Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators by George A. Anastassiou

Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators by George A. Anastassiou

Author:George A. Anastassiou
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham


(6.75)

where continuous.

First we see that is well defined and continuous for any . Following [7], p. 289, because , , for any x there exists , hence . We have that , and , if . Furthermore it holds , all and .

Call , then .

So for we get

(6.76)

Notice here that ,

By [7], p. 297, we get that

(6.77)

, , N is odd.

We get that ()

(6.78)

odd,

are positive sublinear operators, [7], p. 290.

We give

Theorem 6.22

Same assumptions as in Theorem 6.20. Then

(6.79)

It holds

Proof

By Theorem 6.7, choose , use of (6.36) and (6.78). At the left hand side of (6.79) is zero, thus (6.79) is trivially true.



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