-valued functions defined on I.
From (11) and (12), we infer that F 1 : Ω 1 → Y is equicontinuous on a finite closed interval of ( 0, ∞ ).
Step 3. We prove that both F 1 : Ω 1 → Y and F 2 : Ω 2 → X are equicontinuous on a finite closed interval of ( 0, ∞ ).
Suppose that the given closed interval [ a, b ] is partitioned into uniform spacing ( b − a ) / n and nodes t i = a + i h (for i = 0, …, n ).
Continuity of T implies that { T x n } n = 1 ∞ belongs to another bounded closed interval, so { T x n } n = 1 ∞ is bounded.
