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F is a nonlinear contraction mapping of type (Φ1).
Notice also that every generalized α-ψ-contractive mapping of type II is also a generalized α-ψ-contractive mapping of type I.
Then T is of mapping of type (P) if and only if (A_{T}) is monotone.
Then T is a mapping of type (Q) if and only if (A_{T}) is monotone.
Let be a smooth Banach space, a nonempty subset of and a mapping of type (P).
Suppose f is a rational Geraghty contractive mapping of type II.
Now we claim that the operator Φ ′ is a mapping of type ( S + ).
Suppose f is a rational Geraghty contractive mapping of type III.
From Lemma 2.1, we know that A is a mapping of type ((S_)).
By Lemma 3.2, we also know that each S n is a mapping of type (r) of C into X.
Let be a smooth, strictly convex, and reflexive Banach space, a nonempty subset of and a mapping of type (P).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com