Let $(X,d)$ be a complete metric space. The map $T:X\longrightarrow X$ is called a Kannan type contraction if $$d(T x, T y) ≤ α[d(x, T x) + d(y, T y)], ∀x, y ∈ X,\alpha\in[0,\frac{1}{2}).$$
Kannan's fixed point states that $T$ has a unique fixed point in $X$.It is established in the literature that Banach contraction and Kannan contraction are independent, for example, Kannan contraction need not to be continuos.
I am looking for an application of Kannan fixed point theorem to the existence of solution to the integral equations. I know how to apply Banach's fixed point theorem to the integral equations.
More precisely, I am looking for an integral equation which satisfies the Kannan type contraction but not Banach contraction. I will attempt the proof myself if someone gives me an idea of such an integral equation.