Very recently, J. M. Rassias [43] considered the Cauchy difference controlled by the product and sum of powers of norms, that is, θ { ∥ x ∥ p ∥ y ∥ p + ( ∥ x ∥ 2 p + ∥ y ∥ 2 p ) }. Corollary 2.5 Let A be a non-Archimedean Banach algebra over Q 2, ℓ ∈ { − 1, 1 } be fixed and ε, p, q be non-negative real numbers such that ℓ ( p + q ) > 2 ℓ ( ℓ + 2 ).
We extend the results concerning Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Ulam-Găvruta-Rassias stability controlled by the mixed product-sum of powers of norms for equations (1.7) and (1.8).
From Theorem 3.1, we obtain the following corollary concerning the stability of (1.5) in the sense of the JMRassias stability of functional equations controlled by the mixed product-sum of powers of norms introduced by J. M. Rassias [26] and called JMRassias stability by several authors [27 30].
Wild speculation is often made about the sums controlled by the Karmapa Charitable Trust and associated legal entities, but these estimates are largely guesswork.
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The vertical motion is controlled by the collective thrust input, i.e. the sum of the thrusts of the motors.
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In 1978, Rassias [6] attempted to weaken the condition for the bound of Cauchy difference controlled by a sum of unbounded function and provided a generalization of Hyers' theorem.
A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki in [3] and for approximate linear mappings was presented by Rassias in [4] by considering the case when the left-hand side of (1.1) is controlled by a sum of powers of norms.
