What are the conditions for solution of nonlinear Fredholm equations with Banach fixed point theorem?

Question

Consider the nonlinear Fredholm integral equation of the second kind: $$ \varphi(x) = f(x) + \lambda \int_a^b F(x, t, \varphi(t)) \, dt $$ where

• $(f)$ and $(K)$ are given functions,
• $(a, b)$ are constants, and
• $(\varphi)$ is the unknown function.

The associated nonlinear Fredholm operator for this integral equation is: $$ T(\varphi)(x) = f(x) + \lambda \int_a^b K(x, t, \varphi(t)) \, dt $$ Using the Banach Fixed Point Theorem, I want to establish necessary conditions for the existence of a solution of the nonlinear Fredholm equation in this case, where $K:[a, b] \times[a, b] \times[-R, R] \rightarrow \mathbb{R}$.

I know I have to check that Fredholm operator satisfies the following requirements:

1. it is well defined
2. $(T(\varphi)$ is a continuous function for every continuous function $\varphi$
3. $T$ is a contraction, and I checked this since $$ |T(\varphi_1)- T(\varphi_2)|=|\lambda|\left|\int_a^bK(x,t,\varphi_1)-\int_a^bK(x,t,\varphi_2)\right|. $$
4. $T$ is invariant.
5. $T$ satisfies all the hypotheses of the Banach Fixed-Point Theorem.

But now I'm really stuck any help?