Solution of some irregular functional equations and their stability

Subject Areas : Functional analysis

Y. Sayyari ¹ , M. Dehghanian ² , Sh. Nasiri ³

1 - Department of Mathematics, Sirjan University of Technology, P. O. Box 78137-33385, Sirjan, Iran
2 - Department of Mathematics, Sirjan University of Technology, P. O. Box 78137-33385, Sirjan, Iran
3 - Department of Computer Engineering, Sirjan University of Technology P. O. Box 78137-33385, Sirjan, Iran

Received: 2022-11-01 Accepted : 2022-12-27 Published : 2022-12-01

Keywords: Hyers-Ulam stability, Additive functional equation, unital algebra,

Abstract :

‎In this note‎, ‎we study the following functional equations‎:‎\begin{align*}‎‎&L(L(p ,r)+L(‎q‎,r)+p + q ,r)+L(L( p, r)+ p , r)+L(q, r )=0,\\‎‎&L(L( p , r )+ p + q+e, r )+L( p, r)=L( p + q , r )+ p L(q , r)‎\end{align*}‎‎and‎ ‎$‎L( p , q )=L(\zeta p , q), \vert \zeta\vert <1‎$,‎without any regularity assumption for all $ p , q , r \in A$, where $L:A^2\rightarrow A$ is defined by ‎$‎L( p , q ):=g( p + q )-g( p )-g( q )‎$‎ for all $ p , q\in A$. Also, we find general solutions of the above functional equations on algebras, unital algebras and real numbers, respectively. Finally, we investigate the stability of those functional equations in algebras and unital algebras, respectively.

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