I have the following integral equation with symmetric kernel $$g(x)=\cos \pi x +\lambda \int_{0}^{1} k(x,t)g(t)\,dt $$ where $k(x,t)$ is a symmetric kernel given by $$k(x,t)= \begin{cases} (x+1)t, & 0 < x <t \\ (t+1)x, & t < x <1 \end{cases} $$
Please, help me.
I do not want the complete solution but I want the method or some hints.
I have not fully worked through this, so my suggestion here might be a dead end; however, it will give you a place to start. My suggestion is iterate the definition of $g(x)$ and try to discover some pattern. For example,
$$g(x) = \cos(\pi x) + \lambda \int_0^1 k(x,t) \left( \cos(\pi t) + \lambda \int_0^1 k(t,r) g(r) \, dr \right) \,dt \\ = \cos(\pi x) + \lambda \int_0^1 k(x,t) \cos(\pi t) \,dt + \lambda^2 \int_0^1 \int_0^1 k(x,t)k(t,r) g(r) \,drdt $$
Notice now that the first integral is something you can explicitly calculate. By changing the order of integration, the second integral becomes $$ \int_0^1 g(r)\left(\int_0^1 k(x,t) k(t,r) \, dt\right)\,dr $$
You should be able to explicitly calculate the parenthetical integral (perhaps it will be something in terms of the kernel $k$ again, which could be very useful in recognizing a pattern -- though, as I mentioned, I am not sure since I have not explicitly calculated the integral). From here, you can iterate the process ad nauseam. Hope this helps!