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The proof if our main results will be obtained by a suitable applications of an abstract critical point theorem stated in [1].
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A suitable application of Lemma 2.2 to (46) yields (47).
Now a suitable application of Corollary 2.6 to (3.6) yields (3.2).
Then a suitable application of Theorem 2 to (19) yields the desired result.
Then a suitable application of Corollary 2.7 to (69) yields the desired result.
Then a suitable application of Theorem 2.1 to (37) yields the desired result.
A suitable application of Theorem 2.2 to (3.5) yields the desired inequality.
Then a suitable application of Theorem 3.2 yields the desired result.
Treat as one variable, and a suitable application of Theorem 2.1 to (32) yields, which implies.
Then a suitable application of Theorem 3.1 yields the desired inequality (4.2). □.
Now, a suitable application of part (ii) of Theorem 2.3 to (3.6) yields the required estimate in (3.3).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com