Abstract Linear and Nonlinear Volterra Equations Preserving Positivity

Let X be a real or complex Banach space. We study the Volterra equation \[({\text{v}})\qquad u(t) + \int_0^t {a(t - s)Au(s)\,ds} = f(t)\quad (0 \leqq t \leqq T,T > 0),\] where a is a given kernel, A is a bounded or unbounded linear operator from X to X, and f is a given function with values in X. (Of particular importance is the case $f = u_0 + a * g$, $u_0 \in X$, $g \in L^1 (0,T;X)$, where $ * $ denotes the convolution). We establish existence, uniqueness, continuity results and sufficient conditions involving a, A, f which insure that solutions of (v) are positive by using certain representation formulas for solutions of (v). We also discuss the positivity of solutions of (v) when A is a nonlinear (m-accretive) operator and we discuss several examples when A is a partial differential operator.