I am working on a problem involving a bounded continuous function that satisfies a particular integral equation, and I aim to prove that the function has a limit as $ t \to + \infty $. Here are the details:
$g \in C(\mathbb{R})$, and the function $ x + g(x) $ is strictly monotonic. $ u:[0, \infty) \to \mathbb{R} $ is a bounded and continuous function, satisfying the following equation:$$\forall t\ge1,\quad u(t) + \int_{t-1}^{t} g(u(s)) \, \mathrm{d}s = \text{Constant} $$I am to prove that $ \underset{t \to +\infty}{\lim} u(t) $ exists.
My attempt:
I tried to differentiate both sides, as $u^\prime\left( t \right) +g\left( u\left( t \right) \right) -g\left( u\left( t-1 \right) \right) =0$, or define $F(x):=g(x)+x$ and $u^\prime\left( t \right) -u\left( t \right) +u\left( t-1 \right) +F\left( u\left( t \right) \right) -F\left( u\left( t-1 \right) \right) =0$, but it seems not to work.