I am a little rusty in this area so please forgive the slowness. I am trying to prove or disprove the existence of fixed points for the following integral equation. Throughout I am interested in the metric space $C[0,1]$ which is complete under the $\Vert \cdot \Vert_\infty$ norm.
Define $\Phi: C[0,1] \rightarrow C[0,1] $ by
$$\Phi(x)(\theta) = \frac{c(\theta) } {c(\theta) + N \int_0^1 x(t) c(t) dt }. $$ Surely some restrictions on $c(\theta)$ will be needed, but I would rather make them as loose as possible. If it helps, for concreteness put $c(t) = t + t^p$. $N$ and $p$ are known positive parameters.
I am trying to show a solution to the equation $\Phi(x) = x$ exists. I tried to show $\Phi$ is a contraction mapping without success. Moreover, for the particular $c(\cdot)$ above I can see that $\Vert \Phi(\mathbf{1}) - \Phi(\mathbf{0})\Vert = \Vert \mathbf{1} - \mathbf{0} \Vert = 1. $ Is this a correct counterexample to $\Phi$ being a contraction mapping?
If so, do I have much hope of showing existence? I was hoping there was a clean fixed-point theorem I could quote, but I haven't found any so far.