Hyperbolic summation involving certain arithmetic functions and the integer part function

Let f be an arithmetic function satisfying certain conditions. In this paper, we give an asymptotic formula for the sum $$\sum _{n_1{n}_2\cdots n_r\le x}f\left(\lfloor \frac{x}{n_1{n}_2\cdots n_r}\rfloor \right),r\ge 2.\text{,}$$ where $\lfloor \mathrm{.}\rfloor$ denotes the integer part function.