Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\mathbb R$ is bounded with $$\int_0^t k(t,s)\ \mathrm{d}s=-1.$$ In particular, the latter condition tells us $k$ is unbounded. [Arguably the simplest example has $k(t,s)=\tfrac{-1}t$.]
Let me call $x:(0,1]\to\mathbb R$ a solution if it is bounded and continuous and solves the Volterra equation (of the second kind) $$x(t)+\int_0^t k(t,s)x(s)\ \mathrm{d}s=f(t) \ \forall t\in(0,1].$$
Are there known results that will guarantee (possibly with more regularity assumptions on $k,f$) that:
- A solution exists; and
- The solution is unique up to shifting by a constant?
Note that literal uniqueness of this solution is impossible in this environment because adding a constant function to $x$ has no effect on whether the Volterra equation is satisfied.
