So the exercise goes like this: Problem
(a)Prove that the there is a $f:[0,1]$ that satisfies the following condition: $$ f(x)=\cos(x)+ \int^x_0{f(y)e^{-y}dy}\text{ for } x\epsilon[0,1] $$ Hint:Use the Existence and Uniqueness Theorem for Ordinary Differential Equations
Why the solution of $f$ must be defined in the range of $[0,1]$?
So my proof goes like this: In order to use the theorem mentioned earlier we have to write the given equation as a differential equation.So we have the following: $$f'(x)=-\sin(x)+f(x)\cdot e^{-x}$$ Moreover, $$f'(x) \text{ and } \frac {\partial df } {\partial y} \text{are both continious in a rectangle:} [0,1] \times [-M,M], M>0$$ So by the theorem of existence and uniquess for ordinary differential equations there is a $f$ that satisfies the differential equation and the initial condition ($f(0)=1$) in some interval $I$ containing $0$.
My questiions:
1.Is their any simplier proof?
2.What i should write as answer for the (b)?What i mean is there any solid proof like in the part (a)?
Thanks a lot for the help.