Functional inequalities and pairs of hom-derivations and homomorphisms
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In this paper, we introduce and solve the following additive-additive ( π , π‘ ) -functional inequality β 2 π ( π₯ + π¦ 2 ) β π ( π₯ ) β π ( π¦ ) β + β 2 β ( π₯ + π¦ 2 ) + 2 β ( π₯ β π¦ 2 ) β 2 β ( π₯ ) β β€ β π ( π ( π₯ + π¦ ) β π ( π₯ ) β π ( π¦ ) ) β + β π‘ ( β ( π₯ + π¦ ) + β ( π₯ β π¦ ) β 2 β ( π₯ ) ) β , where π and π‘ are fixed nonzero complex numbers with | π | + | π‘ | < 1 . We define a pair of hom-derivation and homomorphism in complex Banach algebras, and using the direct method and the fixed point method, we prove the Hyers-Ulam stability of pairs of hom-derivations and homomorphisms in complex Banach algebras, associated to the additive-additive ( π , π‘ ) -functional inequality ( ??? ) and the following functional inequality β π ( π₯ π¦ ) β π ( π₯ ) β ( π¦ ) β β ( π₯ ) π ( π¦ ) β + β β ( π₯ π¦ ) β β ( π₯ ) β ( π¦ ) β β€ π ( π₯ , π¦ ) .
Abstract
In this paper, we introduce and solve the following additive-additive ( π , π‘ ) -functional inequality β 2 π ( π₯ + π¦ 2 ) β π ( π₯ ) β π ( π¦ ) β + β 2 β ( π₯ + π¦ 2 ) + 2 β ( π₯ β π¦ 2 ) β 2 β ( π₯ ) β β€ β π ( π ( π₯ + π¦ ) β π ( π₯ ) β π ( π¦ ) ) β + β π‘ ( β ( π₯ + π¦ ) + β ( π₯ β π¦ ) β 2 β ( π₯ ) ) β , where π and π‘ are fixed nonzero complex numbers with | π | + | π‘ | < 1 . We define a pair of hom-derivation and homomorphism in complex Banach algebras, and using the direct method and the fixed point method, we prove the Hyers-Ulam stability of pairs of hom-derivations and homomorphisms in complex Banach algebras, associated to the additive-additive ( π , π‘ ) -functional inequality ( ??? ) and the following functional inequality β π ( π₯ π¦ ) β π ( π₯ ) β ( π¦ ) β β ( π₯ ) π ( π¦ ) β + β β ( π₯ π¦ ) β β ( π₯ ) β ( π¦ ) β β€ π ( π₯ , π¦ ) .