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Now, we give an example to validate the main result as follows: Example 4.4 ([28]).
Now, we give an example to illustrate Theorem 2.1 as follows: Example 2.3.
We obtain a Fibonacci-∇-sequence in the groupoid (X, ∇) discussed in Example 3.4 as follows: Example 3.5.
However, the converse is false as follows: Example 35 Consider, X = [0,1], and define f by f(x) = 1 - x2.
Now, we give some real numerical examples in which the conditions satisfy the ones of Theorems 3.1 and 2.1 and some numerical experiment results to explain the main results Theorems 3.1 and 2.1 as follows: Example 5.1.
Remark 4.9 If the spaces L p has a weakly sequentially continuous generalized duality mappings, then we obtain Theorems 4.7 and 4.8 hold for L p spaces with 1 < p < ∞, p ≠ 2. In this section, we give a simple example and some numerical experiment result to explain the convergence of the sequence (3.14) as follows: Example 5.1 Let X = R and C = [ 0, 1 2 ].
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