Intelligent Numerical Methods: Applications to Fractional Calculus by George A. Anastassiou & Ioannis K. Argyros

Author:George A. Anastassiou & Ioannis K. Argyros
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(12.2.9)

(12.2.10)

and

where,

Then, the conclusions of Theorem 12.1 for sequence hold with , , , replacing , q and r, respectively. Moreover, is a solution of the equation .

Proof

Notice that

Therefore, the proof of Theorem 12.1 can apply. Then, in view of the estimate

we deduce by letting that .

Concerning the uniqueness of the solution we have the following result:

Proposition 12.3

Under the hypotheses of Proposition 12.2, further suppose that there exists continuous and nondecreasing such that

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