Let S be a semigroup, Z(S) the center of S and \(\sigma :S\rightarrow S\) is an automorphism that need not be involutive. We determine the complex-valued solutions of the generalization of Van Vleck’s functional equation \(\begin{aligned} \displaystyle \int _{S} f(xyt)d\mu (t)-\tau (y)\displaystyle \int _{S} f(\sigma (y)xt)d\mu (t)= 2f(x)g(y),\ x,y\in S, \end{aligned}\) where \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_i})_{i\in I}\) , such that \(z_i\in Z(S)\) for all \(i\in I\) , and \(\tau :S\rightarrow \mathbb {C}\) is a multiplicative function such that \(\tau (x\sigma (x))=1\) for all \(x\in S\) . This allows us to solve the functional equation \(\begin{aligned} \displaystyle \int _{S} f(x\varphi (y)t)d\mu (t) -\displaystyle \int _{S} f(\psi (y)xt)d\mu (t)= 2f(x)g(y),\ x,y\in S, \end{aligned}\) where \(\varphi ,\psi :S\rightarrow S\) are automorphisms such that \(\varphi \) is involutive and \(\psi \) not necessarily involutive. Some consequences of these results are given.
