I'm trying to solve the following equation
$$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$
where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on $\mathbb{R}$. Does anyone can recommend a method or a book that can lead to solve this kind of equations? Thanks in advance.
The given equation can be written as: $$f(x)=e^{-cx}+\lambda \int_0^x ce^{-cy}f(x-y)\,dy = e^{-cx}+\lambda\int_0^x ce^{-c(x-y)}f(y)\,dy $$ $$\Rightarrow f(x)=e^{-cx}+\lambda e^{-cx}\int_0^x e^{cy}f(y)\,dy \Rightarrow e^{cx}f(x)=1+\lambda c\int_0^xe^{cy}f(y)\,dy$$ Differentiate both the sides with respect to $x$ to obtain: $$e^{cx}f'(x)+ce^{cx}f(x)=\lambda ce^{cx}f(x) \Rightarrow\frac{f'(x)}{f(x)}=c(\lambda-1)$$ The above is a simple differential equation with the solution: $$f(x)=Ae^{c(\lambda-1)x}$$ where $A$ is some constant. To determine $A$, put $x=0$ in the given equation to find that $f(0)=1$. With this value of $f(0)$, $A$ is equal to $1$, hence, the final answer is: $$f(x)= e^{c(\lambda-1)x}$$