Lemma 3.1 Let γ : [ 0, 1 ] → [ 0, 1 ] be a left continuous function and ∗ be a continuous t-norm.
Let S : P ¯ ( γ, w 5 ) → P ¯ ( γ, w 5 ) be a continuous and completely continuous operator.
Let be a continuous function on and a continuous function on.
Notice that every uniformly continuous mapping must be a continuous and closed mapping.
Definition 5.1 Let ξ I ∈ I be a continuous random variable with continuous probability density function p I : I → ( 0, ∞ ).
Let ({X _{I}}in I) be a continuous random variable with continuous probability density function ({p_{I}}:Irightarrow ( {0,infty} ) ).
Assume that ψ is a nonnegative continuous concave functional on C such that ψ ( x ) ≤ ∥ x ∥ for all x ∈ C ¯ ( w 4 ), and let S : C ¯ ( w 4 ) → C ¯ ( w 4 ) be a continuous and completely continuous operator.
Let ∗ and ⋄ be a continuous t-norm and a continuous t-conorm such that a ∗ b ≤ a b ≤ a ⋄ b for all a, b ∈ [ 0, 1 ].
We point out that a basic property of the truncated random variable is as follows: Let ξ I ∈ I be a continuous random variable with continuous probability density function p I : I → ( 0, ∞ ).
We point out a basic property of the truncated random variable as follows [1]: Let ({X _{I}}in I) be a continuous random variable with continuous probability density function ({p_{I}}:Irightarrow ( {0,infty} ) ).
