TY - JOUR
ID - 774
TI - Solutions and stability of variant of Van Vleck's and D'Alembert's functional equations
JO - International Journal of Nonlinear Analysis and Applications
JA - IJNAA
LA - en
SN - 2008-6822
AU - Rassias, Th.M.
AU - Elqorachi, Elhoucien
AU - Redouani, Ahmed
AD - Department of Mathematics, National Technical University of Athens, Zofrafou Campus, 15780 Athens, Greece
AD - Ibn Zohr University, Faculty of Sciences
Department of Mathematic, Agadir, Morocco
Y1 - 2016
PY - 2016
VL - 7
IS - 2
SP - 279
EP - 301
KW - semigroup
KW - d'Alembert's equation
KW - Van Vleck's equation, sine function
KW - involution
KW - multiplicative function, homomorphism, superstability
DO - 10.22075/ijnaa.2017.1803.1472
N2 - In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation$$\int_{S}f(\sigma(y)xt)d\mu(t)-\int_{S}f(xyt)d\mu(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a semigroup, $\sigma$ is an involutive morphism of $S$, and $\mu$ is a complex measure that is linear combinations of Dirac measures $(\delta_{z_{i}})_{i\in I}$, such that for all $i\in I$, $z_{i}$ is contained in the center of $S$. (2) We determine the complex-valued continuous solutions of the following variant of d'Alembert's functional equation$$\int_{S}f(xty)d\upsilon(t)+\int_{S}f(\sigma(y)tx)d\upsilon(t) = 2f(x)f(y), \;x,y\in S,$$ where $S$ is a topological semigroup, $\sigma$ is a continuous involutive automorphism of $S$, and $\upsilon$ is a complex measure with compact support and which is $\sigma$-invariant. (3) We prove the superstability theorems of the first functional equation.
UR - https://ijnaa.semnan.ac.ir/article_774.html
L1 - https://ijnaa.semnan.ac.ir/article_774_ac5ba88e6d8ed3f180cc2ff75a074111.pdf
ER -