Let $A$ be an unital alternative $*$-algebra. Assume that contains a nontrivial symmetric idempotent element $e$ which satisfies $xA \cdot e = 0$ implies $x and (1_A - e) 0$. In this paper, it is shown $\Phi$ nonlinear $*$-Jordan-type derivation on A if only additive $*$-derivation. As application, we get result $W^{*}$-algebras.

Abstract We provide a list of equivalent conditions under which an additive operator acting on space smooth functions compact real interval is multiple the derivation.

In this study, we define the generalized symmetric bi-derivation and its trace on MV-algebras give examples. We investigate some properties of bi-derivation’s trace. introduce isotone bi-additive bi-derivation. show that if A is a linearly ordered MV-algebra, Γ related to D γ , then = 0 or γ1 1

Using fixed pointmethods, we investigate approximately higher ternary Jordan derivations in Banach ternaty algebras via the Cauchy functional equation$$f(lambda_{1}x+lambda_{2}y+lambda_3z)=lambda_1f(x)+lambda_2f(y)+lambda_3f(z)~.$$