I'm going straight to the point. When gathering Volterra equations of the second kind with mixed formulations, one can arrive to the following problem: Let $X,M$ Hilbert spaces and $\mathcal{J}=[0,T]$, with $T<\infty$. Find $(u,\lambda):\mathcal{J}\rightarrow X\times M$ such that
\begin{equation} \label{linear-mixed-problem1} \begin{aligned} Au(t) +B'\lambda(t) &=f(t) + \int_{0}^{t}\big[\widehat{A}(t,\tau)u(\tau)+ \widehat{B}'(t,\tau)\lambda(\tau)\big]d\tau,\\ Bu(t)&= \int_{0}^{t}\widehat{B}(t,\tau)u(\tau)d\tau\hspace{0.1cm}, \end{aligned} \end{equation}
a.e. in $\mathcal{J}$. Here $A\in \mathcal{L}(X;X'), B\in\mathcal{L}(X;M')$ and $B'\in\mathcal{L}(M;X')$ are time-independent linear elliptic partial differential operators acting on an open bounded domain $\Omega\in \mathbb{R}^n$, with $A$ a self-adjoint operator and $B'$ the dual operator of $B$. The functions $f\in X'$ a.e. in $\mathcal{J}$ is a given load, and we assume that appropriate boundary conditions are given. The operators $\widehat{A}(t,\tau), \widehat{B}(t,\tau)$ and $\widehat{B}'(t,\tau)$ satisfies a "similarity" condition to $A,B$ and $B'$, respectively. Observe that a weak formulation can be achieved.
The big question is: Is this problem make sense?. Any clue or references about how to attack it (well-posedness)?. Thanks.