Continuity and compactness of this operator?

Question

Set $X=C([a,b],\mathbb{R})$ with the uniform norm. I want to prove that $T:X\rightarrow X$ given by $$T(x)(t)=\int_a^b K(t,s)f(s,x(s))ds $$ is continuous and compact, where $K:[a,b]\times [a,b]\rightarrow \mathbb{R}$ is continuous and $f:[a,b]\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded. I think I've applied correctly arzela-ascoli theorem in order to prove the compactness of $T$. However I am not pretty sure for the continuity. Some help or hint would be appreciated. Thanks in advance.

Answer 1

Let $\varepsilon >0$. Notice that $f$ is uniformly continuous w.r.t. the first variable, therefore, for all $y_0\in [a,b]$, there is $\delta_{y_0}$ s.t. for all $t,y\in [a,b]$, $$|y-y_0|<\delta \implies \forall t\in [a,b], |f(t,y)-f(t,y_0)|\leq \frac{\varepsilon}{K(b-a)},$$ where $K$ is s.t. $|K(t,s)|\leq K$ for all $t,s\in [a,b]$.

Let $x,\tilde x\in X$ s.t. $$\|x-\tilde x\|_X<\delta .$$ In particular, $$|T(x)(t)-T(\tilde x)(t)|\leq K\int_a^b |f(s,x(x))-f(s,\tilde x(s))|ds\leq \varepsilon .$$

The claim follow.