find $\lambda$ such that the integral has a solution. I have the integral equation:
$u(x) = f(x) + \lambda \int_0^{\frac{1}{2}}u(y)dy$
I have to find $\lambda$ such that the integral has a solution. 
How to approach such problems?

We can rewrite this equation as 
$$
u(x) - f(x) =\lambda \int_0^{1/2}u(y)\,dy
$$
Note that since the integral does not vary over $x$, any solution $u$ must differ from $f$ by a constant.  That is, we may write $u(x) = f(x) + k$ for some constant $k$.
The question then becomes whether a suitable constant $k$ exists.
Substituting $u(x) = f(x) + k$ into the original equation, we have
$$
f(x) + k =  \lambda \int_0^{1/2} f(x)\,dx + \frac 12 \lambda k + f(x) \iff\\
\left(1 - \frac 12 \lambda \right)k = \lambda \int_0^{1/2} f(x)\,dx
$$
When $\lambda \neq 2$, the above equation may be solved for $k$.
For $\lambda = 2$, a solution exists only if $\int_0^{1/2}f(x)\,dx = 0$.
