Similar proof of Peano's Existence Theorem

Question

As many of you will know, Peano's theorem states that if $f(x,y)$ is continuous and bounded in the strip $T: |x-x_0| \le a, |y|\le\infty $. Then the intitial value problem $y'=f(x,y), y(x_0)=y_0$, has at least one solution in $|x-x_0| \le a $.

Now, consider the following problem. Let $J= [0,a]$. Let $g(x)$ be a continuous function. Let $k(x,t,z)$ be a function which is continuous and bounded in {${(x,t,z)\in J\times J\times \mathbb{R}: t\lt x}$}. How can I prove that there is (at least) a solution of the equation

$$y(x)= g(x)+ \int_0 ^x k(x,t,y(t))dt, x \in J $$

Answer 1

Let $\tilde k (x,t,z) = k(z,t,z+g(t))$. This is a continuous function. By the Peano theorem, the equation $z(x)=\int_0^x \tilde k(x,t,z(t))\,dt$ has a solution. Then $y=z+g$ satisfies $$y(x)=g(x)+\int_0^x \tilde k(x,t,y(t)-g(t))\,dt = g(x)+\int_0^x k(x,t,y(t))\,dt $$