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(2) is -Kannan, [10 12], if there exists a real constant such that (1.2). (3) is -times reasonable expansive self-mapping if there exists a real constant such that ;,, [1], (4 Picard's -stability means that if is a complete metric space and Picard's iteration satisfies as for then, that is, q is a fixed point of, [9].
Assume that is monotone and weakly continuous along segments (i.e., weakly as for ), then the dual VIP (1.6) is equivalent to the primal VIP (1.1).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com