Discussion
The resolvent kernel for Volterra integral equation with kernel :
\[ k(t, s) = a^{t-s}, \quad a \gt 0 \]
is :
Answer: C
The resolvent kernel is \(R(t, s; \lambda) = a^{t-s}\, e^{\lambda(t-s)}\).
The iterated kernels of \(k(t, s) = a^{t-s}\) are
\[ k_{n}(t, s) = a^{t-s}\,\frac{(t - s)^{n-1}}{(n - 1)!} \]
(each iteration integrates a constant over \([s, t]\)). Summing the Neumann series,
\[ \begin{aligned} R &= \sum_{n \ge 1} \lambda^{n-1} k_{n} \\ &= a^{t-s} \sum_{m \ge 0} \frac{[\lambda(t - s)]^{m}}{m!} = a^{t-s} e^{\lambda(t-s)} \end{aligned} \]
Options B and D contain \(a^{t+s}\), which is not a function of \(t - s\), although every iterated kernel of a difference kernel is. Option A puts \(\lambda\) in the exponent of a instead of e.
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